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Z. Pretel1, ∗ +1Centro Brasileiro de Pesquisas F´ısicas, Rua Dr. Xavier Sigaud, +150 URCA, Rio de Janeiro CEP 22290-180, RJ, Brazil +(Dated: January 10, 2023) +We construct dark energy stars with Chaplygin-type equation of state (EoS) in the presence of +anisotropic pressure within the framework of Einstein gravity. From the classification established by +Iyer et al. [Class. Quantum Grav. 2, 219 (1985)], we discuss the possible existence of isotropic dark +energy stars as compact objects. However, there is the possibility of constructing ultra-compact stars +for sufficiently large anisotropies. We investigate the stellar stability against radial oscillations, and +we also determine the moment of inertia and tidal deformability of these stars. We find that the usual +static criterion for radial stability dM/dρc > 0 still holds for dark energy stars since the squared +frequency of the fundamental pulsation mode vanishes at the critical central density corresponding +to the maximum-mass configuration. The dependence of the tidal Love number on the anisotropy +parameter α is also examined. We show that the surface gravitational redshift, moment of inertia and +dimensionless tidal deformability undergo significant changes due to anisotropic pressure, primarily +in the high-mass region. Furthermore, in light of the detection of gravitational waves GW190814, +we explore the possibility of describing the secondary component of such event as a stable dark +energy star in the presence of anisotropy. +I. +INTRODUCTION +Different types of observations (such as Type Ia super- +novae, structure formation and CMB anisotropies) indi- +cate that our Universe is not only expanding, it is acceler- +ating. Within the standard ΛCDM model (which is based +on cold dark matter and cosmological constant in Ein- +stein gravity), this cosmic acceleration is due to a smooth +component with large negative pressure and repulsive +gravity, the so-called dark energy. Such a model gives +a good agreement with the recent observational data [1], +but suffers from the well-known coincidence problem and +the fine-tuning problem [2, 3]. The exact physical nature +of dark energy is still a mystery and, consequently, the +possibility that dark matter and dark energy could be +different manifestations of a single substance has been +considered [4–7]. In that regard, it was shown that the in- +homogeneous Chaplygin gas offers a simple unified model +of dark matter and dark energy [8]. It was also argued +that if the Universe is dominated by the Chaplygin gas +a cosmological constant would be ruled out with high +confidence [9]. +Using the Planck 2015 CMB anisotropy, type-Ia su- +pernovae and observed Hubble parameter data sets, the +full parameter space of the modified Chaplygin gas was +measured by Li et al. [10]. Based on recent observations +of high-redshift quasars, Zheng and colleagues [11] inves- +tigated a series of Chaplygin gas models as candidates +for dark matter-energy unification. The application of +the Hamilton-Jacobi formalism for generalized Chaplygin +gas models was carried out in Ref. [12]. Additionally, it is +worth mentioning that Odintsov et al. [13] considered two +different equations of state for dark energy (i.e., power- +law and logarithmic effective corrections to the pressure). +∗ juanzarate@cbpf.br +They showed that the power-law model only yielded some +modest results, achieved under negative values of bulk +viscosity, while the logarithmic scenario provide good fits +in comparison to the ΛCDM model. +Another way to give rise to an accelerated expansion of +the Universe is by modifying the geometry itself [14, 15], +namely, considering higher curvature corrections to the +standard Einstein-Hilbert action. +Under this outlook, +the cosmic acceleration can be modeled in the scope of +a scalar-tensor gravity theory [16, 17]. Moreover, within +the context of the so-called f(R) theories [18, 19], the +quadratic term in the Ricci scalar R leads to an infla- +tionary solution in the early Universe [20], although such +a model does not provide a late-time accelerated expan- +sion. Nevertheless, the late-time acceleration era can be +realized by terms containing inverse powers of R [21], +though it was shown that this is not compatible with +the solar system experiments [22]. For a comprehensive +study on the evolution of the early and present Universe +in f(R) modified gravity, we refer the reader to the re- +view articles [23–25] and references contained therein. +On the other hand, the astrophysical implications due +to the f(R) modified gravitational Lagrangian on com- +pact stars have been intensively investigated in the past +few years [26–33]. +According to the aforementioned works, different dark +energy models have been proposed in order to explain the +mechanisms that lead to the cosmic acceleration. Only +about 4% of the Universe is made of familiar atomic mat- +ter, 20% dark matter, and it turns out that roughly 76% +of the Universe is dark energy [34]. Within the context +of General Relativity, dark energy is an exotic negative +pressure contribution that can lead to the observed accel- +erated expansion. In the absence of consensus regarding a +theoretical description for the current accelerated expan- +sion of the Universe, theorists have proposed using the +Chaplygin gas as a useful phenomenological description +arXiv:2301.03504v1 [gr-qc] 9 Jan 2023 + +2 +[4]. If dark energy is distributed anywhere permeating or- +dinary matter, then it could be present in the interior of a +compact star. Therefore, the purpose of this manuscript +is to investigate the possible existence of compact stars +with dark energy by assuming a Chaplygin-type EoS. For +such stars to exist in nature, they need to be stable under +small radial perturbations. +Adopting a description of dark energy by means of a +phantom (ghost) scalar field, Yazadjiev [35] constructed a +general class of exact interior solutions describing mixed +relativistic stars containing both ordinary matter and +dark energy. +The energy conditions and gravitational +wave echoes of such stars were recently analyzed in +Ref. [36]. Furthermore, the effect of the dynamical scalar +field quintessence dark energy on neutron stars was inves- +tigated in [37]. Panotopoulos and collaborators [38] stud- +ied slowly rotating dark energy stars made of isotropic +matter using the Chaplygin EoS. Bhar [39] proposed a +model for a dark energy star made of dark and ordinary +matter in the Tolman–Kuchowicz spacetime geometry. +For further stellar models with dark energy we also refer +the reader to Refs. [40–48]. +In addition, anisotropy in compact stars may arise due +to strong magnetic fields, pion condensation, phase tran- +sitions, mixture of two fluids, bosonic composition, rota- +tion, etc. Thus, regardless of the specific source of the +anisotropy, it is more natural to think of anisotropic fluids +when studying compact stars at densities above nuclear +saturation density. In that regard, the literature offers +some physically motivated functional relations for the +anisotropy, see for example Refs. [49–55]. However, we +must point out that these anisotropic models are based +on general assumptions (or ansatzes) that do not directly +relate to exotic modifications of matter or gravity. In- +deed, it has been argued that the deformation near the +maximum neutron-star mass comes from the anisotropic +pressure within these stars, which is caused by the distor- +tion of Fermi surface predicted by the equation of state +of the models [56]. Becerra-Vergara et al. [57] showed +that the contribution of the fourth order corrections pa- +rameter (a4) of the QCD perturbation on the radial +and tangential pressure generate significant effects on the +mass-radius relation and the stability of quark stars. It +has also been shown that the stellar structure equations +in Eddington-inspired Born-Infeld theory with isotropic +matter can be recast into GR with a modified (apparent) +anisotropic matter [58]. +Motivated by the several works already mentioned, we +aim to discuss the impact of anisotropy on the macro- +scopic properties of dark energy stars with Chaplygin-like +EoS. We will address the following questions: Do these +stars belong to families of compact or ultra-compact +stars? How does anisotropy affect the compactness and +radial stability of dark energy stars satisfying the causal- +ity condition? In particular, by adopting the phenomeno- +logical ansatz proposed by Horvat et al. [51], we deter- +mine the radius, mass, gravitational redshift, frequency +of the fundamental oscillation mode, moment of inertia +and the dimensionless tidal deformability of anisotropic +dark energy stars. The isotropic solutions are recovered +when the anisotropy parameter vanishes, i.e. when α = 0. +The organization of this paper is as follows: In Sec. II +we start with a brief overview of relativistic stellar struc- +ture, describing the basic equations for radial pulsations, +moment of inertia and tidal deformability. We then in- +troduce the Chaplygin-like EoS and discuss its relation to +the cosmological context in Sec. III, as well as we present +the anisotropy profile. Section IV provides a discussion +of the numerical results for the different physical prop- +erties of dark energy stars. Finally, our conclusions are +summarized in Sec. V. +II. +STELLAR STRUCTURE EQUATIONS +In order to study the basic features of compact stars +with dark energy, in this section we briefly summarize the +stellar structure equations in Einstein gravity. In particu- +lar, we focus on hydrostatic equilibrium structure, radial +pulsations, moment of inertia, and tidal deformability. +The theory of gravity to be used in this work is general +relativity, where the Einstein field equations are given by +Gµν ≡ Rµν − 1 +2gµνR = 8πTµν, +(1) +with Gµν being the Einstein tensor, Rµν the Ricci tensor, +R denotes the scalar curvature, and Tµν is the energy- +momentum tensor. +Since we are interested in isolated +compact stars, we consider that the spacetime can be +described by the spherically symmetric four-dimensional +line element +ds2 = −e2ψdt2 + e2λdr2 + r2(dθ2 + sin2 θdφ2). +(2) +In addition, we model the compact-star matter by an +anisotropic perfect fluid, whose energy-momentum tensor +is given by +Tµν = (ρ + pt)uµuν + ptgµν − σkµkν, +(3) +where ρ is the energy density, σ ≡ pt − pr the anisotropy +factor, pr the radial pressure, pt the tangential pressure, +uµ the four-velocity of the fluid, and kµ is a unit four- +vector. +These four-vectors must satisfy uµuµ = −1, +kµkµ = 1 and uµkµ = 0. Notice that the stellar fluid +becomes isotropic when σ = 0. +A. +TOV equations +When the stellar fluid remains in hydrostatic equilib- +rium, neither metric nor thermodynamic quantities de- +pend on the time coordinate. +This allows us to write +uµ = e−ψδµ +0 and kµ = e−λδµ +1 . Accordingly, the hydro- +static equilibrium of an anisotropic compact star is gov- + +3 +erned by the TOV equations: +dm +dr = 4πr2ρ, +(4) +dpr +dr = −(ρ + pr) +�m +r2 + 4πrpr +� � +1 − 2m +r +�−1 ++ 2σ +r , +(5) +dψ +dr = − +1 +ρ + pr +dpr +dr + +2σ +r(ρ + pr), +(6) +which are obtained from Eqs. (1)-(3) together with the +conservation law ∇µT µ +1 += 0. The metric function λ(r) +is determined from the relation e−2λ = 1 − 2m/r, where +m(r) is the gravitational mass within a sphere of radius +r. +By supplying an EoS for the radial pressure in the form +pr = pr(ρ) and a defined anisotropy relation for σ, the +system of differential equations (4)-(6) is then numeri- +cally integrated from the center at r = 0 to the surface +of the star r = R which correspond to a vanishing pres- +sure. Therefore, the above equations will be solved under +the requirement of the following boundary conditions +ρ(0) = ρc, +m(0) = 0, +ψ(R) = 1 +2 ln +� +1 − 2M +R +� +, (7) +where ρc is the central energy density, and M ≡ m(R) +is the total mass of the star calculated at its surface. +The numerical solution of the TOV equations describes +the equilibrium background and allow us to obtain the +metric components and fluid variables. +B. +Radial oscillations +A rigorous analysis of the radial stability of compact +stars requires the calculation of the frequencies of nor- +mal vibration modes. Such frequencies can be found by +considering small deviations from the hydrostatic equi- +librium state but maintaining the spherical symmetry of +the star. In the linear treatment, where all quadratic (or +higher-order) or mixed terms in the perturbations are +discarded, one assumes that all perturbations in physical +quantities are arbitrarily small. +The fluid element lo- +cated at r in the unperturbed configuration is displaced +to radial coordinate r + ξ(t, r) in the perturbed config- +uration, where ξ is the Lagrangian displacement. +All +perturbations have a harmonic time dependence of the +form ∼ eiνt, where ν is the oscillation frequency to be +determined. Consequently, defining ζ ≡ ξ/r, the adia- +batic1 radial pulsations of anisotropic compact stars are +1 In the adiabatic theory, it is assumed that the fluid elements of +the star neither gain nor lose heat during the oscillation. +governed by the following differential equations [55] +dζ +dr = − 1 +r +� +3ζ + ∆pr +γpr ++ +2σζ +ρ + pr +� ++ dψ +dr ζ, +(8) +d(∆pr) +dr += ζ +� +ν2e2(λ−ψ)(ρ + pr)r − 4dpr +dr +−8π(ρ + pr)e2λrpr + r(ρ + pr) +�dψ +dr +�2 ++2σ +�4 +r + dψ +dr +� ++ 2dσ +dr +� ++ 2σ dζ +dr +− ∆pr +�dψ +dr + 4π(ρ + pr)re2λ +� ++ 2 +r δσ, +(9) +where ∆pr is the Lagrangian perturbation of the radial +pressure and γ = (1+ρ/pr)dpr/dρ is the adiabatic index +at constant specific entropy. +The above first-order time-independent equations (8) +and (9) require boundary conditions set at the center and +surface of the star, similar to a vibrating string fixed at +its ends. Since Eq. (8) has a singularity at the origin, the +following condition must be required +∆pr = − 2σζ +ρ + pr +γpr − 3γζpr +as +r → 0, +(10) +while the Lagrangian perturbation of the radial pressure +at the surface must satisfy +∆pr = 0 +as +r → R. +(11) +C. +Moment of inertia +Suppose a particle is dropped from rest at a great dis- +tance from a rotating star, then it would experience an +ever increasing drag in the direction of rotation as it ap- +proaches the star. Based on this description, we intro- +duce the angular velocity acquired by an observer falling +freely from infinity, denoted by ω(r, θ). Here we will cal- +culate the moment of inertia of an anisotropic dark en- +ergy star under the slowly rotating approximation [59]. +This means that when we consider rotational corrections +only to first order in the angular velocity of the star Ω, +the line element (2) is replaced by its slowly rotating +counterpart, namely +ds2 = − e2ψ(r)dt2 + e2λ(r)dr2 + r2(dθ2 + sin2 θdφ2) +− 2ω(r, θ)r2 sin2 θdtdφ, +(12) +and following Ref. [59], it is pertinent to define the differ- +ence ϖ ≡ Ω−ω as the coordinate angular velocity of the +fluid element at (r, θ) seen by the freely falling observer. +Keep in mind that Ω is the angular velocity of the stel- +lar fluid as seen by an observer at rest at some spacetime +point (t, r, θ, φ), and hence the four-velocity up to linear +terms in Ω can be written as uµ = (e−ψ, 0, 0, Ωe−ψ). To +this order, the spherical symmetry is still preserved and + +4 +it is possible to extend the validity of the TOV equations +(4)-(6). Nonetheless, the 03-component of the field equa- +tions contributes an additional differential equation for +angular velocity. By retaining only first-order terms in +Ω, such component becomes +eψ−λ +r4 +∂ +∂r +� +e−(ψ+λ)r4 ∂ϖ +∂r +� ++ +1 +r2 sin3 θ +∂ +∂θ +� +sin3 θ∂ϖ +∂θ +� += 16π(ρ + pt)ϖ. +(13) +As in the case of isotropic fluids, we follow the same +treatment carried out by Hartle [59, 60] and we assume +that ϖ can be written as +ϖ(r, θ) = +∞ +� +l=1 +ϖl(r) +� −1 +sin θ +dPl +dθ +� +, +(14) +where Pl are Legendre polynomials. Taking this expan- +sion into account, Eq. (13) becomes +eψ−λ +r4 +d +dr +� +e−(ψ+λ)r4 dϖl +dr +� +− l(l + 1) − 2 +r2 +ϖl += 16π(ρ + pt)ϖl. +(15) +At a distance far away from the star, where e−(ψ+λ) +becomes unity, the asymptotic solution of Eq. (15) takes +the form ϖl(r) → a1r−l−2 + a2rl−1. If spacetime is to +be flat at large r, then ω → 2J/r3 (or equivalently, ϖ → +Ω − 2J/r3) for r → ∞, where J is the total angular +momentum of the star [59, 61]. +Therefore, comparing +this with the asymptotic behavior of ϖl(r), we find that +l = 1. As a result, ϖ is a function only of the radial +coordinate, and Eq. (15) reduces to +eψ−λ +r4 +d +dr +� +e−(ψ+λ)r4 dϖ +dr +� += 16π(ρ + pt)ϖ, +(16) +which can be integrated to give +� +r4 dϖ +dr +� +R += 16π +� R +0 +(ρ + pt)r4eλ−ψϖdr. +(17) +In view of Eq. (17), we can obtain the angular mo- +mentum J and hence the moment of inertia I = J/Ω of +a slowly rotating anisotropic star: +I = 8π +3 +� R +0 +(ρ + pr + σ)eλ−ψr4 �ϖ +Ω +� +dr, +(18) +which reduces to the expression given in Ref. [61] for +isotropic compact stars when σ = 0. For an arbitrary +choice of the central value ϖ(0), the appropriate bound- +ary conditions for the differential equation (16) come +from the requirements of regularity at the center of the +star and asymptotic flatness at infinity, namely +dϖ +dr +���� +r=0 += 0, +lim +r→∞ ϖ = Ω. +(19) +Once the solution for ϖ(r) is found, we can then deter- +mine the moment of inertia through the integral (18). It +is remarkable that the above expression for I is referred +to as the “slowly rotating” approximation because it was +obtained to lowest order in the angular velocity Ω [61]. +This means that the stellar structure equations are still +given by the TOV equations (4)-(6). +D. +Tidal deformability +It is well known that the tidal properties of neutron +stars are measurable in gravitational waves emitted from +the inspiral of a binary neutron-star coalescence [62, 63]. +In that regard, here we also study the dimensionless tidal +deformability of individual dark energy stars. To do so, +we follow the procedure carried out by Hinderer et al. [64] +(see also Refs. [65–70] for additional results). The basic +idea is as follows: In a binary system, the deformation +of a compact star due to the tidal effect created by the +companion star is characterized by the tidal deformabil- +ity parameter ¯λ = −Qij/Eij, where Qij is the induced +quadrupole moment tensor and Eij is the tidal field ten- +sor [68]. Namely, the latter describes the tidal field from +the spacetime curvature sourced by the distant compan- +ion. +The tidal parameter is related to the tidal Love number +k2 through the relation2 +¯λ = 2 +3k2R5, +(20) +but it is common in the literature to define the dimen- +sionless tidal deformability Λ = ¯λ/M 5, so in our results +we will focus on Λ. The calculation of ¯λ requires consider- +ing linear quadrupolar perturbations (due to the external +tidal field) to the equilibrium configuration. Thus, the +spacetime metric is given by gµν = g0 +µν + hµν, where g0 +µν +describes the equilibrium configuration and hµν is a lin- +earized metric perturbation. For static and even-parity +perturbations in the Regge-Wheeler gauge [71], the per- +turbed metric can be written as [64] +hµν = +diag +� +−e2ψ(r)H0, e2λ(r)H2, r2K, r2 sin2 θK +� +Y2m(θ, φ), +(21) +where H0, H2 and K are functions of the radial coordi- +nate, and Ylm are the spherical harmonics for l = 2. +Since the perturbed energy-momentum tensor is given +by δT ν +µ = diag(−δρ, δpr, δpt, δpt), the linearized field +2 It should be noted that the tidal deformability parameter is be- +ing denoted by ¯λ in order not to be confused with the metric +component λ. + +5 +equations imply that: +� +� +� +� +� +H0 = −H2 ≡ H +from +δG2 +2 − δG3 +3 = 0, +K′ = 2Hψ′ + H′ +from +δG2 +1 = 0, +δpt = +H +8πre−2λ(λ′ + ψ′)Y2m +from +δG2 +2 = 8πδpt. +In addition, from δG0 +0 − δG1 +1 = −8π(δρ + δpt), we can +obtain the following differential equation [72] +H′′ + PH′ + QH = 0, +(22) +or alternatively, +ry′ = −y2 + (1 − rP)y − r2Q, +(23) +where we have defined +y ≡ rH′ +H , +(24) +P ≡ 2 +r + e2λ +�2m +r2 + 4πr(pr − ρ) +� +, +(25) +Q ≡ 4πe2λ +� +4ρ + 8pr + ρ + pr +Av2sr +(1 + v2 +sr) +� +− 6e2λ +r2 +− 4ψ′2, +(26) +with A ≡ dpt/dpr and vsr being the radial speed of +sound. +By matching the internal solution with the external +solution of the perturbed variable H at the surface of the +star r = R, we obtain the tidal Love number [72] +k2 = 8 +5(1 − 2C)2C5 [2C(yR − 1) − yR + 2] +× +� +2C[4(yR + 1)C4 + (6yR − 4)C3 ++ (26 − 22yR)C2 + 3(5yR − 8)C − 3yR + 6 +� ++ 3(1 − 2C)2 [2C(yR − 1) − yR + 2] log(1 − 2C) +�−1 , +(27) +where C ≡ M/R is the compactness of the star, and +yR ≡ y(R) is obtained by integrating equation (23) from +the origin up to the stellar surface. +III. +EQUATION OF STATE AND ANISOTROPY +MODEL +As it is well known, a possible alternative to the Phan- +tom and Quintessence fields is the Chaplygin gas, where +the EoS assumes the form pr = −B/ρ, with B being a +positive constant (given in m−4 units). In fact, it was ar- +gued that such gas could provide a solution to unify the +effects of dark matter in the early times and dark energy +in late times [4, 11]. Although the literature provides a +more generalized version for such EoS in the context of +the Friedmann-Lemaˆıtre-Robertson-Walker Universe [5– +7, 73–77], here we will use the simplest form plus a linear +term corresponding to a barotropic fluid, namely +pr = Aρ − B +ρ , +(28) +where A is a positive dimensionless constant. Our model +is characterized by two free parameters A and B. Never- +theless, we must emphasize here that Li et al. [10] consid- +ered an equation of state with three degrees of freedom, +specifically p = Aρ − B/ρα, where α is an extra param- +eter. They carried out a statistical treatment of astro- +nomical data in order to constrain the parameter space. +In the light of the Markov chain Monte Carlo method, +they found that at 2σ level, α = −0.0156+0.0982+0.2346 +−0.1380−0.2180 +and A = 0.0009+0.0018+0.0030 +−0.0017−0.0030 from CMB+JLA+CC data +sets. +In other words, the constants α and A are very +close to zero and hence the nature of unified dark matter- +energy model is very similar to the cosmological standard +ΛCDM model. +On the other hand, at astrophysics level, compact stars +obeying the EoS (28) have been investigated by several +authors, see for example Refs. [38, 41, 43–45]. In this +work we will adopt values of A and B for which appre- +ciable changes in the mass-radius diagram can be visu- +alized in order to compare our theoretical results with +observational measurements of massive pulsars. +In order to describe physically realistic compact stars, +the causality condition must be respected throughout the +interior region of the star. In other words, the speed of +sound (defined by vs ≡ +� +dp/dρ) cannot be greater than +the speed of light. Thus, in view of Eq. (28), we have +v2 +sr ≡ dpr +dρ = A + B +ρ2 , +(29) +and since the radial pressure vanishes at the surface of +the star, then B = Aρ2. Thereby, the causality condition +v2 +sr(R) = 2A < 1 implies that A < 0.5. +Besides, it is more realistic to consider stellar models +where there exists a tangential pressure as well as a radial +one, since anisotropies arise at high densities, i.e. above +the nuclear saturation density as considered in this work. +Although the literature offers different functional rela- +tions to model anisotropic pressures at very high densities +inside compact stars [49–54], here we adopt the simplest +model, which was proposed by Horvat and collaborators +[51] +σ = α +�2m +r +� +pr = α +� +1 − e−2λ� +pr, +(30) +where α is a dimensionless parameter that controls the +amount of anisotropy within the stellar fluid. This pa- +rameter can assume positive or negative values of the +order of unity, see Refs. [26, 32, 51, 52, 55, 78–82]. No- +tice that the isotropic solutions are recovered when the +value of α vanishes. Specifically, the anisotropy ansatz +(30) has two important characteristics: (i) the fluid be- +comes isotropic at the center generating regular solutions +and (ii) the effect of anisotropy vanishes in the hydro- +static equilibrium equation in the Newtonian limit. Un- +like this profile, the effect of anisotropy does not van- +ish in the hydrostatic equilibrium equation in the non- +relativistic regime for the Bowers-Liang model [49], which + +6 +could be an unphysical trait as argued in Ref. [79]. For +a broader discussion on the different ways of generating +static spherically symmetric anisotropic fluid solutions, +we refer the reader to the recent review article [83]. +Since the Eulerian perturbation for the metric poten- +tial λ can be written as δλ = −4πr(ρ+pr)e2λξ [55], then +δσ takes the form +δσ = α +� +(1 − e−2λ)δpr − 8πpr(ρ + pr)r2ζ +� +, +(31) +where it should be noted that the relation between the +Eulerian and Lagrangian perturbations for radial pres- +sure is given by ∆pr = δpr + rζp′ +r. The above expression +will be substituted in Eq. (9) when we discuss later the +radial pulsations in the stellar interior for at least some +values of α. +IV. +NUMERICAL RESULTS +A. +Equilibrium configurations +So far we do not know exactly whether the millisecond +pulsars (observed in compact binaries from optical spec- +troscopic and photometric measurements) are hadronic, +quark or hybrid stars. +In fact, it has been theorized +that cold quark matter might exist at the core of heavy +neutron stars [84]. Despite the precise measurements of +masses [85–87] and radii [88–90], such constraints are still +unable to distinguish the theoretical predictions coming +from the different models for strange stars and (hybrid) +neutron stars. This means that the dense matter EoS +within compact stars still remains poorly understood. +Furthermore, a realistic compact star possesses high mag- +netic fields and rotation properties, which significantly +alter its internal structure. For comparison reasons, it is +therefore common to use the observational mass-radius +measurements (in view of the detection of gravitational +waves and electromagnetic signals) on the mass-radius +diagrams for any type of EoS even being of different mi- +croscopic compositions. In that perspective, our theoret- +ical results will be compared with observational measure- +ments. +We begin our discussion of dark energy stars by con- +sidering the isotropic case (i.e., when σ = 0 in the TOV +equations). We numerically integrate Eqs. (4)-(6) from +the center up to the surface of the star through the +boundary conditions (7). As usual, the radius R is de- +termined when the pressure vanishes, and the total mass +M is calculated at the surface. The felt panel of Fig. 1 +exhibits the mass-radius relations of dark energy stars for +different values of parameters A and B in the EoS (28). +Remark that we have adopted values of A less than 0.5 in +order to respect the causality condition. One can observe +that small values of A (see black curve) do not provide +compact stars that fit current observational data. How- +ever, higher values of maximum mass can be obtained for +larger values of A, see for example red and green curves. +For a fixed value of A, the maximum mass decreases as +the parameter B increases. +We perceive that the sec- +ondary component resulting from the gravitational-wave +signal GW190814 [91] can be consistently described as a +compact star with Chaplygin EoS (28) for A = 0.4 and +B ∈ [4, 5]µ. Furthermore, the magenta curve fits very +well with all observational data, but its maximum-mass +value is above 3M⊙. +Another interesting feature of these stars is their com- +pactness, defined by C ≡ M/R. According to the clas- +sification adopted by Iyer et al. [92], the configurations +shown in the mass-radius diagram correspond to compact +stars, see the right plot of Fig. 1. Besides, we can appre- +ciate that the compactness of dark energy stars is of the +order of the compactness of hadronic-matter stars, as is +the case of the SLy EoS [93], despite the fact that the +maximum mass in the magenta configuration sequence +can exceed 3M⊙. Nonetheless, as we will see later, the +introduction of anisotropy can turn such stars into ultra- +compact objects. +Of course, this will depend on the +amount of anisotropy in the stellar interior. +In order to include anisotropic pressures and investi- +gate their effects on the internal structure of dark energy +stars, we will adopt two specific models with the following +parameters +⋆ Model I: A = 0.3, B = 6.0µ , +⋆ Model II: A = 0.4, B = 5.2µ , +which are models favored by observational measurements +according to the left panel of Fig. 1. Moreover, model II +precisely corresponds to the first model considered by +Panotopoulos et al. [38]. +Similar to the isotropic case, we numerically solve the +hydrostatic background equations (4)-(6) with boundary +conditions (7), but taking into account the anisotropy +profile (30). For instance, for the model I and a central +density ρc = 2.0×1018 kg/m3, Fig. 2 illustrates the mass +density, pressure and squared speed of sound as functions +of the radial coordinate for different values of the free pa- +rameter α. We can see that the internal structure of a +dark energy star is affected by the presence of anisotropy. +In effect, the radius of the star increases (decreases) for +more positive (negative) values of α. In addition, we re- +mark that the speed of sound, both radial and tangential, +respect the causality condition. This has also been veri- +fied for other values of central density considered in the +construction of Fig. 1. +Varying the central density, we obtain the mass-radius +diagrams and mass-central density relations for models I +and II, as shown in Fig. 3. +We observe that the sub- +stantial changes introduced by anisotropy in dark en- +ergy stars occur in the high-mass branch (close to the +maximum-mass point), while the effects are irrelevant at +low central densities. The maximum-mass values increase +as the parameter α increases (see also the data in Table +I). Note that model I without anisotropic pressures is +not capable of generating maximum masses above 2M⊙. + +7 +SLy +A = 0.2, B = 6μ +A = 0.3, B = 3μ +A = 0.3, B = 4μ +A = 0.3, B = 5μ +A = 0.3, B = 6μ +A = 0.4, B = 3μ +A = 0.4, B = 4μ +A = 0.4, B = 5μ +A = 0.4, B = 6μ +A = 0.48, B = 3μ +4 +6 +8 +10 +12 +14 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +R [km] +M [M⊙] +C > 1/3 +(ultra-compact objects) +1/6 < C < 1/3 +(compact objects) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +M [M⊙] +C +FIG. 1. Left panel: Mass-radius diagrams for dark energy stars with Chaplygin-like EoS (28) and isotropic pressure (σ = 0) +for several values of the positive parameters A and B. +Here the constant B is given in µ = 10−20 m−4 units. +The gray +horizontal stripe at 2.0M⊙ stands for the two massive NS pulsars J1614-2230 [85] and J0348+0432 [86]. Yellow and blue regions +represent the observational measurements of the masses of the highly massive NS pulsars J0740+6620 [87] and J2215+5135 +[94], respectively. The filled pink band stands for the lower mass of the compact object detected by the GW190814 event +[91], and the cyan area is the mass-radius constraint from the GW170817 event. Moreover, the NICER measurements for PSR +J0030+0451 are displayed by black dots with their respective error bars [95, 96]. Right panel: Variation of the compactness +with total gravitational mass, where the gray and orange stripes represent compact and ultra-compact objects, respectively, +according to the classification given in Ref. [92]. For comparison reasons, we have included the results corresponding to the +SLy EoS [93] by blue curves in both plots. +Nevertheless, the inclusion of anisotropies (see the blue +curve for α = 0.4) allows a significant increase in the +maximum mass and hence a more favorable description +of the compact objects observed in nature. On the other +hand, model II with anisotropies (see orange curves) fits +better with the observational measurements. In particu- +lar, in view of the lower mass of the compact object from +the coalescence GW190814 [91], two curves are partic- +ularly outstanding. In other words, such object can be +well described as an anisotropic dark energy star when +α = 0.2 and α = 0.4. Moreover, model II with negative +anisotropies (such as α = −0.4) favors the description of +the massive pulsar J2215+5135 [94]. +The left panel of Fig. 4 describes the behavior of +compactness as a function of central density. +Positive +anisotropies lead to an increase in compactness, mainly +in the high-central-density branch. Remarkably, for suf- +ficiently large values of α (see purple curve), it is possible +to obtain anisotropic dark energy stars as ultra-compact +objects. +The gravitational redshift, conventionally defined as +the fractional change between observed and emitted +wavelengths compared to emitted wavelength, in the case +of a Schwarzschild star is given by [61] +zsur = eλ(R) − 1 = +� +1 − 2M +R +�−1/2 +− 1. +(32) +In the right plot of Fig. 4, the surface gravitational red- +TABLE I. +Maximum-mass configurations with Chaplygin- +like EoS (28) for model I and II. The energy density values +correspond to the critical central density where the function +M(ρc) is a maximum on the right plot of Fig. 3. +Model +α +ρc [1018 kg/m3] +R [km] +M [M⊙] +−0.4 +2.424 +9.812 +1.786 +−0.2 +2.364 +9.902 +1.852 +I +0 +2.295 +9.994 +1.919 +0.2 +2.219 +10.086 +1.988 +0.4 +2.135 +10.180 +2.059 +−0.4 +1.777 +11.630 +2.320 +−0.2 +1.721 +11.738 +2.402 +II +0 +1.661 +11.845 +2.486 +0.2 +1.594 +11.955 +2.570 +0.4 +1.523 +12.065 +2.565 +shift is plotted as a function of the total mass for both +models I and II. This plot indicates that the gravita- +tional redshift of light emitted at the surface of a dark +energy star is substantially affected by the anisotropy in +the high-mass region, while the changes are negligible for +sufficiently low masses. For a fixed value of central den- +sity, Table II shows that positive (negative) anisotropy +increases (decreases) the value of the redshift. + +8 +TABLE II. +Radius, mass, redshift, fundamental mode frequency (f0 = ν0/2π), moment of inertia and dimensionless tidal +deformability of dark energy stars with central energy density ρc = 1.5 × 1018 kg/m3 as predicted by models I and II for +several values of the anisotropy parameter α. Remarkably, with the exception of the fundamental mode frequency and tidal +deformability, these properties undergo a significant increase as α increases. +Model +α +R [km] +M [M⊙] +zsur +f0 [kHz] +I [1038 kg · m2] +Λ +−0.4 +10.062 +1.713 +0.418 +2.414 +1.695 +13.278 +−0.2 +10.163 +1.781 +0.440 +2.312 +1.820 +10.709 +I +0 +10.263 +1.852 +0.463 +2.201 +1.957 +8.598 +0.2 +10.361 +1.926 +0.489 +2.081 +2.105 +6.868 +0.4 +10.456 +2.003 +0.518 +1.950 +2.265 +5.454 +−0.4 +11.767 +2.310 +0.543 +1.131 +3.298 +4.889 +−0.2 +11.859 +2.395 +0.574 +0.998 +3.531 +3.823 +II +0 +11.944 +2.481 +0.609 +0.840 +3.778 +2.978 +0.2 +12.019 +2.569 +0.647 +0.637 +4.037 +2.309 +0.4 +12.083 +2.656 +0.688 +0.315 +4.303 +1.782 +α = -0.6 +α = -0.3 +α = 0 +α = 0.3 +α = 0.6 +0 +2 +4 +6 +8 +10 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +r [km] +ρ [kg/m3] +Solid lines: pr +Dashed lines: pt +α = -0.6 +α = -0.3 +α = 0 +α = 0.3 +α = 0.6 +0 +2 +4 +6 +8 +10 +0 +1 +2 +3 +4 +5 +r [km] +Pressure [1034 Pa] +Solid lines: vsr +2 +Dashed lines: vst +2 +α = -0.6 +α = -0.3 +α = 0 +α = 0.3 +α = 0.6 +0 +2 +4 +6 +8 +10 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +r [km] +(Speed of sound)2/c2 +FIG. 2. Radial behavior of the mass density (left panel), pressures (middle panel) and squared speed of sound (right panel) +inside an anisotropic dark energy star with central density ρc = 2.0 × 1018 kg/m3 and several values of the parameter α. All +plots correspond to model I and the black curves represent the isotropic solutions. Note that both the radial and tangential +speed of sound obey the causality condition. Furthermore, one can observe that the increase in α leads to larger radii, and the +anisotropy is more pronounced in the intermediate regions. +B. +Oscillation spectrum +A necessary condition (the well-known M(ρc) method) +for stellar stability is that stable stars must lie in the re- +gion where dM/dρc > 0. +According to the right plot +of Fig. 3, the full blue and orange circles on each curve +indicate the onset of instability for each family of equi- +librium solutions. However, a sufficient condition is to +calculate the frequencies of the radial vibration modes +for each central density [61]. Here we will analyze if both +methods are compatible in the case of dark energy stars +including anisotropic pressure. +Once the equilibrium equations (4)-(6) are integrated +from the center to the surface of the star, we then pro- +ceed to solve the radial pulsation equations (8) and (9) +with the corresponding boundary conditions (10) and +(11) using the shooting method. Namely, we integrate +from the origin (where we consider the normalized eigen- +functions ζ(0) = 1) up to the stellar surface for a set +of trial values ν2 satisfying the condition (10). In this +way, the appropriate eigenfrequencies correspond to the +values for which the boundary condition (11) is fulfilled. +For instance, for a central density ρc = 1.5×1018 kg/m3, +α = 0.4 and parameters given by model I, Fig. 5 dis- +plays the radial behavior of the perturbation variables +for the first five squared eigenfrequencies ν2 +n, where n +indicates the number of nodes inside the star. This fre- +quency spectrum forms an infinite discrete sequence, i.e. +ν2 +0 < ν2 +1 < ν2 +2 < · · · , where the eigenvalue corresponding +to n = 0 is the lowest one (or equivalently, the longest +period of all the allowed vibration modes) and it is known +as the fundamental mode. +Such mode has no nodes, + +9 +Model I +Model II +α = -0.4 +α = -0.2 +α = 0 +α = 0.2 +α = 0.4 +7 +8 +9 +10 +11 +12 +0.5 +1.0 +1.5 +2.0 +2.5 +R [km] +M [M⊙] +Model I +Model II +α = -0.4 +α = -0.2 +α = 0 +α = 0.2 +α = 0.4 +17.8 +18.0 +18.2 +18.4 +18.6 +0.5 +1.0 +1.5 +2.0 +2.5 +Log ρc [kg/m3] +M [M⊙] +FIG. 3. Mass-radius diagram (left panel) and mass-central density relation (right panel) for anisotropic dark energy stars as +predicted by model I (blue curves) and II (orange curves) with anisotropy profile (30) for several values of α. The colored +bands in the left plot represent the same as in Fig. 1. Moreover, the full blue and orange circles on the right plot indicate the +maximum-mass points for model I and II, respectively. Note that the maximum-mass values for model II correspond to lower +central densities than those for model I, however, model II allows larger masses (see also Table I). The critical central density +corresponding to the maximum point on the M(ρc) curve is modified by the presence of anisotropy for both models. +Model I +Model II +C > 1/3 +(ultra-compact objects) +1/6 < C < 1/3 +(compact objects) +α = 0.7 +α = -0.4 +α = -0.2 +α = 0 +α = 0.2 +α = 0.4 +17.8 +18.0 +18.2 +18.4 +18.6 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +Log ρc [kg/m3] +C +Model I +Model II +α = -0.4 +α = -0.2 +α = 0 +α = 0.2 +α = 0.4 +0.5 +1.0 +1.5 +2.0 +2.5 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +M [M⊙] +zsur +FIG. 4. +Left panel: Variation of the compactness with central density for several anisotropic dark energy star sequences. +The gray and light-green stripes represent compact and ultra-compact objects, respectively, according to the classification +established by Iyer et al. [92]. Positive anisotropy results in increased compactness for sufficiently high central densities, while +the opposite occurs for negative anisotropy. Note also that dark energy stars would correspond to ultra-compact objects if +α > 0.4 for model II, see for instance the purple curve for α = 0.7. Right panel: Surface gravitational redshift as a function +of the total mass. In the high-redshift region it can be observed that positive (negative) anisotropy increases (decreases) the +value of zsur. Meanwhile, the effect of anisotropy is irrelevant for sufficiently low redshifts. +whereas the first overtone (n = 1) has one node, the +second overtone (n = 2) has two, and so on. Stable stars +are described by their oscillatory behavior so that ν2 +n > 0 +(i.e., νn is purely real). On the other hand, if any of these +is negative for a particular star, the frequency is purely +imaginary and hence the star is unstable. +Since each higher-order mode has a squared eigenfre- +quency that is larger than in the case of the preceding +mode, it is enough to calculate the frequency of the fun- +damental pulsation mode for the equilibrium sequences +presented in Fig. 3. +With this in mind, in Fig. 6 we +plot the squared frequency of the fundamental oscilla- +tion mode as a function of the central density (left panel) +and gravitational mass (right panel). According to the +left plot, the squared frequency of the fundamental mode +is exactly zero at the critical-central-density value corre- +sponding to the maximum-mass configuration as shown +in the right plot of Fig. 3, see the full blue and orange cir- + +10 +cles for both models. Furthermore, according to the right +plot of Fig. 6, the maximum-mass values (that is, when +dM/dρc = 0) can be used as turning points from stability +to dynamical instability. Therefore, we can conclude that +the usual criterion to guarantee stability dM/dρc > 0 is +still valid for the case of anisotropic dark energy stars. +In other words, the conventional M(ρc) method is com- +patible with the calculation of the eigenfrequencies of the +normal vibration modes. +If the anisotropic dark energy star has a central den- +sity higher than one corresponding to the maximum-mass +configuration (indicated by full blue and orange circles +in Figs. 3 and 6), the star will become unstable against +radial perturbations and collapse to form a black hole. +For further details on the dissipative gravitational col- +lapse of compact stellar objects we also refer the reader +to Refs. [55, 97–99]. +Nonetheless, we must point out +that there are EoS models that allow a compact star to +migrate to another branch of stable solutions instead of +forming a black hole when it is subjected to a perturba- +tion. As a matter of fact, the first-order phase transition +between nuclear and quark matter can generate multiple +stable branches in the mass-radius diagram for hybrid +stars [100]. +C. +Moment of inertia +To calculate the moment of inertia of anisotropic dark +energy stars, we first need to solve the differential equa- +tion for the rotational drag (16) with boundary condi- +tions (19). In particular, for model I and central density +ρc = 1.5 × 1018 kg/m3, figure 7 illustrates the angular +velocity everywhere for several values of α. As can be +observed in the right plot, the dragging angular velocity +outside the star has the behavior ω(r) ∼ r−3, so that at +infinity (where spacetime is flat) the distant local inertial +frames do not rotate around the star, namely, ω(r) → 0 +for r → ∞. Moreover, anisotropy significantly affects the +angular velocity of the local inertial frames in the inte- +rior region of the star. More specifically, the dragging +angular velocity increases (decreases) for positive (nega- +tive) values of the anisotropy parameter α. We can then +determine the moment of inertia using the integral given +in Eq. (18). For the above central density, we present the +moment of inertia of some dark energy configurations for +both models in Table II, where it can be noticed that I +increases as the value of α increases. +We can now calculate the moment of inertia for a whole +sequence of dark energy stars by varying the central den- +sity ρc. The left panel of Fig. 8 displays the moment of +inertia as a function of the gravitational mass for both +models. Remarkably, model II provides larger values for +the moment of inertia than model I. Indeed, the maxi- +mum value Imax depends quite sensitively on the free pa- +rameters A and B in the EoS (28). In addition, the main +effect of anisotropy on the moment of inertia for slow ro- +tation occurs in the high-mass region, while its influence +is irrelevant for sufficiently low masses. In order to bet- +ter quantify the changes in the maximum values of the +moment of inertia induced by the anisotropic pressure, +we can define the following relative difference +∆I = Imax,ani − Imax,iso +Imax,iso +, +(33) +where Imax,iso and Imax,ani are the maximum values of the +moment of inertia for isotropic and anisotropic configura- +tions, respectively. In the right plot of Fig. 8 we present +the dependence ∆I against the anisotropy parameter α. +The impact of anisotropy is getting stronger as |α| grows, +reaching variations (with respect to the isotropic case) of +up to ∼ 20% for α = 0.5. We can also note that such +relative variations are almost independent of the model +adopted. +D. +Tidal properties +We will now investigate how the anisotropy parameter +α affects the tidal properties of dark energy stars. Given +a specific value of α, this requires solving the differential +equation (23) for a range of central densities. The left +panel of Fig. 9 is the result of calculating the tidal Love +number (27) for a sequence of stellar configurations by +considering different values of α, where the isotropic case +corresponds to α = 0. Similar to the trends in strange +quark stars, as reported in Ref. [70], the Love number of +dark energy stars grows until it reaches a maximum value +and then decreases as compactness increases. Note also +that the maximum value of k2 is sensitive to the value +of α, indicating that the Love number decreases as the +parameter α increases for both models. Although model +II provides larger maximum masses (as well as redshift +and moment of inertia) than model I, we see that the +behavior is different for the maximum values in the tidal +Love number. +Ultimately, in the right plot of Fig. 9, the dimensionless +tidal deformability Λ = ¯λ/M 5 is plotted as a function of +mass, where it can be observed that smaller masses yield +higher deformabilities. +In each model, the presence of +anisotropy has a negligible effect on Λ for small masses, +while slightly more significant changes take place only in +the high-mass region. +V. +CONCLUSIONS AND OUTLOOK +In this work, we have focused on the equilibrium struc- +ture of dark energy stars by using a Chaplygin-like equa- +tion of state under the presence of both isotropic and +anisotropic pressures within the context of standard GR. +Our goal was to construct stable compact stars whose +characteristics could be compared with the observational +data on the mass-radius diagram. +In this perspective, +the global properties of a compact star such as radius, +mass, redshift, moment of inertia, oscillation spectrum + +11 +n=0 mode +n=1 mode +n=2 mode +n=3 mode +n=4 mode +n=5 mode +0 +2 +4 +6 +8 +10 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +r [km] +ζn (r) +n=0 mode +n=1 mode +n=2 mode +n=3 mode +n=4 mode +n=5 mode +0 +2 +4 +6 +8 +10 +-1.5 +-1.0 +-0.5 +0.0 +r [km] +Δpr,n (r) [1035 Pa] +FIG. 5. Numerical solution of the radial pulsation equations (8) and (9) in the case of an anisotropic dark energy star with +central density ρc = 1.5 × 1018 kg/m3, α = 0.4 and EoS parameters given by model I. The radius, mass and the fundamental +mode frequency for such configuration are found in Table II. The lines with different colors and styles indicate different overtones +so that the solution corresponding to the nth vibration mode contains n nodes in the internal structure of the star. Note that +the eigenfunctions ζn(r) have been normalized assuming ζ = 1 at r = 0, and the Lagrangian perturbation of the radial pressure +∆pr,n(r) obeys the boundary condition (11) at the stellar surface. Since f0 is real, this configuration corresponds to a stable +anisotropic dark energy star. +Model I +Model II +17.8 +18.0 +18.2 +18.4 +18.6 +0.0 +0.5 +1.0 +1.5 +Log ρc [kg/m3] +ν0 +2 [109 s-2] +-0.05 +0. +0.05 +18.12 +18.22 +18.32 +18.42 +Model I +Model II +α = -0.4 +α = -0.2 +α = 0 +α = 0.2 +α = 0.4 +1.0 +1.5 +2.0 +2.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +M [M⊙] +ν0 +2 [109 s-2] +FIG. 6. Left panel: Squared frequency of the fundamental pulsation mode as a function of central mass density for anisotropic +dark energy stars predicted by Einstein gravity. The full blue and orange circles indicate the central density values where +ν2 +0 = 0, whose values precisely correspond to the maximum-mass points on the M(ρc) curves on the right plot of Fig. 3. Right +plot: Squared frequency of the fundamental mode versus gravitational mass, where it can be observed that the maximum-mass +values determine the boundary between stable and unstable stars. +and tidal deformability have been calculated. To describe +the anisotropic pressure within the dark energy fluid we +have adopted the anisotropy profile proposed by Horvat +et al. [51], where a free parameter α measures the degree +of anisotropy. +We have discussed the possibility of observing sta- +ble dark energy stars made of a negative pressure fluid +“−B/ρ” plus a barotropic component “Aρ”. By way of +comparison, the EoS parameters A and B have been cho- +sen in such a way that they agree sufficiently with the +observational data, e.g. the mass-radius constraint from +the GW170817 event. For isotropic configurations, we +have shown that various sets of values {A, B} can be +chosen since they obey the causality condition and con- +sistently describe compact stars observed in the Universe. +Furthermore, we saw that the secondary component re- +sulting from the gravitational-wave signal GW190814 [91] +can be described as a dark energy star using A = 0.4 and + +12 +α = -0.4 +α = -0.2 +α = 0 +α = 0.2 +α = 0.4 +0 +10 +20 +30 +40 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +r [km] +ϖ/Ω +α = -0.4 +α = -0.2 +α = 0 +α = 0.2 +α = 0.4 +0 +10 +20 +30 +40 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +r [km] +ω/Ω +FIG. 7. Left panel: Numerical solution of the differential equation (16) for a dark energy star described by model I and central +density ρc = 1.5 × 1018 kg/m3 in the presence of anisotropy for several values of the free parameter α. The solid and dashed +lines represent the interior and exterior solutions, respectively. Right panel: Ratio of frame-dragging angular velocity to the +angular velocity of the star, namely ω(r)/Ω = 1 − ϖ(r)/Ω. It can be observed that the outer solution behaves asymptotically +at large distances from the surface of the star (this is, ω → 0 for r → ∞). Furthermore, appreciable changes in the angular +velocity due to anisotropy can be noticeable, mainly in the interior region of the star. +Model I +Model II +α = -0.4 +α = -0.2 +α = 0 +α = 0.2 +α = 0.4 +0.5 +1.0 +1.5 +2.0 +2.5 +0 +1 +2 +3 +4 +M [M⊙] +I [1038 kg.m2] +Model I +Model II +-0.4 +-0.2 +0.0 +0.2 +0.4 +-15 +-10 +-5 +0 +5 +10 +15 +20 +α +ΔI [%] +FIG. 8. Left panel: Moment of inertia versus mass for anisotropic dark energy stars, where a higher mass results in larger +moment on inertia for both models. It is observed that the substantial impact of anisotropy on the moment of inertia occurs +predominantly in the high-mass branch. Right panel: Relative deviation (33) as a function of the anisotropy parameter. The +maximum value of the moment of inertia can undergo variations with respect to its isotropic counterpart of up to ∼ 20% for +α = 0.5. +B ∈ [4, 5]µ. +Based on these results, we have established two mod- +els with different values A and B in order to explore +the effects of anisotropy in the interior region of a dark +energy star. In particular, the maximum-mass values in- +crease as the parameter α increases. +We noticed that +model I without anisotropic pressures is not capable of +generating maximum masses above 2M⊙. However, the +inclusion of anisotropies (α = 0.4) allows a significant in- +crease in the maximum mass and thus a more favorable +description of the compact objects observed in nature. +On the other hand, model II with anisotropies fits bet- +ter with the observational measurements, although such +a model can lead to the formation of ultra-compact ob- +jects for sufficiently large values of α. We also calculated +the surface gravitational redshift for such stars, and our +results indicated that zsur is substantially affected by the +anisotropy in the high-mass branch, while the changes +are irrelevant for sufficiently low masses. +A star exists in the Universe only if it is dynamically +stable, so our second task was to investigate whether the +dark energy stars are stable or unstable with respect to an +adiabatic radial perturbation. Our results showed that +the standard criterion for radial stability dM/dρc > 0 +still holds for dark energy stars since the squared fre- +quency of the fundamental pulsation mode (ν2 +0) van- + +13 +Model I +Model II +α = -0.4 +α = -0.2 +α = 0 +α = 0.2 +α = 0.4 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.010 +0.015 +0.020 +C +k2 +Model I +Model II +α = -0.4 +α = -0.2 +α = 0 +α = 0.2 +α = 0.4 +0.5 +1.0 +1.5 +2.0 +2.5 +1 +5 +10 +50 +100 +500 +1000 +M [M⊙] +Λ +FIG. 9. Left panel: Tidal Love number plotted as a function of the compactness C ≡ M/R. Right panel: Dimensionless tidal +deformability versus gravitational mass predicted by each model, where larger masses yield smaller deformabilities. Note also +that the Love number is substantially modified by the anisotropy parameter α for both models, while its greatest effect on tidal +deformability Λ occurs only in the high-mass region. +ishes at the critical central density corresponding to the +maximum-mass configuration. This has been examined +in detail for both isotropic (α = 0) and anisotropic +(α ̸= 0) stellar configurations. +In the slowly rotating approximation, where only first- +order terms in the angular velocity are kept, we have also +determined the moment of inertia of anisotropic dark en- +ergy stars. For this purpose, we first had to calculate the +frame-dragging angular velocity for each central density. +The presence of anisotropic pressure results in a substan- +tial increase (decrease) of the angular velocity ω for more +positive (negative) values of α. We found that the signif- +icant impact of the anisotropy on the moment of inertia +occurs mainly in the high-mass branch for both models. +Furthermore, the maximum value of the moment of in- +ertia can undergo variations of up to ∼ 20% for α = 0.5 +as compared with the isotropic case. +We have analyzed the effect of anisotropic pressure on +the tidal properties of such stars. In particular, our out- +comes revealed that the tidal Love number is sensitive to +moderate variations of the parameter α, indicating that +the maximum value of k2 can increase as α decreases. +In addition, the greatest effect of anisotropy on the di- +mensionless tidal deformability takes place only in the +high-mass region. +Based on the foregoing results, the +present work thereby serves to develop a comprehensive +perspective on the relativistic structure of dark energy +stars in the presence of anisotropy. +Summarizing, we have explored the possible existence +of stable dark energy stars whose masses and radii are +not in disagreement with the current observational data. +The Chaplygin-like EoS predicts maximum-mass values +consistent with observational measurements of highly +massive pulsars. Future research includes the adoption +of widespread versions of Chaplygin gas that best fit +key cosmological parameters. In future studies we will +thereby take further steps in that direction, focusing on +the different types of generalized Chaplygin gas models +as discussed in Ref. [11]. In addition, as carried out in the +case of boson stars [101], it would be interesting to em- +ploy a Fisher matrix analysis in order to distinguish dark +energy stars from black holes and neutron stars from tidal +interactions in inspiraling binary systems. It is also worth +mentioning that Romano [102] has recently discussed the +effects of dark energy on the propagation of gravitational +waves. In that regard, we expect that future electromag- +netic observations of compact binaries and gravitational- +wave astronomy will provide a better understanding of +compact stars in the presence of dark energy, and even +help us answer the most basic question: How did dark +energy form in the Universe? 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Romano, arXiv:2211.05760 [gr-qc] (2022). + diff --git a/0dE1T4oBgHgl3EQf4wVz/content/tmp_files/load_file.txt b/0dE1T4oBgHgl3EQf4wVz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..553bf963471a66db8f3904ed6e95474a9b725950 --- /dev/null +++ b/0dE1T4oBgHgl3EQf4wVz/content/tmp_files/load_file.txt @@ -0,0 +1,1380 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf,len=1379 +page_content='Radial pulsations, moment of inertia and tidal deformability of dark energy stars Juan M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Pretel1, ∗ 1Centro Brasileiro de Pesquisas F´ısicas, Rua Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Xavier Sigaud, 150 URCA, Rio de Janeiro CEP 22290-180, RJ, Brazil (Dated: January 10, 2023) We construct dark energy stars with Chaplygin-type equation of state (EoS) in the presence of anisotropic pressure within the framework of Einstein gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' From the classification established by Iyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Quantum Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 2, 219 (1985)], we discuss the possible existence of isotropic dark energy stars as compact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' However, there is the possibility of constructing ultra-compact stars for sufficiently large anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' We investigate the stellar stability against radial oscillations, and we also determine the moment of inertia and tidal deformability of these stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' We find that the usual static criterion for radial stability dM/dρc > 0 still holds for dark energy stars since the squared frequency of the fundamental pulsation mode vanishes at the critical central density corresponding to the maximum-mass configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The dependence of the tidal Love number on the anisotropy parameter α is also examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' We show that the surface gravitational redshift, moment of inertia and dimensionless tidal deformability undergo significant changes due to anisotropic pressure, primarily in the high-mass region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Furthermore, in light of the detection of gravitational waves GW190814, we explore the possibility of describing the secondary component of such event as a stable dark energy star in the presence of anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' INTRODUCTION Different types of observations (such as Type Ia super- novae, structure formation and CMB anisotropies) indi- cate that our Universe is not only expanding, it is acceler- ating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Within the standard ΛCDM model (which is based on cold dark matter and cosmological constant in Ein- stein gravity), this cosmic acceleration is due to a smooth component with large negative pressure and repulsive gravity, the so-called dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Such a model gives a good agreement with the recent observational data [1], but suffers from the well-known coincidence problem and the fine-tuning problem [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The exact physical nature of dark energy is still a mystery and, consequently, the possibility that dark matter and dark energy could be different manifestations of a single substance has been considered [4–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In that regard, it was shown that the in- homogeneous Chaplygin gas offers a simple unified model of dark matter and dark energy [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' It was also argued that if the Universe is dominated by the Chaplygin gas a cosmological constant would be ruled out with high confidence [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Using the Planck 2015 CMB anisotropy, type-Ia su- pernovae and observed Hubble parameter data sets, the full parameter space of the modified Chaplygin gas was measured by Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Based on recent observations of high-redshift quasars, Zheng and colleagues [11] inves- tigated a series of Chaplygin gas models as candidates for dark matter-energy unification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The application of the Hamilton-Jacobi formalism for generalized Chaplygin gas models was carried out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Additionally, it is worth mentioning that Odintsov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [13] considered two different equations of state for dark energy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=', power- law and logarithmic effective corrections to the pressure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' ∗ juanzarate@cbpf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='br They showed that the power-law model only yielded some modest results, achieved under negative values of bulk viscosity, while the logarithmic scenario provide good fits in comparison to the ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Another way to give rise to an accelerated expansion of the Universe is by modifying the geometry itself [14, 15], namely, considering higher curvature corrections to the standard Einstein-Hilbert action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Under this outlook, the cosmic acceleration can be modeled in the scope of a scalar-tensor gravity theory [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Moreover, within the context of the so-called f(R) theories [18, 19], the quadratic term in the Ricci scalar R leads to an infla- tionary solution in the early Universe [20], although such a model does not provide a late-time accelerated expan- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Nevertheless, the late-time acceleration era can be realized by terms containing inverse powers of R [21], though it was shown that this is not compatible with the solar system experiments [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' For a comprehensive study on the evolution of the early and present Universe in f(R) modified gravity, we refer the reader to the re- view articles [23–25] and references contained therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' On the other hand, the astrophysical implications due to the f(R) modified gravitational Lagrangian on com- pact stars have been intensively investigated in the past few years [26–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' According to the aforementioned works, different dark energy models have been proposed in order to explain the mechanisms that lead to the cosmic acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Only about 4% of the Universe is made of familiar atomic mat- ter, 20% dark matter, and it turns out that roughly 76% of the Universe is dark energy [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Within the context of General Relativity, dark energy is an exotic negative pressure contribution that can lead to the observed accel- erated expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In the absence of consensus regarding a theoretical description for the current accelerated expan- sion of the Universe, theorists have proposed using the Chaplygin gas as a useful phenomenological description arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='03504v1 [gr-qc] 9 Jan 2023 2 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' If dark energy is distributed anywhere permeating or- dinary matter, then it could be present in the interior of a compact star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Therefore, the purpose of this manuscript is to investigate the possible existence of compact stars with dark energy by assuming a Chaplygin-type EoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' For such stars to exist in nature, they need to be stable under small radial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Adopting a description of dark energy by means of a phantom (ghost) scalar field, Yazadjiev [35] constructed a general class of exact interior solutions describing mixed relativistic stars containing both ordinary matter and dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The energy conditions and gravitational wave echoes of such stars were recently analyzed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Furthermore, the effect of the dynamical scalar field quintessence dark energy on neutron stars was inves- tigated in [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Panotopoulos and collaborators [38] stud- ied slowly rotating dark energy stars made of isotropic matter using the Chaplygin EoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Bhar [39] proposed a model for a dark energy star made of dark and ordinary matter in the Tolman–Kuchowicz spacetime geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' For further stellar models with dark energy we also refer the reader to Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [40–48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In addition, anisotropy in compact stars may arise due to strong magnetic fields, pion condensation, phase tran- sitions, mixture of two fluids, bosonic composition, rota- tion, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Thus, regardless of the specific source of the anisotropy, it is more natural to think of anisotropic fluids when studying compact stars at densities above nuclear saturation density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In that regard, the literature offers some physically motivated functional relations for the anisotropy, see for example Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [49–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' However, we must point out that these anisotropic models are based on general assumptions (or ansatzes) that do not directly relate to exotic modifications of matter or gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In- deed, it has been argued that the deformation near the maximum neutron-star mass comes from the anisotropic pressure within these stars, which is caused by the distor- tion of Fermi surface predicted by the equation of state of the models [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Becerra-Vergara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [57] showed that the contribution of the fourth order corrections pa- rameter (a4) of the QCD perturbation on the radial and tangential pressure generate significant effects on the mass-radius relation and the stability of quark stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' It has also been shown that the stellar structure equations in Eddington-inspired Born-Infeld theory with isotropic matter can be recast into GR with a modified (apparent) anisotropic matter [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Motivated by the several works already mentioned, we aim to discuss the impact of anisotropy on the macro- scopic properties of dark energy stars with Chaplygin-like EoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' We will address the following questions: Do these stars belong to families of compact or ultra-compact stars?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' How does anisotropy affect the compactness and radial stability of dark energy stars satisfying the causal- ity condition?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In particular, by adopting the phenomeno- logical ansatz proposed by Horvat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [51], we deter- mine the radius, mass, gravitational redshift, frequency of the fundamental oscillation mode, moment of inertia and the dimensionless tidal deformability of anisotropic dark energy stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The isotropic solutions are recovered when the anisotropy parameter vanishes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' when α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The organization of this paper is as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' II we start with a brief overview of relativistic stellar struc- ture, describing the basic equations for radial pulsations, moment of inertia and tidal deformability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' We then in- troduce the Chaplygin-like EoS and discuss its relation to the cosmological context in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' III, as well as we present the anisotropy profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Section IV provides a discussion of the numerical results for the different physical prop- erties of dark energy stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Finally, our conclusions are summarized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' STELLAR STRUCTURE EQUATIONS In order to study the basic features of compact stars with dark energy, in this section we briefly summarize the stellar structure equations in Einstein gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In particu- lar, we focus on hydrostatic equilibrium structure, radial pulsations, moment of inertia, and tidal deformability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The theory of gravity to be used in this work is general relativity, where the Einstein field equations are given by Gµν ≡ Rµν − 1 2gµνR = 8πTµν, (1) with Gµν being the Einstein tensor, Rµν the Ricci tensor, R denotes the scalar curvature, and Tµν is the energy- momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Since we are interested in isolated compact stars, we consider that the spacetime can be described by the spherically symmetric four-dimensional line element ds2 = −e2ψdt2 + e2λdr2 + r2(dθ2 + sin2 θdφ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' (2) In addition, we model the compact-star matter by an anisotropic perfect fluid, whose energy-momentum tensor is given by Tµν = (ρ + pt)uµuν + ptgµν − σkµkν, (3) where ρ is the energy density, σ ≡ pt − pr the anisotropy factor, pr the radial pressure, pt the tangential pressure, uµ the four-velocity of the fluid, and kµ is a unit four- vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' These four-vectors must satisfy uµuµ = −1, kµkµ = 1 and uµkµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Notice that the stellar fluid becomes isotropic when σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' TOV equations When the stellar fluid remains in hydrostatic equilib- rium, neither metric nor thermodynamic quantities de- pend on the time coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' This allows us to write uµ = e−ψδµ 0 and kµ = e−λδµ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Accordingly, the hydro- static equilibrium of an anisotropic compact star is gov- 3 erned by the TOV equations: dm dr = 4πr2ρ, (4) dpr dr = −(ρ + pr) �m r2 + 4πrpr � � 1 − 2m r �−1 + 2σ r , (5) dψ dr = − 1 ρ + pr dpr dr + 2σ r(ρ + pr), (6) which are obtained from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' (1)-(3) together with the conservation law ∇µT µ 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The metric function λ(r) is determined from the relation e−2λ = 1 − 2m/r, where m(r) is the gravitational mass within a sphere of radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' By supplying an EoS for the radial pressure in the form pr = pr(ρ) and a defined anisotropy relation for σ, the system of differential equations (4)-(6) is then numeri- cally integrated from the center at r = 0 to the surface of the star r = R which correspond to a vanishing pres- sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Therefore, the above equations will be solved under the requirement of the following boundary conditions ρ(0) = ρc, m(0) = 0, ψ(R) = 1 2 ln � 1 − 2M R � , (7) where ρc is the central energy density, and M ≡ m(R) is the total mass of the star calculated at its surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The numerical solution of the TOV equations describes the equilibrium background and allow us to obtain the metric components and fluid variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Radial oscillations A rigorous analysis of the radial stability of compact stars requires the calculation of the frequencies of nor- mal vibration modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Such frequencies can be found by considering small deviations from the hydrostatic equi- librium state but maintaining the spherical symmetry of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In the linear treatment, where all quadratic (or higher-order) or mixed terms in the perturbations are discarded, one assumes that all perturbations in physical quantities are arbitrarily small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The fluid element lo- cated at r in the unperturbed configuration is displaced to radial coordinate r + ξ(t, r) in the perturbed config- uration, where ξ is the Lagrangian displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' All perturbations have a harmonic time dependence of the form ∼ eiνt, where ν is the oscillation frequency to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Consequently, defining ζ ≡ ξ/r, the adia- batic1 radial pulsations of anisotropic compact stars are 1 In the adiabatic theory, it is assumed that the fluid elements of the star neither gain nor lose heat during the oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' governed by the following differential equations [55] dζ dr = − 1 r � 3ζ + ∆pr γpr + 2σζ ρ + pr � + dψ dr ζ, (8) d(∆pr) dr = ζ � ν2e2(λ−ψ)(ρ + pr)r − 4dpr dr −8π(ρ + pr)e2λrpr + r(ρ + pr) �dψ dr �2 +2σ �4 r + dψ dr � + 2dσ dr � + 2σ dζ dr − ∆pr �dψ dr + 4π(ρ + pr)re2λ � + 2 r δσ, (9) where ∆pr is the Lagrangian perturbation of the radial pressure and γ = (1+ρ/pr)dpr/dρ is the adiabatic index at constant specific entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The above first-order time-independent equations (8) and (9) require boundary conditions set at the center and surface of the star, similar to a vibrating string fixed at its ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' (8) has a singularity at the origin, the following condition must be required ∆pr = − 2σζ ρ + pr γpr − 3γζpr as r → 0, (10) while the Lagrangian perturbation of the radial pressure at the surface must satisfy ∆pr = 0 as r → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' (11) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Moment of inertia Suppose a particle is dropped from rest at a great dis- tance from a rotating star, then it would experience an ever increasing drag in the direction of rotation as it ap- proaches the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Based on this description, we intro- duce the angular velocity acquired by an observer falling freely from infinity, denoted by ω(r, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Here we will cal- culate the moment of inertia of an anisotropic dark en- ergy star under the slowly rotating approximation [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' This means that when we consider rotational corrections only to first order in the angular velocity of the star Ω, the line element (2) is replaced by its slowly rotating counterpart, namely ds2 = − e2ψ(r)dt2 + e2λ(r)dr2 + r2(dθ2 + sin2 θdφ2) − 2ω(r, θ)r2 sin2 θdtdφ, (12) and following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [59], it is pertinent to define the differ- ence ϖ ≡ Ω−ω as the coordinate angular velocity of the fluid element at (r, θ) seen by the freely falling observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Keep in mind that Ω is the angular velocity of the stel- lar fluid as seen by an observer at rest at some spacetime point (t, r, θ, φ), and hence the four-velocity up to linear terms in Ω can be written as uµ = (e−ψ, 0, 0, Ωe−ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' To this order, the spherical symmetry is still preserved and 4 it is possible to extend the validity of the TOV equations (4)-(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Nonetheless, the 03-component of the field equa- tions contributes an additional differential equation for angular velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' By retaining only first-order terms in Ω, such component becomes eψ−λ r4 ∂ ∂r � e−(ψ+λ)r4 ∂ϖ ∂r � + 1 r2 sin3 θ ∂ ∂θ � sin3 θ∂ϖ ∂θ � = 16π(ρ + pt)ϖ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' (13) As in the case of isotropic fluids, we follow the same treatment carried out by Hartle [59, 60] and we assume that ϖ can be written as ϖ(r, θ) = ∞ � l=1 ϖl(r) � −1 sin θ dPl dθ � , (14) where Pl are Legendre polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Taking this expan- sion into account, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' (13) becomes eψ−λ r4 d dr � e−(ψ+λ)r4 dϖl dr � − l(l + 1) − 2 r2 ϖl = 16π(ρ + pt)ϖl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' (15) At a distance far away from the star, where e−(ψ+λ) becomes unity, the asymptotic solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' (15) takes the form ϖl(r) → a1r−l−2 + a2rl−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' If spacetime is to be flat at large r, then ω → 2J/r3 (or equivalently, ϖ → Ω − 2J/r3) for r → ∞, where J is the total angular momentum of the star [59, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Therefore, comparing this with the asymptotic behavior of ϖl(r), we find that l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' As a result, ϖ is a function only of the radial coordinate, and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' (15) reduces to eψ−λ r4 d dr � e−(ψ+λ)r4 dϖ dr � = 16π(ρ + pt)ϖ, (16) which can be integrated to give � r4 dϖ dr � R = 16π � R 0 (ρ + pt)r4eλ−ψϖdr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' (17) In view of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' (17), we can obtain the angular mo- mentum J and hence the moment of inertia I = J/Ω of a slowly rotating anisotropic star: I = 8π 3 � R 0 (ρ + pr + σ)eλ−ψr4 �ϖ Ω � dr, (18) which reduces to the expression given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [61] for isotropic compact stars when σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' For an arbitrary choice of the central value ϖ(0), the appropriate bound- ary conditions for the differential equation (16) come from the requirements of regularity at the center of the star and asymptotic flatness at infinity, namely dϖ dr ���� r=0 = 0, lim r→∞ ϖ = Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' (19) Once the solution for ϖ(r) is found, we can then deter- mine the moment of inertia through the integral (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' It is remarkable that the above expression for I is referred to as the “slowly rotating” approximation because it was obtained to lowest order in the angular velocity Ω [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' This means that the stellar structure equations are still given by the TOV equations (4)-(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Tidal deformability It is well known that the tidal properties of neutron stars are measurable in gravitational waves emitted from the inspiral of a binary neutron-star coalescence [62, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In that regard, here we also study the dimensionless tidal deformability of individual dark energy stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' To do so, we follow the procedure carried out by Hinderer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [64] (see also Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [65–70] for additional results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The basic idea is as follows: In a binary system, the deformation of a compact star due to the tidal effect created by the companion star is characterized by the tidal deformabil- ity parameter ¯λ = −Qij/Eij, where Qij is the induced quadrupole moment tensor and Eij is the tidal field ten- sor [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Namely, the latter describes the tidal field from the spacetime curvature sourced by the distant compan- ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The tidal parameter is related to the tidal Love number k2 through the relation2 ¯λ = 2 3k2R5, (20) but it is common in the literature to define the dimen- sionless tidal deformability Λ = ¯λ/M 5, so in our results we will focus on Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The calculation of ¯λ requires consider- ing linear quadrupolar perturbations (due to the external tidal field) to the equilibrium configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Thus, the spacetime metric is given by gµν = g0 µν + hµν, where g0 µν describes the equilibrium configuration and hµν is a lin- earized metric perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' For static and even-parity perturbations in the Regge-Wheeler gauge [71], the per- turbed metric can be written as [64] hµν = diag � −e2ψ(r)H0, e2λ(r)H2, r2K, r2 sin2 θK � Y2m(θ, φ), (21) where H0, H2 and K are functions of the radial coordi- nate, and Ylm are the spherical harmonics for l = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Since the perturbed energy-momentum tensor is given by δT ν µ = diag(−δρ, δpr, δpt, δpt), the linearized field 2 It should be noted that the tidal deformability parameter is be- ing denoted by ¯λ in order not to be confused with the metric component λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 5 equations imply that: � � � � � H0 = −H2 ≡ H from δG2 2 − δG3 3 = 0, K′ = 2Hψ′ + H′ from δG2 1 = 0, δpt = H 8πre−2λ(λ′ + ψ′)Y2m from δG2 2 = 8πδpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In addition, from δG0 0 − δG1 1 = −8π(δρ + δpt), we can obtain the following differential equation [72] H′′ + PH′ + QH = 0, (22) or alternatively, ry′ = −y2 + (1 − rP)y − r2Q, (23) where we have defined y ≡ rH′ H , (24) P ≡ 2 r + e2λ �2m r2 + 4πr(pr − ρ) � , (25) Q ≡ 4πe2λ � 4ρ + 8pr + ρ + pr Av2sr (1 + v2 sr) � − 6e2λ r2 − 4ψ′2, (26) with A ≡ dpt/dpr and vsr being the radial speed of sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' By matching the internal solution with the external solution of the perturbed variable H at the surface of the star r = R, we obtain the tidal Love number [72] k2 = 8 5(1 − 2C)2C5 [2C(yR − 1) − yR + 2] × � 2C[4(yR + 1)C4 + (6yR − 4)C3 + (26 − 22yR)C2 + 3(5yR − 8)C − 3yR + 6 � + 3(1 − 2C)2 [2C(yR − 1) − yR + 2] log(1 − 2C) �−1 , (27) where C ≡ M/R is the compactness of the star, and yR ≡ y(R) is obtained by integrating equation (23) from the origin up to the stellar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' EQUATION OF STATE AND ANISOTROPY MODEL As it is well known, a possible alternative to the Phan- tom and Quintessence fields is the Chaplygin gas, where the EoS assumes the form pr = −B/ρ, with B being a positive constant (given in m−4 units).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In fact, it was ar- gued that such gas could provide a solution to unify the effects of dark matter in the early times and dark energy in late times [4, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Although the literature provides a more generalized version for such EoS in the context of the Friedmann-Lemaˆıtre-Robertson-Walker Universe [5– 7, 73–77], here we will use the simplest form plus a linear term corresponding to a barotropic fluid, namely pr = Aρ − B ρ , (28) where A is a positive dimensionless constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Our model is characterized by two free parameters A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Never- theless, we must emphasize here that Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [10] consid- ered an equation of state with three degrees of freedom, specifically p = Aρ − B/ρα, where α is an extra param- eter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' They carried out a statistical treatment of astro- nomical data in order to constrain the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In the light of the Markov chain Monte Carlo method, they found that at 2σ level, α = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0156+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0982+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2346 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='1380−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2180 and A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0009+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0018+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0030 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0017−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0030 from CMB+JLA+CC data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In other words, the constants α and A are very close to zero and hence the nature of unified dark matter- energy model is very similar to the cosmological standard ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' On the other hand, at astrophysics level, compact stars obeying the EoS (28) have been investigated by several authors, see for example Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [38, 41, 43–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In this work we will adopt values of A and B for which appre- ciable changes in the mass-radius diagram can be visu- alized in order to compare our theoretical results with observational measurements of massive pulsars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In order to describe physically realistic compact stars, the causality condition must be respected throughout the interior region of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In other words, the speed of sound (defined by vs ≡ � dp/dρ) cannot be greater than the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Thus, in view of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' (28), we have v2 sr ≡ dpr dρ = A + B ρ2 , (29) and since the radial pressure vanishes at the surface of the star, then B = Aρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Thereby, the causality condition v2 sr(R) = 2A < 1 implies that A < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Besides, it is more realistic to consider stellar models where there exists a tangential pressure as well as a radial one, since anisotropies arise at high densities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' above the nuclear saturation density as considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Although the literature offers different functional rela- tions to model anisotropic pressures at very high densities inside compact stars [49–54], here we adopt the simplest model, which was proposed by Horvat and collaborators [51] σ = α �2m r � pr = α � 1 − e−2λ� pr, (30) where α is a dimensionless parameter that controls the amount of anisotropy within the stellar fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' This pa- rameter can assume positive or negative values of the order of unity, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [26, 32, 51, 52, 55, 78–82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' No- tice that the isotropic solutions are recovered when the value of α vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Specifically, the anisotropy ansatz (30) has two important characteristics: (i) the fluid be- comes isotropic at the center generating regular solutions and (ii) the effect of anisotropy vanishes in the hydro- static equilibrium equation in the Newtonian limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Un- like this profile, the effect of anisotropy does not van- ish in the hydrostatic equilibrium equation in the non- relativistic regime for the Bowers-Liang model [49], which 6 could be an unphysical trait as argued in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' For a broader discussion on the different ways of generating static spherically symmetric anisotropic fluid solutions, we refer the reader to the recent review article [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Since the Eulerian perturbation for the metric poten- tial λ can be written as δλ = −4πr(ρ+pr)e2λξ [55], then δσ takes the form δσ = α � (1 − e−2λ)δpr − 8πpr(ρ + pr)r2ζ � , (31) where it should be noted that the relation between the Eulerian and Lagrangian perturbations for radial pres- sure is given by ∆pr = δpr + rζp′ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The above expression will be substituted in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' (9) when we discuss later the radial pulsations in the stellar interior for at least some values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' NUMERICAL RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Equilibrium configurations So far we do not know exactly whether the millisecond pulsars (observed in compact binaries from optical spec- troscopic and photometric measurements) are hadronic, quark or hybrid stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In fact, it has been theorized that cold quark matter might exist at the core of heavy neutron stars [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Despite the precise measurements of masses [85–87] and radii [88–90], such constraints are still unable to distinguish the theoretical predictions coming from the different models for strange stars and (hybrid) neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' This means that the dense matter EoS within compact stars still remains poorly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Furthermore, a realistic compact star possesses high mag- netic fields and rotation properties, which significantly alter its internal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' For comparison reasons, it is therefore common to use the observational mass-radius measurements (in view of the detection of gravitational waves and electromagnetic signals) on the mass-radius diagrams for any type of EoS even being of different mi- croscopic compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In that perspective, our theoret- ical results will be compared with observational measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' We begin our discussion of dark energy stars by con- sidering the isotropic case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=', when σ = 0 in the TOV equations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' We numerically integrate Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' (4)-(6) from the center up to the surface of the star through the boundary conditions (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' As usual, the radius R is de- termined when the pressure vanishes, and the total mass M is calculated at the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The felt panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 1 exhibits the mass-radius relations of dark energy stars for different values of parameters A and B in the EoS (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Remark that we have adopted values of A less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 in order to respect the causality condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' One can observe that small values of A (see black curve) do not provide compact stars that fit current observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' How- ever, higher values of maximum mass can be obtained for larger values of A, see for example red and green curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' For a fixed value of A, the maximum mass decreases as the parameter B increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' We perceive that the sec- ondary component resulting from the gravitational-wave signal GW190814 [91] can be consistently described as a compact star with Chaplygin EoS (28) for A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 and B ∈ [4, 5]µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Furthermore, the magenta curve fits very well with all observational data, but its maximum-mass value is above 3M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Another interesting feature of these stars is their com- pactness, defined by C ≡ M/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' According to the clas- sification adopted by Iyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [92], the configurations shown in the mass-radius diagram correspond to compact stars, see the right plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Besides, we can appre- ciate that the compactness of dark energy stars is of the order of the compactness of hadronic-matter stars, as is the case of the SLy EoS [93], despite the fact that the maximum mass in the magenta configuration sequence can exceed 3M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Nonetheless, as we will see later, the introduction of anisotropy can turn such stars into ultra- compact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Of course, this will depend on the amount of anisotropy in the stellar interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In order to include anisotropic pressures and investi- gate their effects on the internal structure of dark energy stars, we will adopt two specific models with the following parameters ⋆ Model I: A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='3, B = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0µ , ⋆ Model II: A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4, B = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2µ , which are models favored by observational measurements according to the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Moreover, model II precisely corresponds to the first model considered by Panotopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Similar to the isotropic case, we numerically solve the hydrostatic background equations (4)-(6) with boundary conditions (7), but taking into account the anisotropy profile (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' For instance, for the model I and a central density ρc = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0×1018 kg/m3, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 2 illustrates the mass density, pressure and squared speed of sound as functions of the radial coordinate for different values of the free pa- rameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' We can see that the internal structure of a dark energy star is affected by the presence of anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In effect, the radius of the star increases (decreases) for more positive (negative) values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In addition, we re- mark that the speed of sound, both radial and tangential, respect the causality condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' This has also been veri- fied for other values of central density considered in the construction of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Varying the central density, we obtain the mass-radius diagrams and mass-central density relations for models I and II, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' We observe that the sub- stantial changes introduced by anisotropy in dark en- ergy stars occur in the high-mass branch (close to the maximum-mass point), while the effects are irrelevant at low central densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The maximum-mass values increase as the parameter α increases (see also the data in Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Note that model I without anisotropic pressures is not capable of generating maximum masses above 2M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 7 SLy A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2, B = 6μ A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='3, B = 3μ A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='3, B = 4μ A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='3, B = 5μ A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='3, B = 6μ A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4, B = 3μ A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4, B = 4μ A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4, B = 5μ A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4, B = 6μ A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='48, B = 3μ 4 6 8 10 12 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 R [km] M [M⊙] C > 1/3 (ultra-compact objects) 1/6 < C < 1/3 (compact objects) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='35 M [M⊙] C FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Left panel: Mass-radius diagrams for dark energy stars with Chaplygin-like EoS (28) and isotropic pressure (σ = 0) for several values of the positive parameters A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Here the constant B is given in µ = 10−20 m−4 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The gray horizontal stripe at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0M⊙ stands for the two massive NS pulsars J1614-2230 [85] and J0348+0432 [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Yellow and blue regions represent the observational measurements of the masses of the highly massive NS pulsars J0740+6620 [87] and J2215+5135 [94], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The filled pink band stands for the lower mass of the compact object detected by the GW190814 event [91], and the cyan area is the mass-radius constraint from the GW170817 event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Moreover, the NICER measurements for PSR J0030+0451 are displayed by black dots with their respective error bars [95, 96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Right panel: Variation of the compactness with total gravitational mass, where the gray and orange stripes represent compact and ultra-compact objects, respectively, according to the classification given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' For comparison reasons, we have included the results corresponding to the SLy EoS [93] by blue curves in both plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Nevertheless, the inclusion of anisotropies (see the blue curve for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4) allows a significant increase in the maximum mass and hence a more favorable description of the compact objects observed in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' On the other hand, model II with anisotropies (see orange curves) fits better with the observational measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In particu- lar, in view of the lower mass of the compact object from the coalescence GW190814 [91], two curves are partic- ularly outstanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In other words, such object can be well described as an anisotropic dark energy star when α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Moreover, model II with negative anisotropies (such as α = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4) favors the description of the massive pulsar J2215+5135 [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 4 describes the behavior of compactness as a function of central density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Positive anisotropies lead to an increase in compactness, mainly in the high-central-density branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Remarkably, for suf- ficiently large values of α (see purple curve), it is possible to obtain anisotropic dark energy stars as ultra-compact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The gravitational redshift, conventionally defined as the fractional change between observed and emitted wavelengths compared to emitted wavelength, in the case of a Schwarzschild star is given by [61] zsur = eλ(R) − 1 = � 1 − 2M R �−1/2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' (32) In the right plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 4, the surface gravitational red- TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Maximum-mass configurations with Chaplygin- like EoS (28) for model I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The energy density values correspond to the critical central density where the function M(ρc) is a maximum on the right plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Model α ρc [1018 kg/m3] R [km] M [M⊙] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='424 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='812 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='786 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='364 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='902 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='852 I 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='295 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='994 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='919 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='219 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='086 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='988 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='135 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='180 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='059 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='777 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='630 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='320 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='721 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='738 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='402 II 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='661 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='845 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='486 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='594 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='955 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='570 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='523 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='065 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='565 shift is plotted as a function of the total mass for both models I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' This plot indicates that the gravita- tional redshift of light emitted at the surface of a dark energy star is substantially affected by the anisotropy in the high-mass region, while the changes are negligible for sufficiently low masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' For a fixed value of central den- sity, Table II shows that positive (negative) anisotropy increases (decreases) the value of the redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 8 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Radius, mass, redshift, fundamental mode frequency (f0 = ν0/2π), moment of inertia and dimensionless tidal deformability of dark energy stars with central energy density ρc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 × 1018 kg/m3 as predicted by models I and II for several values of the anisotropy parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Remarkably, with the exception of the fundamental mode frequency and tidal deformability, these properties undergo a significant increase as α increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Model α R [km] M [M⊙] zsur f0 [kHz] I [1038 kg · m2] Λ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='062 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='713 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='418 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='414 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='695 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='278 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='163 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='781 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='440 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='312 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='820 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='709 I 0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='263 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='852 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='201 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='957 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='598 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='083 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='656 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='688 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='315 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='303 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='782 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='6 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='3 α = 0 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='3 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='6 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 r [km] ρ [kg/m3] Solid lines: pr Dashed lines: pt α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='6 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='3 α = 0 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='3 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='6 0 2 4 6 8 10 0 1 2 3 4 5 r [km] Pressure [1034 Pa] Solid lines: vsr 2 Dashed lines: vst 2 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='6 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='3 α = 0 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='3 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='6 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='8 r [km] (Speed of sound)2/c2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Radial behavior of the mass density (left panel), pressures (middle panel) and squared speed of sound (right panel) inside an anisotropic dark energy star with central density ρc = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 × 1018 kg/m3 and several values of the parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' All plots correspond to model I and the black curves represent the isotropic solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Note that both the radial and tangential speed of sound obey the causality condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Furthermore, one can observe that the increase in α leads to larger radii, and the anisotropy is more pronounced in the intermediate regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Oscillation spectrum A necessary condition (the well-known M(ρc) method) for stellar stability is that stable stars must lie in the re- gion where dM/dρc > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' According to the right plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 3, the full blue and orange circles on each curve indicate the onset of instability for each family of equi- librium solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' However, a sufficient condition is to calculate the frequencies of the radial vibration modes for each central density [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Here we will analyze if both methods are compatible in the case of dark energy stars including anisotropic pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Once the equilibrium equations (4)-(6) are integrated from the center to the surface of the star, we then pro- ceed to solve the radial pulsation equations (8) and (9) with the corresponding boundary conditions (10) and (11) using the shooting method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Namely, we integrate from the origin (where we consider the normalized eigen- functions ζ(0) = 1) up to the stellar surface for a set of trial values ν2 satisfying the condition (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In this way, the appropriate eigenfrequencies correspond to the values for which the boundary condition (11) is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' For instance, for a central density ρc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5×1018 kg/m3, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 and parameters given by model I, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 5 dis- plays the radial behavior of the perturbation variables for the first five squared eigenfrequencies ν2 n, where n indicates the number of nodes inside the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' This fre- quency spectrum forms an infinite discrete sequence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' ν2 0 < ν2 1 < ν2 2 < · · · , where the eigenvalue corresponding to n = 0 is the lowest one (or equivalently, the longest period of all the allowed vibration modes) and it is known as the fundamental mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Such mode has no nodes, 9 Model I Model II α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 α = 0 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 7 8 9 10 11 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 R [km] M [M⊙] Model I Model II α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 α = 0 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 Log ρc [kg/m3] M [M⊙] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Mass-radius diagram (left panel) and mass-central density relation (right panel) for anisotropic dark energy stars as predicted by model I (blue curves) and II (orange curves) with anisotropy profile (30) for several values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The colored bands in the left plot represent the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Moreover, the full blue and orange circles on the right plot indicate the maximum-mass points for model I and II, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Note that the maximum-mass values for model II correspond to lower central densities than those for model I, however, model II allows larger masses (see also Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The critical central density corresponding to the maximum point on the M(ρc) curve is modified by the presence of anisotropy for both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Model I Model II C > 1/3 (ultra-compact objects) 1/6 < C < 1/3 (compact objects) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='7 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 α = 0 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='35 Log ρc [kg/m3] C Model I Model II α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 α = 0 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='7 M [M⊙] zsur FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Left panel: Variation of the compactness with central density for several anisotropic dark energy star sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The gray and light-green stripes represent compact and ultra-compact objects, respectively, according to the classification established by Iyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Positive anisotropy results in increased compactness for sufficiently high central densities, while the opposite occurs for negative anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Note also that dark energy stars would correspond to ultra-compact objects if α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 for model II, see for instance the purple curve for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Right panel: Surface gravitational redshift as a function of the total mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In the high-redshift region it can be observed that positive (negative) anisotropy increases (decreases) the value of zsur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Meanwhile, the effect of anisotropy is irrelevant for sufficiently low redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' whereas the first overtone (n = 1) has one node, the second overtone (n = 2) has two, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Stable stars are described by their oscillatory behavior so that ν2 n > 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=', νn is purely real).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' On the other hand, if any of these is negative for a particular star, the frequency is purely imaginary and hence the star is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Since each higher-order mode has a squared eigenfre- quency that is larger than in the case of the preceding mode, it is enough to calculate the frequency of the fun- damental pulsation mode for the equilibrium sequences presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' With this in mind, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 6 we plot the squared frequency of the fundamental oscilla- tion mode as a function of the central density (left panel) and gravitational mass (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' According to the left plot, the squared frequency of the fundamental mode is exactly zero at the critical-central-density value corre- sponding to the maximum-mass configuration as shown in the right plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 3, see the full blue and orange cir- 10 cles for both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Furthermore, according to the right plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 6, the maximum-mass values (that is, when dM/dρc = 0) can be used as turning points from stability to dynamical instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Therefore, we can conclude that the usual criterion to guarantee stability dM/dρc > 0 is still valid for the case of anisotropic dark energy stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In other words, the conventional M(ρc) method is com- patible with the calculation of the eigenfrequencies of the normal vibration modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' If the anisotropic dark energy star has a central den- sity higher than one corresponding to the maximum-mass configuration (indicated by full blue and orange circles in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 3 and 6), the star will become unstable against radial perturbations and collapse to form a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' For further details on the dissipative gravitational col- lapse of compact stellar objects we also refer the reader to Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [55, 97–99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Nonetheless, we must point out that there are EoS models that allow a compact star to migrate to another branch of stable solutions instead of forming a black hole when it is subjected to a perturba- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' As a matter of fact, the first-order phase transition between nuclear and quark matter can generate multiple stable branches in the mass-radius diagram for hybrid stars [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Moment of inertia To calculate the moment of inertia of anisotropic dark energy stars, we first need to solve the differential equa- tion for the rotational drag (16) with boundary condi- tions (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In particular, for model I and central density ρc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 × 1018 kg/m3, figure 7 illustrates the angular velocity everywhere for several values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' As can be observed in the right plot, the dragging angular velocity outside the star has the behavior ω(r) ∼ r−3, so that at infinity (where spacetime is flat) the distant local inertial frames do not rotate around the star, namely, ω(r) → 0 for r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Moreover, anisotropy significantly affects the angular velocity of the local inertial frames in the inte- rior region of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' More specifically, the dragging angular velocity increases (decreases) for positive (nega- tive) values of the anisotropy parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' We can then determine the moment of inertia using the integral given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' For the above central density, we present the moment of inertia of some dark energy configurations for both models in Table II, where it can be noticed that I increases as the value of α increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' We can now calculate the moment of inertia for a whole sequence of dark energy stars by varying the central den- sity ρc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 8 displays the moment of inertia as a function of the gravitational mass for both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Remarkably, model II provides larger values for the moment of inertia than model I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Indeed, the maxi- mum value Imax depends quite sensitively on the free pa- rameters A and B in the EoS (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In addition, the main effect of anisotropy on the moment of inertia for slow ro- tation occurs in the high-mass region, while its influence is irrelevant for sufficiently low masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In order to bet- ter quantify the changes in the maximum values of the moment of inertia induced by the anisotropic pressure, we can define the following relative difference ∆I = Imax,ani − Imax,iso Imax,iso , (33) where Imax,iso and Imax,ani are the maximum values of the moment of inertia for isotropic and anisotropic configura- tions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In the right plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 8 we present the dependence ∆I against the anisotropy parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The impact of anisotropy is getting stronger as |α| grows, reaching variations (with respect to the isotropic case) of up to ∼ 20% for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' We can also note that such relative variations are almost independent of the model adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Tidal properties We will now investigate how the anisotropy parameter α affects the tidal properties of dark energy stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Given a specific value of α, this requires solving the differential equation (23) for a range of central densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 9 is the result of calculating the tidal Love number (27) for a sequence of stellar configurations by considering different values of α, where the isotropic case corresponds to α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Similar to the trends in strange quark stars, as reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [70], the Love number of dark energy stars grows until it reaches a maximum value and then decreases as compactness increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Note also that the maximum value of k2 is sensitive to the value of α, indicating that the Love number decreases as the parameter α increases for both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Although model II provides larger maximum masses (as well as redshift and moment of inertia) than model I, we see that the behavior is different for the maximum values in the tidal Love number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Ultimately, in the right plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 9, the dimensionless tidal deformability Λ = ¯λ/M 5 is plotted as a function of mass, where it can be observed that smaller masses yield higher deformabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In each model, the presence of anisotropy has a negligible effect on Λ for small masses, while slightly more significant changes take place only in the high-mass region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' CONCLUSIONS AND OUTLOOK In this work, we have focused on the equilibrium struc- ture of dark energy stars by using a Chaplygin-like equa- tion of state under the presence of both isotropic and anisotropic pressures within the context of standard GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Our goal was to construct stable compact stars whose characteristics could be compared with the observational data on the mass-radius diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In this perspective, the global properties of a compact star such as radius, mass, redshift, moment of inertia, oscillation spectrum 11 n=0 mode n=1 mode n=2 mode n=3 mode n=4 mode n=5 mode 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 r [km] ζn (r) n=0 mode n=1 mode n=2 mode n=3 mode n=4 mode n=5 mode 0 2 4 6 8 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 r [km] Δpr,n (r) [1035 Pa] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Numerical solution of the radial pulsation equations (8) and (9) in the case of an anisotropic dark energy star with central density ρc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 × 1018 kg/m3, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 and EoS parameters given by model I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The radius, mass and the fundamental mode frequency for such configuration are found in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The lines with different colors and styles indicate different overtones so that the solution corresponding to the nth vibration mode contains n nodes in the internal structure of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Note that the eigenfunctions ζn(r) have been normalized assuming ζ = 1 at r = 0, and the Lagrangian perturbation of the radial pressure ∆pr,n(r) obeys the boundary condition (11) at the stellar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Since f0 is real, this configuration corresponds to a stable anisotropic dark energy star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Model I Model II 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 Log ρc [kg/m3] ν0 2 [109 s-2] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='05 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='12 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='22 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='32 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='42 Model I Model II α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 α = 0 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 M [M⊙] ν0 2 [109 s-2] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Left panel: Squared frequency of the fundamental pulsation mode as a function of central mass density for anisotropic dark energy stars predicted by Einstein gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The full blue and orange circles indicate the central density values where ν2 0 = 0, whose values precisely correspond to the maximum-mass points on the M(ρc) curves on the right plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Right plot: Squared frequency of the fundamental mode versus gravitational mass, where it can be observed that the maximum-mass values determine the boundary between stable and unstable stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' and tidal deformability have been calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' To describe the anisotropic pressure within the dark energy fluid we have adopted the anisotropy profile proposed by Horvat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [51], where a free parameter α measures the degree of anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' We have discussed the possibility of observing sta- ble dark energy stars made of a negative pressure fluid “−B/ρ” plus a barotropic component “Aρ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' By way of comparison, the EoS parameters A and B have been cho- sen in such a way that they agree sufficiently with the observational data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' the mass-radius constraint from the GW170817 event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' For isotropic configurations, we have shown that various sets of values {A, B} can be chosen since they obey the causality condition and con- sistently describe compact stars observed in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Furthermore, we saw that the secondary component re- sulting from the gravitational-wave signal GW190814 [91] can be described as a dark energy star using A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 and 12 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 α = 0 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 0 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 r [km] ϖ/Ω α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 α = 0 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 0 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='6 r [km] ω/Ω FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Left panel: Numerical solution of the differential equation (16) for a dark energy star described by model I and central density ρc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 × 1018 kg/m3 in the presence of anisotropy for several values of the free parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The solid and dashed lines represent the interior and exterior solutions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Right panel: Ratio of frame-dragging angular velocity to the angular velocity of the star, namely ω(r)/Ω = 1 − ϖ(r)/Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' It can be observed that the outer solution behaves asymptotically at large distances from the surface of the star (this is, ω → 0 for r → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Furthermore, appreciable changes in the angular velocity due to anisotropy can be noticeable, mainly in the interior region of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Model I Model II α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 α = 0 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 0 1 2 3 4 M [M⊙] I [1038 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='m2] Model I Model II 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 15 10 5 0 5 10 15 20 α ΔI [%] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Left panel: Moment of inertia versus mass for anisotropic dark energy stars, where a higher mass results in larger moment on inertia for both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' It is observed that the substantial impact of anisotropy on the moment of inertia occurs predominantly in the high-mass branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Right panel: Relative deviation (33) as a function of the anisotropy parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The maximum value of the moment of inertia can undergo variations with respect to its isotropic counterpart of up to ∼ 20% for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' B ∈ [4, 5]µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Based on these results, we have established two mod- els with different values A and B in order to explore the effects of anisotropy in the interior region of a dark energy star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In particular, the maximum-mass values in- crease as the parameter α increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' We noticed that model I without anisotropic pressures is not capable of generating maximum masses above 2M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' However, the inclusion of anisotropies (α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4) allows a significant in- crease in the maximum mass and thus a more favorable description of the compact objects observed in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' On the other hand, model II with anisotropies fits bet- ter with the observational measurements, although such a model can lead to the formation of ultra-compact ob- jects for sufficiently large values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' We also calculated the surface gravitational redshift for such stars, and our results indicated that zsur is substantially affected by the anisotropy in the high-mass branch, while the changes are irrelevant for sufficiently low masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' A star exists in the Universe only if it is dynamically stable, so our second task was to investigate whether the dark energy stars are stable or unstable with respect to an adiabatic radial perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Our results showed that the standard criterion for radial stability dM/dρc > 0 still holds for dark energy stars since the squared fre- quency of the fundamental pulsation mode (ν2 0) van- 13 Model I Model II α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 α = 0 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='020 C k2 Model I Model II α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 α = 0 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 1 5 10 50 100 500 1000 M [M⊙] Λ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Left panel: Tidal Love number plotted as a function of the compactness C ≡ M/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Right panel: Dimensionless tidal deformability versus gravitational mass predicted by each model, where larger masses yield smaller deformabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Note also that the Love number is substantially modified by the anisotropy parameter α for both models, while its greatest effect on tidal deformability Λ occurs only in the high-mass region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' ishes at the critical central density corresponding to the maximum-mass configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' This has been examined in detail for both isotropic (α = 0) and anisotropic (α ̸= 0) stellar configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In the slowly rotating approximation, where only first- order terms in the angular velocity are kept, we have also determined the moment of inertia of anisotropic dark en- ergy stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' For this purpose, we first had to calculate the frame-dragging angular velocity for each central density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The presence of anisotropic pressure results in a substan- tial increase (decrease) of the angular velocity ω for more positive (negative) values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' We found that the signif- icant impact of the anisotropy on the moment of inertia occurs mainly in the high-mass branch for both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Furthermore, the maximum value of the moment of in- ertia can undergo variations of up to ∼ 20% for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content='5 as compared with the isotropic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' We have analyzed the effect of anisotropic pressure on the tidal properties of such stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In particular, our out- comes revealed that the tidal Love number is sensitive to moderate variations of the parameter α, indicating that the maximum value of k2 can increase as α decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In addition, the greatest effect of anisotropy on the di- mensionless tidal deformability takes place only in the high-mass region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Based on the foregoing results, the present work thereby serves to develop a comprehensive perspective on the relativistic structure of dark energy stars in the presence of anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Summarizing, we have explored the possible existence of stable dark energy stars whose masses and radii are not in disagreement with the current observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The Chaplygin-like EoS predicts maximum-mass values consistent with observational measurements of highly massive pulsars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Future research includes the adoption of widespread versions of Chaplygin gas that best fit key cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In future studies we will thereby take further steps in that direction, focusing on the different types of generalized Chaplygin gas models as discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In addition, as carried out in the case of boson stars [101], it would be interesting to em- ploy a Fisher matrix analysis in order to distinguish dark energy stars from black holes and neutron stars from tidal interactions in inspiraling binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' It is also worth mentioning that Romano [102] has recently discussed the effects of dark energy on the propagation of gravitational waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' In that regard, we expect that future electromag- netic observations of compact binaries and gravitational- wave astronomy will provide a better understanding of compact stars in the presence of dark energy, and even help us answer the most basic question: How did dark energy form in the Universe?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' Anyway, our results sug- gest that dark energy stars deserve further investigation by taking into account the cosmological aspects as well as the gravitational-wave signals from binary mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' ACKNOWLEDGMENTS The author would like to acknowledge the anonymous reviewer for useful constructive feedback and valuable suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' The author would also like to thank Maria F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' da Silva for giving helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE1T4oBgHgl3EQf4wVz/content/2301.03504v1.pdf'} +page_content=' This research work was financially supported by the PCI program of the Brazilian agency “Conselho Nacional de Desenvolvi- mento Cient´ıfico e Tecnol´ogico”–CNPq.' metadata={'source': 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Feintuch, J. Tabrikian, Fellow, IEEE, I. Bilik, Senior Member, IEEE, and H. Permuter, Senior Member, IEEE +Abstract—This work addresses the problem of direction- of- +arrival (DOA) estimation in the presence of non-Gaussian, +heavy-tailed, and spatially-colored interference. Conventionally, +the interference is considered to be Gaussian-distributed and +spatially white. However, in practice, this assumption is not guar- +anteed, which results in degraded DOA estimation performance. +Maximum likelihood DOA estimation in the presence of non- +Gaussian and spatially colored interference is computationally +complex and not practical. Therefore, this work proposes a +neural network (NN) based DOA estimation approach for spa- +tial spectrum estimation in multi-source scenarios with a-priori +unknown number of sources in the presence of non-Gaussian +spatially-colored interference. The proposed approach utilizes +a single NN instance for simultaneous source enumeration and +DOA estimation. It is shown via simulations that the proposed +approach significantly outperforms conventional and NN-based +approaches in terms of probability of resolution, estimation +accuracy, and source enumeration accuracy in conditions of low +SIR, small sample support, and when the angular separation +between the source DOAs and the spatially-colored interference +is small. +Index Terms—Array Processing, DOA Estimation, Source +Enumeration, Spatially-Colored Interference, Non-Gaussian In- +terference, Neural Networks, Deep Learning, Machine Learning, +MVDR, MDL, AIC, Radar. +I. INTRODUCTION +Direction-of-arrival (DOA) estimation using a sensor array +is required in multiple applications, such as radar, sonar, +ultrasonic, wireless communications, and medical imaging [1]. +In real-world applications, the signal received at the sensor +array is a superposition of signals from the sources of interest, +interference, and receiver thermal noise. In radars, the received +signal consists of a target echo, clutter, and thermal noise. In +multiple scenarios, the radar clutter has a spatially-colored, +heavy-tailed non-Gaussian distribution [2], which can signifi- +cantly degrade the performance of conventional estimators. +Minimum-variance-distortionless-response (MVDR) [3], is +a conventional adaptive beamforming approach for DOA es- +timation. MVDR estimates the spatial spectrum and obtains +the source DOAs via a one-dimensional peak search on a +predefined grid. The estimation of signal parameters using +rotational invariance techniques (ESPIRIT) [4], multiple signal +classification (MUSIC) [5], and root-MUSIC (R-MUSIC) [6] +are additional widely used DOA estimation approaches. These +approaches involve received signal autocorrelation matrix +processing, which conventionally is performed via the sam- +ple autocorrelation matrix estimation [3]–[6]. However, the +Stefan Feintuch, Joseph Tabrikian, Igal Bilik, and Haim H. Permuter +are with the School of Electrical and Computer Engineering, Ben Gurion +University of the Negev, Beer Sheva, Israel. (e-mails: stefanfe@post.bgu.ac.il, +joseph@bgu.ac.il, bilik@bgu.ac.il, haimp@bgu.ac.il) +performance of the sample autocorrelation matrix estimator +degrades in small sample support or non-Gaussian scenarios. +Furthermore, these methods use the second-order statistics +only and omit the higher-order statistics on non-Gaussian- +distributed interference. In addition, ESPRIT, MUSIC, and R- +MUSIC approaches require a-priori knowledge of the number +of sources (or targets), which limits their practical use. +The problem of DOA estimation in the presence of non- +Gaussian interference is of great practical interest. The max- +imum likelihood estimator (MLE) for DOA estimation in the +presence of non-Gaussian interference does not have a closed- +form analytical solution [7], [8]. Multiple model-based DOA +estimation approaches have been intensively studied in the +literature [7]–[18]. +Robust covariance matrix-based DOA estimation and source +enumeration methods have been studied in the literature. For +complex elliptically symmetric (CES) distributed data, the +authors in [9] showed that a scatter matrix-based beamformer +is consistent, and the semiparametric lower bound and Slepian- +Bangs formula for DOA estimation were derived in [10]. +In [11], a generalized covariance-based (GC) approach for +the covariance matrix estimation in scenarios with impulsive +alpha-stable noise was proposed for MUSIC DOA estimation. +However, these methods consider a specific family of distri- +butions, such as the CES or alpha-stable, and are therefore, +limited in the case of model mismatch. In [12], a probability +measure transform (MT) based covariance matrix estimator +was proposed for MUSIC-based DOA estimation and mini- +mum descriptive length (MDL) based source enumeration. The +MT-based covariance estimator was also adopted for robust +MVDR beamformer [13]. These methods are usually based +on setting a parameter that determines the tradeoff between +the level of robustness and performance. +The problem of DOA estimation in the presence of a mix- +ture of spatially-white K-distributed and Gaussian-distributed +noise under a deterministic and unknown (conditional) source +model was studied in [7]. An iterative MLE-based approach +for the conditional and joint likelihood of interference distri- +bution’s parameters was derived in [14], [15]. This approach +was further extended in [16] to marginal likelihood function. +However, this approach is computationally complex due to +numerical integral evaluation that involves a 2M dimensional +grid search for M targets [8]. Therefore, [8] proposed a kernel +minimum error entropy-based adaptive estimator and a novel +criterion to reduce the estimator’s computational complexity. +The expectation-maximization (EM) with a partial relaxation- +based DOA estimation algorithm under the conditional model +assumption was proposed in [17]. In [18] a sparse Bayesian +learning (SBL) approach for outlier rejection of impulsive +arXiv:2301.02856v1 [eess.SP] 7 Jan 2023 + +2 +and spatially-white interference was proposed. This EM-based +approach does not require a-priori knowledge of the number +of sources and was shown to resolve highly-correlated and co- +herent sources. However, none of these model-based DOA es- +timation approaches considered an a-priori unknown number +of sources and spatially-colored interference and therefore are +limited for real-world applications. Although source enumera- +tion methods, such as MDL and Akaike information criterion +(AIC) [19] can be used, they assume signal Gaussianity, and +can therefore be inaccurate in non-Gaussian scenarios. +Deep learning and machine learning approaches were re- +cently adopted for radar signal processing. Three types of +NN-based DOA estimation approaches have been introduced +in literature [20]. The first approach assumes a-priori known +number of sources, and uses a NN, which is optimized to +output a vector of the estimated DOAs [21]–[27]. The second +approach does not assume a-priori known number of sources +and uses a NN for source enumeration [25]–[31]. The third +approach uses a NN to estimate source presence probability +at each DOA on a predefined angular grid and obtains the +source DOAs via a peak search [32]–[41]. However, all these +approaches have not addressed non-Gaussian and spatially- +colored interference [20]–[41]. +The cases of non-Gaussian and/or spatially-colored inter- +ference have been addressed using machine learning-based +approaches. For massive MIMO cognitive radar, a reinforce- +ment learning-based approach for multi-target detection un- +der heavy-tailed spatially-colored interference was proposed +in [42]. In [43], authors addressed the MIMO radar target +detection under non-Gaussian spatially-colored interference +by using a CNN architecture that is optimized according to +a novel loss. A radial-basis-function (RBF) NN [44] and a +convolutional neural network (CNN) [45] architectures were +proposed for DOA estimation in the presence of non-Gaussian +impulsive noise. In [46], a CNN-based architecture that in- +cludes denoising NN, source enumeration NN, and DOA esti- +mation sub-NNs, was introduced. However, [44]–[46] consider +spatially-white noise and are suboptimal for scenarios with +spatially-colored interference. +This work addresses the problem of DOA estimation of a- +priori unknown number of sources in the presence of non- +Gaussian, heavy-tailed, spatially-colored interference at a low +signal-to-interference ratio (SIR) and small sample size. The +contribution of this work include: +1) A novel NN-based processing mechanism is used for +array processing within non-Gaussian spatially-colored +interference. The proposed NN architecture utilizes the +structure of information within the set of received com- +plex snapshots. +2) The proposed NN is optimized to output an interference- +mitigated spatial spectrum, and is used for simultaneous +source enumeration and DOA estimation of sources +within non-Gaussian spatially-colored interference. +The proposed approach outperforms conventional adaptive +beamforming and competing straightforward NN-based meth- +ods in terms of probability of resolution and estimation +accuracy in scenarios with non-Gaussian spatially-colored +interference. In addition, the proposed approach outperforms +conventional source enumeration techniques in scenarios char- +acterized by non-Gaussian spatially-colored interference. +The following notations are used throughout the paper. +Roman boldface lower-case and upper-case letters represent +vectors and matrices, respectively while Italic letters stand for +scalars. IN is the identity matrix of size N × N and 1N +is a column vector of length N whose entries are equal to +one. E(·), (·)T , and (·)H are the expectation, transpose, and +Hermitian transpose operators, respectively. Vec(·), diag(·), +and | · | stand for the vectorization, diagonalization, and +absolute value operators, respectively. [a]n and [A]n,m are the +n-th and n, m-th elements of the vector a and the matrix A, +respectively. +The remainder of this paper is organized as follows. The +addressed problem is stated in Section II. Section III intro- +duces the proposed NN-based DOA estimation approach. The +proposed approach is evaluated via simulations in Section IV. +Our conclusions are summarized in Section V. +II. PROBLEM DEFINITION +This work considers the problem of DOA estimation using +an array of L receiving elements and M distinct and unknown +sources with DOAs, Θ = {θ1, . . . , θM}. The measurements +contain K spatial snapshots, {xk}K +k=1: +xk = A (Θ) sk + σcck + nk , +(1) += +M +� +m=1 +a (θm) sk,m + σcck + nk , k = 1, . . . , K , +where A (Θ) = +�a (θ1) +· · · +a (θM)� +, with a (θm) ∈ CL +denoting the steering vector for source at direction θm, +and sk ≜ +�sk,1 +· · · +sk,M +�T is the source signal vector. +We assume an unconditional model [47], where {sk} +i.i.d. +∼ +CN +� +0M, diag +� +σ2 +1, . . . , σ2 +M +�� +, is temporally uncorrelated be- +tween pulses. The targets are assumed to be spatially distinct. +The receiver thermal noise, denoted by nk, is considered to be +complex Gaussian-distributed {nk} +i.i.d. +∼ CN +� +0L, σ2 +nIL +� +. The +heavy-tailed non-Gaussian and spatially-colored interference is +modeled by the interference amplitude σc, and the interference +component ck ∈ CL. The considered compound-Gaussian +distributed interference, {ck} +i.i.d. +∼ K (ν, θc) represents a non- +Gaussian interference with angular spread around an unknown +direction θc, such that c ∼ K (ν, θc) implies +c = √τz , +(2) +τ +|= +z, τ ∼ Γ (ν, ν) , z ∼ CN (0L, Mθc) . +The compound-Gaussian statistical model is conventionally +used in the literature to model heavy-tailed non-Gaussian +interference [7], [8], [14], [16], [43], [48]. The texture com- +ponent, τ ∈ R+, determines the heavy-tailed behavior and +is characterized by, ν. The speckle component, z ∈ CL, +determines the spatial distribution of the interference and +is characterized by the covariance matrix, Mθc. The spatial +covariance matrix of the interference upholds: +E +� +σ2 +cccH� +=σ2 +cE [τ] E +� +zzH� += σ2 +cMθc , +(3) + +3 +where Mθc can be modeled as [14]–[16], [43], [48]: +[Mθc]m,l = ρ|m−l|ej(m−l)π sin θc . +(4) +The model in (3) and (4), represents the spatial interference, +characterized by ρ, with a spread around the interference DOA, +θc. +III. THE PROPOSED DAFC-BASED NEURAL NETWORK +The proposed approach generalizes the NN architecture that +was introduced for linear-frequency-modulated (LFM) radar +target detection in the range-Doppler domain [49]. In the +following, the data pre-processing and the proposed NN-based +processing mechanism are introduced in Subsections III-A and +III-B. The proposed NN architecture and loss function are +detailed in Subsections III-C and III-D, respectively. +A. Pre-Processing +The input matrix, X ∈ CL×K is constructed from the set +of K snapshots in (1), {xk}: +X = +� +x1 +x2 +· · · +xK +� +, +(5) +where the k-th column of X contains the k-th snapshot. +The variation between the columns of X is induced by the +statistical characteristics of the source signal sk, interference +signal ck, and thermal noise nk. Therefore, each column in +X can be interpreted as a complex “feature” vector containing +essential information for DOA estimation. The set of columns +in X can be interpreted as “realizations” of that feature. +The complex-valued matrix, X, is converted into real-valued +representation needed for the NN-based processing. To keep +consistency with [49], we apply a transpose operator to the +input matrix, such that the snapshots are stacked in rows. The +output of the pre-processing denoted by Z0 ∈ CK×2L, is: +Z0 = +� +Re +� +XT � +, Im +� +XT �� +. +(6) +B. Dimensional Alternating Fully-Connected +The dimensional alternating fully-connected (DAFC) block +was introduced to process measurements in a form similar to +the model in Section II [49]. Fig. 1 schematically shows the +DAFC mechanism. +For arbitrary dimensions D1, D2, D3, the formulation of a +general fully-connected (FC) layer applied to each row in a +given matrix Z ∈ RD1×D2 can be represented by the transform +F (·): +F : RD1×D2 → RD1×D3 , +(7) +F (Z) ≜ h +� +ZW + 1D1bT � +. +This matrix-to-matrix transformation is characterized by the +“learnable” weight matrix, W ∈ RD2×D3, the bias vector, +b ∈ RD3, and a scalar element-wise activation function, h(·). +Let Fr (·) and Fc (·) be two separate, and not necessarily +identical instances of F (·) from (7), and Zin be an arbitrary +input matrix. The DAFC mechanism is formulated by the +following operations: +Dimensional Alternating Fully Connected +• Input: Zin ∈ RH×W +Fr : RH×W → RH×W ′ +Fc : RW ′×H → RW ′×H′ +1) Apply a single FC layer to each row in Zin: +Zr = Fr (Zin) +2) Apply a single FC layer to each column in Zr: +Zc = Fc +� +ZT +r +� +3) Transpose to keep orientation: +Zout = ZT +c +• Output: Zout ≜ S (Z) ∈ RH′×W ′ +In the following, three DAFC design principles are detailed. +1) Structured transformation +The input to the first DAFC block is the pre-processed, Z0, +given in (6). Therefore, the first FC layer, Fr, of the first DAFC +block extracts spatial-related features from each row in Z0. +The second FC layer, Fc, of the first DAFC block, introduces +an interaction between transformed rows. This implies that +a) Fr performs “spatial-feature” extraction by transforming +the pre-processed i.i.d. snapshots (the rows of Z0) to a +high-dimensional feature space, and b) the Fc performs a +nonlinear transformation of the extracted features (the columns +of Fr (Z0)) from each snapshot. In this way, the DAFC utilizes +both spatial and statistical information. In addition, it can +exploit high-order statistics-related features. Thus, the DAFC +mechanism can contribute to estimating the source DOAs and +mitigating the interference when incorporated into a NN. +2) Sparsity +Conventional DOA estimation considers the input data as +the collection of measurement vectors (the snapshots {xk}) in +a matrix form. One straightforward approach to processing +the input data using a NN is to reshape it and process it +via an FC-based architecture. In this way, each neuron in the +layer’s output interacts with every neuron in the input. On +the other hand, the DAFC block transforms the data using +a structured transformation, which is significantly sparser in +terms of learnable parameters compared to the straightforward +FC-based approach. +This parameter reduction can be observed in the following +typical case. Consider an input matrix Z1 ∈ RD1×D1, which +is transformed to an output matrix Z2 ∈ RD2×D2. The +number of learnable parameters in the FC- and the proposed +DAFC-based approaches is of the order of O +� +D2 +1D2 +2 +� +, and +O (D1D2), respectively. Notice that the DAFC-based transfor- +mation complexity grows linearly with the number of learnable +parameters compared to the quadratic complexity growth of +the straightforward, FC-based approach. +The contribution of learnable parameters dimension reduc- +tion is twofold. First, the conventional NN optimization is +gradient-based [50]. Therefore, a significant reduction in the +learnable parameter dimension reduces the degrees of freedom +in the optimizable parameter space and improves the gradient- +based learning algorithm convergence rate. Second, reduction + +4 +Figure 1: The DAFC mechanism concept. Each row of dimen- +sion W in Zin, represented by the red color, is transformed by +Fr to a row of dimension W ′ in the middle matrix, represented +by the transparent red color. Next, each column of dimension +H in the middle matrix, represented by the blue color, is +transformed by Fc to a column of dimension H′ in Zout, +represented by the transparent blue color. +in the learnable parameter dimension can be interpreted as +increasing the “inductive bias” of the NN model [51], which +conventionally contributes to the NN statistical efficiency and +generalization ability, thus, reducing the NNs tendency to +overfit the training data. +3) Nonlinearity +The proposed DAFC considers an additional degree of +nonlinearity compared to the straightforward FC-based ap- +proach. A straightforward matrix-to-matrix approach includes +an interaction of every neuron in the output matrix with +every neuron in the input matrix, followed by an element-wise +nonlinear activation function. On the other hand, the proposed +DAFC consists of two degrees of nonlinearity, in Fr and Fc. +Although the weight matrices applied as part of Fr and Fc +are of lower dimension than the weight matrix used in the +straightforward approach, the extra degree of nonlinearity can +increase the NN’s capacity [50]. Therefore, a NN architecture +with the proposed DAFC is capable of learning a more abstract +and rich transformation of the input data. +C. NN Architecture +The continuous DOA space is discretized into a d- +dimensional grid: φ = +�φ1 +φ2 +· · · +φd +�T . This implies +that the entire field-of-view (FOV) is partitioned into d DOAs, +{φi}d +i=1, determined by the selected grid resolution, ∆φ ≜ +φi+1 − φi. The proposed NN is designed to represent a +mapping from the input set of snapshots, {xk} given in (1), +into the probability of source present in the DOAs {φi}d +i=1. +The proposed NN architecture is formulated as follows: +Z0 = P (X) , +(8) +zvec = Vec (S6 (. . . S1 (Z0))) , +ˆy = G3 (G2 (G1 (zvec))) , +Operator +Output +Dimension +Activation +# Parameters +P +K × 2L +- +- +S1 +64 × 256 +tanh- +ReLu +9,536 +S2 +128 × 512 +tanh- +ReLu +139,904 +S3 +256 × 1024 +tanh- +ReLu +558,336 +S4 +64 × 512 +tanh- +ReLu +541,248 +S5 +16 × 256 +tanh- +ReLu +132,368 +S6 +4 × 128 +tanh- +ReLu +32,964 +vec +512 +- +- +G1 +1024 +tanh +525,312 +G2 +256 +tanh +262,400 +G3 +d +sigmoid +31,097 +Table I: +Specification of the proposed NN architecture for +K = 16, L = 16, d = 121. “tanh-ReLu” activation stands +for tanh in Fr and ReLU in Fc of each DAFC block. The +number of total learnable parameters is 2, 233, 165. +where Z0 is the output of the pre-processing procedure, +denoted as P (·) and detailed in Section III-A, and X is the +input matrix in (5). +In the next stage, six DAFC instances, represented by +S1 (·) , . . . , S6 (·), of different dimensions with tanh activa- +tion for the row transform (Fr in Section III-B) and ReLu +activation for the column transform (Fc in Section III-B), are +used to generate the vectorized signal zvec. Our experiments +showed that this configuration of row and column activation +functions provides the best performance. At the last stage, the +signal, zvec, is processed by three FC layers, where the first +two use tanh activation, and the final (output) layer of equal +size to the DOA grid dimension, d, uses sigmoid activation +function to output ˆy ∈ [0, 1]d. Thus, {[ˆy]i}d +i=1 represent the +estimated probabilities of a source presence at {φi}d +i=1. Table I +and Fig. 2 summarize the parameters and architecutre of the +proposed NN-based approach. +The estimated source DOAs are extracted from the spatial +spectrum via peak search and applying 0.5 threshold: +{i1, . . . , i ˆ +N} = peak search +� +{[ˆy]i}d +i=1 +� +(9) +ˆΘ = +� +φin : [ˆy]in > 0.5 +� ˆ +N +n=1 . +Namely, the set of estimated DOAs, ˆΘ, consists of the grid +points corresponding to the peaks of ˆy that exceed the 0.5 +threshold. The number of peaks that exceed this threshold is +used for source enumeration, and therefore the proposed NN +can be utilized as a source enumeration method as well. +The dimensionality of the hidden layers in the proposed + +5 +Figure 2: Proposed NN architecture. The pre-processing P is described in Section III-A and appears in yellow. The purple +matrices denote the concatenation of DAFC blocks, which is detailed in Section III-B. The blue vector represents a vectorization +of the last DAFC output, and the orange vectors stands for FC layers with tanh activation function. The last green vector is +the output of the last FC layer, which consists of sigmoid activation function and yields the estimated spatial spectrum ˆy. +NN architecture expands in the first layers and then reduces. +This trend resembles the NN architecture presented in [49] and +characterizes both the DAFC-based and FC-based processing +stages. This expansion-reduction structure can be explained +by a) the early NN stages need to learn an expressive and +meaningful transformation of the input data by mapping it to +a higher dimensional representation and b) the late stages need +to extract significant features from the early mappings, and are +therefore limited in dimensionality. In addition, the late stages +are adjacent to the output vector and therefore need to be of +similar dimension. +D. Loss Function +The label used for the supervised learning process, y ∈ +{0, 1}d, is defined as a sparse binary vector with the value 1, +at the grid points that correspond to the source DOAs, and +0, otherwise. In practice, the DOAs in Θ do not precisely +correspond to the grid points. Therefore, for each DOA in +Θ, the nearest grid point in {φi}d +i=1 is selected as the +representative grid point in the label. Each training example +is determined by the input-label pair, (X, y). Using the NN +feed-forward in (8), X is used to generate the output spatial +spectrum, ˆy, which is considered as the estimated label. +The loss function, L, is a weighted mean of the binary cross +entropy (BCE) loss computed at each grid point: +L (y, ˆy, t) = 1 +d +d +� +i=1 +w(t) +i BCE ([y]i , [ˆy]i) , +(10) +BCE (y, ˆy) = −y log (ˆy) − (1 − y) log (1 − ˆy) , +where w(t) +i +represents the loss weight of the i-th grid point at +the t-th epoch. The loss value for equally-weighted BCEs eval- +uated per grid point (w(t) +i += 1 in (10)) does not significantly +increase in the case of a large error in source/interference esti- +mated probability, due to the sparsity of the label y. This forces +the NN convergence into a sub-optimal solution that is prone +to “miss” the sources. Therefore, the loss weights, {w(t) +i }d +i=1, +are introduced to “focus” the penalty on source/interference +grid points. +The loss weight of the i-th grid point, w(t) +i , is determined +by the presence of source or interference in the corresponding +label entry [y]i. This relation is defined using the epoch and +label dependent factors e(t) +0 , e(t) +1 , according to: +w(t) +i += +� +1/e(t) +1 , +if φi contains source or interference +1/e(t) +0 , +else +. +(11) +For t = 0, the factor e(0) +1 +is determined by the fraction of +label grid points that contain source or interference out of +the total label grid points in the training set, and e(0) +0 +is the +corresponding complement. For subsequent epochs, the factors +are updated according to a predefined schedule, similarly to a +predefined learning rate schedule. The loss weights are updated +Nw times with spacing of ∆t epochs during training. The +update values are determined by updating e(t) +0 , e(t) +1 , according +to the following decaying rule: +e(t) +q += (1 − β(l))e(l∆t) +q ++ β(l), l∆t ≤ t < (l + 1)∆t +(12) +q = 0, 1, l = 1, . . . , Nw, + +6 +where l is the loss weight update iteration, and {β(l)}Nw +l=1 +represent the loss weight update factors which uphold, 0 ≤ +β(l) ≤ 1. Note that for Nw∆t ≤ t, the weight factor +remains e(Nw∆t) +i +during the rest of the training stage. No- +tice that as β(l) → 1, the corresponding loss weights will +tend to be equally distributed across the grid points, i.e., +e(t) +1 +≈ e(t) +0 . In this case, an erroneously estimated proba- +bility for source/interference containing grid point is equally +weighted to a neither-containing grid point. On the other +hand, as β(l) → 0, the corresponding factors will uphold +e(t) +1 +≪ e(t) +0 , yielding a significantly larger contribution of +source/interference containing grid points to the loss value. +The rule in (12) enables a “transition of focus” throughout +the training. That is, during the early epochs β(l) → 0, which +contributes more weight to the source/interference containing +areas in the estimated label ˆy (i.e., the estimated spatial spec- +trum) to focus the NN to being correct for source/interference. +During the later epochs, β(l) is incrementally increased, which +relaxes the focus on source/interference from early epochs. +Thus, reducing erroneously estimated sources in areas that do +not contain source/interference (i.e. “false-alarms”). +IV. PERFORMANCE EVALUATION +This section evaluates the performance of the proposed +DAFC-based NN approach and compares it to the conventional +approaches, summarized in Subsection IV-A1. The data for +all considered scenarios is simulated using the measurement +model from Section II. +A. Setup & Training +This work considers a uniform linear array (ULA) with +half-wavelength-spaced L elements. Each simulated example +consists of the input-label pair, (X, y), where the input X is +defined in (5), and the label y is defined in Section III-D. +The simulation configurations are detailed in Table II. The +performance of the proposed approach is evaluated using +a single NN instance. Therefore, a single NN model is +used for various signal-to-interference ratios (SIRs), signal- +to-noise ratios (SNRs), interference-to-noise ratios (INRs), +DOAs, interference distribution, and the number of sources for +joint DOA estimation and source enumeration. The following +definitions for the m-th source are used in all experiments: +INR += +E[∥c∥2] +E[∥n∥2] = σ2 +c/σ2 +n , +(13) +SNRm += +E[∥a(θm)sm∥2] +E[∥n∥2] += σ2 +m/σ2 +n , +(14) +SIRm += +E[∥a(θm)sm∥2] +E[∥c∥2] += σ2 +m/σ2 +c . +(15) +The NN optimization for all evaluated architectures is +performed using the loss function in (10) and Adam opti- +mizer [52] with a learning rate of 10−3, and a plateau learning +rate scheduler with a decay of 0.905. The set of loss weight up- +date factors, {β(l)}Nw +l=1, in (12) is chosen as the evenly-spaced +logarithmic scale between 10−5 and 10−2 with Nw = 6, that +is {10−5, 7.25 · 10−5, 5.25 · 10−4, 3.8 · 10−3, 2.78 · 10−2, 0.2}. +The chosen batch size is 512, the number of epochs is 500, +and early stopping is applied according to the last 200 epochs. +Notation +Description +Value +Mmax +Maximal +number +of +sources +4 +L +Number of sensors +16 +K +Number of snapshots +16 +d +Angular grid dimension +121 +∆φ +Angular grid resolution +1◦ +FOV +Field of view +[−60◦, 60◦] +σ2 +n +Thermal noise power +1 +Table II: Simulation Configurations. +1) DOA Estimation Approaches: This subsection briefly +summarizes the conventional DOA estimation approaches. The +performance of the proposed approach is compared to the +conventional MVDR, CNN, and FC-based NN. All the NN- +based approaches were implemented using similar number of +layers and learnable parameters. In addition, the FC-based NN +and CNN were optimized using the same learning algorithm +and configurations. +(a) Conventional Adaptive Beamforming +The MVDR [3] estimator is based on adaptive beamforming, +and it is the maximum likelihood estimator in the presence +of unknown Gaussian interference [53]. The MVDR estimates +DOAs by a peak search on the MVDR spectrum: +PMV DR (φ) = +1 +aH (φ) ˆR−1 +x a (φ) +, +(16) +where ˆRx = +1 +K +�K +k=1 xkxH +k is the sample covariance ma- +trix estimator. Notice that the MVDR spectrum utilizes only +second-order statistics of the received signal xk. For Gaussian- +only interference (i.e. ck = 0 in (1)), the second-order statistics +contains the entire statistical information. However, for non- +Gaussian interference, information from higher-order statistics +is needed. +(b) CNN Architecture +We consider a CNN-based DOA estimation approach using a +CNN architecture that is similar to the architecture provided +in [38]. The input to the CNN of dimension L × L × 3 +consists of the real, imaginary, and angle parts of ˆRx. The +CNN architecture consists of 4 consecutive CNN blocks, +such that each block contains a convolutional layer, a batch +normalization layer, and a ReLu activation. The convolutional +layers consist of [128, 256, 256, 128] filters. Kernel sizes of +3 × 3 for the first block and 2 × 2 for the following three +blocks are used. Similarly to [38], 2 × 2 strides are used +for the first block and 1 × 1 for the following three blocks. +Next, a flatten layer is used to vectorize the hidden tensor, +and 3 FC layers of dimensions 1024, 512, 256 are used +with a ReLu activation and Dropout of 30%. Finally, the +output layer is identical to the proposed DAFC-based NN +as detailed in Subsection III-C. The considered loss function +is identical to the proposed DAFC-based approach in (10). +The number of trainable parameters in the considered CNN + +7 +architecture accounts for 3, 315, 449. Notice that the CNN- +based architecture utilizes the information within the sample +covariance matrix and therefore, is limited to second-order +statistics only. +(c) FC Architecture +A straightforward implementation of an FC-based architecture, +as mentioned in Subsection III-B, was implemented. The +data matrix, X, is vectorized, and the real and imaginary +parts of the values were concatenated to obtain a 2KL- +dimension input vector. The selected hidden layers are of +sizes: [512, 512, 1024, 1024, 512, 256] where each hidden layer +is followed by a tanh activation function. The output layer +is identical to the proposed DAFC-based NN approach as +detailed in Subsection III-C. The considered loss function +is (10), and the number of trainable parameters in the FC- +based NN accounts for 2, 787, 449. Notice that the FC-based +NN architecture utilizes all the measurements by interacting +with all samples in the input data. However, this processing is +not specifically tailored to the structure of information within +the measurements. On the other hand, the proposed DAFC- +based NN utilizes the information structure to process the input +data. Therefore, for the considered DOA estimation problem, +the “inductive bias” [51] for this approach is improper and can +result in under-fitted NN architecture. +2) Performance Evaluation Metrics: This subsection dis- +cusses the criteria for the performance evaluation of the +proposed DOA estimation approach. In this work, similarly +to [38], the DOA estimation accuracy of a set of sources +is evaluated by the Hausdorff distance between sets. The +Hausdorff distance, dH between the sets, A, and B, is defined +as: +dH (A, B) = max {d (A, B) , d (B, A)} , +(17) +d (A, B) = sup {inf {|α − β| : β ∈ B} : α ∈ A} . +Notice that d (A, B) ̸= d (B, A). Let Θ = {θm}M +m=1 and ˆΘ = +{ˆθm} ˆ +M +m=1 be the sets of true and estimated DOAs, respectively. +The estimation error is obtained by evaluating the Hausdorff +distance, dH(Θ, ˆΘ). We define the root mean squared distance +(RMSD) for an arbitrary set of N examples (e.g., test set), +� +X(n), y(n)�N +n=1, with the corresponding true and estimated +DOAs, +� +Θ(n), ˆΘ(n)�N +n=1 as: +RMSD ≜ +� +� +� +� 1 +N +N +� +n=1 +d2 +H +� +Θ(n), ˆΘ(n) +� +. +(18) +Angular resolution is one of the key criteria for DOA +estimation performance. The probability of resolution is com- +monly used as a performance evaluation metric for angular +resolution. In the considered problem, resolution between two +sources and between source and interference are used for +performance evaluation. For an arbitrary example with M +sources, the resolution event Ares is defined as: +Ares +� +Θ, ˆΘ +� +≜ +� +1, +�M +m=1 ξm ≤ 2◦ and | ˆΘ| ≥ M +0, +else +, +(19) +ξm ≜ min +ˆθ∈ ˆΘ +|θm − ˆθ|, m = 1, . . . , M . +For example, a scene with M sources is considered success- +fully resolved if for each true DOA a) there exists a close- +enough estimated DOA, ˆθ ∈ ˆΘ, that is at most 2◦ apart, and +b) there exists at least M DOA estimations. According to (18), +the probability of resolution, can be defined as: +Pres = 1 +N +N +� +n=1 +Ares +� +Θ(n), ˆΘ(n)� +. +(20) +3) Data Sets: This subsection describes the structure and +formation of Training & Test sets. +(a) Training Set +The considered training set contains Ntrain += +10, 000 +examples re-generated at each epoch. For each exam- +ple, i.e. an input-label pair (X, y), the number of DOA +sources, M, is generated from uniform and i.i.d. distribution, +{1, . . . , Mmax}. The training set contains 10% of interference- +free examples and 90% of interference-containing. Out of the +interference-containing examples, 90% generated such that the +source DOAs, {θm}M +m=1, and the interference’s DOA, θc, are +distributed uniformly over the simulated FOV. The remaining +10% are generated such that θc is distributed uniformly over +the FOV, and the source DOAs, {θm}M +m=1, are distributed +uniformly over the interval [θc − 8◦, θc + 8◦]. This data set +formation enables to “focus” the NN training on the chal- +lenging scenarios where the source and interference DOAs are +closely spaced. The generalization capabilities of the proposed +NN to variations in interference statistics are achieved via the +interference angular spread parameter, ρ, from the uniform dis- +tribution, U ([0.7, 0.95]), and the interference spikiness param- +eter, ν, from the uniform distribution, U ([0.1, 1.5]). The INR +for each interference-containing example and {SIRm}M +m=1 or +{SNRm}M +m=1 are drawn independently according to Table III. +(b) Test Set +The test set consists of Ntest = 20, 000 examples. The results +are obtained by averaging the evaluated performance over 50 +independent test set realizations. Considering the low-snapshot +support regime, the number of snapshots is set to K = 16, +except for experiment (c) in IV-B2. Considering heavy-tailed +interference, the spikiness parameter is set to ν = 0.2. The +INR is set to INR = 5 dB, and the interference angular spread +parameter is set to ρ = 0.9. The signal amplitude was set to be +identical for all sources, σ1 = · · · = σm, except for experiment +(b) in IV-B2. +B. Experiments +1) Single Source Within Interference: In this scenario, the +ability to resolve a single source from interference is evaluated. +Let M = 1 with θ1 = 0.55◦, and θc = θ1 + ∆θc such +that ∆θc is the angular separation between the single source +and interference. The 0.55◦ offset is considered to impose +a realistic off-grid condition. Fig. 3 shows the RMSD and +probability of resolution for all evaluated approaches. +Fig. 3a shows that the FC-based NN approach does not +manage to resolve the single source from the interference +for all evaluated angular separations. This result supports the + +8 +(a) +(b) +Figure 3: Scenario with a single source at θ1 = 0.55◦ and interference located at θc = θ1 + ∆θc. (a) probability of resolution +and (b) RMSD. +Notation Description +Value +ρ +Interference angular +spread parameter +∼ U ([0.7, 0.95]) +ν +Interference +spikiness parameter +∼ U ([0.1, 1.5]) +INR +INR +∼ U ([0, 10]) [dB] +SIRm +SIR of m-th source +∼ U ([−10, 10]) [dB] +SNRm +SNR of m-th source +∼ U ([−10, 10]) [dB] +Table III: Training set parameters. SNRm distribution applies +to interference-free examples. +under-fitting limitation of the FC-based NN approach for the +DOA estimation, which can be explained by the architecture +that processes the input data as-is, without any structured +transformation or model-based pre-processing. +The MVDR and CNN performance in terms of the resolu- +tion are similar since both rely only on second-order statistics, +which is sufficient in scenarios with widely separated sources +and interference. Fig. 3a shows that the proposed DAFC-based +NN approach outperforms all other considered approaches in +low angular separation scenarios. This can be explained by the +fact that the DAFC uses the high-order statistics needed for +the resolution of closely spaced sources and interference. +Fig. 3b shows the RMSD of all considered DOA estimation +approaches. The proposed DAFC-based NN approach outper- +forms the other tested approaches in low SIR. At high SIR +and small angular separation, ∆θc = 5◦, the interference +is negligible with respect to the strong source signal, and +therefore, the DAFC-based, CNN, and MVDR approaches +obtain similar performance. For large angular separation, +∆θc = 30◦, the source and the interference are sufficiently +separated, and therefore, DOA estimation errors are mainly +induced by the interference DOA, θc. The MVDR spectrum +contains a peak at θc = 30.55◦, and therefore, MVDR’s +RMSD = 30◦ is approximately constant. The NNs are trained +to output a 0-probability for the interference, therefore, the +NN-based approaches: FC, CNN, and DAFC achieve a smaller +DOA estimation error. The DAFC-based NN and CNN utilize +structured transformations, which better fit the input data, and +therefore, they outperform the FC-based NN approach in terms +of RMSD. +2) Resolving Two Sources from Interference: This subsec- +tion evaluates the performance of the tested DOA estimation +approaches in scenarios with two sources within AWGN and +interference. +(a) Resolution of Equal-Strength Sources +In the following experiment, the resolution between two equal- +power sources, M = 2, with θ1 = − ∆θ +2 + 0.55◦, and θ2 = +∆θ +2 +0.55◦, is evaluated. The off-grid additional 0.55◦ offset to +the ∆θ angular separation between the sources represents the +practical scenario. The interference at θc = 0.55◦ influences +the two sources similarly. Fig. 4 shows the probability of +resolution of the tested approaches in scenarios with (a) the +AWGN only and (b) spatially-colored interference. +The FC-based NN approach does not resolve the two targets +in both evaluated scenarios. Subplot (a) in Fig. 4 shows +that the proposed DAFC-based NN approach outperforms the +MVDR and the CNN at low-SNR and small angular separation +scenarios due to its generalization ability to spatially-white +interference. Subplot (b) in Fig. 4 shows that at low SIR +of SIR = −5 dB, the performances of MVDR and CNN +significantly degrade compared to the proposed DAFC-based +NN approach. Comparing subplots in Fig. 4, notice that at +SIR = −5 dB, the MVDR fails to resolve the sources with +angular separation ∆θ < 20◦ due to the presence of the heavy- +tailed spatially-colored interference in the proximity of the +sources. However, the proposed DAFC-based NN approach +mitigates this interference and resolves the sources, and hence, +outperforms other tested approaches at both SIR = 0 dB and +SIR = −5 dB. +Subplot (b) in Fig. 4 shows the non-monotonic trend of +CNN and MVDR performance at 4◦ < ∆θ < 18◦ and + +1.0 +0.8 +MVDR, SIR=0 +DAFC, SIR=0 +FC, SIR=0 +S +0.6 +res +CNN. SIR=O +P +MVDR, SIR=-5 +DAFC, SIR=-5 +0.4 +FC, SIR=-5 +CNN, SIR=-5 +0.2 +10 +15 +20 +25 +30 +△0c [Deg]RMSD [Deg] +101 +MVDR, △0c=5 +DAFC, A0c=5 +FC, △Qc=5 +CNN, △0c=5 +MVDR, △0c=30 +DAFC, △0c=30 +100 +FC, △0c=30 +CNN, △0.=30 +-10 +-5 +0 +5 +10 +15 +20 +SIR[dB]9 +(a) +(b) +Figure 4: Probability of resolution for two sources located at θ1,2 = θc ± ∆θ/2, and interference located at θc = 0.55◦. (a) +AWGN-only scenario and (b) interference-containing scenario. +(a) FC +(b) MVDR +(c) CNN +(d) DAFC +Figure 5: Spatial spectrum, two sources with SIR = −5 dB +located at θ1,2 = θc ± ∆θ/2 with ∆θ = 12◦ and θc = 0.55◦. +The dashed blue lines represent the mean spatial spectrum, +and the color fill represents the standard deviation around the +mean obtained from 2, 000 i.i.d. examples. The solid vertical +orange lines represent the true source DOAs, and the dashed +vertical green line represents the interference DOA. +SIR = −5 dB. For 4◦ < ∆θ < 8◦ the sources are closer +to the peak of the interference’s lobe and are therefore less +mitigated by it. As ∆θ initially increases, 8◦ < ∆θ < 12◦, +the sources reach DOAs which are in the proximity of the +interference lobe’s “nulls” which explains the reduction in +resolution, and as ∆θ further increases, 16◦ < ∆θ, the +sources are sufficiently separated from the interference such +that the resolution increases. As a result, MVDR and CNN- +based approaches that use second-order statistics only, can not +resolve the sources in the vicinity of a stronger interference. +Fig. 5 shows the average spatial spectrum of all tested +approaches for ∆θ = 12◦ and SIR = −5 dB. The average +spatial spectrum of the FC-based NN approach does not show +two prominent peaks, which results in its poor probability of +resolution in Fig. 4. The MVDR “bell-shaped” spatial spec- +trum does not contain the two prominent peaks at θ1,2 since the +interference “masks” the two sources. The CNN and proposed +DAFC-based NN approaches show two peaks at the average +spatial spectrum. The peaks at the CNN’s average spatial +spectrum are lower, resulting in a low-resolution probability. +The average spatial spectrum of the proposed DAFC-based +NN approach contains two high peaks, resulting in a superior +probability of resolution in Fig. 4. +(b) Resolution of Unequal-Power Sources +Fig. 6 shows the probability of resolution in a scenario +with two sources, M = 2, at θ1 = −∆θ/2 + 0.55◦, and +θ2 = +∆θ/2 + 0.55◦ with interference located between the +sources at θc = 0.55◦. The signal strength of the second +source is set to SIR1 = SIR2 + 10 dB. Comparing Fig. 6 to +Fig. 4b, the competing methods show similar trends, except +the degradation of the CNN’s probability of resolution for the +SIR = 0 dB case. On the other hand, the proposed DAFC- +based NN approach outperforms other tested approaches in +terms of the probability of resolution. Therefore, Fig. 6 demon- +strates the generalization ability of the proposed DAFC-based +NN approach to a variance between source strengths. +(c) Effect of the Number of Snapshots on the Resolution +This experiment investigates the influence of the number +of snapshots, K, on the ability to resolve two proximate +sources from heavy-tailed spatially-colored interference. The +equal-strength resolution scenario is repeated using K = +4, 8, 16, 32, 64 with different instances of NN training for +each K value. Fig. 7 shows the probability of resolution for +two equal-strength sources at θ1,2 = θc ±∆θ/2 for ∆θ = 12◦ +and θc = 0.55◦. +The FC-based NN approach fails to resolve the two sources. +For SIR = 0 dB, the MVDR, CNN, and DAFC-based NN + +DAFC +0.8 +SIR=-5.00 dB +INR=5.00 dB +0.6 +0.4 +0.2 +0.0 +-60 +-40 +-20 +0 +20 +40 +60 +Φ[Deg]1.0 +0.8 ++*+ +MVDR, SNR=O +DAFC, SNR=O +0.6 +FC, SNR=0 +res +CNN, SNR=0 +MVDR, SNR=-5 +P +0.4 +DAFC, SNR=-5 +FC, SNR=-5 +CNN, SNR=-5 +0.2 +0.0 +10 +20 +30 +40 +50 +60 +Ae [Deg]1.0 +0.8 +MVDR, SIR=O +DAFC, SIR=O +0.6 +FC, SIR=0 +res +CNN. SIR=0 +MVDR, SIR=-5 +P +0.4 +DAFC, SIR=-5 +FC, SIR=-5 +CNN, SIR=-5 +0.2 +0.0 +10 +20 +30 +40 +50 +60 +Ae [Deg]0.35 +FC +SIR=-5.00 dB +0.30 +NR=5.00 dB +0.25 +0.20 +0.15 +0.10 +0.05 +0.00 +-60 +-40 +-20 +0 +20 +40 +60 +Φ[Deg]MVDR +1.2 +SIR=-5.00 dB +1.0 +INR=5.00 dB +0.8 +0.6 +0.4 +0.2 +0.0 +-60 +-40 +-20 +0 +20 +40 +60 +Φ[Deg]0.6 +CNN +SIR=-5.00 dB +0.5 +INR=5.00 dB +0.4 +0.3 +0.2 +0.1 +0.0 +-60 +-40 +-20 +0 +20 +40 +60 +Φ[Deg]10 +Figure 6: Probability of resolution for two sources located at +θ1,2 = θc ±∆θ/2, and interference located at θc = 0.55◦. The +SIR in the legend represents the SIR of the first source, SIR1. +The SIR of the second source is set to SIR2 = SIR1 + 10 dB. +approaches achieve a monotonic increasing probability of +resolution with increasing K. The proposed DAFC-based NN +approach slightly outperforms other tested approaches. At low +SIR of SIR = −5 dB, the proposed DAFC-based NN ap- +proach significantly outperforms the other tested approaches. +This can be explained by the fact that increasing K increases +the probability for outliers to be present in the input data +matrix, X. Therefore, the estimated autocorrelation matrix, +ˆRx, is more likely to be biased by the interference-related +outliers, which results in interference “masking” the sources. +The proposed DAFC-based NN approach is immune to these +outliers and successfully exploits the information from the +additional snapshots to improve the probability of resolution. +Figure 7: Probability of resolution for two sources located at +θ1,2 = θc ± ∆θ/2 with ∆θ = 12◦, and interference located at +θc = 0.55◦, as a function of the number of snapshots, K. +Figs. 4, 5, 6, and 7 show the ability of the proposed +DAFC-based NN approach to utilize the information structure +of the input data by exploiting the higher-order statistics +and performing the domain-fitted transformation in order to +provide superior resolution ability in the case of proximate +heavy-tailed spatially-colored interference, low SIR and small +sample size. +3) Multiple Source Localization: The performances of the +tested DOA estimation approaches are evaluated and compared +in a multi-source scenario. Four sources, (M = 4) were +simulated with angular separation, ∆θ: {θ1, θ2, θ3, θ4} = +θc + {−2∆θ, −∆θ, ∆θ, 2∆θ}, where θc = 0.51◦ represents +a realistic off-grid condition. The RMSD of evaluated meth- +ods is depicted in Fig. 8. The proposed DAFC-based NN +approach outperforms the other tested approaches at low SIR +(SIR < 0 dB) for large and small angular separations. For +high SIR and low angular separation, ∆θ = 5◦, the MVDR +achieves the lowest RMSD. The reason is that for this case, the +interference is negligible with respect to the lobe of the strong +source in the MVDR’s spectrum. However, at high angular +separation, ∆θ = 20◦, the proposed DAFC-based NN ap- +proach significantly outperforms the other tested approaches. +This is explained by Fig. 9, that shows the spectrum of the +tested DOA estimation approaches. Notice that the proposed +DAFC-based NN mitigates interference, while the spectra of +other tested approaches contain high peaks at the interference +DOA, θc. These peaks increase the Hausdorff distance in (17), +increasing the RMSD of other tested approaches in Fig. 8. +Figure 8: RMSD in scenarios with M = 4 sources located at +{θ1, θ2, θ3, θ4} = θc + {−2∆θ, −∆θ, ∆θ, 2∆θ}, where θc = +0.51◦. +4) Multiple Source Enumeration: The source enumeration +performance is evaluated in this experiment. The DOAs of +the sources are selected from the set of following values: +{10.51◦, −9.49◦, −19.49◦, 10.51◦} such that for M sources, +the DOAs are selected to be the first M DOAs. The interfer- +ence is located at θc = 0.51◦. The proposed DAFC-based NN +approach is compared to the MDL and AIC [19]. Fig. 10 shows +the source enumeration confusion matrices for the MDL, AIC, +and the proposed DAFC-based NN with SIR = 0 dB. +Figs. 10a, 10b show that in both the MDL and the AIC, the +predicted number of sources has a constant bias for each true +M due to the spatially-colored interference. Fig. 10c shows the +source enumeration performance of the proposed DAFC-based +NN approach in the presence of spatially colored interference. +The DAFC-based NN identifies the interference and does not +count it as one of the sources by outputting a low probability +for angular grid points near θc, resulting in a better source +enumeration performance. + +1.0 +0.8 +MVDR, SIR=O +DAFC, SIR=O +0.6 +FC, SIR=0 +res +CNN. SIR=0 +DP +MVDR, SIR=-5 +0.4 +DAFC, SIR=-5 +FC, SIR=-5 +CNN, SIR=-5 +0.2 +0.0 +10 +20 +30 +40 +50 +60 +Ae [Deg]1.0 +0.8 +0.6 +res +MVDR. SIR=O +0.4 +DAFC, SIR=0 +FC, SIR=0 +CNN, SIR=0 +0.2 +MVDR,SIR=-5 +DAFC, SIR=-5 +FC, SIR=-5 +0.0 +CNN, SIR=-5 +4 +8 +16 +32 +64 +KRMSD [Deg] +101 +MVDR. A0=5 +DAFC, △0=5 +FC, △0=5 +CNN, △0=5 +MVDR,A0=20 +100 +DAFC, △0=20 +FC, △0=20 +CNN, △0=20 +-10 +-5 +0 +5 +10 +15 +20 +SIR[dB]11 +(a) FC +(b) MVDR +(c) CNN +(d) DAFC +Figure 9: Spatial spectrum, four sources with SIR = 0 dB +located at {θ1, θ2, θ3, θ4} = θc + {−2∆θ, −∆θ, ∆θ, 2∆θ}, +where θc = 0.51◦ and ∆θ = 20◦. The dashed blue lines rep- +resent the mean spatial spectrum, and the color fill represents +the standard deviation around the mean obtained from 2, 000 +i.i.d. examples. The solid vertical orange lines represent the +true source DOAs and the dashed vertical green line represents +the interference DOA. +5) Loss Weights: This experiment evaluates the effect of the +loss weight update factors, {β(l)}Nw +l=1, introduced in (12), on +the confidence level in the spatial spectrum. Let �B denote the +set of {β(l)}Nw +l=1 values used in the proposed approach. The +loss weights, {w(t) +i }d +i=1, are defined by the factors e(t) +0 , e(t) +1 +according to (11), and are introduced to provide a trade-off +between the penalty obtained on source/interference and the +penalty obtained for the rest of the output spatial spectrum. +For comparison, we set B0 = {10−6, 3.98 · 10−6, 1.58 · +10−5, 6.31 · 10−5, 2.51 · 10−4, 10−3}, and B1 = {10−3, 3.98 · +10−3, 0.0158, 0.063, 0.25, 0.1} as two sets of loss weight up- +date factors. For B0, the loss weight update factors are closer +to 0, hence the loss weights emphasize the source/interference, +since e(t) +1 +≪ e(t) +0 +which, according to (11), translates to larger +w(t) +i +for source/interference grid points. For B1 the values are +closer to 1, hence the loss weights are more equally distributed +among grid points, since e(t) +1 +≈ e(t) +0 . The experiment in IV-B1 +is repeated here for the DAFC-based NN approach with the +two additional B0, B1 values mentioned above. +Let ˆp1 represent the probability assigned for the source- +containing grid point in the estimated label ˆy. Let ˆp0 represent +the maximum over probabilities assigned for non-source grid +points in ˆy, excluding a 5-grid point guard interval around the +source. Fig. 11 shows ˆp1 and ˆp0 for various angular separations +between the source and interference for SIR = −5 dB. For +B0, the source’s contribution to the loss value is substan- +tially higher, which results in a higher probability for the +(a) MDL +(b) AIC +(c) DAFC +Figure 10: Confusion matrix for source enumeration, SIR = +0 dB, sources located at {10.51◦, −9.49◦, −19.49◦, 10.51◦}. +(a) MDL, (b) AIC, (c) proposed DAFC-based NN. +source-containing grid point. However, this results in a higher +probability obtained for non-source grid points, since their +contribution to the loss value is negligible compared to the +source-containing grid point, increasing “false-alarm” peaks +in the spatial spectrum, subsequently increasing the estimation +error. Correspondingly, for B1 the source’s contribution to the +loss value is less significant, which results in low probability +assigned for the source-containing grid points, as well as low +probability for non-source grid points. +V. CONCLUSION +This work addresses the problem of DOA estimation and +source enumeration of an unknown number of sources within +heavy-tailed, non-Gaussian, and spatially colored interference. +A novel DAFC-based NN approach is proposed for this + +FC +0.4 +SIR=0.00 dB +INR=5.00 dB +0.3 +0.2 +0.1 +0.0 +-60 +-40 +-20 +0 +20 +40 +60 +Φ[Deg]1.6 +1.4 +1.2 +1.0 +MVDR +SIR=0.00 dB +0.8 +INR=5.00 dB +0.6 +0.4 +0.2 +0.0 +-60 +-40 +-20 +0 +20 +40 +60 +Φ[Deg]0.8 +0.7 +0.6 +0.5 +CNN +0.4 +SIR=0.00 dB +NR三5.00 dB +0.3 +0.2 +0.1 +0.0 +-60 +-40 +-20 +0 +20 +40 +60 +Φ[Deg]0.8 +0.6 +DAFC +SIR0.00 dB +0.4 +INR=5.00dB +0.2 +0.0 +-60 +-40 +-20 +0 +20 +40 +60 +Φ[Deg]0 +0 +0 +0 +0 +0 +0 +0 +0.6 +0.5 +0 +0.13 +0.66 +0.2 +0 +0 +0 +True M +- 0.4 +0 +0 +0.16 +0.68 +0.16 +0 +0 +0.3 +0 +0 +0 +0.18 +0.68 +0.14 +0 +3 +0.2 +- 0.1 +0 +0 +0 +0 +0.21 +0.66 +0.12 +4 +- 0.0 +1 +1 +1 +1 +0 +1 +2 +3 +4 +5 +6 +Predicted M0.6 +0 +0 +0 +0 +0 +0 +0 +0 +0.5 +0 +0.06 +0.57 +0.36 +0.01 +0 +0 +- 0.4 +True M +0 +0 +0.07 +0.61 +0.32 +0 +0 +0.3 +0.2 +0 +0 +0 +0.08 +0.63 +0.29 +0 +3 +- 0.1 +0 +0 +0 +0 +0.09 +0.65 +0.26 +4 +- 0.0 +1 +1 +1 +1 +0 +1 +2 +3 +4 +5 +6 +Predicted M0 +0 +0 +0 +0 +0 +0.8 +0 +0.7 +0 +0.83 +0.17 +0.01 +0 +0 +-0.6 +True M +-0.5 +0 +0 +0.85 +0.15 +0 +0 +2 +- 0.4 +0.3 +0 +0 +0.01 +0.89 +0.1 +0 +- 0.2 +0 +0 +0 +0.01 +0.89 +0.1 +- 0.1 +4 +- 0.0 +1 +1 +1 +1 +1 +0 +1 +2 +3 +4 +5 +Predicted M12 +Figure 11: Loss weight update factor impact on probability +levels obtained in the DAFC-based NN’s spatial spectrum, +single target at θ1 = 0.55◦ with interference at θc = θ1 +∆θc, +SIR = −5 dB. ˆp1 represents the probability obtained for +source-containing grid points. ˆp0 represents the probability +obtained for non-source grid points. +problem. The DAFC mechanism applies a structured transfor- +mation capable of exploiting the interference non-Gaussianity +for its mitigation while retaining a low complexity of learnable +parameters. The proposed DAFC-based NN approach is opti- +mized to provide an interference-mitigated spatial spectrum +using a loss weight scheduling routine, performing DOA +estimation and source enumeration using a unified NN. +The performance of the proposed approach is compared to +MVDR, CNN-based, and FC-based approaches. Simulations +showed the superiority of the proposed DAFC-based NN ap- +proach in terms of probability of resolution and estimation ac- +curacy, evaluated by RMSD, especially in weak signal power, +small number of snapshots, and near-interference scenarios. +The source enumeration performance of the proposed DAFC- +based NN approach was compared to the MDL and AIC. 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Krolik, “Relationships between +adaptive minimum variance beamforming and optimal source localiza- +tion,” IEEE Transactions on Signal Processing, vol. 48, no. 1, pp. 1–12, +2000. + diff --git a/79E1T4oBgHgl3EQfBwKM/content/tmp_files/load_file.txt b/79E1T4oBgHgl3EQfBwKM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cab233c45769721202400cbc511e3d69a39fdaa6 --- /dev/null +++ b/79E1T4oBgHgl3EQfBwKM/content/tmp_files/load_file.txt @@ -0,0 +1,1051 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf,len=1050 +page_content='1 Neural Network-Based DOA Estimation in the Presence of Non-Gaussian Interference S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Feintuch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Tabrikian, Fellow, IEEE, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Bilik, Senior Member, IEEE, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Permuter, Senior Member, IEEE Abstract—This work addresses the problem of direction- of- arrival (DOA) estimation in the presence of non-Gaussian, heavy-tailed, and spatially-colored interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Conventionally, the interference is considered to be Gaussian-distributed and spatially white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' However, in practice, this assumption is not guar- anteed, which results in degraded DOA estimation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Maximum likelihood DOA estimation in the presence of non- Gaussian and spatially colored interference is computationally complex and not practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Therefore, this work proposes a neural network (NN) based DOA estimation approach for spa- tial spectrum estimation in multi-source scenarios with a-priori unknown number of sources in the presence of non-Gaussian spatially-colored interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The proposed approach utilizes a single NN instance for simultaneous source enumeration and DOA estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' It is shown via simulations that the proposed approach significantly outperforms conventional and NN-based approaches in terms of probability of resolution, estimation accuracy, and source enumeration accuracy in conditions of low SIR, small sample support, and when the angular separation between the source DOAs and the spatially-colored interference is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Index Terms—Array Processing, DOA Estimation, Source Enumeration, Spatially-Colored Interference, Non-Gaussian In- terference, Neural Networks, Deep Learning, Machine Learning, MVDR, MDL, AIC, Radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' INTRODUCTION Direction-of-arrival (DOA) estimation using a sensor array is required in multiple applications, such as radar, sonar, ultrasonic, wireless communications, and medical imaging [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' In real-world applications, the signal received at the sensor array is a superposition of signals from the sources of interest, interference, and receiver thermal noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' In radars, the received signal consists of a target echo, clutter, and thermal noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' In multiple scenarios, the radar clutter has a spatially-colored, heavy-tailed non-Gaussian distribution [2], which can signifi- cantly degrade the performance of conventional estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Minimum-variance-distortionless-response (MVDR) [3], is a conventional adaptive beamforming approach for DOA es- timation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' MVDR estimates the spatial spectrum and obtains the source DOAs via a one-dimensional peak search on a predefined grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The estimation of signal parameters using rotational invariance techniques (ESPIRIT) [4], multiple signal classification (MUSIC) [5], and root-MUSIC (R-MUSIC) [6] are additional widely used DOA estimation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' These approaches involve received signal autocorrelation matrix processing, which conventionally is performed via the sam- ple autocorrelation matrix estimation [3]–[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' However, the Stefan Feintuch, Joseph Tabrikian, Igal Bilik, and Haim H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Permuter are with the School of Electrical and Computer Engineering, Ben Gurion University of the Negev, Beer Sheva, Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' (e-mails: stefanfe@post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='bgu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='il, joseph@bgu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='il, bilik@bgu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='il, haimp@bgu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='il) performance of the sample autocorrelation matrix estimator degrades in small sample support or non-Gaussian scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Furthermore, these methods use the second-order statistics only and omit the higher-order statistics on non-Gaussian- distributed interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' In addition, ESPRIT, MUSIC, and R- MUSIC approaches require a-priori knowledge of the number of sources (or targets), which limits their practical use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The problem of DOA estimation in the presence of non- Gaussian interference is of great practical interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The max- imum likelihood estimator (MLE) for DOA estimation in the presence of non-Gaussian interference does not have a closed- form analytical solution [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Multiple model-based DOA estimation approaches have been intensively studied in the literature [7]–[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Robust covariance matrix-based DOA estimation and source enumeration methods have been studied in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' For complex elliptically symmetric (CES) distributed data, the authors in [9] showed that a scatter matrix-based beamformer is consistent, and the semiparametric lower bound and Slepian- Bangs formula for DOA estimation were derived in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' In [11], a generalized covariance-based (GC) approach for the covariance matrix estimation in scenarios with impulsive alpha-stable noise was proposed for MUSIC DOA estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' However, these methods consider a specific family of distri- butions, such as the CES or alpha-stable, and are therefore, limited in the case of model mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' In [12], a probability measure transform (MT) based covariance matrix estimator was proposed for MUSIC-based DOA estimation and mini- mum descriptive length (MDL) based source enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The MT-based covariance estimator was also adopted for robust MVDR beamformer [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' These methods are usually based on setting a parameter that determines the tradeoff between the level of robustness and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The problem of DOA estimation in the presence of a mix- ture of spatially-white K-distributed and Gaussian-distributed noise under a deterministic and unknown (conditional) source model was studied in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' An iterative MLE-based approach for the conditional and joint likelihood of interference distri- bution’s parameters was derived in [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' This approach was further extended in [16] to marginal likelihood function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' However, this approach is computationally complex due to numerical integral evaluation that involves a 2M dimensional grid search for M targets [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Therefore, [8] proposed a kernel minimum error entropy-based adaptive estimator and a novel criterion to reduce the estimator’s computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The expectation-maximization (EM) with a partial relaxation- based DOA estimation algorithm under the conditional model assumption was proposed in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' In [18] a sparse Bayesian learning (SBL) approach for outlier rejection of impulsive arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='02856v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='SP] 7 Jan 2023 2 and spatially-white interference was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' This EM-based approach does not require a-priori knowledge of the number of sources and was shown to resolve highly-correlated and co- herent sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' However, none of these model-based DOA es- timation approaches considered an a-priori unknown number of sources and spatially-colored interference and therefore are limited for real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Although source enumera- tion methods, such as MDL and Akaike information criterion (AIC) [19] can be used, they assume signal Gaussianity, and can therefore be inaccurate in non-Gaussian scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Deep learning and machine learning approaches were re- cently adopted for radar signal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Three types of NN-based DOA estimation approaches have been introduced in literature [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The first approach assumes a-priori known number of sources, and uses a NN, which is optimized to output a vector of the estimated DOAs [21]–[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The second approach does not assume a-priori known number of sources and uses a NN for source enumeration [25]–[31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The third approach uses a NN to estimate source presence probability at each DOA on a predefined angular grid and obtains the source DOAs via a peak search [32]–[41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' However, all these approaches have not addressed non-Gaussian and spatially- colored interference [20]–[41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The cases of non-Gaussian and/or spatially-colored inter- ference have been addressed using machine learning-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' For massive MIMO cognitive radar, a reinforce- ment learning-based approach for multi-target detection un- der heavy-tailed spatially-colored interference was proposed in [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' In [43], authors addressed the MIMO radar target detection under non-Gaussian spatially-colored interference by using a CNN architecture that is optimized according to a novel loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' A radial-basis-function (RBF) NN [44] and a convolutional neural network (CNN) [45] architectures were proposed for DOA estimation in the presence of non-Gaussian impulsive noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' In [46], a CNN-based architecture that in- cludes denoising NN, source enumeration NN, and DOA esti- mation sub-NNs, was introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' However, [44]–[46] consider spatially-white noise and are suboptimal for scenarios with spatially-colored interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' This work addresses the problem of DOA estimation of a- priori unknown number of sources in the presence of non- Gaussian, heavy-tailed, spatially-colored interference at a low signal-to-interference ratio (SIR) and small sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The contribution of this work include: 1) A novel NN-based processing mechanism is used for array processing within non-Gaussian spatially-colored interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The proposed NN architecture utilizes the structure of information within the set of received com- plex snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 2) The proposed NN is optimized to output an interference- mitigated spatial spectrum, and is used for simultaneous source enumeration and DOA estimation of sources within non-Gaussian spatially-colored interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The proposed approach outperforms conventional adaptive beamforming and competing straightforward NN-based meth- ods in terms of probability of resolution and estimation accuracy in scenarios with non-Gaussian spatially-colored interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' In addition, the proposed approach outperforms conventional source enumeration techniques in scenarios char- acterized by non-Gaussian spatially-colored interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The following notations are used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Roman boldface lower-case and upper-case letters represent vectors and matrices, respectively while Italic letters stand for scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' IN is the identity matrix of size N × N and 1N is a column vector of length N whose entries are equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' E(·), (·)T , and (·)H are the expectation, transpose, and Hermitian transpose operators, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Vec(·), diag(·), and | · | stand for the vectorization, diagonalization, and absolute value operators, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' [a]n and [A]n,m are the n-th and n, m-th elements of the vector a and the matrix A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The addressed problem is stated in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Section III intro- duces the proposed NN-based DOA estimation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The proposed approach is evaluated via simulations in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Our conclusions are summarized in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' PROBLEM DEFINITION This work considers the problem of DOA estimation using an array of L receiving elements and M distinct and unknown sources with DOAs, Θ = {θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' , θM}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The measurements contain K spatial snapshots, {xk}K k=1: xk = A (Θ) sk + σcck + nk , (1) = M � m=1 a (θm) sk,m + σcck + nk , k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' , K , where A (Θ) = �a (θ1) · · a (θM)� , with a (θm) ∈ CL denoting the steering vector for source at direction θm, and sk ≜ �sk,1 · · sk,M �T is the source signal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' We assume an unconditional model [47], where {sk} i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' ∼ CN � 0M, diag � σ2 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' , σ2 M �� , is temporally uncorrelated be- tween pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The targets are assumed to be spatially distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The receiver thermal noise, denoted by nk, is considered to be complex Gaussian-distributed {nk} i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' ∼ CN � 0L, σ2 nIL � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The heavy-tailed non-Gaussian and spatially-colored interference is modeled by the interference amplitude σc, and the interference component ck ∈ CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The considered compound-Gaussian distributed interference, {ck} i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' ∼ K (ν, θc) represents a non- Gaussian interference with angular spread around an unknown direction θc, such that c ∼ K (ν, θc) implies c = √τz , (2) τ |= z, τ ∼ Γ (ν, ν) , z ∼ CN (0L, Mθc) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The compound-Gaussian statistical model is conventionally used in the literature to model heavy-tailed non-Gaussian interference [7], [8], [14], [16], [43], [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The texture com- ponent, τ ∈ R+, determines the heavy-tailed behavior and is characterized by, ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The speckle component, z ∈ CL, determines the spatial distribution of the interference and is characterized by the covariance matrix, Mθc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The spatial covariance matrix of the interference upholds: E � σ2 cccH� =σ2 cE [τ] E � zzH� = σ2 cMθc , (3) 3 where Mθc can be modeled as [14]–[16], [43], [48]: [Mθc]m,l = ρ|m−l|ej(m−l)π sin θc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' (4) The model in (3) and (4), represents the spatial interference, characterized by ρ, with a spread around the interference DOA, θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' THE PROPOSED DAFC-BASED NEURAL NETWORK The proposed approach generalizes the NN architecture that was introduced for linear-frequency-modulated (LFM) radar target detection in the range-Doppler domain [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' In the following, the data pre-processing and the proposed NN-based processing mechanism are introduced in Subsections III-A and III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The proposed NN architecture and loss function are detailed in Subsections III-C and III-D, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Pre-Processing The input matrix, X ∈ CL×K is constructed from the set of K snapshots in (1), {xk}: X = � x1 x2 · · xK � , (5) where the k-th column of X contains the k-th snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The variation between the columns of X is induced by the statistical characteristics of the source signal sk, interference signal ck, and thermal noise nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Therefore, each column in X can be interpreted as a complex “feature” vector containing essential information for DOA estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The set of columns in X can be interpreted as “realizations” of that feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The complex-valued matrix, X, is converted into real-valued representation needed for the NN-based processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' To keep consistency with [49], we apply a transpose operator to the input matrix, such that the snapshots are stacked in rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The output of the pre-processing denoted by Z0 ∈ CK×2L, is: Z0 = � Re � XT � , Im � XT �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' (6) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Dimensional Alternating Fully-Connected The dimensional alternating fully-connected (DAFC) block was introduced to process measurements in a form similar to the model in Section II [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 1 schematically shows the DAFC mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' For arbitrary dimensions D1, D2, D3, the formulation of a general fully-connected (FC) layer applied to each row in a given matrix Z ∈ RD1×D2 can be represented by the transform F (·): F : RD1×D2 → RD1×D3 , (7) F (Z) ≜ h � ZW + 1D1bT � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' This matrix-to-matrix transformation is characterized by the “learnable” weight matrix, W ∈ RD2×D3, the bias vector, b ∈ RD3, and a scalar element-wise activation function, h(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Let Fr (·) and Fc (·) be two separate, and not necessarily identical instances of F (·) from (7), and Zin be an arbitrary input matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The DAFC mechanism is formulated by the following operations: Dimensional Alternating Fully Connected Input: Zin ∈ RH×W Fr : RH×W → RH×W ′ Fc : RW ′×H → RW ′×H′ 1) Apply a single FC layer to each row in Zin: Zr = Fr (Zin) 2) Apply a single FC layer to each column in Zr: Zc = Fc � ZT r � 3) Transpose to keep orientation: Zout = ZT c Output: Zout ≜ S (Z) ∈ RH′×W ′ In the following, three DAFC design principles are detailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 1) Structured transformation The input to the first DAFC block is the pre-processed, Z0, given in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Therefore, the first FC layer, Fr, of the first DAFC block extracts spatial-related features from each row in Z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The second FC layer, Fc, of the first DAFC block, introduces an interaction between transformed rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' This implies that a) Fr performs “spatial-feature” extraction by transforming the pre-processed i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' snapshots (the rows of Z0) to a high-dimensional feature space, and b) the Fc performs a nonlinear transformation of the extracted features (the columns of Fr (Z0)) from each snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' In this way, the DAFC utilizes both spatial and statistical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' In addition, it can exploit high-order statistics-related features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Thus, the DAFC mechanism can contribute to estimating the source DOAs and mitigating the interference when incorporated into a NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 2) Sparsity Conventional DOA estimation considers the input data as the collection of measurement vectors (the snapshots {xk}) in a matrix form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' One straightforward approach to processing the input data using a NN is to reshape it and process it via an FC-based architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' In this way, each neuron in the layer’s output interacts with every neuron in the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' On the other hand, the DAFC block transforms the data using a structured transformation, which is significantly sparser in terms of learnable parameters compared to the straightforward FC-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' This parameter reduction can be observed in the following typical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Consider an input matrix Z1 ∈ RD1×D1, which is transformed to an output matrix Z2 ∈ RD2×D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The number of learnable parameters in the FC- and the proposed DAFC-based approaches is of the order of O � D2 1D2 2 � , and O (D1D2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Notice that the DAFC-based transfor- mation complexity grows linearly with the number of learnable parameters compared to the quadratic complexity growth of the straightforward, FC-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The contribution of learnable parameters dimension reduc- tion is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' First, the conventional NN optimization is gradient-based [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Therefore, a significant reduction in the learnable parameter dimension reduces the degrees of freedom in the optimizable parameter space and improves the gradient- based learning algorithm convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Second, reduction 4 Figure 1: The DAFC mechanism concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Each row of dimen- sion W in Zin, represented by the red color, is transformed by Fr to a row of dimension W ′ in the middle matrix, represented by the transparent red color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Next, each column of dimension H in the middle matrix, represented by the blue color, is transformed by Fc to a column of dimension H′ in Zout, represented by the transparent blue color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' in the learnable parameter dimension can be interpreted as increasing the “inductive bias” of the NN model [51], which conventionally contributes to the NN statistical efficiency and generalization ability, thus, reducing the NNs tendency to overfit the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 3) Nonlinearity The proposed DAFC considers an additional degree of nonlinearity compared to the straightforward FC-based ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' A straightforward matrix-to-matrix approach includes an interaction of every neuron in the output matrix with every neuron in the input matrix, followed by an element-wise nonlinear activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' On the other hand, the proposed DAFC consists of two degrees of nonlinearity, in Fr and Fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Although the weight matrices applied as part of Fr and Fc are of lower dimension than the weight matrix used in the straightforward approach, the extra degree of nonlinearity can increase the NN’s capacity [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Therefore, a NN architecture with the proposed DAFC is capable of learning a more abstract and rich transformation of the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' NN Architecture The continuous DOA space is discretized into a d- dimensional grid: φ = �φ1 φ2 · · φd �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' This implies that the entire field-of-view (FOV) is partitioned into d DOAs, {φi}d i=1, determined by the selected grid resolution, ∆φ ≜ φi+1 − φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The proposed NN is designed to represent a mapping from the input set of snapshots, {xk} given in (1), into the probability of source present in the DOAs {φi}d i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The proposed NN architecture is formulated as follows: Z0 = P (X) , (8) zvec = Vec (S6 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' S1 (Z0))) , ˆy = G3 (G2 (G1 (zvec))) , Operator Output Dimension Activation # Parameters P K × 2L S1 64 × 256 tanh- ReLu 9,536 S2 128 × 512 tanh- ReLu 139,904 S3 256 × 1024 tanh- ReLu 558,336 S4 64 × 512 tanh- ReLu 541,248 S5 16 × 256 tanh- ReLu 132,368 S6 4 × 128 tanh- ReLu 32,964 vec 512 G1 1024 tanh 525,312 G2 256 tanh 262,400 G3 d sigmoid 31,097 Table I: Specification of the proposed NN architecture for K = 16, L = 16, d = 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' “tanh-ReLu” activation stands for tanh in Fr and ReLU in Fc of each DAFC block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The number of total learnable parameters is 2, 233, 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' where Z0 is the output of the pre-processing procedure, denoted as P (·) and detailed in Section III-A, and X is the input matrix in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' In the next stage, six DAFC instances, represented by S1 (·) , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' , S6 (·), of different dimensions with tanh activa- tion for the row transform (Fr in Section III-B) and ReLu activation for the column transform (Fc in Section III-B), are used to generate the vectorized signal zvec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Our experiments showed that this configuration of row and column activation functions provides the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' At the last stage, the signal, zvec, is processed by three FC layers, where the first two use tanh activation, and the final (output) layer of equal size to the DOA grid dimension, d, uses sigmoid activation function to output ˆy ∈ [0, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Thus, {[ˆy]i}d i=1 represent the estimated probabilities of a source presence at {φi}d i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Table I and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 2 summarize the parameters and architecutre of the proposed NN-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The estimated source DOAs are extracted from the spatial spectrum via peak search and applying 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='5 threshold: {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' , i ˆ N} = peak search � {[ˆy]i}d i=1 � (9) ˆΘ = � φin : [ˆy]in > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='5 � ˆ N n=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Namely, the set of estimated DOAs, ˆΘ, consists of the grid points corresponding to the peaks of ˆy that exceed the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='5 threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The number of peaks that exceed this threshold is used for source enumeration, and therefore the proposed NN can be utilized as a source enumeration method as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The dimensionality of the hidden layers in the proposed 5 Figure 2: Proposed NN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The pre-processing P is described in Section III-A and appears in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The purple matrices denote the concatenation of DAFC blocks, which is detailed in Section III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The blue vector represents a vectorization of the last DAFC output, and the orange vectors stands for FC layers with tanh activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The last green vector is the output of the last FC layer, which consists of sigmoid activation function and yields the estimated spatial spectrum ˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' NN architecture expands in the first layers and then reduces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' This trend resembles the NN architecture presented in [49] and characterizes both the DAFC-based and FC-based processing stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' This expansion-reduction structure can be explained by a) the early NN stages need to learn an expressive and meaningful transformation of the input data by mapping it to a higher dimensional representation and b) the late stages need to extract significant features from the early mappings, and are therefore limited in dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' In addition, the late stages are adjacent to the output vector and therefore need to be of similar dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Loss Function The label used for the supervised learning process, y ∈ {0, 1}d, is defined as a sparse binary vector with the value 1, at the grid points that correspond to the source DOAs, and 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' In practice, the DOAs in Θ do not precisely correspond to the grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Therefore, for each DOA in Θ, the nearest grid point in {φi}d i=1 is selected as the representative grid point in the label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Each training example is determined by the input-label pair, (X, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Using the NN feed-forward in (8), X is used to generate the output spatial spectrum, ˆy, which is considered as the estimated label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The loss function, L, is a weighted mean of the binary cross entropy (BCE) loss computed at each grid point: L (y, ˆy, t) = 1 d d � i=1 w(t) i BCE ([y]i , [ˆy]i) , (10) BCE (y, ˆy) = −y log (ˆy) − (1 − y) log (1 − ˆy) , where w(t) i represents the loss weight of the i-th grid point at the t-th epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The loss value for equally-weighted BCEs eval- uated per grid point (w(t) i = 1 in (10)) does not significantly increase in the case of a large error in source/interference esti- mated probability, due to the sparsity of the label y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' This forces the NN convergence into a sub-optimal solution that is prone to “miss” the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Therefore, the loss weights, {w(t) i }d i=1, are introduced to “focus” the penalty on source/interference grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The loss weight of the i-th grid point, w(t) i , is determined by the presence of source or interference in the corresponding label entry [y]i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' This relation is defined using the epoch and label dependent factors e(t) 0 , e(t) 1 , according to: w(t) i = � 1/e(t) 1 , if φi contains source or interference 1/e(t) 0 , else .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' (11) For t = 0, the factor e(0) 1 is determined by the fraction of label grid points that contain source or interference out of the total label grid points in the training set, and e(0) 0 is the corresponding complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' For subsequent epochs, the factors are updated according to a predefined schedule, similarly to a predefined learning rate schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The loss weights are updated Nw times with spacing of ∆t epochs during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The update values are determined by updating e(t) 0 , e(t) 1 , according to the following decaying rule: e(t) q = (1 − β(l))e(l∆t) q + β(l), l∆t ≤ t < (l + 1)∆t (12) q = 0, 1, l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' , Nw, 6 where l is the loss weight update iteration, and {β(l)}Nw l=1 represent the loss weight update factors which uphold, 0 ≤ β(l) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Note that for Nw∆t ≤ t, the weight factor remains e(Nw∆t) i during the rest of the training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' No- tice that as β(l) → 1, the corresponding loss weights will tend to be equally distributed across the grid points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=', e(t) 1 ≈ e(t) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' In this case, an erroneously estimated proba- bility for source/interference containing grid point is equally weighted to a neither-containing grid point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' On the other hand, as β(l) → 0, the corresponding factors will uphold e(t) 1 ≪ e(t) 0 , yielding a significantly larger contribution of source/interference containing grid points to the loss value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The rule in (12) enables a “transition of focus” throughout the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' That is, during the early epochs β(l) → 0, which contributes more weight to the source/interference containing areas in the estimated label ˆy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=', the estimated spatial spec- trum) to focus the NN to being correct for source/interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' During the later epochs, β(l) is incrementally increased, which relaxes the focus on source/interference from early epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Thus, reducing erroneously estimated sources in areas that do not contain source/interference (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' “false-alarms”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' PERFORMANCE EVALUATION This section evaluates the performance of the proposed DAFC-based NN approach and compares it to the conventional approaches, summarized in Subsection IV-A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The data for all considered scenarios is simulated using the measurement model from Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Setup & Training This work considers a uniform linear array (ULA) with half-wavelength-spaced L elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Each simulated example consists of the input-label pair, (X, y), where the input X is defined in (5), and the label y is defined in Section III-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The simulation configurations are detailed in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The performance of the proposed approach is evaluated using a single NN instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Therefore, a single NN model is used for various signal-to-interference ratios (SIRs), signal- to-noise ratios (SNRs), interference-to-noise ratios (INRs), DOAs, interference distribution, and the number of sources for joint DOA estimation and source enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The following definitions for the m-th source are used in all experiments: INR = E[∥c∥2] E[∥n∥2] = σ2 c/σ2 n , (13) SNRm = E[∥a(θm)sm∥2] E[∥n∥2] = σ2 m/σ2 n , (14) SIRm = E[∥a(θm)sm∥2] E[∥c∥2] = σ2 m/σ2 c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' (15) The NN optimization for all evaluated architectures is performed using the loss function in (10) and Adam opti- mizer [52] with a learning rate of 10−3, and a plateau learning rate scheduler with a decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The set of loss weight up- date factors, {β(l)}Nw l=1, in (12) is chosen as the evenly-spaced logarithmic scale between 10−5 and 10−2 with Nw = 6, that is {10−5, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='25 · 10−5, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='25 · 10−4, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='8 · 10−3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='78 · 10−2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The chosen batch size is 512, the number of epochs is 500, and early stopping is applied according to the last 200 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Notation Description Value Mmax Maximal number of sources 4 L Number of sensors 16 K Number of snapshots 16 d Angular grid dimension 121 ∆φ Angular grid resolution 1◦ FOV Field of view [−60◦, 60◦] σ2 n Thermal noise power 1 Table II: Simulation Configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 1) DOA Estimation Approaches: This subsection briefly summarizes the conventional DOA estimation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The performance of the proposed approach is compared to the conventional MVDR, CNN, and FC-based NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' All the NN- based approaches were implemented using similar number of layers and learnable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' In addition, the FC-based NN and CNN were optimized using the same learning algorithm and configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' (a) Conventional Adaptive Beamforming The MVDR [3] estimator is based on adaptive beamforming, and it is the maximum likelihood estimator in the presence of unknown Gaussian interference [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The MVDR estimates DOAs by a peak search on the MVDR spectrum: PMV DR (φ) = 1 aH (φ) ˆR−1 x a (φ) , (16) where ˆRx = 1 K �K k=1 xkxH k is the sample covariance ma- trix estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Notice that the MVDR spectrum utilizes only second-order statistics of the received signal xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' For Gaussian- only interference (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' ck = 0 in (1)), the second-order statistics contains the entire statistical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' However, for non- Gaussian interference, information from higher-order statistics is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' (b) CNN Architecture We consider a CNN-based DOA estimation approach using a CNN architecture that is similar to the architecture provided in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The input to the CNN of dimension L × L × 3 consists of the real, imaginary, and angle parts of ˆRx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The CNN architecture consists of 4 consecutive CNN blocks, such that each block contains a convolutional layer, a batch normalization layer, and a ReLu activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The convolutional layers consist of [128, 256, 256, 128] filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Kernel sizes of 3 × 3 for the first block and 2 × 2 for the following three blocks are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Similarly to [38], 2 × 2 strides are used for the first block and 1 × 1 for the following three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Next, a flatten layer is used to vectorize the hidden tensor, and 3 FC layers of dimensions 1024, 512, 256 are used with a ReLu activation and Dropout of 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Finally, the output layer is identical to the proposed DAFC-based NN as detailed in Subsection III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The considered loss function is identical to the proposed DAFC-based approach in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The number of trainable parameters in the considered CNN 7 architecture accounts for 3, 315, 449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Notice that the CNN- based architecture utilizes the information within the sample covariance matrix and therefore, is limited to second-order statistics only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' (c) FC Architecture A straightforward implementation of an FC-based architecture, as mentioned in Subsection III-B, was implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The data matrix, X, is vectorized, and the real and imaginary parts of the values were concatenated to obtain a 2KL- dimension input vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The selected hidden layers are of sizes: [512, 512, 1024, 1024, 512, 256] where each hidden layer is followed by a tanh activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The output layer is identical to the proposed DAFC-based NN approach as detailed in Subsection III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The considered loss function is (10), and the number of trainable parameters in the FC- based NN accounts for 2, 787, 449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Notice that the FC-based NN architecture utilizes all the measurements by interacting with all samples in the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' However, this processing is not specifically tailored to the structure of information within the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' On the other hand, the proposed DAFC- based NN utilizes the information structure to process the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Therefore, for the considered DOA estimation problem, the “inductive bias” [51] for this approach is improper and can result in under-fitted NN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 2) Performance Evaluation Metrics: This subsection dis- cusses the criteria for the performance evaluation of the proposed DOA estimation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' In this work, similarly to [38], the DOA estimation accuracy of a set of sources is evaluated by the Hausdorff distance between sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The Hausdorff distance, dH between the sets, A, and B, is defined as: dH (A, B) = max {d (A, B) , d (B, A)} , (17) d (A, B) = sup {inf {|α − β| : β ∈ B} : α ∈ A} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Notice that d (A, B) ̸= d (B, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Let Θ = {θm}M m=1 and ˆΘ = {ˆθm} ˆ M m=1 be the sets of true and estimated DOAs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The estimation error is obtained by evaluating the Hausdorff distance, dH(Θ, ˆΘ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' We define the root mean squared distance (RMSD) for an arbitrary set of N examples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=', test set), � X(n), y(n)�N n=1, with the corresponding true and estimated DOAs, � Θ(n), ˆΘ(n)�N n=1 as: RMSD ≜ � � � � 1 N N � n=1 d2 H � Θ(n), ˆΘ(n) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' (18) Angular resolution is one of the key criteria for DOA estimation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The probability of resolution is com- monly used as a performance evaluation metric for angular resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' In the considered problem, resolution between two sources and between source and interference are used for performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' For an arbitrary example with M sources, the resolution event Ares is defined as: Ares � Θ, ˆΘ � ≜ � 1, �M m=1 ξm ≤ 2◦ and | ˆΘ| ≥ M 0, else , (19) ξm ≜ min ˆθ∈ ˆΘ |θm − ˆθ|, m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' , M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' For example, a scene with M sources is considered success- fully resolved if for each true DOA a) there exists a close- enough estimated DOA, ˆθ ∈ ˆΘ, that is at most 2◦ apart, and b) there exists at least M DOA estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' According to (18), the probability of resolution, can be defined as: Pres = 1 N N � n=1 Ares � Θ(n), ˆΘ(n)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' (20) 3) Data Sets: This subsection describes the structure and formation of Training & Test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' (a) Training Set The considered training set contains Ntrain = 10, 000 examples re-generated at each epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' For each exam- ple, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' an input-label pair (X, y), the number of DOA sources, M, is generated from uniform and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' distribution, {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' , Mmax}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The training set contains 10% of interference- free examples and 90% of interference-containing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Out of the interference-containing examples, 90% generated such that the source DOAs, {θm}M m=1, and the interference’s DOA, θc, are distributed uniformly over the simulated FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The remaining 10% are generated such that θc is distributed uniformly over the FOV, and the source DOAs, {θm}M m=1, are distributed uniformly over the interval [θc − 8◦, θc + 8◦].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' This data set formation enables to “focus” the NN training on the chal- lenging scenarios where the source and interference DOAs are closely spaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The generalization capabilities of the proposed NN to variations in interference statistics are achieved via the interference angular spread parameter, ρ, from the uniform dis- tribution, U ([0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='95]), and the interference spikiness param- eter, ν, from the uniform distribution, U ([0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The INR for each interference-containing example and {SIRm}M m=1 or {SNRm}M m=1 are drawn independently according to Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' (b) Test Set The test set consists of Ntest = 20, 000 examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The results are obtained by averaging the evaluated performance over 50 independent test set realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Considering the low-snapshot support regime, the number of snapshots is set to K = 16, except for experiment (c) in IV-B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Considering heavy-tailed interference, the spikiness parameter is set to ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The INR is set to INR = 5 dB, and the interference angular spread parameter is set to ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The signal amplitude was set to be identical for all sources, σ1 = · · · = σm, except for experiment (b) in IV-B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Experiments 1) Single Source Within Interference: In this scenario, the ability to resolve a single source from interference is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Let M = 1 with θ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='55◦, and θc = θ1 + ∆θc such that ∆θc is the angular separation between the single source and interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='55◦ offset is considered to impose a realistic off-grid condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 3 shows the RMSD and probability of resolution for all evaluated approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 3a shows that the FC-based NN approach does not manage to resolve the single source from the interference for all evaluated angular separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' This result supports the 8 (a) (b) Figure 3: Scenario with a single source at θ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='55◦ and interference located at θc = θ1 + ∆θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' (a) probability of resolution and (b) RMSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Notation Description Value ρ Interference angular spread parameter ∼ U ([0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='95]) ν Interference spikiness parameter ∼ U ([0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='5]) INR INR ∼ U ([0, 10]) [dB] SIRm SIR of m-th source ∼ U ([−10, 10]) [dB] SNRm SNR of m-th source ∼ U ([−10, 10]) [dB] Table III: Training set parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' SNRm distribution applies to interference-free examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' under-fitting limitation of the FC-based NN approach for the DOA estimation, which can be explained by the architecture that processes the input data as-is, without any structured transformation or model-based pre-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The MVDR and CNN performance in terms of the resolu- tion are similar since both rely only on second-order statistics, which is sufficient in scenarios with widely separated sources and interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 3a shows that the proposed DAFC-based NN approach outperforms all other considered approaches in low angular separation scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' This can be explained by the fact that the DAFC uses the high-order statistics needed for the resolution of closely spaced sources and interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 3b shows the RMSD of all considered DOA estimation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The proposed DAFC-based NN approach outper- forms the other tested approaches in low SIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' At high SIR and small angular separation, ∆θc = 5◦, the interference is negligible with respect to the strong source signal, and therefore, the DAFC-based, CNN, and MVDR approaches obtain similar performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' For large angular separation, ∆θc = 30◦, the source and the interference are sufficiently separated, and therefore, DOA estimation errors are mainly induced by the interference DOA, θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The MVDR spectrum contains a peak at θc = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='55◦, and therefore, MVDR’s RMSD = 30◦ is approximately constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The NNs are trained to output a 0-probability for the interference, therefore, the NN-based approaches: FC, CNN, and DAFC achieve a smaller DOA estimation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The DAFC-based NN and CNN utilize structured transformations, which better fit the input data, and therefore, they outperform the FC-based NN approach in terms of RMSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 2) Resolving Two Sources from Interference: This subsec- tion evaluates the performance of the tested DOA estimation approaches in scenarios with two sources within AWGN and interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' (a) Resolution of Equal-Strength Sources In the following experiment, the resolution between two equal- power sources, M = 2, with θ1 = − ∆θ 2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='55◦, and θ2 = ∆θ 2 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='55◦, is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The off-grid additional 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='55◦ offset to the ∆θ angular separation between the sources represents the practical scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The interference at θc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='55◦ influences the two sources similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 4 shows the probability of resolution of the tested approaches in scenarios with (a) the AWGN only and (b) spatially-colored interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The FC-based NN approach does not resolve the two targets in both evaluated scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Subplot (a) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 4 shows that the proposed DAFC-based NN approach outperforms the MVDR and the CNN at low-SNR and small angular separation scenarios due to its generalization ability to spatially-white interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Subplot (b) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 4 shows that at low SIR of SIR = −5 dB, the performances of MVDR and CNN significantly degrade compared to the proposed DAFC-based NN approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Comparing subplots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 4, notice that at SIR = −5 dB, the MVDR fails to resolve the sources with angular separation ∆θ < 20◦ due to the presence of the heavy- tailed spatially-colored interference in the proximity of the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' However, the proposed DAFC-based NN approach mitigates this interference and resolves the sources, and hence, outperforms other tested approaches at both SIR = 0 dB and SIR = −5 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Subplot (b) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 4 shows the non-monotonic trend of CNN and MVDR performance at 4◦ < ∆θ < 18◦ and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='8 MVDR, SIR=0 DAFC, SIR=0 FC, SIR=0 S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='6 res CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' SIR=O P MVDR, SIR=-5 DAFC, SIR=-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='4 FC, SIR=-5 CNN, SIR=-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='2 10 15 20 25 30 △0c [Deg]RMSD [Deg] 101 MVDR, △0c=5 DAFC, A0c=5 FC, △Qc=5 CNN, △0c=5 MVDR, △0c=30 DAFC, △0c=30 100 FC, △0c=30 CNN, △0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='=30 10 5 0 5 10 15 20 SIR[dB]9 (a) (b) Figure 4: Probability of resolution for two sources located at θ1,2 = θc ± ∆θ/2, and interference located at θc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='55◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' (a) AWGN-only scenario and (b) interference-containing scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' (a) FC (b) MVDR (c) CNN (d) DAFC Figure 5: Spatial spectrum, two sources with SIR = −5 dB located at θ1,2 = θc ± ∆θ/2 with ∆θ = 12◦ and θc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='55◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The dashed blue lines represent the mean spatial spectrum, and the color fill represents the standard deviation around the mean obtained from 2, 000 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The solid vertical orange lines represent the true source DOAs, and the dashed vertical green line represents the interference DOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' SIR = −5 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' For 4◦ < ∆θ < 8◦ the sources are closer to the peak of the interference’s lobe and are therefore less mitigated by it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' As ∆θ initially increases, 8◦ < ∆θ < 12◦, the sources reach DOAs which are in the proximity of the interference lobe’s “nulls” which explains the reduction in resolution, and as ∆θ further increases, 16◦ < ∆θ, the sources are sufficiently separated from the interference such that the resolution increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' As a result, MVDR and CNN- based approaches that use second-order statistics only, can not resolve the sources in the vicinity of a stronger interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 5 shows the average spatial spectrum of all tested approaches for ∆θ = 12◦ and SIR = −5 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The average spatial spectrum of the FC-based NN approach does not show two prominent peaks, which results in its poor probability of resolution in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The MVDR “bell-shaped” spatial spec- trum does not contain the two prominent peaks at θ1,2 since the interference “masks” the two sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The CNN and proposed DAFC-based NN approaches show two peaks at the average spatial spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The peaks at the CNN’s average spatial spectrum are lower, resulting in a low-resolution probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The average spatial spectrum of the proposed DAFC-based NN approach contains two high peaks, resulting in a superior probability of resolution in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' (b) Resolution of Unequal-Power Sources Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 6 shows the probability of resolution in a scenario with two sources, M = 2, at θ1 = −∆θ/2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='55◦, and θ2 = +∆θ/2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='55◦ with interference located between the sources at θc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='55◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The signal strength of the second source is set to SIR1 = SIR2 + 10 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Comparing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 6 to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 4b, the competing methods show similar trends, except the degradation of the CNN’s probability of resolution for the SIR = 0 dB case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' On the other hand, the proposed DAFC- based NN approach outperforms other tested approaches in terms of the probability of resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Therefore, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 6 demon- strates the generalization ability of the proposed DAFC-based NN approach to a variance between source strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' (c) Effect of the Number of Snapshots on the Resolution This experiment investigates the influence of the number of snapshots, K, on the ability to resolve two proximate sources from heavy-tailed spatially-colored interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The equal-strength resolution scenario is repeated using K = 4, 8, 16, 32, 64 with different instances of NN training for each K value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 7 shows the probability of resolution for two equal-strength sources at θ1,2 = θc ±∆θ/2 for ∆θ = 12◦ and θc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='55◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The FC-based NN approach fails to resolve the two sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' For SIR = 0 dB, the MVDR, CNN, and DAFC-based NN DAFC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='8 SIR=-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='00 dB INR=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='00 dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='0 60 40 20 0 20 40 60 Φ[Deg]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='8 +*+ MVDR, SNR=O DAFC, SNR=O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='6 FC, SNR=0 res CNN, SNR=0 MVDR, SNR=-5 P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='4 DAFC, SNR=-5 FC, SNR=-5 CNN, SNR=-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='0 10 20 30 40 50 60 Ae [Deg]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='8 MVDR, SIR=O DAFC, SIR=O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='6 FC, SIR=0 res CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' SIR=0 MVDR, SIR=-5 P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='4 DAFC, SIR=-5 FC, SIR=-5 CNN, SIR=-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='0 10 20 30 40 50 60 Ae [Deg]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='35 FC SIR=-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='00 dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='30 NR=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='00 dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='00 60 40 20 0 20 40 60 Φ[Deg]MVDR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='2 SIR=-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='00 dB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='0 INR=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='00 dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='0 60 40 20 0 20 40 60 Φ[Deg]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='6 CNN SIR=-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='00 dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='5 INR=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='00 dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='0 60 40 20 0 20 40 60 Φ[Deg]10 Figure 6: Probability of resolution for two sources located at θ1,2 = θc ±∆θ/2, and interference located at θc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='55◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The SIR in the legend represents the SIR of the first source, SIR1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The SIR of the second source is set to SIR2 = SIR1 + 10 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' approaches achieve a monotonic increasing probability of resolution with increasing K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The proposed DAFC-based NN approach slightly outperforms other tested approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' At low SIR of SIR = −5 dB, the proposed DAFC-based NN ap- proach significantly outperforms the other tested approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' This can be explained by the fact that increasing K increases the probability for outliers to be present in the input data matrix, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Therefore, the estimated autocorrelation matrix, ˆRx, is more likely to be biased by the interference-related outliers, which results in interference “masking” the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The proposed DAFC-based NN approach is immune to these outliers and successfully exploits the information from the additional snapshots to improve the probability of resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Figure 7: Probability of resolution for two sources located at θ1,2 = θc ± ∆θ/2 with ∆θ = 12◦, and interference located at θc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='55◦, as a function of the number of snapshots, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 4, 5, 6, and 7 show the ability of the proposed DAFC-based NN approach to utilize the information structure of the input data by exploiting the higher-order statistics and performing the domain-fitted transformation in order to provide superior resolution ability in the case of proximate heavy-tailed spatially-colored interference, low SIR and small sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 3) Multiple Source Localization: The performances of the tested DOA estimation approaches are evaluated and compared in a multi-source scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Four sources, (M = 4) were simulated with angular separation, ∆θ: {θ1, θ2, θ3, θ4} = θc + {−2∆θ, −∆θ, ∆θ, 2∆θ}, where θc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='51◦ represents a realistic off-grid condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The RMSD of evaluated meth- ods is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The proposed DAFC-based NN approach outperforms the other tested approaches at low SIR (SIR < 0 dB) for large and small angular separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' For high SIR and low angular separation, ∆θ = 5◦, the MVDR achieves the lowest RMSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The reason is that for this case, the interference is negligible with respect to the lobe of the strong source in the MVDR’s spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' However, at high angular separation, ∆θ = 20◦, the proposed DAFC-based NN ap- proach significantly outperforms the other tested approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' This is explained by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 9, that shows the spectrum of the tested DOA estimation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Notice that the proposed DAFC-based NN mitigates interference, while the spectra of other tested approaches contain high peaks at the interference DOA, θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' These peaks increase the Hausdorff distance in (17), increasing the RMSD of other tested approaches in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Figure 8: RMSD in scenarios with M = 4 sources located at {θ1, θ2, θ3, θ4} = θc + {−2∆θ, −∆θ, ∆θ, 2∆θ}, where θc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='51◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 4) Multiple Source Enumeration: The source enumeration performance is evaluated in this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The DOAs of the sources are selected from the set of following values: {10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='51◦, −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='49◦, −19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='49◦, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='51◦} such that for M sources, the DOAs are selected to be the first M DOAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The interfer- ence is located at θc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='51◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The proposed DAFC-based NN approach is compared to the MDL and AIC [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 10 shows the source enumeration confusion matrices for the MDL, AIC, and the proposed DAFC-based NN with SIR = 0 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 10a, 10b show that in both the MDL and the AIC, the predicted number of sources has a constant bias for each true M due to the spatially-colored interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 10c shows the source enumeration performance of the proposed DAFC-based NN approach in the presence of spatially colored interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The DAFC-based NN identifies the interference and does not count it as one of the sources by outputting a low probability for angular grid points near θc, resulting in a better source enumeration performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='8 MVDR, SIR=O DAFC, SIR=O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='6 FC, SIR=0 res CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' SIR=0 DP MVDR, SIR=-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='4 DAFC, SIR=-5 FC, SIR=-5 CNN, SIR=-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='0 10 20 30 40 50 60 Ae [Deg]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='6 res MVDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' SIR=O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='4 DAFC, SIR=0 FC, SIR=0 CNN, SIR=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='2 MVDR,SIR=-5 DAFC, SIR=-5 FC, SIR=-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='0 CNN, SIR=-5 4 8 16 32 64 KRMSD [Deg] 101 MVDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' A0=5 DAFC, △0=5 FC, △0=5 CNN, △0=5 MVDR,A0=20 100 DAFC, △0=20 FC, △0=20 CNN, △0=20 10 5 0 5 10 15 20 SIR[dB]11 (a) FC (b) MVDR (c) CNN (d) DAFC Figure 9: Spatial spectrum, four sources with SIR = 0 dB located at {θ1, θ2, θ3, θ4} = θc + {−2∆θ, −∆θ, ∆θ, 2∆θ}, where θc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='51◦ and ∆θ = 20◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The dashed blue lines rep- resent the mean spatial spectrum, and the color fill represents the standard deviation around the mean obtained from 2, 000 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The solid vertical orange lines represent the true source DOAs and the dashed vertical green line represents the interference DOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 5) Loss Weights: This experiment evaluates the effect of the loss weight update factors, {β(l)}Nw l=1, introduced in (12), on the confidence level in the spatial spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Let �B denote the set of {β(l)}Nw l=1 values used in the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The loss weights, {w(t) i }d i=1, are defined by the factors e(t) 0 , e(t) 1 according to (11), and are introduced to provide a trade-off between the penalty obtained on source/interference and the penalty obtained for the rest of the output spatial spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' For comparison, we set B0 = {10−6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='98 · 10−6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='58 · 10−5, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='31 · 10−5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='51 · 10−4, 10−3}, and B1 = {10−3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='98 · 10−3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='0158, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='063, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='1} as two sets of loss weight up- date factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' For B0, the loss weight update factors are closer to 0, hence the loss weights emphasize the source/interference, since e(t) 1 ≪ e(t) 0 which, according to (11), translates to larger w(t) i for source/interference grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' For B1 the values are closer to 1, hence the loss weights are more equally distributed among grid points, since e(t) 1 ≈ e(t) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The experiment in IV-B1 is repeated here for the DAFC-based NN approach with the two additional B0, B1 values mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Let ˆp1 represent the probability assigned for the source- containing grid point in the estimated label ˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Let ˆp0 represent the maximum over probabilities assigned for non-source grid points in ˆy, excluding a 5-grid point guard interval around the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' 11 shows ˆp1 and ˆp0 for various angular separations between the source and interference for SIR = −5 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' For B0, the source’s contribution to the loss value is substan- tially higher, which results in a higher probability for the (a) MDL (b) AIC (c) DAFC Figure 10: Confusion matrix for source enumeration, SIR = 0 dB, sources located at {10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='51◦, −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='49◦, −19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='49◦, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='51◦}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' (a) MDL, (b) AIC, (c) proposed DAFC-based NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' source-containing grid point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' However, this results in a higher probability obtained for non-source grid points, since their contribution to the loss value is negligible compared to the source-containing grid point, increasing “false-alarm” peaks in the spatial spectrum, subsequently increasing the estimation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Correspondingly, for B1 the source’s contribution to the loss value is less significant, which results in low probability assigned for the source-containing grid points, as well as low probability for non-source grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' CONCLUSION This work addresses the problem of DOA estimation and source enumeration of an unknown number of sources within heavy-tailed, non-Gaussian, and spatially colored interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' A novel DAFC-based NN approach is proposed for this FC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='4 SIR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='00 dB INR=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='00 dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='2 0.' metadata={'source': 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Predicted M12 Figure 11: Loss weight update factor impact on probability levels obtained in the DAFC-based NN’s spatial spectrum, single target at θ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content='55◦ with interference at θc = θ1 +∆θc, SIR = −5 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' ˆp1 represents the probability obtained for source-containing grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' ˆp0 represents the probability obtained for non-source grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The DAFC mechanism applies a structured transfor- mation capable of exploiting the interference non-Gaussianity for its mitigation while retaining a low complexity of learnable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The proposed DAFC-based NN approach is opti- mized to provide an interference-mitigated spatial spectrum using a loss weight scheduling routine, performing DOA estimation and source enumeration using a unified NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The performance of the proposed approach is compared to MVDR, CNN-based, and FC-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' Simulations showed the superiority of the proposed DAFC-based NN ap- proach in terms of probability of resolution and estimation ac- curacy, evaluated by RMSD, especially in weak signal power, small number of snapshots, and near-interference scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' The source enumeration performance of the proposed DAFC- based NN approach was compared to the MDL and AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' It was shown that in the considered scenarios, the proposed approach outperforms the MDL and the AIC in the source enumeration accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfBwKM/content/2301.02856v1.pdf'} +page_content=' REFERENCES [1] H.' metadata={'source': 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a/8NE2T4oBgHgl3EQfPgby/content/tmp_files/2301.03761v1.pdf.txt b/8NE2T4oBgHgl3EQfPgby/content/tmp_files/2301.03761v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fe0a68d590e054060e150e8dae864dd1bf925ba6 --- /dev/null +++ b/8NE2T4oBgHgl3EQfPgby/content/tmp_files/2301.03761v1.pdf.txt @@ -0,0 +1,1718 @@ +1 +Tensor Denoising via Amplification and Stable +Rank Methods +Jonathan Gryak1, Kayvan Najarian2,3,4,5,6, and Harm Derksen7 +Abstract—Tensors in the form of multilinear arrays are ubiq- +uitous in data science applications. Captured real-world data, +including video, hyperspectral images, and discretized physical +systems, naturally occur as tensors and often come with attendant +noise. Under the additive noise model and with the assumption +that the underlying clean tensor has low rank, many denoising +methods have been created that utilize tensor decomposition +to effect denoising through low rank tensor approximation. +However, all such decomposition methods require estimating the +tensor rank, or related measures such as the tensor spectral and +nuclear norms, all of which are NP-hard problems. +In this work we adapt the previously developed framework of +tensor amplification, which provides good approximations of the +spectral and nuclear tensor norms, to denoising synthetic tensors +of various sizes, ranks, and noise levels, along with real-world +tensors derived from physiological signals. We also introduce de- +noising methods based on two variations of rank estimates called +stable X-rank and stable slice rank. The experimental results +show that in the low rank context, tensor-based amplification +provides comparable denoising performance in high signal-to- +noise ratio (SNR) settings and superior performance in noisy +(i.e., low SNR) settings, while the stable X-rank method achieves +superior denoising performance on the physiological signal data. +Index Terms—Tensors, Denoising, Tensor Amplification, Stable +Rank Methods +I. INTRODUCTION +T +ENSORS in the form of multilinear arrays are ubiquitous +in data science applications. Captured real-world data, +including color and hyperspectral images (HSIs), video, and +discretized physical systems, naturally occur as tensors and +often come with attendant noise. As is common in other +signal processing applications, the captured tensor T +∈ +Rp1×p2×···×pd is modeled as T = D + N, where D is a pure +or “clean” tensor D that has been corrupted by additive noise +D, which is typically assumed to be Gaussian. Additionally, +the clean tensor D is assumed to be low rank. +Under this framework, tensor denoising can be achieved by +utilizing tensor decompositions methods, such as the canonical +1Department of Computer Science, Queens College, City University of New +York, New York, NY, USA +2Department of Computational Medicine and Bioinformatics, University of +Michigan, Ann Arbor, MI, USA +3Department of Emergency Medicine, University of Michigan, Ann Arbor, +MI, USA +4Electrical and Computer Engineering, College of Engineering, University +of Michigan, Ann Arbor, MI, USA +5Michigan Institute for Data Science, University of Michigan, Ann Arbor, +MI, USA +6Max Harry Weil Institute for Critical Care Research and Innovation, +University of Michigan, Ann Arbor, MI, USA +7Department of Mathematics, Northeastern University, Boston, MA, USA +polyadic (CP) [1], [2] and Tucker [3], [4] decompositions, to +determine a low-rank approximation of the observed tensor. +These decomposition algorithms require a pre-specified rank +to compute an approximation, however, determining the rank +of a tensor is NP-hard [5]. Thus, tensor decomposition-based +methods utilize some estimate of the tensor rank to effect +tensor denoising. +CP decomposition has been frequently used for HSI de- +noising, such as in Liu et al., [6], which estimated the tensor +rank using covariance matrices of the n-model flattenings; in +Veganzones et al. [7], which used a non-negative variant of +CP decomposition; and in [8], in which a CP decomposition +regularized by the nuclear norm of clustered 3D patches of +the HSI was employed. Tucker decomposition based denoising +include two works by Rajwade et al. that utilized higher order +singular value decomposition [9], a tensor analog of matrix +SVD, to denoise video [10] and images [11]; as well Lee +et al. [12], which focused on denoising tensors with ordinal +values. More recently, a tensor train (matrix product state) +decomposition [13] was used for denoising of HSIs [14]. +A general framework for understanding tensor denoising in +the additive model was developed in [15], that relates the +problem of denoising in the low rank context to the mini- +mization of dual norms ∥D∥X and ∥N∥Y , such as the nuclear +∥·∥⋆ and spectral ∥·∥σ norms, respectively. The calculation +of these norms for tensors is also NP-hard [16],[5], thus in +order make use of the denoising framework in [15] the co- +authors developed the method of tensor amplification [17], +which provides good approximations of the tensor spectral +norm and its dual the nuclear norm. +In this work, we utilize the general framework of [15] and +the approximations of the spectral norm [17] to devise three +novel tensor denoising methods - amplification-based, stable +slice rank, and stable X-rank denoising, the latter two methods +based on their eponymous rank estimates. The performance of +these methods is compared to several standard decomposition- +based denoising methods on synthetic tensors of various sizes, +ranks, and noise levels, along with real-world tensors derived +from physiological signals. The experimental results show that +in the low rank context, tensor-based amplification provides +comparable denoising performance in high signal-to-noise +ratio (SNR) settings and superior performance in noisy (i.e., +low SNR) settings, while the stable X-rank method achieves +superior denoising performance on the physiological signal +data. +arXiv:2301.03761v1 [cs.LG] 10 Jan 2023 + +2 +II. PRELIMINARIES AND RELATED WORK +A. Basic Notation +Let T ∈ Rp1×p2×···×pd denote a real-valued tensor of order +d. In the denoising experiments that are performed in this study +we will assume that the tensor T is the noisy version of a +pure tensor D ∈ Rp1×p2×···×pd corrupted by additive noise +N ∈ Rp1×p2×···×pd, that is, +T = D + N. +(1) +The Frobenius norm of T is denoted ∥T ∥ and defined as +∥T ∥ = +� +� +� +� +p1 +� +i1 +p2 +� +i2 +· · · +pd +� +id +t2 +i1i2···id, +(2) +while the tensor inner product of two tensors T , S of matching +order and dimension is defined as +⟨T , S⟩ = +p1 +� +i1 +p2 +� +i2 +· · · +pd +� +id +ti1i2···idsi1i2···id. +(3) +The induced norm of the tensor inner product is the Frobenius +norm defined above, with the typical relation ⟨T , T ⟩ = ∥T ∥2. +Given a tensor T and a permutation q = ⟨q1, . . . , qd⟩ of +the indices 1 : d, the q-transpose of T is the tensor T ⟨q⟩ ∈ +Rpq1×pq2×···×pqd with entries +(T ⟨q⟩)i1i2...id = tiq1iq2...iqd . +(4) +At times we will need to matricize the tensors under con- +sideration by rearranging their entries in specific ways, as well +as employ various tensor-tensor, tensor-matrix, and matrix- +matrix products. In the definitions below and throughout the +manuscript we will primarily follow the notational conventions +introduced by Kolda and Bader in [18]. +The mode-n flattening or unfolding of the tensor T is the +matrix T(n) ∈ Rpn×N/pn, where N = � +i pi, whose columns +are the mode-n fibers of T . +The n-mode product of a tensor T and a matrix A ∈ RJ×pn +is the tensor T ×n A of size p1 ×p2 ×· · ·×pn−1 ×J ×pn+1 × +· · · × pd with entries +(T ×n A)i1i2...in−1jin+1...id = +pn +� +in=1 +ti1i2...idujin. +(5) +If S = T ×n A, then the n-mode product as defined above is +equivalent to S(n) = AT(n). +The Kronecker product of two matrices A ∈ RI×J and +B ∈ RK×L is the matrix A ⊗ B ∈ RIK×JL defined by +A ⊗ B = +� +���� +a11B +a12B +· · · +a1JB +a21B +a22B +· · · +a2JB +... +... +... +... +aI1B +aI2B +· · · +aIJB +� +���� . +(6) +Finally, given two tensors T +∈ Rp1×···×pd and S +∈ +Rq1×···×qe, their outer product is the tensor T ◦ S of size +p1 × · · · × pd × q1 × · · · × qe with entries +(T ◦ S)i1i2...idj1j2...je = ti1ti2 . . . tidsj1sj2 . . . sje. +(7) +B. Decomposition-based Denoising +Tensor decomposition methods seek to represent a given +tensor by decomposing it into factors such as simple tensors or +matrices and whose combination results in a “good” approx- +imation of the original tensor. In the context of denoising, +it is typical to assume that a noisy signal is sparse, in the +sense that its ℓ1 norm is small. In the case of matrices and +tensors, this assumption corresponds to the original tensor +having low rank, with the high rank components corresponding +to additive noise. Thus, computing a low rank approximation +of the original tensor via tensor decomposition is a means to +effect tensor denoising. +In the case of matrices (order two tensors), singular value +decomposition yields the exact rank r of the matrix and its +decomposition into r factor, with the best low rank approxi- +mation for a given rank l < r provided by choosing the factors +corresponding to the l largest singular values. For higher order +tensors, calculating the exact rank is NP hard [5]. Moreover, +unlike the matrix case, the factors used to create the best rank +r − 1 approximation need not be those used to produce the +best rank r approximation [19], and for degenerate tensors, +the best rank r approximation may not even exist [20]. +Despite these theoretical limitations, in practice one can +utilize tensor decomposition methods to effect denoising by +creating decompositions for a range of rank values, then choos- +ing the best rank r decomposition D that best approximates +the original tensor T , e.g., min ∥T − D∥. This strategy for +tensor denoising was evaluated using three common tensor +decomposition methods: canonical polyadic decomposition, +higher-order orthogonal iteration, and multiway Wiener filters. +1) CP Decomposition via Alternating Least Squares (CP- +ALS): Let U (j) = [uj,1uj,2 . . . uj,r] ∈ Rpj×r, 1 ≤ j ≤ d. CP +decomposition factorizes a d-way tensor into d factor matrices +and a vector Λ = [λ1, λ2, . . . , λr] ∈ Rr: +S = +r +� +i=1 +λiu1,i ◦ u2,i ◦ . . . ◦ ud,i. +(8) +The best rank r approximation problem for a tensor T +∈ +Rp1×p2×...×pd can be given as: +min +Λ,U (1),...,U (d) ∥T − S∥ where S = [Λ ; U (1), U (2), . . . , U (d)]. +(9) +This can be found by employing alternating least squares +(ALS), wherein each iteration of the algorithm an approxima- +tion of the flattening for one mode is found by fixing all other +modes of the tensors and solving a least squares problem. +This process is repeated, cycling through all modes, until +convergence or a maximum number of iterations is reached. In +this work, the implementation of CP-ALS from TensorToolbox +[21] was utilized with the default level of tolerance (10−4) +and maximum number of iterations (50). CP-ALS was run +for specified rank values r ∈ [1, min(pi)], with the rank r∗ +approximation +Dr∗ = min +r +∥T − Dr∥ +(10) +chosen as the best denoised tensor. + +3 +2) Higher-Order Orthogonal Iteration (HOOI): For matri- +ces, orthogonal iteration produces a sequence orthonormal +bases for each subspace of the vector space. De Lathauwer +et al. [22] extended this to tensors, developing the technique +known as higher-order orthogonal iteration (HOOI). This +method uses ALS to estimate the best rank-[r1, . . . , rd] ap- +proximation for a tensor, and is achieved by iteratively solving +the optimization problem +argminU(i)|ri ∥T − G ×1 U(1)|r1 ×2 U(2)|r2 × . . . ×N U(d)|rd∥, +(11) +where G is a core tensor of size r1 × . . . × rd and each U(i)|ri +is a matrix comprised of the ri leftmost singular vectors of +the singular value decomposition of the modal flattening U(i). +HOOI-based +denoising +was +implemented +using +the +tucker_als method in TensorToolbox [21] to determine +the best rank [r∗ +1, . . . , r∗ +d] approximation, where each ri was +chosen equally and uniformly from r ∈ [1, min(pi)], with the +rank r∗ approximation (Eq. 10) chosen as the best denoised +tensor. +3) Multiway Wiener Filter: For a discrete signal y[n] and +filter output ˆy[n], the Wiener filter h[n] is the filter that +minimizes the mean squared error between ˆy[n] and y[n]: +argminh[·]E[(ˆy[n] − y[n])2)]. +(12) +Wiener filters have been used in a variety of denoising appli- +cations, such as for images [23], [24], physiological signals +[25], [26] and speech [27], [28]. +Muti et al. [29] created a multiway Wiener filter that can be +used to denoise tensors of arbitrary size. Given a noisy tensor +T , their method uses an ALS approach to learn Wiener filters +{Hn} for each mode n so that the mean squared error between +T and the denoised tensor D is minimized, where +D = T ×1 H1 ×2 H2 ×3 · · · ×d Hd. +(13) +The implementation of the multiway Wiener filter utilized in +this study and the exposition below follows [30]. The filters +Hn in each mode n are initialized to the identity matrix +of Rpn. At each stage k of the algorithm, the filter Hk +n is +computed for each mode as +Hk +n = VnΛnV ⊺ +n , +(14) +where Vn is a matrix containing the Kn orthonormal basis +vectors of the signal subspace in the column space of T(n), +the mode-n flattening of T , and +Λn = diag +� +λγ +1 − ˆσγ2 +n +λΓ +1 +, . . . , λγ +Kn − ˆσγ2 +n +λΓ +Kn +� +, +(15) +where {λγ +i , i = 1, . . . , Kn} and {λΓ +i , i = 1, . . . , Kn} are +respectively the Kn largest eigenvalues of the matrices γn and +Γn, defined as +γn = +E +� +T(n)qnT(n) +⊺� +(16) +Γn = +E +� +T(n)QnT(n) +⊺� +(17) +with +qn += +d +� +i̸=n +Hi +(18) +Qn += +d +� +i̸=n +H⊺ +i Hi. +(19) +The values ˆσγ2 +n in Equation 15 are estimates of the pn − Kn +smallest eigenvalues of γn, calculated as +ˆσγ2 +n = +1 +pn − Kn +pn +� +i=Kn+1 +λγ +i . +(20) +Following [31], the optimal Kn for mode n is estimated using +the Akaike Information Criterion (AIC). Please refer to [30] +for further details. +III. AMPLIFICATION AND STABLE RANK DENOISING +In this section we introduce three different denoising meth- +ods – Amplification-based, Stable Slice Rank, and Stable X- +Rank denoising – that leverage the general framework for +denoising based on dual norms as introduced in Derksen [15], +to effect denoising on tensors. +A. A Framework for Denoising Using Dual Norms +The model T = D +N utilized in this work can be viewed +as an instance of the additive noise model c = a + b, where +a, b, c ∈ V are elements of a vector space V . In Derksen +[15], a general framework for understanding the denoising of +vectors under the additive model was developed that relates the +problem of denoising the vector c to the minimization of ∥a∥X +and ∥b∥Y , where ∥ · ∥X and ∥ · ∥Y are dual norms. Moreover, +the framework makes the assumptions that the original vector +(or tensor) a is sparse, e.g., that it has few non-zero values or +is of low rank, while the additive noise b is dense or of high +rank. Thus, the norms ∥ · ∥X and ∥ · ∥Y can be interpreted as +respectively measuring the sparsity and noise of the vector (or +tensor) under consideration. The prototypical ∥ · ∥X norm is +the nuclear norm, which for a matrix is the sum of its singular +values, while for a tensor the tensor nuclear norm ∥T ∥⋆, is +defined as +∥T ∥⋆ = min +r +� +i=1 +∥vi∥2, +where v1, . . . , vr are rank-1 tensors and T = �r +i=1 vi. +The prototypical ∥·∥Y norm and dual to ∥·∥X is the spectral +norm, which for a matrix is the absolute value of its largest +singular value, while for a tensor the tensor spectral norm +∥T ∥σ is defined as +∥T ∥σ = sup |T · u1 ⊗ u2 ⊗ . . . ⊗ ud|, +where uj ∈ Rpj and ∥uj∥ = 1 for 1 ≤ j ≤ d. +If V is also an inner product space we also have the induced +norm +� +⟨c, c⟩ that corresponds to the standard Euclidean norm +∥c∥2 for vectors or the Frobenius norm ∥ · ∥, introduced in +Section II-A, for matrices and tensors. As shown in [15], +the denoising of a vector c via a decomposition c = a + b + +4 +that simultaneously minimizes the values ∥a∥X and ∥b∥Y is +governed by the Pareto frontier, which models the competing +objectives of minimizing the two norms in terms of Pareto +efficiency, and the above XY -decomposition that achieves this +is deemed Pareto efficient. Moreover, [15] defines the related +notion of the Pareto subfrontier, which relates the three norms +∥ · ∥X, ∥ · ∥Y , ∥ · ∥2 and their induced decompositions XY , +X2, and 2Y , describing the conditions under which these +decompositions can achieve Pareto efficiency. +B. Amplification-based Denoising +To make use of the denoising framework introduced in +[15] requires the calculations of various norms for the vec- +tors of interest. While the Frobenius norm of a tensor is +easily obtained, computing either the nuclear norm [16] or +the spectral norm [5] for tensors is NP-hard. In order to +obtain an approximation to the tensor spectral norm, the co- +authors developed the methodology of tensor amplification +[17]. For a matrix A with singular values λ1, . . . , λr, the +function φ : A → AA⊺A produces the matrix AA⊺A whose +singular values are λ3 +1, . . . , λ3 +r. Repeated applications of φ(·) +will amplify the larger singular values, which correspond to +the sparse or low rank components of the matrix, while +minimizing smaller singular values that likely correspond to +noise. +Analogously, tensor amplification utilizes degree d polyno- +mial functions on tensors to amplify the low rank structure. +Moreover, for each amplification map Φσ′ there exists a cor- +responding norm ∥·∥σ′,d that approximates the tensor spectral +norm, in the sense that limd→∞ ∥T ∥σ′,d = ∥T ∥σ. Two such +amplification maps – Φσ,4 and Φ# – were introduced for +order 3 tensors in [17], with Φ# being show to be a better +approximation to the tensor spectral norm than Φσ,4. +The method presented in Algorithm 1 utilizes the 2Y - +decomposition framework of [15] and the tensor spectral norm +approximations Φ to denoise a given tensor T . The algorithm +allows for the choice of amplification map as well as the +number of amplifications per round. For third order tensors +the amplification map Φ# was used, while for fourth order +tensors a compatible version of Φσ,4 was employed as there +is no currently known analogue of the Φ# map for fourth +order tensors. Multiple experiments were performed with m, +the number of amplifications per round, ranging from 1 to +10, with m = 5 being found to produce the best denoising +performance. +C. Stable Slice Rank Denoising +Slice rank was introduced in [32] in relation to the cap set +problem. Following Tao [33], the slice rank of a tensor T +is the least non-negative integer srk such that T is a sum +of tensors with slice rank 1, i.e., T = �r +i=1 Ti, where Ti is +contained in the tensor product space +V1 ◦ · · · Vi−1 ◦ s ◦ Vi−1 ◦ · · · Vd, +(21) +where Vj are vector spaces and s is a vector in some Vi. In +[34], the notion of a stable rank for matrices was introduced, +in which the matrix rank function rank(A), is replaced by the +Algorithm 1 Amplification-based tensor denoising. +D ← DENOISE AMPLIFICATION(T , Φ, m) +ϵ ← ∥T ∥ +N ← T +while true do +A ← Φm(N) +A ← +A +∥A∥ +N ← N − ⟨A, N⟩A +Break if ∥N∥ < ϵ +end while +D ← T − N +numerical rank function, +∥A∥2 +∥A∥2σ , or the related stable nuclear +rank +∥A∥2 +⋆ +∥A∥2 . These ranks are stable in the sense that small +perturbations of the values of the matrix A will not change +their value. Extending this methodology to tensors, the stable +slice rank is defined as +�d +i=1 ∥T(i)∥2 +⋆ +∥T ∥2 +, +(22) +where T(i) are the mode-i flattenings of T . +For a given value of the parameter λ, the stable slice rank +(SliceRank) method denoises a tensor by finding a decompo- +sition T = D + N that minimizes the Frobenius norm of +D = � +i Si under the constraints that the nuclear norms of +the flattenings of N are all ≤ λ. The method also produces +a decomposition D = � +i Si that minimizes the sum of the +nuclear norms of S(i), the mode-i flattenings of Si. Typically, +the S(i) have low rank. +Algorithm 2 Stable SliceRank denoising. +(D, {Si}, ssrk) ← DENOISE SLICERANK(T , λ, acc) +Si ← 0 ∈ Rp1×p2×···×pd +curr acc ← 0 +while curr acc < acc do +for i ← 1 : d do +A ← T − � +j̸=i Sj +q ← CIRCSHIFT([1, . . . , d], −(i − 1)) +A ← A⟨q⟩ +(U, D, V ) ← SVD(A(i)) +Ei ← MAX(D − λ, 0) +ei ← DIAG(Ei) +F ← U · Ei · V T +F ← RESHAPE(F, pi, . . . , pd, p1, . . . , pi−1) +q ← CIRCSHIFT([1, . . . , d], (i − 1)) +Si ← F⟨q⟩ +end for +curr acc ← +⟨T − �d +j=1 Sj, �d +j=1 Sj⟩ +λ �d +j=1 ∥A(j)∥⋆ +end while +D ← �d +i=1 Si +ssrk ← +�d +i=1 ∥A(i)∥2 +⋆ +∥D∥2 +Algorithm 2 depicts the implementation of SliceRank de- +noising, which utilizes a number of auxiliary functions from + +5 +MATLAB [35]: circshift performs a cyclic permutation of +an index set [1, . . . , d], with the second parameter determining +the number of forward or backwards shifts; reshape is used +to flatten a tensor into a matrix with the specified dimensions; +and diag returns a vector comprising the entries on the +main diagonal of the specified matrix. The algorithm takes +as hyperparameters λ as described above and acc ∈ (0, 1], +the specified accuracy level that once achieved the algorithm +terminates. The algorithm returns the denoised tensor D, the +decomposition factors Si, and ssrk, the stable slice rank of +D. The hyperparameters were optimized via grid search over +the ranges λ ∈ {10−2, 0.1, 1, 10} and acc ∈ {0.90, 0.95}. +D. Stable X-Rank Denoising +As noted in Section II-B, a degenerate tensor T of rank r +may not have a best rank k < r approximation for a given +rank k. In such cases, a tensor may be approximated to any +desired precision by rank j < k tensors. This is due to the set +of all tensors for a given rank r not being Zariski closed [20]. +In Derksen [36], the G-stable rank of a tensor was introduced +that, among its other advantages, is Zariski closed. Thus, every +tensor T has a best G-stable rkG < r approximation. The G- +stable α rank of a tensor can be defined as +rkG +α (T ) = sup +g∈G +min +i +αi∥g · T ∥2 +∥ (g · T )(i) ∥2σ +, +(23) +where α = (α1, . . . , αd) and g is an element of a reductive +group G, i.e., g ∈ SL(Rp1) × · · · × SL(Rpd). +Using the above definition we can define the related concept +of stable X-rank, which is +sXrkG(T ) = max +α +rkG +α (T ), +(24) +where α is subject to the restriction that � +i αi = d. Algorithm +4 depicts the implementation of the stable X-Rank (XRank) +denoising method. Like SliceRank, the method imposes a con- +straint on the nuclear norm of the flattenings of N. However, +in the XRank method, this cutoff is determined automatically +using Algorithm 3. The hyperparameters were optimized via +grid search over the ranges λ ∈ {10−2, 0.1, 1, 10} and acc ∈ +{0.90, 0.95}. +IV. EXPERIMENTAL RESULTS AND DISCUSSION +In order to evaluate the various denoising methods under +consideration, two sets of synthetic tensors were generated +with varying orders, ranks, and dimensions, resulting in 512 +parameter combinations. For each combination, one hundred +(100) tensors were generated. For all synthetic tensors, varying +amounts of noise were added from a standard Gaussian +distribution N(0, 1), with the resulting noisy tensors having +signal-to-noise ratios (SNR) ranging from 20 dB to −20 dB. +The full range of parameters is provided in Table I. +Additionally, two sets of tensors were extracted from elec- +trocardiogram (ECG) signals to which Gaussian noise was +added prior to tensor extraction, using the same range of +resultant SNRs as those employed in the generation of the +synthetic tensors. +Algorithm 3 Determine the nuclear norm cutoff for XRank +denoising. +c ← FIND CUTOFF(f = [λ1, . . . , λr]⊺, λ) +r ← |f| +t ← 0 ∈ Rr +for i ← 1 : r do +ti ← λ +�i +j=1 λj +1 + λ · i +end for +S ← 0 ∈ Rr×r +for i ← 1 : r do +for j ← 1 : r do +sij ← MAX(fi − tj, 0) +end for +end for +v ← 0 ∈ Rr +for j ← 1 : r do +vj ← �r +i=1(fij − sij)2 + λ �r +i=1(sij)2 +end for +k ← ARGMIN(v) +c ← tk +Algorithm 4 Stable XRank denoising. +(D, {Si}, sxrk) ← DENOISE XRANK(T , λ, acc) +Si ← 0 ∈ Rp1×p2×···×pd +curr acc ← 0 +while curr acc < acc do +for i ← 1 : d do +A ← T − � +j̸=i Sj +q ← CIRCSHIFT([1, . . . , d], −(i − 1)) +A ← A⟨q⟩ +(U, D, V ) ← SVD(A(i)) +c ← FIND CUTOFF(DIAG(D),λ) +Ei ← MAX(D − c, 0) +ei ← DIAG(Ei) +F ← U · Ei · V T +F ← RESHAPE(F, pi, . . . , pd, p1, . . . , pi−1) +q ← CIRCSHIFT([1, . . . , d], (i − 1)) +Si ← F⟨q⟩ +end for +Scurr ← �d +i=1 Si +T S = T − Scurr +y ← 0 +for i ← 1 : d do +q ← CIRCSHIFT([1, . . . , d], −(i − 1)) +B ← T S⟨q⟩ +(U, D, V ) ← SVD(B(i)) +y ← y + d2 +1 +end for +y ← √y +curr acc ← +⟨T , Scurr⟩ +y · +��d +j=1 ∥A(j)∥2⋆ +end while +D ← �d +i=1 Si +sxrk ← +�d +j=1 ∥A(j)∥2 +⋆ +∥D∥2 + +6 +TABLE I: Parameters and their respective values used to +generate the synthetic tensor datasets. +Parameter +Range/Values +Distribution +Normal N(0, 1) +Order +3, 4 +Rank +[1, 5], 10, 20, 25 +Size +5, 10, 25, 50 +SNR +20, 10, 5, 1, −1, −5, −10, −20 +1) Uniform Synthetic Tensors: In this dataset, the dimen- +sions of a given tensor are chosen uniformly across each mode. +To generate synthetic tensors from a distribution D of a given +rank r, size s, and order d, scalar values λ1, . . . , λr are chosen +from D, then for each mode j, r random vectors xj,i ∈ Rs +are chosen from D. The synthetic tensor is then +r +� +i=1 +λix1,i ◦ x2,i ◦ · · · ◦ xd,i. +2) Non-Uniform Synthetic Tensors: In this dataset, one +mode mk of a given tensor is “stretched” to a different +dimension dk by choosing a number uniformly in the range +dk = [s, min(500, sd)], i.e., the lower bound is the dimension +of the other models while the upper bound is the product of +the dimensions of each mode or 500, whichever is lower. After +choosing the stretch mode and its dimension the tensors are +generated in the same manner as for the uniform tensors above. +3) ECG Waveform Tensors: The PTB Diagnostic ECG +Database [37] is comprised of high resolution (1 kHz) digitized +recordings of electrocardiograms (ECGs) from patients with +various cardiovascular diseases, including myocardial infarc- +tion, heart failure, and arrhythmia, as well as healthy controls. +The database is publicly available via Physionet [38]. +Tensor-based methods have been shown to be effective for +a number of ECG analytical tasks, a survey of such methods +can be found in [39]. Given the utility of tensor-based methods +in this context and that such that recordings of physiological +signals may be corrupted by noise yields a natural application +of the proposed denoising methods. In order to evaluate these +methods, we first must construct tensors from the ECGs. In +forming these tensors, one has to consider the amount of +signal over which to perform subsequent signal processing and +feature extraction: two methods were employed. In the first, +ninety (90) seconds of a patient’s ECG recording was sampled +across all twelve ECG leads, while in the second method three +windowed samples of thirty (30) seconds each were extracted. +Using these two sampling strategies we adapted the tensor +formation method introduced in [40] that has been shown +to be effective for subsequent applications of machine learn- +ing for prognosticating severe cardiovascular conditions [41], +[42]. In this method, each ECG signal is preprocessed using +the taut string method, which produces a piecewise linear +approximation of a given signal, parametrized by ϵ, which +controls the coarseness of the approximation. Given a discrete +signal x = (x1, . . . , xn) one can define the finite difference +D(x) = (x2 − x1, . . . , xn − xn−1). For a fixed ϵ > 0, the +taut string estimate of x is the unique function y such that +∥x − y∥∞ ≤ ϵ and ∥D(y)∥2 is minimal. The taut string +approximation can be found efficiently using the method in +[43]. +After the taut string approximation for a given signal is +found, six morphological and statistical features are extracted +following [40]. This process is repeated for five values of +epsilon: (0.0100, 0.1575, 0.3050, 0.4525, 0.6000). As each pa- +tient’s ECG recording is comprised of the standard 12 leads, +the approximation of each 90 second ECG sample via taut +string and the extraction of taut string features yields third +order tensors of size 5 × 6 × 12 for each patient. For the +windowed samples, fourth order tensors were formed of size +5 × 6 × 12 × 3, with the fourth mode corresponding to the +features extracted in each window. +4) Adding Noise: For every generated synthetic tensor, a +set of noisy tensors was created by adding Gaussian noise +(N(0, 1)) so that the resultant tensors had SNRs in the +range [20, 10, 5, 1, −1, −5, −10, −20]. For the ECG waveform +tensors, Gaussian noise was added to each ECG signal to +produce a set of noisy signals with the same SNR range as +for the synthetic tensors. However, this was performed prior +to tensor formation given that in practical applications ECG +signals themselves may come with some intrinsic amount of +noise, rather than noise being introduced to the tensors directly. +A. Results: Synthetic Data +The overall denoising performance for third order tensors +across various ranks and tensors sizes are presented in Table II. +The performance statistics for only one order are presented due +to the incomparable sizes of the non-uniform tensors across +orders, please see Table IV in Appendix A for the fourth order +4. The best performing denoising algorithms for uniformly +sized tensors, as depicted in Table II (a), varied by noise level. +For cleaner tensors (20 and 10 dB), the multiway Wiener +filter performed best overall, achieving mean and standard +deviations in denoised SNRs of 20.95 (13.19) and 20.22 (10.1) +dBs, respectively. For moderately noisy tensors (5, 1, and +−1 dB), ALS was the best performing denoising method, +achieving denoised SNRs of 15.11 (8.26), 11.12 (7.98), and +9.1 (7.88) dBs. For nosier tensors with starting SNRs of −5 +and −10, tensor amplification produced the best denoised +SNRs of 3.37 (5.69) and −2.25 (4.57) dBs. Finally for tensors +with starting SNRs of −20 dB, the noisiest tensors evaluated, +XRank produced on average the highest denoised SNR of +−9.22 (4.79) dB. The results for non-uniformly sized tensors, +as depicted in Table II (b), are much clearer, with ALS +achieving the best denoised SNRs across all starting SNRs, +ranging from 30.81 (12.7) dBs for tenors with starting SNRs +of 20 dB to −6.6 (8.57) dB for the noisiest tensors (starting +SNRs of −20 dB). +The relationship between tensor size (dimension of each +mode) and achieved denoised SNRs is depicted in Figure 1. +With the exception of HOOI, all other denoising algorithms +see improvements in achieved SNR as the size of the tensor +increases. The multiway Wiener filter maintains the best de- +noising performance as size increases, followed by ALS. Both +amplification and XRank have similar denoising performances, +while slice rank and HOOI having the lowest performance +overall. + +7 +TABLE II: Mean (SD) SNR, in decibels, after tensor denoising across all parameters. +Starting SNR +HOOI +ALS +Wiener +Amp +SliceRank +XRank +20 +19.57 (3.32) +28.67 (11.1) +29.05 (13.19) +10.0 (14.13) +18.79 (1.59) +10.89 (5.64) +10 +10.59 (1.12) +19.91 (8.88) +20.22 (10.1) +8.65 (11.09) +13.21 (2.21) +9.84 (4.15) +5 +5.81 (0.75) +15.11 (8.26) +14.94 (8.59) +7.85 (9.64) +8.18 (2.52) +8.56 (3.13) +1 +1.87 (0.7) +11.12 (7.98) +10.34 (7.64) +7.01 (8.56) +3.54 (2.25) +6.91 (2.49) +-1 +-0.12 (0.7) +9.1 (7.88) +7.96 (7.01) +6.48 (8.1) +1.18 (2.03) +5.86 (2.36) +-5 +-4.12 (0.69) +4.99 (7.77) +3.37 (5.69) +5.17 (7.45) +-3.46 (1.57) +3.24 (2.6) +-10 +-9.12 (0.69) +-0.19 (7.65) +-2.25 (4.57) +2.85 (7.35) +-9.02 (1.12) +-0.58 (3.43) +-20 +-19.11 (0.7) +-10.38 (7.39) +-11.17 (7.38) +-11.79 (6.98) +-19.61 (0.6) +-9.22 (4.79) +(a) Uniformly sized tensors. +Starting SNR +HOOI +ALS +Wiener +Amp +SliceRank +XRank +20 +23.91 (7.33) +30.81 (12.7) +29.46 (13.3) +11.68 (14.81) +18.7 (1.53) +11.85 (5.83) +10 +15.45 (4.71) +22.75 (10.29) +21.39 (10.79) +10.33 (11.77) +13.28 (2.11) +10.75 (4.34) +5 +10.99 (3.94) +18.25 (9.55) +16.28 (9.86) +9.45 (10.34) +8.26 (2.44) +9.4 (3.3) +1 +7.27 (3.67) +14.49 (9.23) +11.84 (9.03) +8.25 (9.36) +3.63 (2.19) +7.68 (2.6) +-1 +5.38 (3.63) +12.57 (9.12) +9.54 (8.41) +7.32 (9.03) +1.28 (2.01) +6.6 (2.42) +-5 +1.5 (3.64) +8.66 (8.99) +5.38 (6.92) +4.89 (9.02) +-3.37 (1.61) +3.95 (2.56) +-10 +-3.51 (3.61) +3.65 (8.86) +0.08 (5.37) +1.65 (9.72) +-8.92 (1.29) +-0.13 (3.39) +-20 +-13.59 (3.51) +-6.6 (8.57) +-7.94 (7.44) +-11.74 (8.56) +-19.56 (0.64) +-9.01 (4.66) +(b) Non-uniformly sized tensors. +(a) +(b) +Fig. 1: Denoising performance with respect to tensor size. +The relationship between tensor rank and achieved de- +noised SNRs is depicted in Figure 2. Tensor amplification +achieves the best rank-1 performance for both uniformly +and non-uniformly sized tensors, followed by the multiway +Wiener filter. The denoising performance of both methods +decreases as tensor rank increases, with the multiway Wiener +filter maintaining denoising performance for higher ranks +than amplification. ALS has lower performance at low ranks +but generally maintains its denoising performance as rank +increases, ultimately achieving the best results by rank 20. +XRank has greater performance than SliceRank for uniformly +sized tensors, but their denoising performances converge prior +to rank 20 for non-uniformly sized ones. HOOI has the +lowest denoising performance for uniformly sized tensors, but +performs slightly better than the amplification and SliceRank +methods for non-uniformly sized tensors with ranks greater +than 5. +The results depicted in Table III provide a further investiga- +tion into the denoising performance for low rank (ranks 1 and +2) and high noise (SNRs ≤ 1) tensors. From these results one +can observe that amplification achieves the best performance +for uniformly sized tensors, with the multiway Wiener filter +achieving the second-best denoising performance. In the case +of non-uniformly sized tensors, the Wiener filter and amplifi- +cation achieve comparable results for starting SNRs of 1 and +−1 dB, while amplification achieving better performance for +starting SNRs of −5,−10, and −20 dB. +B. Results: Real Data - ECG Waveform Tensors +Figure 3 shows the denoising performance on the tensors +derived from the PTB dataset. For the tensors derived from +90 second samples of ECG signal (Figure 3 a), only the +stable rank methods (XRank and SliceRank) were able to +achieve any effective denoising, with XRank achieving the +best denoising performance with a modest ≈ 4 dB denoised +SNR for tensors whose signals had SNR ratios of 20 dB +prior to tensor formation. This performance was maintained +for tensors from signals with starting SNRs down to 5 dB, +after which the denoising performance of XRank declines. The +SliceRank method does not yield any tensor denoising until the +starting signal SNR dropped below −5 dB, after which it too +experiences a continued decline in denoising performance. All + +Uniform Synthetic Tensors +1OOH +14 - +ALS +Wiener +12 - +Amp +Denoised SNR (dB) +10 +SliceRank +XRank +8 +6 · +4 +2 - +5 +10 +25 +50 +SizeNon-uniform Synthetic Tensors +16 +IOOH +ALS +14 - +Wiener +12 +Amp +Denoised SNR (dB) +SliceRank +10 +XRank +8 +6 - +4 - +2 +01 +5 +10 +25 +50 +Size8 +(a) +(b) +Fig. 2: Denoising performance with respect to tensor rank. +TABLE III: Mean (SD) denoised SNR, in decibels, for low rank and noisy tensors. +Starting SNR +HOOI +ALS +Wiener +Amp +SliceRank +XRank +1 +1.59 (0.36) +5.97 (2.93) +12.21 (4.05) +14.98 (7.72) +5.64 (2.29) +8.89 (1.91) +-1 +-0.42 (0.35) +3.92 (2.92) +10.31 (3.85) +13.66 (7.28) +3.33 (2.11) +7.34 (2.01) +-5 +-4.43 (0.32) +-0.17 (2.91) +6.42 (3.78) +10.68 (6.90) +-1.52 (1.70) +3.77 (2.34) +-10 +-9.45 (0.30) +-5.24 (2.93) +1.90 (3.84) +6.16 (7.47) +-7.49 (1.39) +-0.79 (2.88) +-20 +-19.45 (0.30) +-15.29 (2.91) +-8.86 (5.90) +-4.36 (7.84) +-18.79 (1.03) +-10.31 (3.48) +(a) Uniformly sized tensors. +Starting SNR +HOOI +ALS +Wiener +Amp +SliceRank +XRank +1 +4.59 (1.82) +8.32 (4.32) +16.96 (8.56) +16.15 (8.84) +5.85 (2.30) +9.68 (1.96) +-1 +2.58 (1.82) +6.25 (4.30) +14.99 (8.41) +14.88 (8.34) +3.51 (2.24) +8.25 (2.03) +-5 +-1.46 (1.84) +2.12 (4.29) +10.77 (8.21) +12.06 (7.72) +-1.25 (2.06) +4.89 (2.39) +-10 +-6.50 (1.87) +-2.98 (4.31) +4.07 (6.49) +7.84 (7.93) +-7.19 (1.87) +0.21 (3.09) +-20 +-16.52 (1.87) +-13.05 (4.31) +-7.18 (6.61) +-1.79 (9.00) +-18.86 (0.84) +-9.05 (3.87) +(b) Non-uniformly sized tensors. +other methods - HOOI, ALS, and amplification - introduced +noise into the tensors across all starting signal SNRs. No +appreciable difference was observed in tensors derived from +the two patient cohorts (healthy and unhealthy). The denoising +results for the tensors derived from windowed samples (Figure +3 b) are essentially the same as those derived from the +90 second samples, with the only exception being a slight +increase in SliceRank’s denoising performance for tensors +corresponding to unhealthy patients with a starting signal SNR +of −5 dB. + +Uniform Synthetic Tensors +1OOH +17.5 +ALS +Wiener +15.0 +Amp +Denoised SNR (dB) +SliceRank +12.5 +XRank +10.0 +7.5 +5.0 +2.5 +0.0 +10 +20 +25 +RankNon-uniform Synthetic Tensors +IOOH +20 +ALS +Wiener +Amp +Denoised SNR (dB) +15 +SliceRank +XRank +10 +5 +0 +10 +20 +25 +4. +5 +Rank9 +(a) +(b) +Fig. 3: Denoising performance on the PTB tensors formed +from a) 90 second samples, and b) windowed samples. +C. Discussion +Overall alternating least squares (ALS) was the best per- +forming method for denoising synthetic tensors across all +tensor orders, sizes, ranks, and starting noise levels, with the +multiway Wiener filter (MWF) also performing well across +all parameters. Amplification-based denoising performed well +for low ranked tensors as well as very noisy (< 0 dB) +tensors. The performance of amplification at low ranks and +its decreased performance at higher ranks is to be expected, +as the amplification maps correspond to approximations of the +spectral norm, which only measures the highest singular value +for a given tensor. Amplification-based denoising for higher +rank tensors may be improved through the development of +a decomposition method that can find successively smaller +singular values and their corresponding rank 1 components, +such as through a gradient-based descent optimization method. +For the tensors derived from physiological signals, only the +XRank method had any appreciable denoising performance. +Such a method may find applications as a preprocessing step +in a machine learning pipeline that utilizes tensorial data, such +as that used for the prediction of hemodynamic decomposition +in [40]. One limitation of the amplification-based denois- +ing method is that amplification requires determining order- +specific amplification maps; currently only those for orders +three and four have been computed. Other tested methods have +no such restriction. However, many real-world data modalities, +such as images (order 3) and video (order 4) can potentially +be denoised using current amplification maps. +V. CONCLUSION +In this work, we utilize the general framework of tensor +denoising introduced [15] and previously developed approxi- +mations of the spectral norm [17] to devise three novel tensor +denoising methods based on tensor amplification and two +notions of tensor rank related to the G-stable rank [36] - stable +slice rank and stable X-rank. The performance of these meth- +ods was compared to several standard decomposition-based +denoising methods on synthetic tensors of various sizes, ranks, +and noise levels, along with real-world tensors derived from +electrocardiogram (ECG) signals. The experimental results +show that in the low rank context, tensor-based amplification +provides comparable denoising performance in high signal-to- +noise ratio (SNR) settings (> 0 dB) and superior performance +in noisy (< 1 dB) settings, while the stable X-rank method +achieves superior denoising performance on the ECG signal +data. Future work will seek to improve the performance of +amplification-based methods for higher rank tensors. +ACKNOWLEDGMENT +This work was partially supported by the National Science +Foundation under Grant No. 1837985 and by the Department +of Defense under Grant No. BA150235. +APPENDIX A +ORDER 4 DENOISING RESULTS + +Healthy Patients +Unhealthy Patients +5 +5 +0 +-0 +-5 - +(dB) +Denoised SNR +-10 +-10 +-15 +-15 +1O0H +20 +-20 +ALS +Wiener +-25 +Amp +-25 +SliceRank +XRank +-30 +-30 +20 +10 +5 +1-1 -5 +-10 +-20 +20 +10 +5 +1 -1 +-5 +-10 +-20 +Starting SNR (dB) +Starting SNR (dB)Healthy Patients +Unhealthy Patients +5 +IOOH +ALS +0 +Wiener +0 +Amp +SliceRank +-5 + XRank +-5 +(dB) +Denoised SNR ( +-10 +-10 +-15 +-15 +-20 +-20 +-25 - +-25 +-30 - +-30 +20 +10 +5 +1-1 -5 +-10 +-20 +20 +10 +5 +1 -1 +-5 +-10 +-20 +Starting SNR (dB) +Starting SNR (dB)10 +TABLE IV: Mean (SD) SNR, in decibels, after tensor denoising across all parameters. +Starting SNR +HOOI +ALS +Wiener +Amp +SliceRank +XRank +20 +19.68 (3.70) +33.04 (12.81) +32.12 (15.50) +10.93 (15.73) +18.96 (1.43) +9.82 (4.10) +10 +10.82 (1.35) +24.54 (9.92) +22.99 (11.98) +9.60 (12.65) +13.31 (2.18) +9.20 (3.60) +5 +6.11 (0.88) +20.04 (8.73) +17.08 (10.24) +8.81 (11.16) +8.29 (2.63) +8.31 (3.10) +1 +2.20 (0.82) +16.21 (8.10) +11.73 (9.21) +7.92 (10.06) +3.62 (2.38) +7.05 (2.67) +-1 +0.22 (0.81) +14.24 (7.85) +8.90 (8.47) +7.36 (9.60) +1.23 (2.12) +6.22 (2.58) +-5 +-3.78 (0.80) +10.14 (7.64) +3.68 (6.62) +6.06 (8.98) +-3.44 (1.61) +4.10 (2.77) +-10 +-8.79 (0.79) +4.89 (7.52) +-2.18 (4.03) +3.93 (8.78) +-9.03 (1.09) +0.69 (3.64) +-20 +-18.78 (0.81) +-5.44 (7.21) +-9.70 (6.52) +-16.67 (3.64) +-19.64 (0.49) +-7.44 (5.15) +(a) Uniformly sized tensors of order 4. +Starting SNR +HOOI +ALS +Wiener +Amp +SliceRank +XRank +20 +25.83 (8.53) +35.74 (14.42) +32.02 (15.29) +14.20 (15.98) +18.82 (1.44) +11.16 (4.57) +10 +17.69 (5.21) +28.12 (11.06) +24.55 (12.18) +12.83 (12.86) +13.34 (2.14) +10.41 (3.85) +5 +13.47 (3.77) +24.00 (9.59) +18.98 (11.13) +11.83 (11.43) +8.32 (2.49) +9.31 (3.17) +1 +10.00 (2.87) +20.60 (8.58) +13.77 (10.31) +10.14 (10.68) +3.67 (2.21) +7.90 (2.67) +-1 +8.21 (2.55) +18.86 (8.14) +10.99 (9.52) +8.70 (10.62) +1.29 (1.99) +6.97 (2.49) +-5 +4.49 (2.20) +15.20 (7.47) +6.25 (7.26) +4.99 (11.29) +-3.41 (1.49) +4.63 (2.69) +-10 +-0.49 (2.06) +10.30 (7.00) +0.78 (4.80) +0.76 (12.49) +-8.95 (1.16) +0.91 (3.77) +-20 +-10.64 (1.95) +-0.10 (6.65) +-6.69 (6.44) +-18.46 (2.99) +-19.53 (0.67) +-7.47 (5.21) +(b) Non-uniformly sized tensors of order 4. +(a) +(b) +Fig. 4: Denoising performance with respect to tensor rank for fourth-order tensors. +TABLE V: Mean (SD) denoised SNR, in decibels, for low rank and noisy tensors. +Starting SNR +HOOI +ALS +Wiener +Amp +SliceRank +XRank +1 +1.59 (0.36) +5.97 (2.93) +12.21 (4.05) +14.98 (7.72) +5.64 (2.29) +8.89 (1.91) +-1 +-0.42 (0.35) +3.92 (2.92) +10.31 (3.85) +13.66 (7.28) +3.33 (2.11) +7.34 (2.01) +-5 +-4.43 (0.32) +-0.17 (2.91) +6.42 (3.78) +10.68 (6.90) +-1.52 (1.70) +3.77 (2.34) +-10 +-9.45 (0.30) +-5.24 (2.93) +1.90 (3.84) +6.16 (7.47) +-7.49 (1.39) +-0.79 (2.88) +-20 +-19.45 (0.30) +-15.29 (2.91) +-8.86 (5.90) +-4.36 (7.84) +-18.79 (1.03) +-10.31 (3.48) +(a) Uniformly sized fourth-order tensors. +Starting SNR +HOOI +ALS +Wiener +Amp +SliceRank +XRank +1 +10.68 (2.08) +22.45 (7.05) +24.99 (11.39) +23.47 (14.61) +5.82 (2.20) +10.97 (2.55) +-1 +8.67 (2.08) +20.36 (7.02) +21.36 (11.26) +22.36 (13.82) +3.43 (2.08) +9.71 (2.59) +-5 +4.65 (2.07) +16.19 (6.95) +14.20 (9.11) +20.10 (12.42) +-1.59 (1.63) +6.94 (2.98) +-10 +-0.38 (2.05) +11.03 (6.88) +4.95 (5.79) +17.07 (10.93) +-7.48 (1.42) +3.01 (4.21) +-20 +-10.56 (1.98) +0.71 (6.97) +-6.40 (5.90) +-15.23 (4.45) +-18.66 (0.87) +-5.51 (5.53) +(b) Non-uniformly sized fourth-order tensors. + +Uniform Synthetic Tensors, Order 4 +IOOH +ALS +25 +Wiener +Amp +SliceRank +20 +XRank +Denoised SNR (dB) +15 +10 +5 +0 +3 +4 +5 +10 +20 +25 +RankNon-uniform Synthetic Tensors, Order 4 +IOOH +30 +ALS +Wiener +Amp +25 +SliceRank +XRank +Denoised SNR (dB) +20 +15 +10 +5 +0 +2 +3 +4 +5 +10 +20 +25 +Rank11 +REFERENCES +[1] J. 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Kovac, “Local extremes, runs, strings and multires- +olution,” The Annals of Statistics, vol. 29, no. 1, pp. 1–65, 2001. + diff --git a/8NE2T4oBgHgl3EQfPgby/content/tmp_files/load_file.txt b/8NE2T4oBgHgl3EQfPgby/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d2c95ae26ccfe413bbb3dd198bc11b82963ab2df --- /dev/null +++ b/8NE2T4oBgHgl3EQfPgby/content/tmp_files/load_file.txt @@ -0,0 +1,1503 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf,len=1502 +page_content='1 Tensor Denoising via Amplification and Stable Rank Methods Jonathan Gryak1, Kayvan Najarian2,3,4,5,6, and Harm Derksen7 Abstract—Tensors in the form of multilinear arrays are ubiq- uitous in data science applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Captured real-world data, including video, hyperspectral images, and discretized physical systems, naturally occur as tensors and often come with attendant noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Under the additive noise model and with the assumption that the underlying clean tensor has low rank, many denoising methods have been created that utilize tensor decomposition to effect denoising through low rank tensor approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' However, all such decomposition methods require estimating the tensor rank, or related measures such as the tensor spectral and nuclear norms, all of which are NP-hard problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' In this work we adapt the previously developed framework of tensor amplification, which provides good approximations of the spectral and nuclear tensor norms, to denoising synthetic tensors of various sizes, ranks, and noise levels, along with real-world tensors derived from physiological signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' We also introduce de- noising methods based on two variations of rank estimates called stable X-rank and stable slice rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The experimental results show that in the low rank context, tensor-based amplification provides comparable denoising performance in high signal-to- noise ratio (SNR) settings and superior performance in noisy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=', low SNR) settings, while the stable X-rank method achieves superior denoising performance on the physiological signal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Index Terms—Tensors, Denoising, Tensor Amplification, Stable Rank Methods I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' INTRODUCTION T ENSORS in the form of multilinear arrays are ubiquitous in data science applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Captured real-world data, including color and hyperspectral images (HSIs), video, and discretized physical systems, naturally occur as tensors and often come with attendant noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' As is common in other signal processing applications, the captured tensor T ∈ Rp1×p2×···×pd is modeled as T = D + N, where D is a pure or “clean” tensor D that has been corrupted by additive noise D, which is typically assumed to be Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Additionally, the clean tensor D is assumed to be low rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Under this framework,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' tensor denoising can be achieved by utilizing tensor decompositions methods,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' such as the canonical 1Department of Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Queens College,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' City University of New York,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' New York,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' NY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' USA 2Department of Computational Medicine and Bioinformatics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Ann Arbor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' MI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' USA 3Department of Emergency Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Ann Arbor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' MI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' USA 4Electrical and Computer Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' College of Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Ann Arbor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' MI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' USA 5Michigan Institute for Data Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Ann Arbor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' MI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' USA 6Max Harry Weil Institute for Critical Care Research and Innovation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Ann Arbor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' MI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' USA 7Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Northeastern University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Boston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' USA polyadic (CP) [1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' [2] and Tucker [3],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' [4] decompositions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' to determine a low-rank approximation of the observed tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' These decomposition algorithms require a pre-specified rank to compute an approximation, however, determining the rank of a tensor is NP-hard [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Thus, tensor decomposition-based methods utilize some estimate of the tensor rank to effect tensor denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' CP decomposition has been frequently used for HSI de- noising, such as in Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=', [6], which estimated the tensor rank using covariance matrices of the n-model flattenings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' in Veganzones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' [7], which used a non-negative variant of CP decomposition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' and in [8], in which a CP decomposition regularized by the nuclear norm of clustered 3D patches of the HSI was employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Tucker decomposition based denoising include two works by Rajwade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' that utilized higher order singular value decomposition [9], a tensor analog of matrix SVD, to denoise video [10] and images [11];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' as well Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' [12], which focused on denoising tensors with ordinal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' More recently, a tensor train (matrix product state) decomposition [13] was used for denoising of HSIs [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' A general framework for understanding tensor denoising in the additive model was developed in [15], that relates the problem of denoising in the low rank context to the mini- mization of dual norms ∥D∥X and ∥N∥Y , such as the nuclear ∥·∥⋆ and spectral ∥·∥σ norms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The calculation of these norms for tensors is also NP-hard [16],[5], thus in order make use of the denoising framework in [15] the co- authors developed the method of tensor amplification [17], which provides good approximations of the tensor spectral norm and its dual the nuclear norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' In this work, we utilize the general framework of [15] and the approximations of the spectral norm [17] to devise three novel tensor denoising methods - amplification-based, stable slice rank, and stable X-rank denoising, the latter two methods based on their eponymous rank estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The performance of these methods is compared to several standard decomposition- based denoising methods on synthetic tensors of various sizes, ranks, and noise levels, along with real-world tensors derived from physiological signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The experimental results show that in the low rank context, tensor-based amplification provides comparable denoising performance in high signal-to-noise ratio (SNR) settings and superior performance in noisy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=', low SNR) settings, while the stable X-rank method achieves superior denoising performance on the physiological signal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='03761v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='LG] 10 Jan 2023 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' PRELIMINARIES AND RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Basic Notation Let T ∈ Rp1×p2×···×pd denote a real-valued tensor of order d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' In the denoising experiments that are performed in this study we will assume that the tensor T is the noisy version of a pure tensor D ∈ Rp1×p2×···×pd corrupted by additive noise N ∈ Rp1×p2×···×pd, that is, T = D + N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' (1) The Frobenius norm of T is denoted ∥T ∥ and defined as ∥T ∥ = � � � � p1 � i1 p2 � i2 · · pd � id t2 i1i2···id, (2) while the tensor inner product of two tensors T , S of matching order and dimension is defined as ⟨T , S⟩ = p1 � i1 p2 � i2 · · pd � id ti1i2···idsi1i2···id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' (3) The induced norm of the tensor inner product is the Frobenius norm defined above, with the typical relation ⟨T , T ⟩ = ∥T ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Given a tensor T and a permutation q = ⟨q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , qd⟩ of the indices 1 : d, the q-transpose of T is the tensor T ⟨q⟩ ∈ Rpq1×pq2×···×pqd with entries (T ⟨q⟩)i1i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='id = tiq1iq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='iqd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' (4) At times we will need to matricize the tensors under con- sideration by rearranging their entries in specific ways, as well as employ various tensor-tensor, tensor-matrix, and matrix- matrix products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' In the definitions below and throughout the manuscript we will primarily follow the notational conventions introduced by Kolda and Bader in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The mode-n flattening or unfolding of the tensor T is the matrix T(n) ∈ Rpn×N/pn, where N = � i pi, whose columns are the mode-n fibers of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The n-mode product of a tensor T and a matrix A ∈ RJ×pn is the tensor T ×n A of size p1 ×p2 ×· · ·×pn−1 ×J ×pn+1 × · · × pd with entries (T ×n A)i1i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='in−1jin+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='id = pn � in=1 ti1i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='idujin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' (5) If S = T ×n A, then the n-mode product as defined above is equivalent to S(n) = AT(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The Kronecker product of two matrices A ∈ RI×J and B ∈ RK×L is the matrix A ⊗ B ∈ RIK×JL defined by A ⊗ B = � ���� a11B a12B · · a1JB a21B a22B · · a2JB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' aI1B aI2B · · aIJB � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' (6) Finally, given two tensors T ∈ Rp1×···×pd and S ∈ Rq1×···×qe, their outer product is the tensor T ◦ S of size p1 × · · · × pd × q1 × · · · × qe with entries (T ◦ S)i1i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='idj1j2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='je = ti1ti2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' tidsj1sj2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' sje.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' (7) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Decomposition-based Denoising Tensor decomposition methods seek to represent a given tensor by decomposing it into factors such as simple tensors or matrices and whose combination results in a “good” approx- imation of the original tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' In the context of denoising, it is typical to assume that a noisy signal is sparse, in the sense that its ℓ1 norm is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' In the case of matrices and tensors, this assumption corresponds to the original tensor having low rank, with the high rank components corresponding to additive noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Thus, computing a low rank approximation of the original tensor via tensor decomposition is a means to effect tensor denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' In the case of matrices (order two tensors), singular value decomposition yields the exact rank r of the matrix and its decomposition into r factor, with the best low rank approxi- mation for a given rank l < r provided by choosing the factors corresponding to the l largest singular values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' For higher order tensors, calculating the exact rank is NP hard [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Moreover, unlike the matrix case, the factors used to create the best rank r − 1 approximation need not be those used to produce the best rank r approximation [19], and for degenerate tensors, the best rank r approximation may not even exist [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Despite these theoretical limitations, in practice one can utilize tensor decomposition methods to effect denoising by creating decompositions for a range of rank values, then choos- ing the best rank r decomposition D that best approximates the original tensor T , e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=', min ∥T − D∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' This strategy for tensor denoising was evaluated using three common tensor decomposition methods: canonical polyadic decomposition, higher-order orthogonal iteration, and multiway Wiener filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' 1) CP Decomposition via Alternating Least Squares (CP- ALS): Let U (j) = [uj,1uj,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' uj,r] ∈ Rpj×r, 1 ≤ j ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' CP decomposition factorizes a d-way tensor into d factor matrices and a vector Λ = [λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , λr] ∈ Rr: S = r � i=1 λiu1,i ◦ u2,i ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' ◦ ud,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' (8) The best rank r approximation problem for a tensor T ∈ Rp1×p2×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='×pd can be given as: min Λ,U (1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=',U (d) ∥T − S∥ where S = [Λ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' U (1), U (2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , U (d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' (9) This can be found by employing alternating least squares (ALS), wherein each iteration of the algorithm an approxima- tion of the flattening for one mode is found by fixing all other modes of the tensors and solving a least squares problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' This process is repeated, cycling through all modes, until convergence or a maximum number of iterations is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' In this work, the implementation of CP-ALS from TensorToolbox [21] was utilized with the default level of tolerance (10−4) and maximum number of iterations (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' CP-ALS was run for specified rank values r ∈ [1, min(pi)], with the rank r∗ approximation Dr∗ = min r ∥T − Dr∥ (10) chosen as the best denoised tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' 3 2) Higher-Order Orthogonal Iteration (HOOI): For matri- ces, orthogonal iteration produces a sequence orthonormal bases for each subspace of the vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' De Lathauwer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' [22] extended this to tensors, developing the technique known as higher-order orthogonal iteration (HOOI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' This method uses ALS to estimate the best rank-[r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , rd] ap- proximation for a tensor, and is achieved by iteratively solving the optimization problem argminU(i)|ri ∥T − G ×1 U(1)|r1 ×2 U(2)|r2 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' ×N U(d)|rd∥, (11) where G is a core tensor of size r1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' × rd and each U(i)|ri is a matrix comprised of the ri leftmost singular vectors of the singular value decomposition of the modal flattening U(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' HOOI-based denoising was implemented using the tucker_als method in TensorToolbox [21] to determine the best rank [r∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , r∗ d] approximation, where each ri was chosen equally and uniformly from r ∈ [1, min(pi)], with the rank r∗ approximation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' 10) chosen as the best denoised tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' 3) Multiway Wiener Filter: For a discrete signal y[n] and filter output ˆy[n], the Wiener filter h[n] is the filter that minimizes the mean squared error between ˆy[n] and y[n]: argminh[·]E[(ˆy[n] − y[n])2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' (12) Wiener filters have been used in a variety of denoising appli- cations, such as for images [23], [24], physiological signals [25], [26] and speech [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Muti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' [29] created a multiway Wiener filter that can be used to denoise tensors of arbitrary size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Given a noisy tensor T , their method uses an ALS approach to learn Wiener filters {Hn} for each mode n so that the mean squared error between T and the denoised tensor D is minimized, where D = T ×1 H1 ×2 H2 ×3 · · · ×d Hd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' (13) The implementation of the multiway Wiener filter utilized in this study and the exposition below follows [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The filters Hn in each mode n are initialized to the identity matrix of Rpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' At each stage k of the algorithm, the filter Hk n is computed for each mode as Hk n = VnΛnV ⊺ n , (14) where Vn is a matrix containing the Kn orthonormal basis vectors of the signal subspace in the column space of T(n), the mode-n flattening of T , and Λn = diag � λγ 1 − ˆσγ2 n λΓ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , λγ Kn − ˆσγ2 n λΓ Kn � , (15) where {λγ i , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , Kn} and {λΓ i , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , Kn} are respectively the Kn largest eigenvalues of the matrices γn and Γn, defined as γn = E � T(n)qnT(n) ⊺� (16) Γn = E � T(n)QnT(n) ⊺� (17) with qn = d � i̸=n Hi (18) Qn = d � i̸=n H⊺ i Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' (19) The values ˆσγ2 n in Equation 15 are estimates of the pn − Kn smallest eigenvalues of γn, calculated as ˆσγ2 n = 1 pn − Kn pn � i=Kn+1 λγ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' (20) Following [31], the optimal Kn for mode n is estimated using the Akaike Information Criterion (AIC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Please refer to [30] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' AMPLIFICATION AND STABLE RANK DENOISING In this section we introduce three different denoising meth- ods – Amplification-based, Stable Slice Rank, and Stable X- Rank denoising – that leverage the general framework for denoising based on dual norms as introduced in Derksen [15], to effect denoising on tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' A Framework for Denoising Using Dual Norms The model T = D +N utilized in this work can be viewed as an instance of the additive noise model c = a + b, where a, b, c ∈ V are elements of a vector space V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' In Derksen [15], a general framework for understanding the denoising of vectors under the additive model was developed that relates the problem of denoising the vector c to the minimization of ∥a∥X and ∥b∥Y , where ∥ · ∥X and ∥ · ∥Y are dual norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Moreover, the framework makes the assumptions that the original vector (or tensor) a is sparse, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=', that it has few non-zero values or is of low rank, while the additive noise b is dense or of high rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Thus, the norms ∥ · ∥X and ∥ · ∥Y can be interpreted as respectively measuring the sparsity and noise of the vector (or tensor) under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The prototypical ∥ · ∥X norm is the nuclear norm, which for a matrix is the sum of its singular values, while for a tensor the tensor nuclear norm ∥T ∥⋆, is defined as ∥T ∥⋆ = min r � i=1 ∥vi∥2, where v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , vr are rank-1 tensors and T = �r i=1 vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The prototypical ∥·∥Y norm and dual to ∥·∥X is the spectral norm, which for a matrix is the absolute value of its largest singular value, while for a tensor the tensor spectral norm ∥T ∥σ is defined as ∥T ∥σ = sup |T · u1 ⊗ u2 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' ⊗ ud|, where uj ∈ Rpj and ∥uj∥ = 1 for 1 ≤ j ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' If V is also an inner product space we also have the induced norm � ⟨c, c⟩ that corresponds to the standard Euclidean norm ∥c∥2 for vectors or the Frobenius norm ∥ · ∥, introduced in Section II-A, for matrices and tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' As shown in [15], the denoising of a vector c via a decomposition c = a + b 4 that simultaneously minimizes the values ∥a∥X and ∥b∥Y is governed by the Pareto frontier, which models the competing objectives of minimizing the two norms in terms of Pareto efficiency, and the above XY -decomposition that achieves this is deemed Pareto efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Moreover, [15] defines the related notion of the Pareto subfrontier, which relates the three norms ∥ · ∥X, ∥ · ∥Y , ∥ · ∥2 and their induced decompositions XY , X2, and 2Y , describing the conditions under which these decompositions can achieve Pareto efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Amplification-based Denoising To make use of the denoising framework introduced in [15] requires the calculations of various norms for the vec- tors of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' While the Frobenius norm of a tensor is easily obtained, computing either the nuclear norm [16] or the spectral norm [5] for tensors is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' In order to obtain an approximation to the tensor spectral norm, the co- authors developed the methodology of tensor amplification [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' For a matrix A with singular values λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , λr, the function φ : A → AA⊺A produces the matrix AA⊺A whose singular values are λ3 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , λ3 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Repeated applications of φ(·) will amplify the larger singular values, which correspond to the sparse or low rank components of the matrix, while minimizing smaller singular values that likely correspond to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Analogously, tensor amplification utilizes degree d polyno- mial functions on tensors to amplify the low rank structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Moreover, for each amplification map Φσ′ there exists a cor- responding norm ∥·∥σ′,d that approximates the tensor spectral norm, in the sense that limd→∞ ∥T ∥σ′,d = ∥T ∥σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Two such amplification maps – Φσ,4 and Φ# – were introduced for order 3 tensors in [17], with Φ# being show to be a better approximation to the tensor spectral norm than Φσ,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The method presented in Algorithm 1 utilizes the 2Y - decomposition framework of [15] and the tensor spectral norm approximations Φ to denoise a given tensor T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The algorithm allows for the choice of amplification map as well as the number of amplifications per round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' For third order tensors the amplification map Φ# was used, while for fourth order tensors a compatible version of Φσ,4 was employed as there is no currently known analogue of the Φ# map for fourth order tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Multiple experiments were performed with m, the number of amplifications per round, ranging from 1 to 10, with m = 5 being found to produce the best denoising performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Stable Slice Rank Denoising Slice rank was introduced in [32] in relation to the cap set problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Following Tao [33], the slice rank of a tensor T is the least non-negative integer srk such that T is a sum of tensors with slice rank 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=', T = �r i=1 Ti, where Ti is contained in the tensor product space V1 ◦ · · · Vi−1 ◦ s ◦ Vi−1 ◦ · · · Vd, (21) where Vj are vector spaces and s is a vector in some Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' In [34], the notion of a stable rank for matrices was introduced, in which the matrix rank function rank(A), is replaced by the Algorithm 1 Amplification-based tensor denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' D ← DENOISE AMPLIFICATION(T , Φ, m) ϵ ← ∥T ∥ N ← T while true do A ← Φm(N) A ← A ∥A∥ N ← N − ⟨A, N⟩A Break if ∥N∥ < ϵ end while D ← T − N numerical rank function, ∥A∥2 ∥A∥2σ , or the related stable nuclear rank ∥A∥2 ⋆ ∥A∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' These ranks are stable in the sense that small perturbations of the values of the matrix A will not change their value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Extending this methodology to tensors, the stable slice rank is defined as �d i=1 ∥T(i)∥2 ⋆ ∥T ∥2 , (22) where T(i) are the mode-i flattenings of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' For a given value of the parameter λ, the stable slice rank (SliceRank) method denoises a tensor by finding a decompo- sition T = D + N that minimizes the Frobenius norm of D = � i Si under the constraints that the nuclear norms of the flattenings of N are all ≤ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The method also produces a decomposition D = � i Si that minimizes the sum of the nuclear norms of S(i), the mode-i flattenings of Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Typically, the S(i) have low rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Algorithm 2 Stable SliceRank denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' (D, {Si}, ssrk) ← DENOISE SLICERANK(T , λ, acc) Si ← 0 ∈ Rp1×p2×···×pd curr acc ← 0 while curr acc < acc do for i ← 1 : d do A ← T − � j̸=i Sj q ← CIRCSHIFT([1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , d], −(i − 1)) A ← A⟨q⟩ (U, D, V ) ← SVD(A(i)) Ei ← MAX(D − λ, 0) ei ← DIAG(Ei) F ← U · Ei · V T F ← RESHAPE(F, pi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , pd, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , pi−1) q ← CIRCSHIFT([1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , d], (i − 1)) Si ← F⟨q⟩ end for curr acc ← ⟨T − �d j=1 Sj, �d j=1 Sj⟩ λ �d j=1 ∥A(j)∥⋆ end while D ← �d i=1 Si ssrk ← �d i=1 ∥A(i)∥2 ⋆ ∥D∥2 Algorithm 2 depicts the implementation of SliceRank de- noising, which utilizes a number of auxiliary functions from 5 MATLAB [35]: circshift performs a cyclic permutation of an index set [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , d], with the second parameter determining the number of forward or backwards shifts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' reshape is used to flatten a tensor into a matrix with the specified dimensions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' and diag returns a vector comprising the entries on the main diagonal of the specified matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The algorithm takes as hyperparameters λ as described above and acc ∈ (0, 1], the specified accuracy level that once achieved the algorithm terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The algorithm returns the denoised tensor D, the decomposition factors Si, and ssrk, the stable slice rank of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The hyperparameters were optimized via grid search over the ranges λ ∈ {10−2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='1, 1, 10} and acc ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='90, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='95}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Stable X-Rank Denoising As noted in Section II-B, a degenerate tensor T of rank r may not have a best rank k < r approximation for a given rank k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' In such cases, a tensor may be approximated to any desired precision by rank j < k tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' This is due to the set of all tensors for a given rank r not being Zariski closed [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' In Derksen [36], the G-stable rank of a tensor was introduced that, among its other advantages, is Zariski closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Thus, every tensor T has a best G-stable rkG < r approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The G- stable α rank of a tensor can be defined as rkG α (T ) = sup g∈G min i αi∥g · T ∥2 ∥ (g · T )(i) ∥2σ , (23) where α = (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , αd) and g is an element of a reductive group G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=', g ∈ SL(Rp1) × · · · × SL(Rpd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Using the above definition we can define the related concept of stable X-rank, which is sXrkG(T ) = max α rkG α (T ), (24) where α is subject to the restriction that � i αi = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Algorithm 4 depicts the implementation of the stable X-Rank (XRank) denoising method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Like SliceRank, the method imposes a con- straint on the nuclear norm of the flattenings of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' However, in the XRank method, this cutoff is determined automatically using Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The hyperparameters were optimized via grid search over the ranges λ ∈ {10−2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='1, 1, 10} and acc ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='90, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='95}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' EXPERIMENTAL RESULTS AND DISCUSSION In order to evaluate the various denoising methods under consideration, two sets of synthetic tensors were generated with varying orders, ranks, and dimensions, resulting in 512 parameter combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' For each combination, one hundred (100) tensors were generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' For all synthetic tensors, varying amounts of noise were added from a standard Gaussian distribution N(0, 1), with the resulting noisy tensors having signal-to-noise ratios (SNR) ranging from 20 dB to −20 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The full range of parameters is provided in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Additionally, two sets of tensors were extracted from elec- trocardiogram (ECG) signals to which Gaussian noise was added prior to tensor extraction, using the same range of resultant SNRs as those employed in the generation of the synthetic tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Algorithm 3 Determine the nuclear norm cutoff for XRank denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' c ← FIND CUTOFF(f = [λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , λr]⊺, λ) r ← |f| t ← 0 ∈ Rr for i ← 1 : r do ti ← λ �i j=1 λj 1 + λ · i end for S ← 0 ∈ Rr×r for i ← 1 : r do for j ← 1 : r do sij ← MAX(fi − tj, 0) end for end for v ← 0 ∈ Rr for j ← 1 : r do vj ← �r i=1(fij − sij)2 + λ �r i=1(sij)2 end for k ← ARGMIN(v) c ← tk Algorithm 4 Stable XRank denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' (D, {Si}, sxrk) ← DENOISE XRANK(T , λ, acc) Si ← 0 ∈ Rp1×p2×···×pd curr acc ← 0 while curr acc < acc do for i ← 1 : d do A ← T − � j̸=i Sj q ← CIRCSHIFT([1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , d], −(i − 1)) A ← A⟨q⟩ (U, D, V ) ← SVD(A(i)) c ← FIND CUTOFF(DIAG(D),λ) Ei ← MAX(D − c, 0) ei ← DIAG(Ei) F ← U · Ei · V T F ← RESHAPE(F, pi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , pd, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , pi−1) q ← CIRCSHIFT([1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , d], (i − 1)) Si ← F⟨q⟩ end for Scurr ← �d i=1 Si T S = T − Scurr y ← 0 for i ← 1 : d do q ← CIRCSHIFT([1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , d], −(i − 1)) B ← T S⟨q⟩ (U, D, V ) ← SVD(B(i)) y ← y + d2 1 end for y ← √y curr acc ← ⟨T , Scurr⟩ y · ��d j=1 ∥A(j)∥2⋆ end while D ← �d i=1 Si sxrk ← �d j=1 ∥A(j)∥2 ⋆ ∥D∥2 6 TABLE I: Parameters and their respective values used to generate the synthetic tensor datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Parameter Range/Values Distribution Normal N(0, 1) Order 3, 4 Rank [1, 5], 10, 20, 25 Size 5, 10, 25, 50 SNR 20, 10, 5, 1, −1, −5, −10, −20 1) Uniform Synthetic Tensors: In this dataset, the dimen- sions of a given tensor are chosen uniformly across each mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' To generate synthetic tensors from a distribution D of a given rank r, size s, and order d, scalar values λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , λr are chosen from D, then for each mode j, r random vectors xj,i ∈ Rs are chosen from D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The synthetic tensor is then r � i=1 λix1,i ◦ x2,i ◦ · · · ◦ xd,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' 2) Non-Uniform Synthetic Tensors: In this dataset, one mode mk of a given tensor is “stretched” to a different dimension dk by choosing a number uniformly in the range dk = [s, min(500, sd)], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=', the lower bound is the dimension of the other models while the upper bound is the product of the dimensions of each mode or 500, whichever is lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' After choosing the stretch mode and its dimension the tensors are generated in the same manner as for the uniform tensors above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' 3) ECG Waveform Tensors: The PTB Diagnostic ECG Database [37] is comprised of high resolution (1 kHz) digitized recordings of electrocardiograms (ECGs) from patients with various cardiovascular diseases, including myocardial infarc- tion, heart failure, and arrhythmia, as well as healthy controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The database is publicly available via Physionet [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Tensor-based methods have been shown to be effective for a number of ECG analytical tasks, a survey of such methods can be found in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Given the utility of tensor-based methods in this context and that such that recordings of physiological signals may be corrupted by noise yields a natural application of the proposed denoising methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' In order to evaluate these methods, we first must construct tensors from the ECGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' In forming these tensors, one has to consider the amount of signal over which to perform subsequent signal processing and feature extraction: two methods were employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' In the first, ninety (90) seconds of a patient’s ECG recording was sampled across all twelve ECG leads, while in the second method three windowed samples of thirty (30) seconds each were extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Using these two sampling strategies we adapted the tensor formation method introduced in [40] that has been shown to be effective for subsequent applications of machine learn- ing for prognosticating severe cardiovascular conditions [41], [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' In this method, each ECG signal is preprocessed using the taut string method, which produces a piecewise linear approximation of a given signal, parametrized by ϵ, which controls the coarseness of the approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Given a discrete signal x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , xn) one can define the finite difference D(x) = (x2 − x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' , xn − xn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' For a fixed ϵ > 0, the taut string estimate of x is the unique function y such that ∥x − y∥∞ ≤ ϵ and ∥D(y)∥2 is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The taut string approximation can be found efficiently using the method in [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' After the taut string approximation for a given signal is found, six morphological and statistical features are extracted following [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' This process is repeated for five values of epsilon: (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='0100, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='1575, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='3050, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='4525, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='6000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' As each pa- tient’s ECG recording is comprised of the standard 12 leads, the approximation of each 90 second ECG sample via taut string and the extraction of taut string features yields third order tensors of size 5 × 6 × 12 for each patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' For the windowed samples, fourth order tensors were formed of size 5 × 6 × 12 × 3, with the fourth mode corresponding to the features extracted in each window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' 4) Adding Noise: For every generated synthetic tensor, a set of noisy tensors was created by adding Gaussian noise (N(0, 1)) so that the resultant tensors had SNRs in the range [20, 10, 5, 1, −1, −5, −10, −20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' For the ECG waveform tensors, Gaussian noise was added to each ECG signal to produce a set of noisy signals with the same SNR range as for the synthetic tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' However, this was performed prior to tensor formation given that in practical applications ECG signals themselves may come with some intrinsic amount of noise, rather than noise being introduced to the tensors directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Results: Synthetic Data The overall denoising performance for third order tensors across various ranks and tensors sizes are presented in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The performance statistics for only one order are presented due to the incomparable sizes of the non-uniform tensors across orders, please see Table IV in Appendix A for the fourth order 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The best performing denoising algorithms for uniformly sized tensors, as depicted in Table II (a), varied by noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' For cleaner tensors (20 and 10 dB), the multiway Wiener filter performed best overall, achieving mean and standard deviations in denoised SNRs of 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='95 (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='19) and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='22 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='1) dBs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' For moderately noisy tensors (5, 1, and −1 dB), ALS was the best performing denoising method, achieving denoised SNRs of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='11 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='26), 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='12 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='98), and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='1 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='88) dBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' For nosier tensors with starting SNRs of −5 and −10, tensor amplification produced the best denoised SNRs of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='37 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='69) and −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='25 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='57) dBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Finally for tensors with starting SNRs of −20 dB, the noisiest tensors evaluated, XRank produced on average the highest denoised SNR of −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='22 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='79) dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The results for non-uniformly sized tensors, as depicted in Table II (b), are much clearer, with ALS achieving the best denoised SNRs across all starting SNRs, ranging from 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='81 (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='7) dBs for tenors with starting SNRs of 20 dB to −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='6 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='57) dB for the noisiest tensors (starting SNRs of −20 dB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The relationship between tensor size (dimension of each mode) and achieved denoised SNRs is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' With the exception of HOOI, all other denoising algorithms see improvements in achieved SNR as the size of the tensor increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The multiway Wiener filter maintains the best de- noising performance as size increases, followed by ALS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Both amplification and XRank have similar denoising performances, while slice rank and HOOI having the lowest performance overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' 7 TABLE II: Mean (SD) SNR, in decibels, after tensor denoising across all parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Starting SNR HOOI ALS Wiener Amp SliceRank XRank 20 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='57 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='32) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='67 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='1) 29.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='66) (b) Non-uniformly sized tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' 1: Denoising performance with respect to tensor size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The relationship between tensor rank and achieved de- noised SNRs is depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Tensor amplification achieves the best rank-1 performance for both uniformly and non-uniformly sized tensors, followed by the multiway Wiener filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The denoising performance of both methods decreases as tensor rank increases, with the multiway Wiener filter maintaining denoising performance for higher ranks than amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' ALS has lower performance at low ranks but generally maintains its denoising performance as rank increases, ultimately achieving the best results by rank 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' XRank has greater performance than SliceRank for uniformly sized tensors, but their denoising performances converge prior to rank 20 for non-uniformly sized ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' HOOI has the lowest denoising performance for uniformly sized tensors, but performs slightly better than the amplification and SliceRank methods for non-uniformly sized tensors with ranks greater than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The results depicted in Table III provide a further investiga- tion into the denoising performance for low rank (ranks 1 and 2) and high noise (SNRs ≤ 1) tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' From these results one can observe that amplification achieves the best performance for uniformly sized tensors, with the multiway Wiener filter achieving the second-best denoising performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' In the case of non-uniformly sized tensors, the Wiener filter and amplifi- cation achieve comparable results for starting SNRs of 1 and −1 dB, while amplification achieving better performance for starting SNRs of −5,−10, and −20 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Results: Real Data - ECG Waveform Tensors Figure 3 shows the denoising performance on the tensors derived from the PTB dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' For the tensors derived from 90 second samples of ECG signal (Figure 3 a), only the stable rank methods (XRank and SliceRank) were able to achieve any effective denoising, with XRank achieving the best denoising performance with a modest ≈ 4 dB denoised SNR for tensors whose signals had SNR ratios of 20 dB prior to tensor formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' This performance was maintained for tensors from signals with starting SNRs down to 5 dB, after which the denoising performance of XRank declines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The SliceRank method does not yield any tensor denoising until the starting signal SNR dropped below −5 dB, after which it too experiences a continued decline in denoising performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' All Uniform Synthetic Tensors 1OOH 14 - ALS Wiener 12 - Amp Denoised SNR (dB) 10 SliceRank XRank 8 6 · 4 2 - 5 10 25 50 SizeNon-uniform Synthetic Tensors 16 IOOH ALS 14 - Wiener 12 Amp Denoised SNR (dB) SliceRank 10 XRank 8 6 - 4 - 2 01 5 10 25 50 Size8 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' 2: Denoising performance with respect to tensor rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' TABLE III: Mean (SD) denoised SNR, in decibels, for low rank and noisy tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Starting SNR HOOI ALS Wiener Amp SliceRank XRank 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='59 (0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='05 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='87) (b) Non-uniformly sized tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' other methods - HOOI, ALS, and amplification - introduced noise into the tensors across all starting signal SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' No appreciable difference was observed in tensors derived from the two patient cohorts (healthy and unhealthy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The denoising results for the tensors derived from windowed samples (Figure 3 b) are essentially the same as those derived from the 90 second samples, with the only exception being a slight increase in SliceRank’s denoising performance for tensors corresponding to unhealthy patients with a starting signal SNR of −5 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Uniform Synthetic Tensors 1OOH 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='5 ALS Wiener 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='0 Amp Denoised SNR (dB) SliceRank 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='5 XRank 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='0 10 20 25 RankNon-uniform Synthetic Tensors IOOH 20 ALS Wiener Amp Denoised SNR (dB) 15 SliceRank XRank 10 5 0 10 20 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' 5 Rank9 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' 3: Denoising performance on the PTB tensors formed from a) 90 second samples, and b) windowed samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Discussion Overall alternating least squares (ALS) was the best per- forming method for denoising synthetic tensors across all tensor orders, sizes, ranks, and starting noise levels, with the multiway Wiener filter (MWF) also performing well across all parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Amplification-based denoising performed well for low ranked tensors as well as very noisy (< 0 dB) tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The performance of amplification at low ranks and its decreased performance at higher ranks is to be expected, as the amplification maps correspond to approximations of the spectral norm, which only measures the highest singular value for a given tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Amplification-based denoising for higher rank tensors may be improved through the development of a decomposition method that can find successively smaller singular values and their corresponding rank 1 components, such as through a gradient-based descent optimization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' For the tensors derived from physiological signals, only the XRank method had any appreciable denoising performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Such a method may find applications as a preprocessing step in a machine learning pipeline that utilizes tensorial data, such as that used for the prediction of hemodynamic decomposition in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' One limitation of the amplification-based denois- ing method is that amplification requires determining order- specific amplification maps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' currently only those for orders three and four have been computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Other tested methods have no such restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' However, many real-world data modalities, such as images (order 3) and video (order 4) can potentially be denoised using current amplification maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' CONCLUSION In this work, we utilize the general framework of tensor denoising introduced [15] and previously developed approxi- mations of the spectral norm [17] to devise three novel tensor denoising methods based on tensor amplification and two notions of tensor rank related to the G-stable rank [36] - stable slice rank and stable X-rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The performance of these meth- ods was compared to several standard decomposition-based denoising methods on synthetic tensors of various sizes, ranks, and noise levels, along with real-world tensors derived from electrocardiogram (ECG) signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' The experimental results show that in the low rank context, tensor-based amplification provides comparable denoising performance in high signal-to- noise ratio (SNR) settings (> 0 dB) and superior performance in noisy (< 1 dB) settings, while the stable X-rank method achieves superior denoising performance on the ECG signal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Future work will seek to improve the performance of amplification-based methods for higher rank tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' ACKNOWLEDGMENT This work was partially supported by the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' 1837985 and by the Department of Defense under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' BA150235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='APPENDIX A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='ORDER 4 DENOISING RESULTS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='Healthy Patients ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='Unhealthy Patients ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='5 - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='(dB) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='Denoised SNR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='1O0H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='ALS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='Wiener ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='Amp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='SliceRank ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='XRank ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='1-1 -5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='1 -1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='Starting SNR (dB) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='Starting SNR (dB)Healthy Patients ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='Unhealthy Patients ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='IOOH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='ALS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='Wiener ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='Amp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='SliceRank ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='XRank ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='(dB) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='Denoised SNR ( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='25 - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='30 - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='1-1 -5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='1 -1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='Starting SNR (dB) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='Starting SNR (dB)10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content='TABLE IV: Mean (SD) SNR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' in decibels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' after tensor denoising across all parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Starting SNR HOOI ALS Wiener Amp SliceRank XRank 20 19.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Davies and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' Kovac, “Local extremes, runs, strings and multires- olution,” The Annals of Statistics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' 29, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} +page_content=' 1–65, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQfPgby/content/2301.03761v1.pdf'} diff --git a/8NE5T4oBgHgl3EQfQg5R/content/tmp_files/load_file.txt b/8NE5T4oBgHgl3EQfQg5R/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0da8ac431b23ec03efc6cffb29d6477eab901611 --- /dev/null +++ b/8NE5T4oBgHgl3EQfQg5R/content/tmp_files/load_file.txt @@ -0,0 +1,1062 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf,len=1061 +page_content='1 Exploring the substrate-driven morphological changes in Nd0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='6Sr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='4MnO3 thin films R S Mrinaleni 1, 2, E P Amaladass1, 2*, S Amirthapandian 1, 2, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Sathyanarayana 1, 2, Jegadeesan P 1, 2, Ganesan K 1, 2, R M Sarguna 1, 2, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Rao 3, Pooja Gupta 3, 4, T Geetha Kumary1, 2, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Rai 3, 4, Awadhesh Mani1, 2 1Material Science Group, Indira Gandhi Centre for Atomic Research, Kalpakkam, 603102, India 2Homi Bhabha National Institute, Indira Gandhi Centre for Atomic Research, Kalpakkam 603102, India 3Synchrotrons Utilisation Section, Raja Ramanna Centre for Advanced Technology, PO RRCAT, Indore, Madhya Pradesh 452013, India 4Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai, Maharashtra 400094, India Corresponding author: edward@igcar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='in ABSTRACT Manganite thin films are promising candidates for studying the strongly correlated electron systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Understanding the growth-and morphology-driven changes in the physical properties of manganite thin films is vital for their applications in oxitronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' This work reports the morphological, structural, and electrical transport properties of nanostructured Nd0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='6Sr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='4MnO3 (NSMO) thin films fabricated using the pulsed laser deposition technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Scanning electron microscopy (SEM) imaging of the thin films revealed two prominent surface morphologies: a granular and a unique crossed-nano-rod-type morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' From X-ray diffraction (XRD) and atomic force microscopy (AFM) analysis, we found that the observed nanostructures resulted from altered growth modes occurring on the terraced substrate surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Furthermore, investigations on the electrical-transport properties of thin films revealed that the films with crossed-nano-rod type morphology showed a sharp resistive transition near the metal-to-insulator transition (MIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' An enhanced temperature coefficient of resistance (TCR) of up to one order of magnitude was also observed compared to the films with granular morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Such enhancement in TCR % by tuning the morphology makes these thin films promising candidates for developing oxide-based temperature sensors and detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 2 INTRODUCTION Nd0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='6Sr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='4MnO3 (NSMO) belongs to the class of magnetic oxides RE1-xAxMnO3 (where RE= La3+, Nd3+, Pr3+, Sm3+, and A = Ca2+, Sr2+, Ba2+, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=') with perovskite (ABO3) structure which exhibits a variety of magnetic phases by tuning the dopant concentration x (x = 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='9)1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Manganites are known for their exotic properties such as the Colossal magnetoresistive (CMR) phenomenon4, Metal-insulator-transition (MIT) accompanied by a magnetic transition from paramagnetic (PM) to ferromagnetic (FM) state5, half-metallicity6, and tuneable in-plane and out of plane magnetic anisotropy7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' These properties are exploited for potential spintronics applications such as spin injection devices8, Magnetic tunnel junctions9–11, and magnetic storage devices (MRAMs)12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' In recent times, the perovskite-manganite systems are the ideal oxide candidates for developing superlattices, self-assembled nano-arrays13, nano-ribbons14, nano-wires, vertically aligned nanocomposite (VAN) thin films15–19, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' which offer enhanced Low-field magnetoresistance (LFMR), switchable magnetic anisotropy and for studying other interesting interface effects such as magnetic exchange bias20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Focus on growth dynamics is required to tune exclusive nano-architectures in the thin film as it offers additional handles to tailor its physical properties such as a high CMR %, high Curie & MIT temperature, high- temperature coefficient of resistance (TCR %), and enhanced magnetoresistive (MR) phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The manganite system is highly sensitive to external perturbations due to the strong connection between the spin-charge and lattice degrees of freedom21,22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' This poses a major challenge in obtaining epitaxial/patterned thin films for useful applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The pulsed laser deposition (PLD) technique has been extensively used to fabricate oxide- based manganite thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' This is because it offers good stoichiometric transfer of the target material onto the substrate in addition to deposition in an oxygen background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Various studies have been carried out to obtain epitaxial thin films by tuning the deposition parameters such as the oxygen partial pressure, substrate temperature, laser energy density, and repetition rate, affecting its growth and physical properties23,24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Additionally, the growth of the thin film is influenced by the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The strain offered by the substrate affects the surface morphology and microstructure of the manganite thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Different methodologies such as i) varying the substrates for different lattice matching25–27 (ii) choice of substrates with different crystallographic orientations with corresponding chemical terminations14 iii) varying the thickness of the thin films28, and iv) high-temperature annealing17 are adopted to tune the strain and morphology of the thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Therefore, thin films with unique morphology and long- range ordered nanostructures can be obtained by fine-tuning the growth parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Compared 3 to the previous works on VAN and other nanostructures of the popular manganite system La- Sr-Mn-O, we have observed a granular nanostructure and another distinct nanostructure with crossed-nano-rods in our thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' We have synthesized NSMO thin films using the PLD technique on single-crystal SrTiO3 (100) oriented substrates (STO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The effects of PLD parameters and annealing conditions on the surface morphology were investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Using SEM, AFM, and XRD techniques, the growth mechanism leading to a specific type of nano- structuring in the NSMO thin films is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Additionally, the morphology-driven changes in the temperature dependence of resistivity are investigated, and we observed a signature trend in the MIT corresponding to the particular morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' EXPERIMENTAL METHODS The NSMO thin films were fabricated using the PLD technique using a commercial NSMO pellet as the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Before deposition, SrTiO3 (STO) (1 0 0) single crystals substrate was cleaned by boiling in de-ionized(DI) water for 3 minutes, followed by ultra-sonication in DI water, acetone, and iso-propyl alcohol followed by rinsing in DI water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' With the water leaching procedure, the SrO terminations present in the substrate surface can be effectively dissolved and removed with DI at elevated temperatures > 60 oC followed by ultra-sonication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' A KrF Excimer laser source (λ = 248 nm) operated with a laser energy density of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='75 J/cm2 at 3Hz was used to ablate the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The films were deposited in an oxygen partial of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='36 mbar with substrate temperature fixed at 750 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' After deposition, the films were in situ annealed at 750 oC for 2h, and the PLD chamber was maintained with O2 background pressure of 0 to 1 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Further, the films were ex-situ annealed in a tube furnace at 950 oC in an oxygen atmosphere with a flow rate of ~ 20 sccm for 2h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The surface morphology of the thin films was examined using a Scanning electron microscope (SEM) from Carl Zeiss, crossbeam 340, and the images were collected in inlens-duo mode at 3-5 kV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Atomic force microscopy (AFM) was used for 2D and 3D visualization of the surface of substrates and the films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' XRD studies have been carried out at Engineering Applications Beamline, BL-02, Indus-2 synchrotron source, India using beam energy of 15 keV for the structural characterization of the films29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The Grazing incidence (GI) and ω-2θ scans were performed, and data were collected using the Dectris detector (MYTHEN2 X 1K) in reflection geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' In the GI-scan, the incident angle is kept fixed at ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5o, and the detector moves along the given 2θ range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The monochromatic high-resolution mode of the beamline was used, 4 keeping the beam energy at 15 keV (λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='826 Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The peaks were indexed with reference to the ICDD data30 (ICDD number - 01-085-6743) RESULTS AND DISCUSSION: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Morphology studies of the nanostructured thin films: The NSMO thin films prepared under the above conditions possessed two prominent surface morphology – granular and rod-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Two representative films with granular nanostructure and crossed-rod nanostructure were chosen to study the physical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' These two systems will be referred to as NS-G and NS-R, where NS stands for NSMO thin film, and ‘G’/’R’ stands for the type of morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The thickness of NS-G and NS-R thin films is determined to be ~ 100 nm by cross-sectional SEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Figure 1(a) shows the SEM image of NS-G thin films with granular morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The film is uniformly covered with multifaceted grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Figure 1(c) (i) shows the average grain size 0 20 40 60 80 100 0 100 200 300 400 Frequency (count) Grain size (nm) Frequency count Lognormal fitting Avg grain size = 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='9 nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='1 nm 0 200 400 600 800 1000 0 100 200 300 400 500 Frequency (count) Length of rod (nm) Frequency count Lognormal fitting Avg length of rod = 188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='7 nm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='7 nm 0 20 40 60 80 100 120 0 100 200 300 Frequency (count) Width of rod (nm) Frequency count Lognormal fitting Avg width of rod = 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='6 nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='2 nm c) (i) (ii) (iii) Figure 1: Scanning electron microscopy images of NSMO thin films on STO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' a) NS-G - granular morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' b) NS-R – self-aligned-crossed-Nano-rod-morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' c) The histograms illustrate the grain size calculation for NS-G and NS-R thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' (i) Average grain size estimated for NS-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' (ii) Average rod length estimated for NS-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' (iii) Average rod width estimated for NS-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 100 nm100 nm5 estimated to be 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='9 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Figure 1(b) shows the SEM image of NS-R thin films with unique surface morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The thin film surface is uniformly covered with nano-rods crossed at right angles embedded in a matrix of NSMO containing square/rectangular pits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' In NS-R, the average rod length is estimated to be 188 nm with an average width of 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='6 nm, as shown in the Figure 1(c), (ii) and (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Further, AFM measurements have been carried out on the NS-G and NS-R thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The 2D and 3D AFM scan in Figure 2 show columnar/island-type features in the NS-G thin film and crossed-rod features in the NS-R thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Structural analysis of the thin film: The bulk NSMO compound has an orthorhombic crystal structure belonging to the Pbnm space group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' In the pseudo-cubic (pc) representation, the unit cell parameter is given by apc ≈ c / 2 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='849 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The substrate STO has a cubic crystal structure with a lattice constant aSTO = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='905 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' NSMO grown on the STO substrate experiences a tensile strain due to the lattice mismatch Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 2: a), b), 2D and c), d) 3D AFM scans of grain-type NSMO thin film (left) and rod- type sample NSMO thin film on STO substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' a) b) c) d) Figure 2: a), b), 2D, and c), d) 3D AFM scans of grain-type NSMO thin film (left) and rod-type sample NSMO thin film on STO substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' nm 20 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='8 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The GI-XRD and high-resolution XRD (HR-XRD) reflections of the films are shown in Figure 3(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The presence of multiple reflections in the GI-XRD scan of NS-G in Figure 3(a) reveals that the granular thin film is polycrystalline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' In NS-R, the reflections of NSMO are absent, as seen in Figure 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' This may be due to its out-of-plane orientation with respect to the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' At the high 2θ angle ≈ 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='1o , the (3 1 0) STO plane gets aligned, resulting in high STO (3 1 0) reflection along with the NSMO (2 4 0) peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' This shows that the films are well-oriented, mirroring the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Though NS-G is oriented, the crystallographic difference between NS-G and NS-R is attributed to the type of nano-structuring in the films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Figure 3: GI-XRD scans of NSMO thin films a) NS-G b) NS-R indexed using ICDD data (* - STO peaks) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Effect of ex-situ annealing on morphology: To gain insight into the type of growth across these films, we compare the morphological changes in the in-situ annealed and ex-situ annealed samples in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' In granular thin films, no significant changes have been observed after in-situ and ex-situ annealing, apart from a minor increase in grain size, as seen in Figure 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Whereas the sample with rod-type morphology obtained after ex-situ annealing in Figure 4(d) exhibits facetted droplets embedded in a matrix with rectangular holes and rod features in the in-situ annealed 5 10 15 20 25 30 35 40 45 50 1E+02 1E+03 1E+04 5 10 15 20 25 30 35 40 45 50 1E+02 1E+03 1E+04 (310) (240),(332) (040), (224) (024), (132) (200) (220), (004) (202), (022) (020), (112) (002), (110) Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' units) 2\uf071\uf020\uf020(deg) NS-G (a) (b) (100) (240),(332) (310) (200) 2\uf071\uf020\uf020(deg) NS-R 7 case, Figure 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' It is evident that once the initial growth mode is set, the ex-situ annealing aids in increasing grain size, relieving the strain in thin films in addition to decreasing oxygen defects in NSMO thin films23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' We inspect the HR-XRD scans of the NS-G and NS-R thin films in the in-situ and ex-situ annealed cases to verify this claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Figure 5(a) and (b) show the HR-XRD scan performed over a range of 2θ (10o – 40o) for the films NS-G and NS-R after in-situ and ex-situ annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' It is observed that the (0 0 4) NSMO peak is absent in the in-situ annealed NS-G thin film, whereas upon ex-situ annealing, NS-G shows improved texturing with the (0 0 4) NSMO peak close to the (0 0 2) substrate peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' In the case of NS-R, along with the substrate’s (002) reflection, corresponding (0 0 l) pseudo-cubic reflections from NSMO are present with significant intensity even in the in-situ annealed condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Further, as we compare HR-XRD scans of NS-G and NS-R after ex-situ annealing, the NS-R thin film has increased relative intensity compared with the NS-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' a) b) Pristine thin film after in-situ annealing Further upon ex-situ annealing a) b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 3: Effect ex-situ annealing on NSMO thin films with a) granular b)rod-type surface morphology c) d) Figure 4: Illustration of the effect of ex-situ annealing on NSMO thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' a) SEM image of in-situ annealed granular thin film b) SEM image of the granular thin film after ex-situ annealing c) SEM image of the thin film after in-situ annealing showing rods and squared blocks in the encircled regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' d) SEM image of the same thin film after ex-situ annealing showing crossed-rod type morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 100nm100 nm100 nm100nm8 Figure 5: High resolution-XRD scan of NSMO thin films around the STO-(200) reflection inset: fine scan of NSMO (004) of NS-R sample showing double peaks – P1 and P2 (* - STO peaks) Therefore, NS-R is highly oriented and more crystalline, which can be attributed to its epitaxial nature of growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Additionally, the HRXRD scan of NS-R thin films after ex-situ annealing shows a doublet feature at its (004) reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' A high-resolution fine scan was performed on the NS-R thin film to confirm the double peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Referring to the literature, we found that a similar doublet feature has been reported due to strain relaxation in PSMO thin films on STO substrate31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' By fitting the peaks using the pseudo-Voigt function, as shown in figure S1 of supplementary information, the peaks were de-convoluted to evaluate the out-of- plane lattice parameter (tabulated in table T1 – supplementary information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The first peak was at 2θ=24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='88o with a c-lattice constant of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='66 Å, and the second peak was at 2θ=24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='97o with a c-lattice constant of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='63 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The reduction in the c-lattice constant of the second peak shows that there is compression of the lattice along the c-axis because of the tensile strain experienced by the thin film due to the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Such a splitting in the peak was absent in films of thickness < 80 nm, indicating that this double peak is due to partial strain relaxation in the thicker film initiated by ex-situ annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Thus, from the detailed XRD studies and discussions in the previous section, it is inferred that difference in initial-growth mode, and subsequent ex-situ annealing has prominently tuned the resulting surface morphology of the NSMO thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The granular thin film NS-G has multiple orientations similar to a polycrystalline system, whereas NS-R shows improved crystallinity and orientation mirroring the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The parameters affecting the initial growth are discussed in the upcoming section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 22 24 26 28 10 1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 22 24 26 28 10 1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 b) Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' units) 2\uf071\uf020(deg) Ex-situ annealed In-situ annealed (200) (004) NS- R P1 a) NS- G 2\uf071\uf020(deg) Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' units) Ex-situ annealed In-situ annealed (200) (004) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='8 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='2 1000 10000 P2 2\uf071\uf020(deg) 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Effect of PLD parameters in tuning the morphology: PLD Parameters like laser energy density, oxygen partial pressure, and substrate temperature highly influence the type of growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Changes in these parameters lead to variations in the energy of the ad-atoms deposited on the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' To understand the role of O2 partial pressure and laser energy density during the deposition, we have prepared NSMO thin films by varying these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Post deposition, the films were in-situ annealed at 750 oC for 2h in an oxygen background pressure of 1 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Ex-situ annealing was carried out subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Figure 6 presents the morphology of films deposited under different laser energy densities varied from 1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='75 J/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' During deposition, the oxygen partial pressure and substrate temperature were maintained at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='36 mbar and 750 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Figure 7 presents the morphology of films obtained at different oxygen partial pressure of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='3mbar, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='4 mbar, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5 mbar while the laser energy density and substrate temperature were maintained at 1 J/cm2 and 750 oC during deposition, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' We found that changes in oxygen partial pressure and laser energy density did not influence the surface morphology, as both type of morphologies have been observed in different deposition runs with the same parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Further, as we have obtained granular and rod-type films for the same substrate temperature of 750 oC, the role of substrate temperature is also ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Thus, irrespective of changes in the parameters mentioned above, thin films of either granular or crossed-rod nanostructure were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Therefore we suspect the a) b) c) d) e) f) 1 J/cm2 1 J/cm2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5 J/cm2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5 J/cm2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='75 J/cm2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='75 J/cm2 Figure 6: The SEM images of NSMO thin films with granular morphology (a), (b), and (c) and rod morphology from (d), (e), and (f) obtained at corresponding laser energy density- 1 J/cm2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5 J/cm2, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='75 J/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 200 nm200nm200nm200nm200nm200nm10 substrate and the strain it offers to plays a vital role in altering the growth mode of the thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Effect of miscut angle in tuning the morphology: The commercial STO substrates used here are one-sided polished, and their surface was found to have a miscut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' In commercially purchased wafers, the occurrence of a miscut in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='05o-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='3o is well known and unavoidable due to mechanical cutting and polishing of single crystal STO wafers14,32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' In Figure 7(a), the as-received STO substrate, after cleaning, shows clear terrace features in the AFM scan, confirming the presence of miscut on the substrate surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' In a given wafer, the miscut can be in-plane or out-of-plane or both (some works refer to this as miscut directions φ and θ instead of in-plane and out-of-plane, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The miscut angle and direction can alter the growth mode as the lattice strain is anisotropic along the substrate surface and step edges6, thus resulting in different surface morphology by forming anisotropic structural domains33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Several works are available in literature 33–35 on the growth of manganite thin film on STO substrate with miscut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' These reports claim that the value of the miscut angle and appropriate adjustments in growth conditions can control the number of structural domains in the thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' As we have already ruled out the possibility of growth conditions influencing the resulting morphology, we tried to evaluate the value of miscut present in our STO substrates to see if it has affected the resulting morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' a) b) c) d) e) f) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='3 mbar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='4 mbar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5 mbar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5 mbar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='4 mbar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='3 mbar Figure 7: The SEM images of NSMO thin films prepared at oxygen partial pressure of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='3 mbar, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='4 mbar, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5 mbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' a), b), and c) are granular NSMO thin, and films with rod morphology are shown in d), e), and f) at corresponding oxygen partial pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 200 nm200nm200nm200 nm200 nm200 nm11 To determine the value of miscut present in the substrates, we have followed the XRD- protocol from literature36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' This was carried out in a BRUKER D8, Lab source XRD setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' According to the protocol, a low incident angle (~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='2o) rocking-scan was initially performed to ensure that the sample was aligned with the X-ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' This was done to optimize the angle of the sample holder, and the offset in the 2θ value (~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='4o) was noted as ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Following that, a rocking scan was performed around the (200) peak of STO (46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='483o), and phi & chi scans were done to orient the wafer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Further, the rocking scan around the (200) peak of STO was repeated, fixing the X-ray tube position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Finally, a detector scan was performed around the (200) peak of STO and this time the offset in 2θ was noted as ζ’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The difference δζ, between ζ and ζ’, gives the estimate of miscut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Next, the sample was rotated by 90o, and the scans mentioned above are repeated in the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The difference between the offsets obtained this time was denoted as δξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Finally, the out-of-plane miscut angle was evaluated using equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' After determining miscut on various STO wafers, we found that the value out of plane miscut angle varies from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='13o up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='48o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 𝜃𝑜𝑢𝑡−𝑜𝑓−𝑝𝑙𝑎𝑛𝑒 = 𝑎𝑟𝑐𝑡𝑎𝑛 √tan2(𝛿𝜁) + tan2(𝛿𝜉) (1) Table 1 : This table illustrates the morphology of NSMO thin films obtained on STO substrates with different values of miscut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Sample Miscut angle Granular Morphology Sample Miscut angle Rod type Morphology NS-G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='48 o NS-R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='31 o G1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='31 o R1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='19 o G2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='30 o R2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='25 o Table T1 : This table illustrates the morphology of NSMO thin films obtained on STO substrates with different values of miscut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 100nm100nm100nm100 nm100nm100nm12 We see from table T1 that, both granular and rod-type morphology was observed on substrates with miscut angle varying from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='13o up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='48o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Sample G1 with granular morphology and NS-R with rod-type morphology, possess the same miscut angle of ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='3o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' This is very interesting, as the value of the miscut angle has not influenced the altered growth modes present in our samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Therefore to comprehend the resulting morphology, we have further investigated the type of growth occurring on the terraced surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Thin film growth on the terraced surface: A miscut on the substrate is useful for epitaxial thin films37 as the steps and terrace edges act as nucleation centres and result in a step flow growth mode38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' But the actual processes governing the step-flow growth are more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The basic parameters driving this type of growth are the coefficient of diffusion and the height of the Ehrlich-Schwoebel (ES) barrier39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The diffusion of the adatoms on the surface and their incorporation into the crystal structure govern the formation of different morphologies at the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Additionally, the ES barrier at the terrace/step edges introduces an asymmetry in the potential energy at the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' An adatom, reaching the terrace, either nucleates or descends into the step depending on the ES barrier height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Similarly, an adatom reaching below the step experiences an inverse step barrier which prevents the particles from attaching to the step from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' If the barrier height is appropriate, ad-atoms can properly attach themselves to the step edges resulting in a step flow growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' However, the existence of the barrier makes the growth on the stepped surfaces highly unstable resulting in modified surface features such as step meandering, nano-columns/wire formation, spirals/mound formations, and faceted pits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' In a recent work by Magdalena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='40, a simulation using the Cellular Automaton model in (2+1)D gave rise to different patterns of surface morphology on vicinal surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' According to the simulation, different processes occurred depending on the values assigned to the barrier height at step edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The adatom could either attach to the step to build the crystal by jumping/ descending at the step edge or scatter away from the barrier resulting in the formation of islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' For a fixed adatom flux, diffusion of adatoms takes place on the vicinal surface, and probabilities are assigned for each of the processes mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Depending on the probability value, various surface patterns were simulated for three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' In case (i), for a high ES barrier, the three-dimensional surface formation resulted in square/rectangular islands following the cubic lattice symmetry at the middle of the terraces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' In case (ii), with a reduced 13 ES barrier height, more atoms were trapped at the top of the step, and a new pattern of nanocolumns emerged consisting of cubic formations with deep narrow cubic pits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Finally, in case (iii), when the height of the barrier was adjusted such that the probability of the adatoms descending the step is equal/of the same order as the probability of the adatoms jumping up to the step from below, it resulted with nano-wire or a columnar growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Further, the presence of additional local sinks that alters the potential barrier also resulted in nano-columns/islands at random positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Thus, we can understand that our resulting granular morphology on the miscut STO substrate is precisely similar to the surface morphology resulting from the case (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' In the STO susbtrate, the presence of disoriented terraces and improperly removed SRO terminations may have altered the ES barrier resulting in local sinks at the substrate surface, thus resulting in the island/columnar growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Finally, the surface morphology of the NS-R thin film resembles the As received substrate after cleaning After treatment for TiO2 termination a) b) c) d) Figure 8: AFM scan of the STO substrate a) as-received commercial substrate after cleaning b) the same substrate after TiO2 termination obtained after heat treatment method with a step height of ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='4 Å (one-unit cell height of STO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' c), d) NSMO thin films grown on the corresponding substrates 10-3nm 400 um um 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='2 010-3nm 600 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='0 um wn100 nm100 nm14 morphology they obtained in case (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' This fact can be verified from a close inspection of the surface of NS-R at high magnification in Figure 9(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The surface morphology clearly shows layer-by-layer growth with squared pits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Further, in attempting to reduce the local sinks, NSMO thin films were synthesized on pure TiO2 terminated substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The substrates are treated with DI water and then annealed at high temperatures according to the protocol for TiO2-termination41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The treatment produced clear step and terrace characteristics in the substrate, as observed in the AFM scan shown in Figure 8(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' NSMO thin films were deposited on these substrates and, subsequently, ex-situ annealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The SEM imaging revealed that they exhibited similar rod-type morphology where the rods are self-aligned and crossed at right angles embedded in a matrix of NSMO with rectangular features, shown in figure 8(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' This procedure was repeated on several TiO2 - terminated STO substrates, and we could reproduce the same morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' This is because the complete removal of SrO assures the absence of local sinks and suppresses the island/columnar growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' However, rods in the thin film are believed to arise from droplets deposited due to high laser energy density (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='75 J/cm2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' This is verified in the SEM images of in-situ annealed NSMO thin film shown in Figure 4(b), where the droplets are elongated into rods upon ex-situ annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Figure 9: SEM images of NSMO thin films under high magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=" a) NSMO thin film grown on as-received, cleaned STO substrate deposited b) NSMO thin film grown with laser fluence on TiO2 terminated STO substrate Lastly, to obtain smoother films, we have synthesized NSMO thin films on fully TiO2 terminated STO substrate at low laser energy density (1 J/cm2), reducing droplets' density." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' As expected, we obtained thin films with reduced density of rods with the same type of morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The SEM image of the film is shown in Figure 9(b), free of nano-rods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The films have rectangular faceted pits, and layer-by-layer growth is evident through the holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' a) b) 100 nm100 nm15 Therefore the ES barrier plays a significant role in vicinal surfaces and can result in the spontaneous ordering of adatoms resulting in unique surface nanostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Thus we emphasize that when films are grown on a commercial substrate, the resulting morphology can be either granular or rod-type depending on the potential energy landscape that depends upon a wide range of parameters, including the size, shape of terraces, and type of terminations present at the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Electrical-transport measurements: The nanostructure plays a vital role in the transport behaviour of a manganite thin film system30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' To understand the transport behaviour of the nanostructured NSMO thin films, the resistivity measurements are carried out using the standard 4-probe geometry 42 and plotted as a function of temperature in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' It is observed that the granular film NS-G has higher resistivity as compared to NS-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Both thin films, NS-G and NS-R, exhibit the insulator-to- metal transition (MIT), and the transition temperature TMIT is found to be 147 K for sample N- G and 135 K for NS-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The transition into the metallic regime is sharper in the case of NS-R compared to NS-G thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The electrical transport behaviour has been analysed using different theoretical models and fitted in the corresponding temperature regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The best fit in each region is chosen based on the reduced \uf0632 value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 𝜌 (𝑇) = 𝜌𝑅 𝑇𝑒𝑥𝑝 (𝐸𝑎 𝜅𝐵𝑇 ⁄ ) …(2) 𝜌 (𝑇) = 𝜌0 𝑒𝑥𝑝 (𝑇0 𝑇 ⁄ ) 1 4 ⁄ …(3) 𝐸ℎ𝑜𝑝𝑝𝑖𝑛𝑔 = 𝜅𝐵 𝑇𝑜 1 4 ⁄ 𝑇 3 4 ⁄ 4 …(4) The high-temperature insulating phase is studied using the small polaron hopping (SPH) model and the variable range hopping (VRH) mechanism given by equations (2) and (3), and hopping energy is calculated from equation (4) 42,43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The VRH model better fits the high-temperature region (≈ 195 K to 300 K) for both films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The hopping energy in the case of NS-G is 128 meV and 125 meV for NS-R, in agreement with the order of value reported for manganite thin films (~100 meV) 43,44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The resistivity in the metallic region below TMIT is generally fitted with an empirical equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' At low temperatures, in addition to the temperature-independent scattering effects from defects and grain boundaries (GBs) (ρo), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=', 16 scattering effects due to electron-electron (ρ2), electron-magnon (ρ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5) and electron-phonon (ρP) dominate along with the strong correlation effects (ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' A low-temperature resistive upturn is observed below 50 K in both films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' In figure S3 of supplementary information, the resistive upturn in the low-temperature region from 4 K up to 60 K is fitted using equation (6), which considers all the scattering mechanisms mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' An enhanced resistive upturn is observed at low temperatures in NS-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' This is due to the enhanced GB-scattering effect and the contribution from other scattering mechanisms at low-temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The contributions from different scattering mechanisms are analysed, and the values are tabulated in supplementary information Table T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The intermediate temperature regime from 90 K to 134 K in the ferromagnetic-metallic state is fitted using the equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The addition of the polaronic term to the resistivity gives a better fitting in this region as theoretical models claim the formation of polaron near the MIT46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 𝜌 (𝑇) = 𝜌𝑜 + 𝜌𝑚𝑇𝑚 (5) 0 50 100 150 200 250 300 0 5 10 15 20 25 30 0 100 200 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5 0 100 200 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='6 46 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='0 \uf0bb \uf044 \uf044 T \uf072 \uf072\uf02f\uf020\uf072\uf028\uf054\uf03d\uf033\uf030\uf030\uf04b\uf029 Temperature (K) NS-G NS-R Linear fit from 110K to 125K- NS-G Linear fit from 110K to 125K- NS-R 33 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='1 \uf0bb \uf044 \uf044 T \uf072 (b) Resistivity - NS -G VRH fit FM-metallic fit Low-temp uptrun TIMT \uf0bb 147 K \uf072\uf020(\uf057\uf02dcm) Temperature (K) (a) (c) Resistivity - NS-R VRH fit FM-metallic fit Low-temp uptrun TIMT \uf0bb 135 K \uf072\uf020(\uf057\uf02dcm) Temperature (K) Figure 10: a), b) Resistivity vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' temperature curve of NSMO thin films – NS-G and NS-R showing insulator to metal transition with decreasing temperature and fitted according to theoretical models in different temperature regimes c) Normalized resistivity plot of NS-G and NS-R thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Inset: Plot of variation of TCR with respect to temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 17 An interesting feature is observed in the resistivity plots of the NS-G and NS-R thin films apart from the low-temperature resistive upturn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' In Figure 10(c), the resistivity of both the thin films (NS-G and NS-R) has been normalized with their resistivity at 300 K, and a linear fitting in the metallic region below TMIT (110 K to 125 K) is carried out to determine the slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The resistivity slope of samples with rod morphology differs from samples with granular morphology up to an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The increase in slope value below the transition temperature indicates the sharpness of the resistive transition for the samples with rod morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' This characteristic increase in slope up to an order is evident in all our thin films with rod-type morphology (see supplementary figure S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' To characterize the sensitivity of resistance with respect to changes in temperature, the temperature-coefficient of resistance (TCR) has been evaluated using equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' It was found that NS-R has a higher value of TCR %, ~ 12 %, compared to NS-G with TCR %, ~ 7 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Additionally, the samples with rod morphology were found to have enhanced TCR% (supplementary information figure S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' To comprehend this result, we discuss the effect of GBs on the conduction mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The manganite system undergoes a disorder-induced phase transition from PM to FM state with decreasing temperature21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Due to phase co-existence during the transition, the conduction channel is presumed to have filamentary FM paths in the PM matrix 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Conduction takes place through the percolation of current across the well-connected FM regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' In addition to the FM filamentary path, the GBs also play a significant role in the conduction mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' We refer to Verutruyen et al.’s 48 work which explores the effect of a single GB in the La-Ca-Mn-O (LCMO) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' They showed that the resistivity falls sharply at the transition temperature when measured on a single grain of LCMO (free of GBs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' However, when measured across a single GB, the resistivity initially decreased, followed by a broad resistive feature near the transition temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Thus, in a granular system, though the conduction takes place through the percolation paths of well-connected FM regions, the GBs cause increased resistivity due to increased spin-dependent scattering across the GB47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The above explanation is consistent with our results, where the thin film with granular morphology (NS-G) shows a broad resistive transition below the transition temperature with reduced TCR 𝜌 (𝑇) = 𝜌𝑜 + 𝜌2𝑇2 + 𝜌4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5𝑇4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5 + 𝜌𝑃𝑇5 + 𝜌0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5𝑇0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5 (6) 𝜌 (𝑇) = 𝜌𝑜 + 𝜌2𝑇2 + 𝜌4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5𝑇4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5 + 𝜌𝑃𝑇5 + 𝜌0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5𝑇0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5 + 𝜌7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5𝑇7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5 (7) 𝑇𝐶𝑅 % = 1 𝜌 ( 𝑑𝜌 𝑑𝑇) x 100 (8) 18 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' If the connectivity is enhanced between the grains, a sharper decrease in the resistivity can occur in the metallic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Remarkably, we observe that all of our thin films with rod- morphology show sharp resistive transition near MIT irrespective of the thickness of the film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Thus, this nanostructure aids improved conduction in the FM metallic phase, leading to the sharp resistive transition with enhanced TCR % comparable to that of a highly-crystalline system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Attempts to enhance the TCR % have been carried out by doping with elements such as Ag, as high TCR % is required for applications in sensors and infrared detectors49,50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' These elements precipitate as nanocomposite in the manganite system and improve the conductivity, leading to a sharper resistive transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' However, in our study, we have substantiated that the enhancement of TCR % is possible with proper tuning of the nanostructured morphology of thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' CONCLUSION: In conclusion, the PLD-grown NSMO thin films were observed to have two prominent surface morphologies – granular and crossed-nano rods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The metal-to-insulator transition (MIT) temperature, TMIT, was found to be 147 K for a granular NSMO (NS-G) thin film and 135 K for a thin film with crossed-rod morphology (NS-R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The nature of the resistive transition is broad in the former, whereas the latter exhibits a sharp MIT feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The temperature coefficient of resistance (TCR) was evaluated, and NS-R thin film has a higher value of TCR %, ~ 12 %, compared to NS-G with TCR % ~ 7 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Additionally, we have observed that all the films with rod-type morphology exhibit a significant enhancement in TCR% up to one order of magnitude compared to the granular thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Thus, we have demonstrated that TCR % can be enhanced with proper tuning of the nanostructures in thin films, which is relevant for technological applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The reason for such nano-structuring is explored in great detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' It was found that parameters like laser energy density, O2 partial pressure, and the substrate miscut angle had minimal effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' At the same time, the difference in the potential landscape of the Ehrlich-Schwoebel (ES) barrier is believed to play a vital role in the growth dynamics of the films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Films grown with reduced laser energy density (1 J/cm2) on the TiO2 terminated substrates exhibited highly reproducible layer-by-layer growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' This substantiates the presence of reduced local sinks and ES barrier height, resulting in epitaxial growth of NSMO thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Therefore, a fine-tuning of a wide range of parameters, including strain and surface terminations, is required to obtain a fine control of the ES barrier that influences the growth process of thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' This paves the way for investigation into the role of the ES barrier in manganite thin film growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Using RHEED and in-situ STM techniques, a 19 few groups have already attempted to experimentally determine the value of the ES barrier on SrTiO3 substrates for the growth of La-Ca-Mn-O manganite system51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' It would be interesting to explore the relationship between the value of the ES-barrier and the type of morphology experimentally in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Author contributions The division of work is as follows: NSMO thin film samples were prepared by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' SEM imaging was carried out by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='A, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' AFM measurements were carried out by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' XRD measurements were carried out by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='S, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='R, PG, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Magneto-transport measurements were carried out by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='M and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Analysis were done by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='M, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='A, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Writing was carried out by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='M, and all authors discussed the results and commented on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='K and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' supervised this research work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Conflict of interest: The authors declare no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Acknowledgments One of the authors (R S Mrinaleni) would like to acknowledge the Department of Atomic Energy, India for the provision of experimental facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' We thank UGC-DAE CSR, Kalpakkam node, for providing access to magnetic and magnetotransport measurement systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The authors are grateful to RRCAT, Indore, for beam line facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Funding statement: One of the authors (R S Mrinaleni) would like to acknowledge the funding support from the Department of Atomic Energy, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' References: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Dagotto.' 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anisotropy at the interface of an oxide heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' B 104, (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=', Lindfors-Vrejoiu, I.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='67 Sr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='33 MnO 3 (LSMO) Thin Films Integrated on Mica Substrates toward Flexible Spintronics and Electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' ACS Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Interfaces 10, 42698–42705 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 27.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='ceramint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Jin, F.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='jmat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Gianfrancesco, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=', Tselev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=', Baddorf, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=', Kalinin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' V & Vasudevan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The Ehrlich–Schwoebel barrier on an oxide surface: a combined Monte-Carlo and in situ scanning tunneling microscopy approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Nanotechnology 26, 455705 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' SUPPLEMENTARY INFORMATION I- Deconvolution of NSMO (004) reflection: Figure S1: The double peak in the HR-XRD scan of NS-R thin film is confirmed by a HR-fine scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The individual peak positions are noted as the centre of the fitted peaks P1 and P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Table ST1: The table illustrates the values of c-lattice parameters evaluated from the (004) NSMO reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='4 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='6 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='8 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='2 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='4 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='6 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='0x10 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='0x10 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5x10 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='0x10 5 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' units) 2\uf071\uf020(deg) NS-R - fine scan Fit Peak 1 Fit Peak 2 Cumulative Fit Peak Sample 2θ for (004) reflection (o deg) Calculated c-lattice parameter (Å) NS-G 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='05o 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='61 NS-R 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='88 o – P1 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='97 o – P2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='66 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='63 25 II- Transport studies on NSMO thin films Three samples with granular morphology G-A, G-B, G-C, and rod morphology R-A, R-B, R- C, were selected and their resistivity was measured using 4-probe technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The normalized- resistivity plot for the selected NSMO thin films are shown Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The value of resistivity is different across the NSMO thin films, since they are deposited under slightly different PLD conditions but all of them exhibited MIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Observing the nature of MIT transition in these selected samples, G-A, G-B, G-C with granular morphology have a broad resistive transition below their MIT temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The samples R-A, R-B, R-C with rod-morphology show a sharp resistive transition in the FM-metallic state below their MIT temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The value of slope is evaluated from the linear fit in the metallic region and it shows that samples with rod-type morphology have increased slope up to one order as compared to the granular samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Temperature coefficient of resistance (TCR) is evaluated for these films and it is found that samples G-A, G-B, G-C have peak-TCR % of 5 %, 4 %, and 8 % at 105 K, 77 K, and 121 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' An enhanced TCRpeak % is obtained for samples with rod-morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The samples R-A, R-B, R-C have peak-TCR % of 21 %, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5 %, and 18 % at 98 K, 80 K, and 100 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 26 Figure S2: Plots of normalized resistivity vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' temperature of NSMO films with granular and rod-type morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' (a),(c),(e): Samples with granular morphology G-A, G-B, G-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' (b),(d),(f): Samples with rod morphology R-A, R-B, R-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' A linear fit in the FM-metallic region give the rate of change of resistivity with respect to temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' III- Low-temperature studies on NSMO thin films – NS-G and NS-R To study the low-temperature transport across the thin films with different morphology, the plot of low- temperature resistivity of the granular thin film NS-G and rod-type thin film NS-R is shown in figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' An enhanced low-temperature resistive upturn is observed in NS-G from figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Using the low- temperature transport equation the resistivity data is fit and the fitting parameters are summarized in table ST2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The first term, ρo which represents the contribution from grain-boundary (GB) scattering is 0 50 100 150 200 250 300 0 2 4 6 8 0 50 100 150 200 250 300 0 30 60 90 0 50 100 150 200 250 300 0 2 4 6 8 10 12 14 16 0 50 100 150 200 250 300 0 30 60 90 120 0 50 100 150 200 250 300 0 2 4 6 8 10 12 14 0 50 100 150 200 250 300 0 5 10 15 20 25 15 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='0 \uf0bb \uf044 \uf044 T \uf072 Sample: G - A Linear fit in FM-metallic region \uf072\uf02f\uf020\uf072\uf028\uf054\uf03d\uf033\uf030\uf030\uf04b\uf029 Temperature (K) TCRpeak % \uf0bb 5 % \uf0bb TCRpeak % \uf0bb 21 % 74 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='5 \uf0bb \uf044 \uf044 T \uf072 Sample: R - A Linear fit in FM-metallic region \uf072\uf02f\uf020\uf072\uf028\uf054\uf03d\uf033\uf030\uf030\uf04b\uf029 Temperature (K) (b) TCRpeak % \uf0bb 4 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='28 \uf0bb \uf044 \uf044 T \uf072 \uf020\uf072\uf02f\uf020\uf072\uf028\uf054\uf03d\uf033\uf030\uf030\uf04b\uf029 Temperature (K) Sample: G - B Linear fit in FM-metallic region (c) TCRpeak % \uf0bb 14 % 76 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='6 \uf0bb \uf044 \uf044 T \uf072 \uf072\uf02f\uf020\uf072\uf028\uf054\uf03d\uf033\uf030\uf030\uf04b\uf029 Temperature (K) Sample: R - B Linear fit in FM-metallic region (d) TCRpeak % \uf0bb 8 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='35 \uf0bb \uf044 \uf044 T \uf072 Sample: G - C Linear fit in FM-metallic region \uf072\uf02f\uf020\uf072\uf028\uf054\uf03d\uf033\uf030\uf030\uf04b\uf029 Temperature (K) (e) TCRpeak % \uf0bb 18 % 20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='1 \uf0bb \uf044 \uf044 T \uf072 \uf072\uf02f\uf020\uf072\uf028\uf054\uf03d\uf033\uf030\uf030\uf04b\uf029 Temperature (K) Sample: R - C Linear fit in FM-metallic region (a) (f) 27 found to be higher by more than one-order in NS-G as compared to NS-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' This is expected as NS-G has a granular morphology and increased contribution from GB scattering affects the transport mechanism even at low-temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Additionally, ρo’s value is higher by orders of magnitude as compared to the other coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' This shows that GB scattering effects dominate the transport mechanism compared to other contributions to the electronic transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' Figure S3: Low-temperature resistive up-turn is observed in the NSMO thin films NS-G and NS-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' The temperature regime from 4 K up to 60K is fit using the low-temperature transport equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content=' 0 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='10 0 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='0030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='0035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='0040 Resistivity - NS-G Low-temp uptrun TIMT \uf0bb 147 K \uf072\uf020(\uf057\uf02dcm) Temperature (K) TIMT \uf0bb 135 K Resistivity - NS-R Low-temp uptrun \uf072\uf020(\uf057\uf02dcm) Temperature (K) Sample 𝝆𝒐 𝝆𝟐 𝝆𝟒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='𝟓 𝝆𝑷 𝝆𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='𝟓 R2 (%) NS-G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='09272 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='16E-5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='00E-9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='33E-10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='0038 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='99 NS-R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='00305 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='53E-7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='90E-11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='90E-12 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='38E-5 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} +page_content='99 Table ST2: The table illustrates the values of coefficients of low-temperature transport after fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE5T4oBgHgl3EQfQg5R/content/2301.05513v1.pdf'} diff --git a/8dFLT4oBgHgl3EQfBS4r/vector_store/index.pkl b/8dFLT4oBgHgl3EQfBS4r/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..91b80afb99101846b3c1990609f212f25fc8dc5f --- /dev/null +++ b/8dFLT4oBgHgl3EQfBS4r/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8f57cba4ab03adcbbede5f87aaa2d6ada1897cdcab2d2d594e31a75e551e5e6c +size 85657 diff --git a/A9AyT4oBgHgl3EQf3_rL/vector_store/index.faiss b/A9AyT4oBgHgl3EQf3_rL/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..9239873ba06cefb1ed296e405c36660832a506f5 --- /dev/null +++ b/A9AyT4oBgHgl3EQf3_rL/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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a/C9FQT4oBgHgl3EQfPDYj/content/tmp_files/2301.13277v1.pdf.txt b/C9FQT4oBgHgl3EQfPDYj/content/tmp_files/2301.13277v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..aeef2a69b7e0e6b16173a8c25cb1e81895c1103b --- /dev/null +++ b/C9FQT4oBgHgl3EQfPDYj/content/tmp_files/2301.13277v1.pdf.txt @@ -0,0 +1,1080 @@ +arXiv:2301.13277v1 [cond-mat.soft] 30 Jan 2023 +Transport properties in liquids from first principles: the case of +liquid water and liquid Argon +Pier Luigi Silvestrelli +Dipartimento di Fisica e Astronomia “G. Galilei”, +Universit`a di Padova, via Marzolo 8, I-35131 Padova, Italy +(Dated: February 1, 2023) +Abstract +Shear and bulk viscosity of liquid water and Argon are evaluated from first principles in the Den- +sity Functional Theory (DFT) framework, by performing Molecular Dynamics simulations in the +NVE ensemble and using the Kubo-Greenwood equilibrium approach. Standard DFT functional is +corrected in such a way to allow for a reasonable description of van der Waals (vdW) effects. For +liquid Argon the thermal conductivity has been also calculated. Concerning liquid water, to our +knowledge this is the first estimate of the bulk viscosity and of the shear-viscosity/bulk-viscosity +ratio from first principles. By analyzing our results we can conclude that our first-principles sim- +ulations, performed at a nominal average temperature of 366 K to guarantee that the systems is +liquid-like, actually describe the basic dynamical properties of liquid water at about 330 K. In +comparison with liquid water, the normal, monatomic liquid Ar is characterized by a much smaller +bulk-viscosity/shear-viscosity ratio (close to unity) and this feature is well reproduced by our first- +principles approach which predicts a value of the ratio in better agreement with experimental +reference data than that obtained using the empirical Lennard-Jones potential. The computed +thermal conductivity of liquid Argon is also in good agreement with the experimental value. +1 + +I. +INTRODUCTION +Transport properties are among the most important and useful features of condensed- +matter systems, particularly for characterizing the dynamical behavior of liquids, since they +play an important role in many technical and natural processes. Therefore their estimate +represents one of the most relevant goal of Molecular Dynamics (MD) simulation techniques +which become particularly useful in cases where experimental data are not available or +difficult to obtain. Different theoretical approaches can be adopted with a varying degree of +accuracy (see, for instance, refs. 1–27, and further references therein). +Basically, in MD simulation transport properties can be evaluated either through a gen- +uine nonequilibrium approach by applying an explicit external perturbation (such as a shear +flow or a temperature gradient), which is clearly direct and intuitive but is affected by non- +trivial technical issues (in particular the need to generate nonequilibrium steady states in +typical systems characterized by finite-size supercells with periodic boundary conditions and +to extrapolate to the limit of zero driving force). Alternatively, the transport coefficients +can be more easily estimated from equilibrium MD simulations by using the Green-Kubo +relations28–30 of statistical mechanics (dissipation-fluctuation theorem) which allow the cal- +culation of transport coefficients by integration of suitable autocorrelation functions. This +latter approach is simpler because standard equilibrium MD simulations can be easily car- +ried out and estimated transport coefficients exhibit a weaker system-size dependence.26 An +equivalent17 equilibrium method exploits the Einstein–Helfand expressions2 to get transport +coefficients directly from the particle displacements and velocities;18 for instance, the shear +viscosity can be computed in terms of the mean-square x displacement of the center of y mo- +mentum, while the thermal conductivity is proportional to the mean square x displacement +of the center of energy. +The shear viscosity describes the resistance of a fluid to shear forces and is a measure +of the shear stress induced by an applied velocity gradient,1 while the bulk viscosity refers +to the resistance to dilatation of an infinitesimal volume element at constant shape and +measures the resistance of a fluid to compression. It is closely connected with absorption +and dispersion of ultrasonic waves in a fluid, so it can provide valuable information about +intermolecular forces. Moreover, the role of the bulk viscosity is acquiring more and more +importance, for instance in the area of surface and interface-related phenomena and for +2 + +the interpretation of acoustic sensor data.31 In spite of its relevance, bulk viscosity has +received less experimental and theoretical attention, partly due to the greater difficulties in +obtaining accurate measurements and estimates. In principle it should be evaluated in the +microcanonical (NVE) ensemble where there is no need to evaluate an additional term which +would be required if, for instance, the canonical NVT ensemble were used.13,31 Moreover, +bulk viscosity is subject to much larger statistical error caused by the fact that it must +be calculated by the regression of fluctuations about a nonzero mean.3 While the shear +viscosity is associated with changes in water Hydrogen-bond network connectivity and is +mostly related to translational molecular motion, the bulk viscosity is associated with local +density fluctuations and reflects the relaxation of both rotational and vibrational modes.32,33 +The thermal conductivity describes instead the capability of a substance to allow molecular +transport of energy driven by temperature gradients. +In general dynamical properties such as the transport coefficients are much more depen- +dent on the simulation size and timescale than structural properties.23 One must also point +out that shear and bulk viscosities, and thermal conductivity are even more difficult to be +evaluated accurately than, for instance, the diffusion coefficient (a single-particle property) +since they are collective transport properties involving all the particles.14 In fact, for esti- +mating the diffusion coefficient one can perform a statistical average over the particles in +addition to the average over time because every particle diffuses individually but any stress +or energy fluctuation is an event involving the system as a whole. As a consequence, in +order to obtain the same statistical accuracy, collective properties need much longer runs +than single particle properties by a factor proportional to the size of the system.12 +We here estimate from first principles simulations, in the framework of the Density Func- +tional Theory (DFT), the shear and bulk viscosity of liquid water and Argon. For liquid +Argon the thermal conductivity is also calculated. By analyzing our results we can con- +clude that our first-principles simulations, performed at a nominal average temperature of +366 K to guarantee that the systems is liquid-like, actually describe the basic dynamical +properties of liquid water at about 330 K. Our approach is also able to reproduce well the +bulk-viscosity/shear-viscosity ratio of liquid Ar which is much smaller than that of liquid +water. +3 + +II. +METHOD +We have performed first principles MD simulations of liquid water using the CPMD +package,34 at constant volume, considering the experimental density of water at room tem- +perature. The computations were performed at the Γ-point only of the Brillouin zone, using +norm-conserving pseudopotentials35 and a basis set of plane waves to expand the wavefunc- +tions with an energy cutoff of 250 Ry; we have explicitly tested that this energy cutoff, much +higher than that used in standard DFT simulations of liquid water, is required to have a +good convergence also for the stress tensor components. +We have adopted the gradient-corrected BLYP36 density functional augmented by van +der Waals (vdW) corrections, hereafter referred to as DFT-D2(BLYP).37 This choice is +motivated both by the fact that BLYP has been shown38–42 to give an acceptable description +of Hydrogen bonding in water, and because it represents a good reference DFT functional to +add vdW corrections.43–46 A good description of Hydrogen bonding is essential here since, in +liquid water, the shear viscosity mostly originates from covalent interactions in the Hydrogen- +bond dynamics of water molecules.19 Moreover, vdW corrections to BLYP are important +because it was shown that BLYP significantly underestimates (by 25%) the equilibrium +density of liquid water; the experimental density can be recovered by adding the vdW +corrections proposed by Grimme,37 which have the further effect of making the oxygen- +oxygen radial distribution function in better agreement with experiment.47,48 Our system +consists of 64 water molecules contained in a supercell with simple-cubic symmetry and +periodic boundary conditions. Hydrogen nuclei have been treated as classical particles with +the mass of the deuterium isotope which allows us to use larger time steps. The effective +mass determining the time scale of the fictitious dynamics of the electrons was 700 a.u. and +the equations of motion were integrated with a time step of 3 a.u. (=0.073 fs). +Our simulation consisted of an initial equilibration phase, lasting about 0.15 ps, in which +the ionic temperature was simply controlled by velocity rescaling, followed by a much longer +(about 22 ps) canonical (NVT) MD simulation (using suitable thermostats for a Nos´e-Hoover +dynamics), followed by a final 22 ps microcanonical (NVE) production MD run. A common +drawback of most standard DFT functionals applied to liquid water at room temperature +is their tendency to ”freeze” the system which therefore exhibits an ice-like behavior. By +applying vdW corrections the problem is reduced but it still present. In particular, since the +4 + +melting temperature of water estimated by DFT-D2(BLYP) is 360 K49 (while it is 411 K with +BLYP), following a common strategy, we performed NVT simulations with an average ionic +temperature of 380 K to be sure that the system is indeed liquid-like. This use of artificially +increased temperature also serves to mimic Nuclear Quantum Effects in simulations of liquid +water.23 The average ionic temperature of the subsequent NVE MD simulation was 366 +K. Several data (atomic coordinates, velocities, stress-tensor components,...) relevant for +characterizing structural and dynamical properties of the system were recorded every 20 +steps in the production stage. +As far as liquid Ar is concerned, before starting MD simulations, we have performed exten- +sive preliminary calculations to choose optimal parameters and a suitable DFT functional. +Clearly in this case even an empirical Lennard-Jones potential reference could probably give +reasonable results but here we are interested in studying transport properties using DFT +functionals in a first-principle framework, which has the advantage of explicitly accounting +for the electronic structure of matter. Application to the face-centered cubic (fcc) Ar crystal +(considering a fcc supercell with 32 Ar atoms) and comparison with experimental reference +values for the equilibrium Ar-Ar distance and the cohesive energy, suggests that, among +many tested, vdW-corrected DFT functionals, DFT-D2(PBE)37,50 is the most adequate to +describe extended systems made by Ar atoms, hence we mainly use it for the MD simula- +tions of liquid Ar. In this case we have checked that a suitable energy cutoff to get a good +convergence for the stress tensor components is 110 Ry. +The liquid Ar sample was prepared starting from an initial (unfavorable) simple cubic +lattice configuration with 64 Ar atoms and considering the experimental Ar density (1.4 +g/cm3) at melting point (84 K). Then the systems was heated by gradually increasing the +ionic temperature (by velocity rescaling) to 500 K (in a time of 1.3 ps) to be sure that the +system was truly melted; then the temperature was gradually decreased (in 1.0 ps) to 150 +K, which is a temperature sufficiently higher than the experimental melting point that it +can be assumed that the system is indeed in a liquid phase; this has been explicitly checked +looking at the translational order parameter.1 +Then a 60 ps canonical (NVT) MD simulation (with a ionic temperature of 150 K) was +performed, followed by a 60 ps microcanonical (NVE) MD production runs with an average +ionic temperature of 129 K. In this case the electronic effective mass was 700 a.u. and the +equations of motion were integrated with a time step of 5 a.u. (=0.121 fs). Data (atomic +5 + +coordinates, velocities, stress-tensor components,...) relevant for structural and dynamical +properties of the system were recorded every 10 steps in the production stage. +As mentioned above, different approaches exist for the calculation of shear, ηS, and +bulk, ηB, viscosity from MD simulations.1,8–11,13 The most used technique is based on the +evaluation of the autocorrelation functions of stress-tensor components; in particular,1 +ηS = +V +kBT +� ∞ +0 +dt⟨Pαβ(0)Pαβ(t)⟩ , +(1) +ηB = +V +9kBT +� ∞ +0 +dt⟨δPαα(0)δPββ(t)⟩ = +V +kBT +� ∞ +0 +dt⟨δP(0)δP(t)⟩ , +(2) +where, in practice the upper limit of integration (∞) is replaced by a reasonably-long +simulation time, tmax, ⟨...⟩ denotes average over different time origins, V is the system +volume, T the ionic temperature, kB the Boltzmann constant, Pαβ quantities denote the +components of the stress tensor, the instantaneous pressure is given by P(t) = 1/3 � +α Pαα +(that is the average of the diagonal elements of the stress tensor), and the fluctuations are +defined as: +δPαα(t) = Pαα(t) − ⟨Pαα⟩ = Pαα(t) − P , δP(t) = P(t) − ⟨P⟩ = P(t) − P , +(3) +where P is the system pressure obtained as the ensemble average of P(t). In isotropic +fluids (with rotational invariance) there are only 5 independent (and equivalent) components +of the traceless stress tensor: Pxy, Pyz, Pzx, (Pxx − Pyy)/2, and (Pyy − Pzz)/2, so that it is +convenient to compute the shear viscosity ηS by averaging over these 5 components to get +better statistics. +Instead, the bulk viscosity ηB has only one component, moreover the diagonal stresses +must be evaluated carefully since a non-vanishing equilibrium average must be subtracted. +In oder to get more accurate evaluations of transport properties and also reliable estimates +of the associated statistical errors, we adopt the block-average technique,51 which consists +of dividing the whole simulation into a sequence of several shorter intervals (“blocks”), +each with an equal number of samples; then block averages are calculated which allow to +estimate means and variances.15 In the case of the bulk-viscosity calculation, to reduce the +error, it is convenient to take for the system pressure the average value of the pressures +over all blocks.13 Clearly the choice of the block size must be made with care; in fact, +6 + +samples become uncorrelated as the block size increases so for small block sizes, the error is +underestimated while for large block sizes the error estimate is inaccurate due to insufficient +sampling (see detailed discussion below). +Typically transport coefficients are estimated from classical MD simulations based on +empirical interatomic potentials. The practical feasibility of calculating transport coefficients +in liquids using instead first principles MD simulations, was demonstrated by D. Alf´e and +M. J. Gillan,12 who used the Green-Kubo relations to compute the shear viscosity of liquid +iron and aluminum, with a statistical error of about 5%. However, the simulations of D. Alf´e +and M. J. Gillan12 were performed in the NVT ensemble, while our simulations have been +carried out using the NVE ensemble, since the NVE simulations also allow the evaluation +of the bulk viscosity without any correction term (see above). +A simpler alternative method exists (valid for temperatures that are not too low52) to +obtain an approximate estimate of the shear viscosity, by exploiting its connection with the +self-diffusion coefficient D via the Stokes-Einstein relation:4,12 +ηS = kBT +2πaD , +(4) +where a is an effective atomic diameter. Such relation is exact for the Brownian motion +of a macroscopic particle of diameter a in a liquid of shear viscosity ηS, but it is only +approximate when applied to atoms; however if a is chosen to be the radius r1 of the first +peak in the radial distribution function, the relation usually predicts ηS to within 40%.12 +Here we take for r1 the position of the first peak in the O-O and Ar-Ar radial distribution +function for liquid water and liquid Ar, respectively, while the diffusion coefficient D can +be computed1 from the mean square displacement of the oxygen atoms (for liquid water) +or Ar atoms (for liquid Ar). The validity of the Stokes–Einstein relation has been recently +discussed in detail by Herrero et al.52 who also explored the connection between structural +properties and transport coefficients. +For liquid Argon the thermal conductivity has been also calculated, using the formula:1 +λT = +V +kBT 2 +� ∞ +0 +dt⟨jE +α (0)jE +α (t)⟩ , +(5) +where jE +α is the α component of the energy current defined as the time derivative of +7 + +δEα = 1 +V +� +i +riα(Ei − ⟨Ei⟩) , +(6) +and Ei is the energy of the i−th Ar atom (located at coordinates rix, riy, riz), which can +be evaluated as +Ei = p2 +i /2mi + 1/2 +� +j̸=i +v(rij) , +(7) +by assuming a pairwise interatomic potential. In order to obtain a pair potential for +evaluating the thermal conductivity of liquid Ar using configurational data from our first- +principles DFT simulations, we have adopted a strategy similar to that proposed in ref. 26: +we assume for the pair potential a Lennard-Jones analytical form: +v(r) = a(b2/r12 − b/r6) , +(8) +where a and b are parameters optimized by fitting the potential-energy curve of the Ar +dimer (at different interatomic distances) obtained by using our DFT approach. +III. +RESULTS AND DISCUSSION +In Fig. 1 and 2 we plot the behavior of the temperature and pressure as a function of time +in the NVE simulation for liquid water and Ar, respectively. As can be seen, these quantities +turn out to be stable and exhibit only moderate oscillations around the average values, which +are, for liquid water, 0.132 GPa and 366 K for the pressure and the temperature, respectively, +while for liquid Ar the values are 0.173 GPa and 129 K. +In Fig. 3 and 4 we instead plot the auto-correlation functions (ACFs), corresponding to +the integrands (considering the average over the components for the shear viscosity) of eqs. +(1) and (2). Differently from what observed in monatomic systems (such as liquid Ar) or +in classical MD simulations where waters are modeled by rigid molecules, in first-principles +simulations of liquid water, high-frequency intermolecular vibrations lead to corresponding +high-frequency oscillations in the pressure and in related ACFs. In order to better appreciate +the global decay behavior of ACFs, in the case of liquid water, high-frequency components +have been cut by Fourier-transforming the ACFs. +A quantitative estimate of the ACFs +relaxation times can be obtained assuming a global exponential decay (≃ e−t/τ) of the + +0 +5 +10 +15 +20 +time (ps) +0 +0 +100 +100 +200 +200 +300 +300 +400 +400 +T(K) +P (10 MPa) +FIG. 1: Temperature and pressure of liquid water plotted as a function of the simulation time. +integrands and computing: +τS = +� ∞ +0 +dt ⟨Pαβ(0)Pαβ(t)⟩ +⟨Pαβ(0)Pαβ(0)⟩ , +(9) +and +τB = +� ∞ +0 +dt ⟨δPαα(0)δPββ(t)⟩ +⟨δPαα(0)δPββ(0)⟩ +(10) +For liquid water we find τS ≃ 6 fs and τB ≃ 4 fs, while for liquid Ar τS ≃ 340 fs and +τB ≃ 410 fs. + +0 +10 +20 +30 +40 +50 +60 +time (ps) +0 +0 +50 +50 +100 +100 +150 +150 +T(K) +P (10 MPa) +FIG. 2: Temperature and pressure of liquid Ar plotted as a function of the simulation time. +The shear and bulk viscosity, computed using eqs. (1) and (2), are plotted as a function +of the upper limit of the integrals in Fig. 5 and 6, while the thermal conductivity of liquid +Ar is reported in Fig. 7. From these curves an approximate estimate of the shear and +bulk viscosity can be obtained considering the values of the quantities corresponding to the +position of the first pronounced maximum-plateau; in fact this indicates that the running +integral starts becoming nearly independent of time implying that the corresponding ACF +has decayed to zero and is fluctuating along the horizontal time axis. Clearly, considering + +0 +2 +4 +6 +8 +10 +time (ps) +-0,005 +0 +0,005 +0,01 +0,015 +0,02 +ACF (Pa ) +for shear viscosity +for bulk viscosity +2 +FIG. 3: Auto-correlation functions (ACFs) used for the evaluation of the shear and bulk viscosities +of liquid water (see text) plotted as a function of the simulation time. +longer times only introduces additional noise to the signal and the beginning of a plateau +represents the desired value of the viscosity with the smallest uncertainty. As can be seen, +the maximum-plateau is reached at about t = 0.8 ps for both the shear and bulk viscosity of +liquid water, while the corresponding values for liquid Ar are 3.0, 5.0 ps, and 0.5 ps for the +shear viscosity, the bulk viscosity, and the thermal conductivity, respectively. As expected, +these times are much larger than the corresponding relaxation times τS and τB estimated + +0 +2 +4 +6 +8 +10 +time (ps) +0 +0,0002 +0,0004 +0,0006 +0,0008 +ACF (Pa ) +for shear viscosity +for bulk viscosity +2 +FIG. 4: Auto-correlation functions (ACFs) used for the evaluation of the shear and bulk viscosities +of liquid Ar (see text) plotted as a function of the simulation time. +above. +As already discussed, a more accurate evaluation, with also a reliable estimate of the +associated statistical error, can be obtained by adopting a block-average technique. +In +this case a proper choice of the block size is crucial: with many, small-size blocks, the +statistical error is small but the blocks are probably correlated and the viscosity is typically +underestimated (not yet converged); on the contrary, with just a few, large-size blocks, these + +0 +0,5 +1 +1,5 +2 +time (ps) +0 +5 +10 +( Pa s) +bulk viscosity +shear viscosity +10-4 +FIG. 5: Shear and bulk viscosity of liquid water plotted as a function of the upper limit of the +integrals of the ACFs. +are probably uncorrelated and the viscosity is converged but the statistical error is large. +In Figs. 8, 9, 10, and 11 we plot the values of the shear and bulk viscosity of liquid +water and Ar evaluated by using different numbers of blocks (keeping constant the total +number of configurations) with the relative statistical errors. The dashed horizontal lines +indicate the corresponding values inferred by considering the maximas-plateaus of the curves +in Figs. 5 and 6. As can be seen, in the case of liquid water, the maxima of the shear and + +0 +1 +2 +3 +4 +5 +6 +time (ps) +0 +1 +2 +3 +4 + ( Pa s) +bulk viscosity +shear viscosity +10-4 +FIG. 6: Shear and bulk viscosity of liquid Ar plotted as a function of the upper limit of the integrals +of the ACFs. +bulk viscosities are obtained considering 16 blocks, each equivalent to a simulation time +of about 1.4 ps. Interestingly, taking statistical uncertainties into account, these maxima +are compatible with the rough estimates obtained before and, for the shear viscosity, also +with the values obtained using the Stokes-Einstein formula (Eq. (4)). As already described +above, in the Stokes-Einstein estimate the shear viscosity is obtained in terms of the diffusion +coefficient D and the radius of the first peak in the radial distribution function (see Eq. (4), + +0 +1 +2 +3 +4 +time (ps) +0 +0,05 +0,1 +thermal conductivity (W/m K) +FIG. 7: Thermal conductivity of liquid Ar plotted as a function of the upper limit of the integral +of the ACF. +for liquid water we have considered the first peak in the oxygen-oxygen radial distribution +function, see below). Actually our reported Stokes-Einstein estimated values are corrected +by finite-size effects: in fact D can be extrapolated to infinite size of the simulation box (see, +for instance, ref. 52) by just considering the shear-viscosity value: + +D∞ = D + 2.837 kBT +6πηSL , +(11) +where L is the size of the cubic simulation box. Therefore, by simultaneously taking +into account Eqs. (4) and (11), one can get a “self-consistent”, finite-size corrected Stokes- +Einstein estimate for ηS : +η∗ +S = kBT +2πaD − 2.837 kBT +6πLD . +(12) +Quantitative data are collected in Table I where they are also compared with some the- +oretical and experimental reference values. +As far as the shear viscosity is concerned, for liquid water our estimated value, obtained +from the NVE simulation at an average temperature of 366 K, agrees with the experimental +reference data at a lower temperature of about 330 K. This is in line with the performances +of other DFT functionals; for instance (see Table I), in recent simulations52 of liquid water +based on the SCAN functional,59 the shear viscosity estimate is close to that obtained from +a force-field approach (that, for this quantity, well reproduces the experimental behavior) +only between 330 and 360 K, while it is severely overestimated at 300 K. With the OPTB88- +vdW functional60 reasonable agreement with experimental data at room temperature is only +found52 at 360 K. +For liquid Ar the behavior is qualitatively similar (see Figs. 10, 11, and 12 for the shear +viscosity, the bulk viscosity, and the thermal conductivity, respectively). In this case both the +maxima of the shear and bulk viscosities are obtained considering 5 blocks, each equivalent +to a simulation time of about 12.0 ps. Even in this case, taking statistical uncertainties into +account, these maxima are compatible with the plateau positions and, for the shear viscosity, +also with the estimate from the Stokes-Einstein formula. The maximum of the thermal +conductivity is instead reached with 25 blocks, each equivalent to a simulation time of about +2.4 ps, and its value (0.11±0.02 W/m K) is again compatible with that estimated considering +the maximum-plateau position and in good agreement with the literature reference value +(0.12 W/m K) at 90 K61 and that obtained by classical MD simulations based on the +Lennard-Jones potential (0.119 W/m K).62 +An interesting physical quantity is represented by the ratio between bulk and shear +viscosity, which can be related to the ratio of observed to classical absorption coefficients in + +0 +5 +10 +15 +20 +25 +30 +35 +# of blocks +0 +2 +4 +6 +8 +shear viscosity ( Pa s) +* +10-4 +expt. 303 K +expt. 323 K +expt. 333 K +FIG. 8: Shear viscosity of liquid water evaluated by using different numbers of blocks (the smaller +is the block number the larger is the number of configurations of each block) with the relative sta- +tistical errors. The dashed horizontal line indicates the position of the first-pronounced maximum- +plateau of the corresponding curve of Fig. +5. +The asterisk denotes the value obtained by the +Stokes-Einstein formula (Eq.12), while the triangles indicate experimental estimates at different +temperatures. + +TABLE I: Shear and bulk viscosity of liquid water and Ar, in 10−4 Pa s, compared with theoretical +and experimental reference data. +Statistical errors are in parenthesis. +η∗ +S indicates the shear +viscosity estimate obtained by the Stokes-Einstein relation (see text). +system +ηS +η∗ +S +ηB +ηB/ηS 3/4ηB/ηS + 1 +water (366 K) +4.8(0.7) 5.7 11.3(2.9) 2.4(0.8) +2.8(0.6) +water DFT SCANa (300K) +23 +— +— +— +— +water DFT SCANa (330K) +6 +— +— +— +— +water DFT SCANa (360K) +5 +— +— +— +— +water DFT OPTB88-vdWa (300K) +30 +— +— +— +— +water DFT OPTB88-vdWa (330K) +15 +— +— +— +— +water DFT OPTB88-vdWa (360K) +8 +— +— +— +— +water force fielda (300K) +8 +— +— +— +— +water force fielda (330K) +5 +— +— +— +— +water force fielda (360K) +3.5 +— +— +— +— +water force fieldb (303K) +6.5(0.4) — 15.5(1.6) 2.4(0.3) +2.8(0.2) +water expt.c (298 K) +8.90 +— +— +— +— +water expt.b,d,e (303 K) +7.97 +— +21.5 +2.7 +3.0 +water expt.f (323 K) +5.47 +— +14.8 +2.7 +3.0 +water expt.c (333 K) +4.67 +— +— +— +— +Ar (129 K) +3.7(1.6) 2.0 4.0(2.2) 1.1(0.8) +1.8(0.6) +Ar expt.g (90 K) +2.33 +— +1.82 +0.8 +1.6 +Ar expt.h (90 K) +2.57 +— +— +— +aref.52. +bref.13. +cref.55. +dref.53. +eref.54. +fref.56. +gref.57. +href.58. + +0 +5 +10 +15 +20 +25 +30 +35 +# of blocks +0 +5 +10 +15 +bulk viscosity ( Pa s) +10-4 +FIG. 9: Bulk viscosity of liquid water evaluated by using different numbers of blocks (the smaller +is the block number the larger is the number of configurations of each block) with the relative sta- +tistical errors. The dashed horizontal line indicates the position of the first-pronounced maximum- +plateau of the corresponding curve of Fig. 5. +ultrasonic absorption experiments.13 In fact, under the condition that the heat conductivity +contribution to the ultrasonic absorption may be neglected, + +0 +10 +20 +30 +40 +50 +# of blocks +0 +1 +2 +3 +4 +5 +6 +shear viscosity ( Pa s) +* +10-4 +expt. 90 K +FIG. 10: Shear viscosity of liquid Ar evaluated by using different numbers of blocks (the smaller is +the block number the larger is the number of configurations of each block) with the relative statis- +tical errors. The dashed horizontal line indicates the position of the first-pronounced maximum- +plateau of the corresponding curve of Fig. +6. +The asterisk denotes the value obtained by the +Stokes-Einstein formula (Eq.4), while the triangle indicates the experimental estimate at 90 K. +α +αclass += 3/4ηB +ηS ++ 1 , +(13) +and water belongs to the group of the so-called ”associated liquids”, characterized by a +ratio from 1 to 3, where structural relaxation is dominant. +Classical MD simulations based on the SPC/E semiempirical potential predict13 a ηB +ηS +ratio of 2.4, leading to a +α +αclass ratio of 2.79, in reasonable agreement with the experimental + +0 +10 +20 +30 +40 +50 +# of blocks +0 +1 +2 +3 +4 +5 +6 +7 +bulk viscosity ( Pa s) +10-4 +FIG. 11: Bulk viscosity of liquid Ar evaluated by using different numbers of blocks (the smaller is +the block number the larger is the number of configurations of each block) with the relative statis- +tical errors. The dashed horizontal line indicates the position of the first-pronounced maximum- +plateau of the corresponding curve of Fig. 6. +value of 3.0.53 Instead normal liquids, such as monatomic liquids (for instance liquid Ar) +are characterized by a ratio no greater than 1.2. Although in general the ratio varies with +temperature and pressure, in liquid water it is found to remain constant within 20% in +the temperature range 0-90 C (273-363 K).63 By taking statistical errors into account, our +estimated value of the +α +αclass ratio (2.8 ± 0.6) is compatible with the available experimental + +0 +10 +20 +30 +40 +50 +# of blocks +0 +0,05 +0,1 +0,15 +0,2 +0,25 +thermal conductivity (W/m K) +FIG. 12: Thermal conductivity of liquid Ar evaluated by using different numbers of blocks (the +smaller is the block number the larger is the number of configurations of each block) with the relative +statistical errors. The dashed horizontal line indicates the position of the maximum-plateau of the +corresponding curve in Fig. 7. +data at ambient temperature (3.0). This is a remarkable result, considering that most of the +reported classical MD simulations13 predict a bulk viscosity lower than the the experimental +one, leading to an underestimated value of the +α +αclass ratio. +One should also point out that a proper comparison with experimental data requires a +careful analysis taking into account the pronounced temperature dependence of shear and + +bulk viscosity. In fact, according to a common empirical model,56,64 the viscosity strongly de- +creases with increasing temperature following an exponential decay. By fitting experimental +data56 with an exponential function and taking statistical errors into account, our estimated +values of the shear and bulk viscosity of liquid water are compatible with experimental +data in the temperature range of 323-344 K. One should also consider that also the bulk- +viscosity/shear-viscosity ratio for liquid water tends to decrease slightly with temperature,56 +suggesting an even better agreement between our estimated value and the experimental +data.56 We remind that our simulations have been carried out at temperatures higher than +ambient temperature to guarantee that the systems is liquid-like. By considering that our +estimate (after finite-size correction) for the diffusion coefficient, D = 5.02 × 10−5 cm2/s, +corresponds to the experimental value measured at about 336 K,65 we can conclude that, +our DFT simulations based on the DFT-D2(BLYP) functional and performed at a nominal +average temperature of 366 K, actually describe the basic dynamical properties of liquid +water at about 330 K. One should also mention that bulk-viscosity measurements are in- +direct and affected by considerable errors.13,27,33,56,66,67 In summary, we can conclude that +our adopted BLYP-D2 functional is able to describe reasonably well the density fluctuations +of liquid water; the discrepancy with respect to experimental data at ambient conditions +can be to a large extend explained in terms of the pronounced temperature dependence of +both shear and bulk viscosity and the need to perform first-principles MD simulations at +temperatures higher than ambient temperature. +As far as liquid Ar is concerned, our shear and bulk viscosities, computed by first- +principles at a nominal average simulation temperature of 129 K, turn out to be some- +how overestimated with respect to the reference experimental values at 90 K, although +they are compatible with them if statistical errors are taken into account. Moreover our +bulk-viscosity/shear-viscosity ratio (close to unity) agrees well with the reference estimate, +while interestingly this is not the case if a standard Lennard-Jones empirical potential is +adopted using classical MD simulations that predict instead a very low value13,62 of the ratio +(0.17-0.35 at high densities), thus showing that this popular potential cannot properly re- +produce all the dynamical properties of liquid Ar and underlining once again the superiority +of first-principles approaches. +We conclude our study by reporting some basic structural properties of our investigated +systems. In particular, in Fig. 13, for liquid water we plot our computed O-O pair correlation + +2 +3 +4 +5 +6 +7 +r (A) +0 +0,5 +1 +1,5 +2 +2,5 +3 +g(r) +FIG. 13: O-O pair correlation function, gOO(r), compared with that obtained experimentally from +X-ray diffraction measurements at ambient conditions.68–70 +function, gOO(r), compared with that obtained experimental from X-ray diffraction measure- +ments at ambient conditions.68–70 The main features of the gOO(r) curves are reported in +Table II. As can be seen, there is a good agreement between the two curves; the fact the +oscillations of our computed curve are slightly reduced with respect to the experimental one +can again be related to the higher effective temperature of our simulation. + +2 +3 +4 +5 +6 +7 +r (A) +0 +0,5 +1 +1,5 +2 +2,5 +3 +3,5 +g(r) +FIG. 14: Ar-Ar pair correlation function, g(r), compared with that obtained experimentally from +neutron-scattering measurements at 85K.73 +In Fig. 14, for liquid Ar our computed Ar-Ar pair correlation function, g(r), is compared +with that obtained experimentally from neutron-scattering measurements at 85 K,73 while +again the main features of the g(r) curves are reported in Table II. Even in this case there +is a reasonable agreement between the simulation and experimental curve, by considering +that simulations for liquid Ar have been performed at significantly higher temperature (129 +K) than experiments (85 K) (note that the experimental melting and boiling points of Ar +are at 84 and 87 K, respectively). After applying the same finite-size correction adopted + +TABLE II: Main features of the O-O pair correlation function, gOO(r), of liquid water and of +the Ar-Ar pair correlation function, g(r) of liquid Ar compared with experimental reference data, +obtained from X-ray diffraction measurements at ambient conditions for liquid water and neutron- +scattering measurements for liquid Ar. rmax and rmin indicate the position of the first maximum +(the main peak) and the first minimum of gOO(r) and g(r), respectively, and gmax and gmin the +corresponding values of the gOO(r) and g(r) functions. +system +rmax(˚A) +gmax +rmin(˚A) +gmin +water (366 K) +2.79 +2.42 +3.66 +0.88 +water expt.a (293 K) 2.80(1) 2.55(5) 3.41(4) 0.85(2) +Ar (129 K) +3.70 +2.80 +5.29 +0.64 +Ar expt.b (85 K) +3.68 +3.05 +5.18 +0.56 +aref.68–70. +bref.73. +above for liquid water, our estimated diffusion coefficient for liquid Ar, D = 3.82 × 10−5 +cm2/s, evaluated at a nominal simulation temperature of 129 K is significantly higher than +the reference value (1.6 × 10−5 cm2/s) reported at 84 K.71 Again this discrepancy can be +explained in terms of the higher temperature of the liquid Ar simulation. +IV. +CONCLUSIONS +Shear and bulk viscosity of liquid water and Argon have been evaluated, together with +other structural and dynamical properties, from first principles by adopting a vdW-corrected +DFT approach, by performing Molecular Dynamics simulations in the NVE ensemble and +using the Kubo-Greenwood equilibrium approach. For liquid Argon the thermal conductivity +has been also calculated. Concerning liquid water, to our knowledge this is the first estimate +of the bulk viscosity and of the shear-viscosity/bulk-viscosity ratio from first principles. By +analyzing our results and comparing then with reference experimental data, we can conclude +that our first-principles simulations, performed at a nominal average temperature of 366 +K to guarantee that the systems is liquid-like, actually describe well the basic dynamical + +properties of liquid water at about 330 K. In comparison with liquid water, the normal, +monatomic liquid Ar is characterized by a much smaller bulk-viscosity/shear-viscosity ratio +(close to unity) and this feature is well reproduced by our first-principles approach which +predicts a value of the ratio in better agreement with experimental reference data than that +obtained using the empirical Lennard-Jones potential. The computed thermal conductivity +of liquid Argon is also in good agreement with the experimental value. +V. +ACKNOWLEDGEMENTS +We acknowledge funding from Fondazione Cariparo, Progetti di Eccellenza 2017, relative +to the project: ”Engineering van der Waals Interactions: Innovative paradigm for the control +of Nanoscale Phenomena”. +VI. +DATA AVAILABILITY +The data that support the findings of this study are available from the corresponding +author upon reasonable request. +1 M. P. Allen and D. J. Tildesley, Computer Simulations of Liquids (Oxford Science Publications, +Clarendon Press, Oxford 1987). +2 E. Helfand, ”Transport Coefficients from Dissipation in a Canonical Ensemble”, Phys. Rev. +119, 1 (1960). +3 B. J. Alder, D. M. Gass, T. E. Wainwright, ”Studies in Molecular Dynamics. VIII. The Trans- +port Coefficients for a Hard-Sphere Fluid”, J. Chem. Phys. 53, 3813 (1970). +4 E. M. Gosling, I. R. McDonald, K. 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A 7, 2130 (1973). + diff --git a/C9FQT4oBgHgl3EQfPDYj/content/tmp_files/load_file.txt b/C9FQT4oBgHgl3EQfPDYj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c57ee5aa6f21cf721a1b0e995855d1909d295b2c --- /dev/null +++ b/C9FQT4oBgHgl3EQfPDYj/content/tmp_files/load_file.txt @@ -0,0 +1,948 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf,len=947 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='13277v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='soft] 30 Jan 2023 Transport properties in liquids from first principles: the case of liquid water and liquid Argon Pier Luigi Silvestrelli Dipartimento di Fisica e Astronomia “G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Galilei”, Universit`a di Padova, via Marzolo 8, I-35131 Padova, Italy (Dated: February 1, 2023) Abstract Shear and bulk viscosity of liquid water and Argon are evaluated from first principles in the Den- sity Functional Theory (DFT) framework, by performing Molecular Dynamics simulations in the NVE ensemble and using the Kubo-Greenwood equilibrium approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Standard DFT functional is corrected in such a way to allow for a reasonable description of van der Waals (vdW) effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' For liquid Argon the thermal conductivity has been also calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Concerning liquid water, to our knowledge this is the first estimate of the bulk viscosity and of the shear-viscosity/bulk-viscosity ratio from first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' By analyzing our results we can conclude that our first-principles sim- ulations, performed at a nominal average temperature of 366 K to guarantee that the systems is liquid-like, actually describe the basic dynamical properties of liquid water at about 330 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' In comparison with liquid water, the normal, monatomic liquid Ar is characterized by a much smaller bulk-viscosity/shear-viscosity ratio (close to unity) and this feature is well reproduced by our first- principles approach which predicts a value of the ratio in better agreement with experimental reference data than that obtained using the empirical Lennard-Jones potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' The computed thermal conductivity of liquid Argon is also in good agreement with the experimental value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' INTRODUCTION Transport properties are among the most important and useful features of condensed- matter systems, particularly for characterizing the dynamical behavior of liquids, since they play an important role in many technical and natural processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Therefore their estimate represents one of the most relevant goal of Molecular Dynamics (MD) simulation techniques which become particularly useful in cases where experimental data are not available or difficult to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Different theoretical approaches can be adopted with a varying degree of accuracy (see, for instance, refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 1–27, and further references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Basically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' in MD simulation transport properties can be evaluated either through a gen- uine nonequilibrium approach by applying an explicit external perturbation (such as a shear flow or a temperature gradient),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' which is clearly direct and intuitive but is affected by non- trivial technical issues (in particular the need to generate nonequilibrium steady states in typical systems characterized by finite-size supercells with periodic boundary conditions and to extrapolate to the limit of zero driving force).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Alternatively, the transport coefficients can be more easily estimated from equilibrium MD simulations by using the Green-Kubo relations28–30 of statistical mechanics (dissipation-fluctuation theorem) which allow the cal- culation of transport coefficients by integration of suitable autocorrelation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' This latter approach is simpler because standard equilibrium MD simulations can be easily car- ried out and estimated transport coefficients exhibit a weaker system-size dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='26 An equivalent17 equilibrium method exploits the Einstein–Helfand expressions2 to get transport coefficients directly from the particle displacements and velocities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='18 for instance, the shear viscosity can be computed in terms of the mean-square x displacement of the center of y mo- mentum, while the thermal conductivity is proportional to the mean square x displacement of the center of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' The shear viscosity describes the resistance of a fluid to shear forces and is a measure of the shear stress induced by an applied velocity gradient,1 while the bulk viscosity refers to the resistance to dilatation of an infinitesimal volume element at constant shape and measures the resistance of a fluid to compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' It is closely connected with absorption and dispersion of ultrasonic waves in a fluid, so it can provide valuable information about intermolecular forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Moreover, the role of the bulk viscosity is acquiring more and more importance, for instance in the area of surface and interface-related phenomena and for 2 the interpretation of acoustic sensor data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='31 In spite of its relevance, bulk viscosity has received less experimental and theoretical attention, partly due to the greater difficulties in obtaining accurate measurements and estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' In principle it should be evaluated in the microcanonical (NVE) ensemble where there is no need to evaluate an additional term which would be required if, for instance, the canonical NVT ensemble were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='13,31 Moreover, bulk viscosity is subject to much larger statistical error caused by the fact that it must be calculated by the regression of fluctuations about a nonzero mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='3 While the shear viscosity is associated with changes in water Hydrogen-bond network connectivity and is mostly related to translational molecular motion, the bulk viscosity is associated with local density fluctuations and reflects the relaxation of both rotational and vibrational modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='32,33 The thermal conductivity describes instead the capability of a substance to allow molecular transport of energy driven by temperature gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' In general dynamical properties such as the transport coefficients are much more depen- dent on the simulation size and timescale than structural properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='23 One must also point out that shear and bulk viscosities, and thermal conductivity are even more difficult to be evaluated accurately than, for instance, the diffusion coefficient (a single-particle property) since they are collective transport properties involving all the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='14 In fact, for esti- mating the diffusion coefficient one can perform a statistical average over the particles in addition to the average over time because every particle diffuses individually but any stress or energy fluctuation is an event involving the system as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' As a consequence, in order to obtain the same statistical accuracy, collective properties need much longer runs than single particle properties by a factor proportional to the size of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='12 We here estimate from first principles simulations, in the framework of the Density Func- tional Theory (DFT), the shear and bulk viscosity of liquid water and Argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' For liquid Argon the thermal conductivity is also calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' By analyzing our results we can con- clude that our first-principles simulations, performed at a nominal average temperature of 366 K to guarantee that the systems is liquid-like, actually describe the basic dynamical properties of liquid water at about 330 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Our approach is also able to reproduce well the bulk-viscosity/shear-viscosity ratio of liquid Ar which is much smaller than that of liquid water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 3 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' METHOD We have performed first principles MD simulations of liquid water using the CPMD package,34 at constant volume, considering the experimental density of water at room tem- perature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' The computations were performed at the Γ-point only of the Brillouin zone, using norm-conserving pseudopotentials35 and a basis set of plane waves to expand the wavefunc- tions with an energy cutoff of 250 Ry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' we have explicitly tested that this energy cutoff, much higher than that used in standard DFT simulations of liquid water, is required to have a good convergence also for the stress tensor components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' We have adopted the gradient-corrected BLYP36 density functional augmented by van der Waals (vdW) corrections, hereafter referred to as DFT-D2(BLYP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='37 This choice is motivated both by the fact that BLYP has been shown38–42 to give an acceptable description of Hydrogen bonding in water, and because it represents a good reference DFT functional to add vdW corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='43–46 A good description of Hydrogen bonding is essential here since, in liquid water, the shear viscosity mostly originates from covalent interactions in the Hydrogen- bond dynamics of water molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='19 Moreover, vdW corrections to BLYP are important because it was shown that BLYP significantly underestimates (by 25%) the equilibrium density of liquid water;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' the experimental density can be recovered by adding the vdW corrections proposed by Grimme,37 which have the further effect of making the oxygen- oxygen radial distribution function in better agreement with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='47,48 Our system consists of 64 water molecules contained in a supercell with simple-cubic symmetry and periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Hydrogen nuclei have been treated as classical particles with the mass of the deuterium isotope which allows us to use larger time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' The effective mass determining the time scale of the fictitious dynamics of the electrons was 700 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' and the equations of motion were integrated with a time step of 3 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' (=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='073 fs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Our simulation consisted of an initial equilibration phase, lasting about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='15 ps, in which the ionic temperature was simply controlled by velocity rescaling, followed by a much longer (about 22 ps) canonical (NVT) MD simulation (using suitable thermostats for a Nos´e-Hoover dynamics), followed by a final 22 ps microcanonical (NVE) production MD run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' A common drawback of most standard DFT functionals applied to liquid water at room temperature is their tendency to ”freeze” the system which therefore exhibits an ice-like behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' By applying vdW corrections the problem is reduced but it still present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' In particular, since the 4 melting temperature of water estimated by DFT-D2(BLYP) is 360 K49 (while it is 411 K with BLYP), following a common strategy, we performed NVT simulations with an average ionic temperature of 380 K to be sure that the system is indeed liquid-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' This use of artificially increased temperature also serves to mimic Nuclear Quantum Effects in simulations of liquid water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='23 The average ionic temperature of the subsequent NVE MD simulation was 366 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Several data (atomic coordinates, velocities, stress-tensor components,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=') relevant for characterizing structural and dynamical properties of the system were recorded every 20 steps in the production stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' As far as liquid Ar is concerned, before starting MD simulations, we have performed exten- sive preliminary calculations to choose optimal parameters and a suitable DFT functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Clearly in this case even an empirical Lennard-Jones potential reference could probably give reasonable results but here we are interested in studying transport properties using DFT functionals in a first-principle framework, which has the advantage of explicitly accounting for the electronic structure of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Application to the face-centered cubic (fcc) Ar crystal (considering a fcc supercell with 32 Ar atoms) and comparison with experimental reference values for the equilibrium Ar-Ar distance and the cohesive energy, suggests that, among many tested, vdW-corrected DFT functionals, DFT-D2(PBE)37,50 is the most adequate to describe extended systems made by Ar atoms, hence we mainly use it for the MD simula- tions of liquid Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' In this case we have checked that a suitable energy cutoff to get a good convergence for the stress tensor components is 110 Ry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' The liquid Ar sample was prepared starting from an initial (unfavorable) simple cubic lattice configuration with 64 Ar atoms and considering the experimental Ar density (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='4 g/cm3) at melting point (84 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Then the systems was heated by gradually increasing the ionic temperature (by velocity rescaling) to 500 K (in a time of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='3 ps) to be sure that the system was truly melted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' then the temperature was gradually decreased (in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='0 ps) to 150 K, which is a temperature sufficiently higher than the experimental melting point that it can be assumed that the system is indeed in a liquid phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' this has been explicitly checked looking at the translational order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='1 Then a 60 ps canonical (NVT) MD simulation (with a ionic temperature of 150 K) was performed, followed by a 60 ps microcanonical (NVE) MD production runs with an average ionic temperature of 129 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' In this case the electronic effective mass was 700 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' and the equations of motion were integrated with a time step of 5 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' (=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='121 fs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Data (atomic 5 coordinates, velocities, stress-tensor components,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=') relevant for structural and dynamical properties of the system were recorded every 10 steps in the production stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' As mentioned above, different approaches exist for the calculation of shear, ηS, and bulk, ηB, viscosity from MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='1,8–11,13 The most used technique is based on the evaluation of the autocorrelation functions of stress-tensor components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' in particular,1 ηS = V kBT � ∞ 0 dt⟨Pαβ(0)Pαβ(t)⟩ , (1) ηB = V 9kBT � ∞ 0 dt⟨δPαα(0)δPββ(t)⟩ = V kBT � ∞ 0 dt⟨δP(0)δP(t)⟩ , (2) where, in practice the upper limit of integration (∞) is replaced by a reasonably-long simulation time, tmax, ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='⟩ denotes average over different time origins, V is the system volume, T the ionic temperature, kB the Boltzmann constant, Pαβ quantities denote the components of the stress tensor, the instantaneous pressure is given by P(t) = 1/3 � α Pαα (that is the average of the diagonal elements of the stress tensor), and the fluctuations are defined as: δPαα(t) = Pαα(t) − ⟨Pαα⟩ = Pαα(t) − P , δP(t) = P(t) − ⟨P⟩ = P(t) − P , (3) where P is the system pressure obtained as the ensemble average of P(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' In isotropic fluids (with rotational invariance) there are only 5 independent (and equivalent) components of the traceless stress tensor: Pxy, Pyz, Pzx, (Pxx − Pyy)/2, and (Pyy − Pzz)/2, so that it is convenient to compute the shear viscosity ηS by averaging over these 5 components to get better statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Instead, the bulk viscosity ηB has only one component, moreover the diagonal stresses must be evaluated carefully since a non-vanishing equilibrium average must be subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' In oder to get more accurate evaluations of transport properties and also reliable estimates of the associated statistical errors, we adopt the block-average technique,51 which consists of dividing the whole simulation into a sequence of several shorter intervals (“blocks”), each with an equal number of samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' then block averages are calculated which allow to estimate means and variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='15 In the case of the bulk-viscosity calculation, to reduce the error, it is convenient to take for the system pressure the average value of the pressures over all blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='13 Clearly the choice of the block size must be made with care;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' in fact, 6 samples become uncorrelated as the block size increases so for small block sizes, the error is underestimated while for large block sizes the error estimate is inaccurate due to insufficient sampling (see detailed discussion below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Typically transport coefficients are estimated from classical MD simulations based on empirical interatomic potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' The practical feasibility of calculating transport coefficients in liquids using instead first principles MD simulations, was demonstrated by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Alf´e and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Gillan,12 who used the Green-Kubo relations to compute the shear viscosity of liquid iron and aluminum, with a statistical error of about 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' However, the simulations of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Alf´e and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Gillan12 were performed in the NVT ensemble, while our simulations have been carried out using the NVE ensemble, since the NVE simulations also allow the evaluation of the bulk viscosity without any correction term (see above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' A simpler alternative method exists (valid for temperatures that are not too low52) to obtain an approximate estimate of the shear viscosity, by exploiting its connection with the self-diffusion coefficient D via the Stokes-Einstein relation:4,12 ηS = kBT 2πaD , (4) where a is an effective atomic diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Such relation is exact for the Brownian motion of a macroscopic particle of diameter a in a liquid of shear viscosity ηS, but it is only approximate when applied to atoms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' however if a is chosen to be the radius r1 of the first peak in the radial distribution function, the relation usually predicts ηS to within 40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='12 Here we take for r1 the position of the first peak in the O-O and Ar-Ar radial distribution function for liquid water and liquid Ar, respectively, while the diffusion coefficient D can be computed1 from the mean square displacement of the oxygen atoms (for liquid water) or Ar atoms (for liquid Ar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' The validity of the Stokes–Einstein relation has been recently discussed in detail by Herrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='52 who also explored the connection between structural properties and transport coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' For liquid Argon the thermal conductivity has been also calculated, using the formula:1 λT = V kBT 2 � ∞ 0 dt⟨jE α (0)jE α (t)⟩ , (5) where jE α is the α component of the energy current defined as the time derivative of 7 δEα = 1 V � i riα(Ei − ⟨Ei⟩) , (6) and Ei is the energy of the i−th Ar atom (located at coordinates rix, riy, riz), which can be evaluated as Ei = p2 i /2mi + 1/2 � j̸=i v(rij) , (7) by assuming a pairwise interatomic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' In order to obtain a pair potential for evaluating the thermal conductivity of liquid Ar using configurational data from our first- principles DFT simulations, we have adopted a strategy similar to that proposed in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 26: we assume for the pair potential a Lennard-Jones analytical form: v(r) = a(b2/r12 − b/r6) , (8) where a and b are parameters optimized by fitting the potential-energy curve of the Ar dimer (at different interatomic distances) obtained by using our DFT approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' RESULTS AND DISCUSSION In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 1 and 2 we plot the behavior of the temperature and pressure as a function of time in the NVE simulation for liquid water and Ar, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' As can be seen, these quantities turn out to be stable and exhibit only moderate oscillations around the average values, which are, for liquid water, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='132 GPa and 366 K for the pressure and the temperature, respectively, while for liquid Ar the values are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='173 GPa and 129 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 3 and 4 we instead plot the auto-correlation functions (ACFs), corresponding to the integrands (considering the average over the components for the shear viscosity) of eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Differently from what observed in monatomic systems (such as liquid Ar) or in classical MD simulations where waters are modeled by rigid molecules, in first-principles simulations of liquid water, high-frequency intermolecular vibrations lead to corresponding high-frequency oscillations in the pressure and in related ACFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' In order to better appreciate the global decay behavior of ACFs, in the case of liquid water, high-frequency components have been cut by Fourier-transforming the ACFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' A quantitative estimate of the ACFs relaxation times can be obtained assuming a global exponential decay (≃ e−t/τ) of the 0 5 10 15 20 time (ps) 0 0 100 100 200 200 300 300 400 400 T(K) P (10 MPa) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 1: Temperature and pressure of liquid water plotted as a function of the simulation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' integrands and computing: τS = � ∞ 0 dt ⟨Pαβ(0)Pαβ(t)⟩ ⟨Pαβ(0)Pαβ(0)⟩ , (9) and τB = � ∞ 0 dt ⟨δPαα(0)δPββ(t)⟩ ⟨δPαα(0)δPββ(0)⟩ (10) For liquid water we find τS ≃ 6 fs and τB ≃ 4 fs, while for liquid Ar τS ≃ 340 fs and τB ≃ 410 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 0 10 20 30 40 50 60 time (ps) 0 0 50 50 100 100 150 150 T(K) P (10 MPa) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 2: Temperature and pressure of liquid Ar plotted as a function of the simulation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' The shear and bulk viscosity, computed using eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' (1) and (2), are plotted as a function of the upper limit of the integrals in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 5 and 6, while the thermal conductivity of liquid Ar is reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' From these curves an approximate estimate of the shear and bulk viscosity can be obtained considering the values of the quantities corresponding to the position of the first pronounced maximum-plateau;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' in fact this indicates that the running integral starts becoming nearly independent of time implying that the corresponding ACF has decayed to zero and is fluctuating along the horizontal time axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Clearly, considering 0 2 4 6 8 10 time (ps) 0,005 0 0,005 0,01 0,015 0,02 ACF (Pa ) for shear viscosity for bulk viscosity 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 3: Auto-correlation functions (ACFs) used for the evaluation of the shear and bulk viscosities of liquid water (see text) plotted as a function of the simulation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' longer times only introduces additional noise to the signal and the beginning of a plateau represents the desired value of the viscosity with the smallest uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' As can be seen, the maximum-plateau is reached at about t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='8 ps for both the shear and bulk viscosity of liquid water, while the corresponding values for liquid Ar are 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='0, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='0 ps, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='5 ps for the shear viscosity, the bulk viscosity, and the thermal conductivity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' As expected, these times are much larger than the corresponding relaxation times τS and τB estimated 0 2 4 6 8 10 time (ps) 0 0,0002 0,0004 0,0006 0,0008 ACF (Pa ) for shear viscosity for bulk viscosity 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 4: Auto-correlation functions (ACFs) used for the evaluation of the shear and bulk viscosities of liquid Ar (see text) plotted as a function of the simulation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' As already discussed, a more accurate evaluation, with also a reliable estimate of the associated statistical error, can be obtained by adopting a block-average technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' In this case a proper choice of the block size is crucial: with many, small-size blocks, the statistical error is small but the blocks are probably correlated and the viscosity is typically underestimated (not yet converged);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' on the contrary, with just a few, large-size blocks, these 0 0,5 1 1,5 2 time (ps) 0 5 10 ( Pa s) bulk viscosity shear viscosity 10-4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 5: Shear and bulk viscosity of liquid water plotted as a function of the upper limit of the integrals of the ACFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' are probably uncorrelated and the viscosity is converged but the statistical error is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 8, 9, 10, and 11 we plot the values of the shear and bulk viscosity of liquid water and Ar evaluated by using different numbers of blocks (keeping constant the total number of configurations) with the relative statistical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' The dashed horizontal lines indicate the corresponding values inferred by considering the maximas-plateaus of the curves in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' As can be seen, in the case of liquid water, the maxima of the shear and 0 1 2 3 4 5 6 time (ps) 0 1 2 3 4 ( Pa s) bulk viscosity shear viscosity 10-4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 6: Shear and bulk viscosity of liquid Ar plotted as a function of the upper limit of the integrals of the ACFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' bulk viscosities are obtained considering 16 blocks, each equivalent to a simulation time of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='4 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Interestingly, taking statistical uncertainties into account, these maxima are compatible with the rough estimates obtained before and, for the shear viscosity, also with the values obtained using the Stokes-Einstein formula (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' (4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' As already described above, in the Stokes-Einstein estimate the shear viscosity is obtained in terms of the diffusion coefficient D and the radius of the first peak in the radial distribution function (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' (4), 0 1 2 3 4 time (ps) 0 0,05 0,1 thermal conductivity (W/m K) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 7: Thermal conductivity of liquid Ar plotted as a function of the upper limit of the integral of the ACF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' for liquid water we have considered the first peak in the oxygen-oxygen radial distribution function, see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Actually our reported Stokes-Einstein estimated values are corrected by finite-size effects: in fact D can be extrapolated to infinite size of the simulation box (see, for instance, ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 52) by just considering the shear-viscosity value: D∞ = D + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='837 kBT 6πηSL , (11) where L is the size of the cubic simulation box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Therefore, by simultaneously taking into account Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' (4) and (11), one can get a “self-consistent”, finite-size corrected Stokes- Einstein estimate for ηS : η∗ S = kBT 2πaD − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='837 kBT 6πLD .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' (12) Quantitative data are collected in Table I where they are also compared with some the- oretical and experimental reference values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' As far as the shear viscosity is concerned, for liquid water our estimated value, obtained from the NVE simulation at an average temperature of 366 K, agrees with the experimental reference data at a lower temperature of about 330 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' This is in line with the performances of other DFT functionals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' for instance (see Table I), in recent simulations52 of liquid water based on the SCAN functional,59 the shear viscosity estimate is close to that obtained from a force-field approach (that, for this quantity, well reproduces the experimental behavior) only between 330 and 360 K, while it is severely overestimated at 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' With the OPTB88- vdW functional60 reasonable agreement with experimental data at room temperature is only found52 at 360 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' For liquid Ar the behavior is qualitatively similar (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 10, 11, and 12 for the shear viscosity, the bulk viscosity, and the thermal conductivity, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' In this case both the maxima of the shear and bulk viscosities are obtained considering 5 blocks, each equivalent to a simulation time of about 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='0 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Even in this case, taking statistical uncertainties into account, these maxima are compatible with the plateau positions and, for the shear viscosity, also with the estimate from the Stokes-Einstein formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' The maximum of the thermal conductivity is instead reached with 25 blocks, each equivalent to a simulation time of about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='4 ps, and its value (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='11±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='02 W/m K) is again compatible with that estimated considering the maximum-plateau position and in good agreement with the literature reference value (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='12 W/m K) at 90 K61 and that obtained by classical MD simulations based on the Lennard-Jones potential (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='119 W/m K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='62 An interesting physical quantity is represented by the ratio between bulk and shear viscosity, which can be related to the ratio of observed to classical absorption coefficients in 0 5 10 15 20 25 30 35 # of blocks 0 2 4 6 8 shear viscosity ( Pa s) 10-4 expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 303 K expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 323 K expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 333 K FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 8: Shear viscosity of liquid water evaluated by using different numbers of blocks (the smaller is the block number the larger is the number of configurations of each block) with the relative sta- tistical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' The dashed horizontal line indicates the position of the first-pronounced maximum- plateau of the corresponding curve of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' The asterisk denotes the value obtained by the Stokes-Einstein formula (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='12), while the triangles indicate experimental estimates at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' TABLE I: Shear and bulk viscosity of liquid water and Ar, in 10−4 Pa s, compared with theoretical and experimental reference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Statistical errors are in parenthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' η∗ S indicates the shear viscosity estimate obtained by the Stokes-Einstein relation (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' system ηS η∗ S ηB ηB/ηS 3/4ηB/ηS + 1 water (366 K) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='8(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='7) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='7 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='3(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='9) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='4(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='8) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='8(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='6) water DFT SCANa (300K) 23 — — — — water DFT SCANa (330K) 6 — — — — water DFT SCANa (360K) 5 — — — — water DFT OPTB88-vdWa (300K) 30 — — — — water DFT OPTB88-vdWa (330K) 15 — — — — water DFT OPTB88-vdWa (360K) 8 — — — — water force fielda (300K) 8 — — — — water force fielda (330K) 5 — — — — water force fielda (360K) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='5 — — — — water force fieldb (303K) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='5(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='4) — 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='5(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='6) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='4(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='8(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='2) water expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='c (298 K) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='90 — — — — water expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='b,d,e (303 K) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='97 — 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='0 water expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='f (323 K) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='47 — 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='0 water expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='c (333 K) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='67 — — — — Ar (129 K) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='7(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='6) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='0(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='1(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='8) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='8(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='6) Ar expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='g (90 K) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='33 — 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='6 Ar expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='h (90 K) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='57 — — — aref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' bref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' cref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' dref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' eref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' fref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' gref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' href.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 0 5 10 15 20 25 30 35 # of blocks 0 5 10 15 bulk viscosity ( Pa s) 10-4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 9: Bulk viscosity of liquid water evaluated by using different numbers of blocks (the smaller is the block number the larger is the number of configurations of each block) with the relative sta- tistical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' The dashed horizontal line indicates the position of the first-pronounced maximum- plateau of the corresponding curve of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' ultrasonic absorption experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='13 In fact, under the condition that the heat conductivity contribution to the ultrasonic absorption may be neglected, 0 10 20 30 40 50 # of blocks 0 1 2 3 4 5 6 shear viscosity ( Pa s) 10-4 expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 90 K FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 10: Shear viscosity of liquid Ar evaluated by using different numbers of blocks (the smaller is the block number the larger is the number of configurations of each block) with the relative statis- tical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' The dashed horizontal line indicates the position of the first-pronounced maximum- plateau of the corresponding curve of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' The asterisk denotes the value obtained by the Stokes-Einstein formula (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='4), while the triangle indicates the experimental estimate at 90 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' α αclass = 3/4ηB ηS + 1 , (13) and water belongs to the group of the so-called ”associated liquids”, characterized by a ratio from 1 to 3, where structural relaxation is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Classical MD simulations based on the SPC/E semiempirical potential predict13 a ηB ηS ratio of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='4, leading to a α αclass ratio of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='79, in reasonable agreement with the experimental 0 10 20 30 40 50 # of blocks 0 1 2 3 4 5 6 7 bulk viscosity ( Pa s) 10-4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 11: Bulk viscosity of liquid Ar evaluated by using different numbers of blocks (the smaller is the block number the larger is the number of configurations of each block) with the relative statis- tical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' The dashed horizontal line indicates the position of the first-pronounced maximum- plateau of the corresponding curve of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' value of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='53 Instead normal liquids, such as monatomic liquids (for instance liquid Ar) are characterized by a ratio no greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Although in general the ratio varies with temperature and pressure, in liquid water it is found to remain constant within 20% in the temperature range 0-90 C (273-363 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='63 By taking statistical errors into account, our estimated value of the α αclass ratio (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='6) is compatible with the available experimental 0 10 20 30 40 50 # of blocks 0 0,05 0,1 0,15 0,2 0,25 thermal conductivity (W/m K) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 12: Thermal conductivity of liquid Ar evaluated by using different numbers of blocks (the smaller is the block number the larger is the number of configurations of each block) with the relative statistical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' The dashed horizontal line indicates the position of the maximum-plateau of the corresponding curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' data at ambient temperature (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' This is a remarkable result, considering that most of the reported classical MD simulations13 predict a bulk viscosity lower than the the experimental one, leading to an underestimated value of the α αclass ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' One should also point out that a proper comparison with experimental data requires a careful analysis taking into account the pronounced temperature dependence of shear and bulk viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' In fact, according to a common empirical model,56,64 the viscosity strongly de- creases with increasing temperature following an exponential decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' By fitting experimental data56 with an exponential function and taking statistical errors into account, our estimated values of the shear and bulk viscosity of liquid water are compatible with experimental data in the temperature range of 323-344 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' One should also consider that also the bulk- viscosity/shear-viscosity ratio for liquid water tends to decrease slightly with temperature,56 suggesting an even better agreement between our estimated value and the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='56 We remind that our simulations have been carried out at temperatures higher than ambient temperature to guarantee that the systems is liquid-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' By considering that our estimate (after finite-size correction) for the diffusion coefficient, D = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='02 × 10−5 cm2/s, corresponds to the experimental value measured at about 336 K,65 we can conclude that, our DFT simulations based on the DFT-D2(BLYP) functional and performed at a nominal average temperature of 366 K, actually describe the basic dynamical properties of liquid water at about 330 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' One should also mention that bulk-viscosity measurements are in- direct and affected by considerable errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='13,27,33,56,66,67 In summary, we can conclude that our adopted BLYP-D2 functional is able to describe reasonably well the density fluctuations of liquid water;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' the discrepancy with respect to experimental data at ambient conditions can be to a large extend explained in terms of the pronounced temperature dependence of both shear and bulk viscosity and the need to perform first-principles MD simulations at temperatures higher than ambient temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' As far as liquid Ar is concerned, our shear and bulk viscosities, computed by first- principles at a nominal average simulation temperature of 129 K, turn out to be some- how overestimated with respect to the reference experimental values at 90 K, although they are compatible with them if statistical errors are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Moreover our bulk-viscosity/shear-viscosity ratio (close to unity) agrees well with the reference estimate, while interestingly this is not the case if a standard Lennard-Jones empirical potential is adopted using classical MD simulations that predict instead a very low value13,62 of the ratio (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='17-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='35 at high densities), thus showing that this popular potential cannot properly re- produce all the dynamical properties of liquid Ar and underlining once again the superiority of first-principles approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' We conclude our study by reporting some basic structural properties of our investigated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' In particular, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 13, for liquid water we plot our computed O-O pair correlation 2 3 4 5 6 7 r (A) 0 0,5 1 1,5 2 2,5 3 g(r) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 13: O-O pair correlation function, gOO(r), compared with that obtained experimentally from X-ray diffraction measurements at ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='68–70 function, gOO(r), compared with that obtained experimental from X-ray diffraction measure- ments at ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='68–70 The main features of the gOO(r) curves are reported in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' As can be seen, there is a good agreement between the two curves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' the fact the oscillations of our computed curve are slightly reduced with respect to the experimental one can again be related to the higher effective temperature of our simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 2 3 4 5 6 7 r (A) 0 0,5 1 1,5 2 2,5 3 3,5 g(r) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 14: Ar-Ar pair correlation function, g(r), compared with that obtained experimentally from neutron-scattering measurements at 85K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='73 In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 14, for liquid Ar our computed Ar-Ar pair correlation function, g(r), is compared with that obtained experimentally from neutron-scattering measurements at 85 K,73 while again the main features of the g(r) curves are reported in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Even in this case there is a reasonable agreement between the simulation and experimental curve, by considering that simulations for liquid Ar have been performed at significantly higher temperature (129 K) than experiments (85 K) (note that the experimental melting and boiling points of Ar are at 84 and 87 K, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' After applying the same finite-size correction adopted TABLE II: Main features of the O-O pair correlation function, gOO(r), of liquid water and of the Ar-Ar pair correlation function, g(r) of liquid Ar compared with experimental reference data, obtained from X-ray diffraction measurements at ambient conditions for liquid water and neutron- scattering measurements for liquid Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' rmax and rmin indicate the position of the first maximum (the main peak) and the first minimum of gOO(r) and g(r), respectively, and gmax and gmin the corresponding values of the gOO(r) and g(r) functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' system rmax(˚A) gmax rmin(˚A) gmin water (366 K) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='79 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='42 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='88 water expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='a (293 K) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='80(1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='55(5) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='41(4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='85(2) Ar (129 K) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='80 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='64 Ar expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='b (85 K) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='68 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='56 aref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='68–70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' bref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' above for liquid water, our estimated diffusion coefficient for liquid Ar, D = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='82 × 10−5 cm2/s, evaluated at a nominal simulation temperature of 129 K is significantly higher than the reference value (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='6 × 10−5 cm2/s) reported at 84 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content='71 Again this discrepancy can be explained in terms of the higher temperature of the liquid Ar simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' CONCLUSIONS Shear and bulk viscosity of liquid water and Argon have been evaluated, together with other structural and dynamical properties, from first principles by adopting a vdW-corrected DFT approach, by performing Molecular Dynamics simulations in the NVE ensemble and using the Kubo-Greenwood equilibrium approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' For liquid Argon the thermal conductivity has been also calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Concerning liquid water, to our knowledge this is the first estimate of the bulk viscosity and of the shear-viscosity/bulk-viscosity ratio from first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' By analyzing our results and comparing then with reference experimental data, we can conclude that our first-principles simulations, performed at a nominal average temperature of 366 K to guarantee that the systems is liquid-like, actually describe well the basic dynamical properties of liquid water at about 330 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' In comparison with liquid water, the normal, monatomic liquid Ar is characterized by a much smaller bulk-viscosity/shear-viscosity ratio (close to unity) and this feature is well reproduced by our first-principles approach which predicts a value of the ratio in better agreement with experimental reference data than that obtained using the empirical Lennard-Jones potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' The computed thermal conductivity of liquid Argon is also in good agreement with the experimental value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We acknowledge funding from Fondazione Cariparo, Progetti di Eccellenza 2017, relative to the project: ”Engineering van der Waals Interactions: Innovative paradigm for the control of Nanoscale Phenomena”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 1 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Allen and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Tildesley, Computer Simulations of Liquids (Oxford Science Publications, Clarendon Press, Oxford 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Helfand, ”Transport Coefficients from Dissipation in a Canonical Ensemble”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 119, 1 (1960).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' 3 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Alder, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Gass, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Wainwright, ”Studies in Molecular Dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' The Trans- port Coefficients for a Hard-Sphere Fluid”, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Phys.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} +page_content=' A 7, 2130 (1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FQT4oBgHgl3EQfPDYj/content/2301.13277v1.pdf'} diff --git a/CNA0T4oBgHgl3EQfAP_i/content/tmp_files/2301.01961v1.pdf.txt b/CNA0T4oBgHgl3EQfAP_i/content/tmp_files/2301.01961v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e64234b084be6f4e48fbf7f41442242594d18920 --- /dev/null +++ b/CNA0T4oBgHgl3EQfAP_i/content/tmp_files/2301.01961v1.pdf.txt @@ -0,0 +1,1182 @@ +arXiv:2301.01961v1 [math.AG] 5 Jan 2023 +SOME MORE FANO THREEFOLDS WITH A MULTIPLICATIVE +CHOW–K ¨UNNETH DECOMPOSITION +ROBERT LATERVEER +ABSTRACT. We exhibit several families of Fano threefolds with a multiplicative Chow–K¨unneth +decomposition, in the sense of Shen–Vial. As a consequence, a certain tautological subring of the +Chow ring of powers of these threefolds injects into cohomology. As a by-product of the argu- +ment, we observe that double covers of projective spaces admit a multiplicative Chow–K¨unneth +decomposition. +1. INTRODUCTION +Given a smooth projective variety Y over C, let Ai(Y ) := CHi(Y )Q denote the Chow groups +of Y (i.e. the groups of codimension i algebraic cycles on Y with Q-coefficients, modulo rational +equivalence). The intersection product defines a ring structure on A∗(Y ) = � +i Ai(Y ), the Chow +ring of Y [14]. +In the special case of K3 surfaces, this ring structure has remarkable properties: +Theorem 1.1 (Beauville–Voisin [3]). Let S be a projective K3 surface. The Q-subalgebra +� +A1(S), cj(S) +� +⊂ A∗(S) +injects into cohomology under the cycle class map. +Theorem 1.2 (Voisin [54], Yin [57]). Let S be a projective K3 surface, and m ∈ N. The Q- +subalgebra +R∗(Sm) := +� +A1(S), ∆S +� +⊂ A∗(Sm) +(generated by pullbacks of divisors and pullbacks of the diagonal ∆S ⊂ S × S) injects into +cohomology under the cycle class map for all m ≤ 2 dim H2 +tr(S, Q)+1 (where H2 +tr(S, Q) denotes +the transcendental part of cohomology). Moreover, R∗(Sm) injects into cohomology for all +m ∈ N if and only if S is Kimura finite-dimensional. +The Chow ring of abelian varieties also has an interesting property: there is a multiplicative +splitting, defined by the Fourier transform [1]. +Motivated by the particular behaviour of K3 surfaces and abelian varieties, Beauville [2] has +conjectured that for certain special varieties, the Chow ring should admit a multiplicative split- +ting. In the wake of Beauville’s “splitting property conjecture”, Shen–Vial [47] have introduced +the concept of multiplicative Chow–K¨unneth decomposition (we will abbreviate this to “MCK +Key words and phrases. Algebraic cycles, Chow group, motive, Beauville’s “splitting property” conjecture, mul- +tiplicative Chow–K¨unneth decomposition, Fano threefolds, tautological ring. +2020 Mathematics Subject Classification: 14C15, 14C25, 14C30. +Supported by ANR grant ANR-20-CE40-0023. +1 + +2 +ROBERT LATERVEER +decomposition”). With the concept of MCK decomposition, it is possible to make concrete sense +of this elusive “splitting property conjecture” of Beauville. +It is hard to understand precisely which varieties admit an MCK decomposition. To give an +idea of what is known: hyperelliptic curves have an MCK decomposition [47, Example 8.16], +but the very general curve of genus ≥ 3 does not have an MCK decomposition [12, Example +2.3]; K3 surfaces have an MCK decomposition, but certain high degree surfaces in P3 do not +have an MCK decomposition (cf. the examples given in [43], cf. also section 2 below). +In this note, we will focus on Fano threefolds and ask the following question: +Question 1.3. Let X be a Fano threefold with Picard number 1. Does X admit an MCK decom- +position ? +The restriction on the Picard number is necessary to rule out a counterexample of Beauville +[2, Examples 9.1.5]. The answer to Question 1.3 is affirmative for cubic threefolds [8], [12], for +intersections of 2 quadrics [32], for intersections of a quadric and a cubic [34], and for prime +Fano threefolds of genus 8 [37] and of genus 10 [38]. +The main result of this paper answers Question 1.3 for several more families of Fano three- +folds: +Theorem (=Theorem 4.1). The following smooth Fano threefolds have a multiplicative Chow– +K¨unneth decomposition: +• hypersurfaces of weighted degree 6 in weighted projective space P(13, 2, 3); +• quartic double solids; +• sextic double solids; +• double covers of a quadric in P4 branched along the intersection with a quartic; +• special Gushel–Mukai threefolds. +In Table 1 (at the end of this paper), we have listed all Fano threefolds of Picard number 1 and +what is known about MCK for them. +To prove Theorem 4.1, we provide a general criterion (Proposition 3.3), that may be useful in +other situations. For example, using this criterion we also prove the following: +Proposition (=Proposition 3.6). Let X be a smooth projective variety such that X → Pn is a +double cover ramified along a smooth divisor D ⊂ Pn of degree d > n. Then X admits an MCK +decomposition. +As a consequence of Theorem 4.1, we obtain an injectivity result similar to Theorem 1.2: +Corollary (cf. Theorem 5.1). Let Y be a Fano threefold as in Theorem 4.1, and m ∈ N. Let +R∗(Y m) := +� +h, ∆Y +� +⊂ +A∗(Y m) +be the Q-subalgebra generated by pullbacks of the polarization h ∈ A1(Y ) and pullbacks of the +diagonal ∆Y ∈ A3(Y × Y ). The cycle class map induces injections +R∗(Y m) ֒→ H∗(Y m, Q) for all m ∈ N . + +SOME MORE FANO THREEFOLDS WITH AN MCK DECOMPOSITION +3 +Conventions. In this paper, the word variety will refer to a reduced irreducible scheme of finite +type over C. A subvariety is a (possibly reducible) reduced subscheme which is equidimensional. +All Chow groups will be with rational coefficients: we will denote by Aj(Y ) the Chow group +of j-dimensional cycles on Y with Q-coefficients; for Y smooth of dimension n the notations +Aj(Y ) and An−j(Y ) are used interchangeably. The notation Aj +hom(Y ) will be used to indicate +the subgroup of homologically trivial cycles. For a morphism f : X → Y , we will write Γf ∈ +A∗(X × Y ) for the graph of f. +The contravariant category of Chow motives (i.e., pure motives with respect to rational equiv- +alence as in [46], [41]) will be denoted Mrat. +2. MCK DECOMPOSITION +Definition 2.1 (Murre [40]). Let X be a smooth projective variety of dimension n. We say that +X has a CK decomposition if there exists a decomposition of the diagonal +∆X = π0 +X + π1 +X + · · · + π2n +X +in An(X × X) , +such that the πi +X are mutually orthogonal idempotents and (πi +X)∗H∗(X, Q) = Hi(X, Q). +(NB: “CK decomposition” is shorthand for “Chow–K¨unneth decomposition”.) +Remark 2.2. Murre has conjectured that any smooth projective variety should have a CK de- +composition [40], [20]. +Definition 2.3 (Shen–Vial [47]). Let X be a smooth projective variety of dimension n, and let +∆sm +X ∈ A2n(X × X × X) denote the class of the small diagonal +∆sm +X := +� +(x, x, x) | x ∈ X +� +⊂ X × X × X . +An MCK decomposition is defined as a CK decomposition {πi +X} of X that is multiplicative, i.e. +it satisfies +πk +X ◦ ∆sm +X ◦ (πi +X × πj +X) = 0 in A2n(X × X × X) for all i + j ̸= k . +(NB: “MCK decomposition” is shorthand for “multiplicative Chow–K¨unneth decomposition”.) +Remark 2.4. The small diagonal (when considered as a correspondence from X × X to X) +induces the multiplication morphism +∆sm +X : +h(X) ⊗ h(X) → h(X) in Mrat . +Let us assume X has a CK decomposition +h(X) = +2n +� +i=0 +hi(X) in Mrat . +By definition, this decomposition is multiplicative if for any i, j the composition +hi(X) ⊗ hj(X) → h(X) ⊗ h(X) +∆sm +X +−−→ h(X) in Mrat +factors through hi+j(X). + +4 +ROBERT LATERVEER +If X has an MCK decomposition, then setting +Ai +(j)(X) := (π2i−j +X +)∗Ai(X) , +one obtains a bigraded ring structure on the Chow ring: that is, the intersection product sends +Ai +(j)(X) ⊗ Ai′ +(j′)(X) to Ai+i′ +(j+j′)(X). +It is conjectured that for any X with an MCK decomposition, one has +Ai +(j)(X) +??= 0 for j < 0 , +Ai +(0)(X) ∩ Ai +hom(X) +??= 0 ; +this is related to Murre’s conjectures B and D, that have been formulated for any CK decompo- +sition [40]. +For more background on the concept of MCK, and for examples of varieties with an MCK +decomposition, we refer to [47, Section 8], as well as [53], [48], [13], [28], [39], [29], [30], [31], +[12], [33], [34], [36], [42]. +3. A GENERAL CRITERION +We develop a general criterion for having an MCK. The criterion hinges on the Franchetta +property for families of varieties, which is defined as follows: +Definition 3.1. Let X → B be a smooth projective morphism, where X , B are smooth quasi- +projective varieties, and let us write Xb for the fiber over b ∈ B. We say that X → B has the +Franchetta property in codimension j if the following holds: for every Γ ∈ Aj(X ) such that the +restriction Γ|Xb is homologically trivial for the very general b ∈ B, the restriction Γ|b is zero in +Aj(Xb) for all b ∈ B. +We say that X → B has the Franchetta property if X → B has the Franchetta property in +codimension j for all j. +This property is studied in [45], [5], [10], [11]. +Definition 3.2. Given a family X → B as in Definition 3.1, we use the shorthand +GDAj +B(Xb) := Im +� +Aj(X ) → Aj(Xb) +� +⊂ Aj(Xb) +(GDA∗() stands for the “generically defined cycles”). +The Franchetta property for X → B means that the generically defined cycles inject into +cohomology. +Proposition 3.3. Let X → B be a family of smooth projective varieties of relative dimension n, +with fiber Xb. Assume the following: +(i) the family X ×B X → B has the Franchetta property; +(ii) there exists a projective quotient variety P (i.e. P = P ′/G where P ′ is smooth projective +and G ⊂ Aut(P ′) is a finite cyclic group) with trivial Chow groups (i.e. A∗ +hom(P) = 0), such +that Xb → P is a double cover with branch locus a smooth ample divisor, for all b ∈ B. +Then Xb admits an MCK decomposition, for all b ∈ B. +Proof. We have the following Lefschetz-type result in cohomology: + +SOME MORE FANO THREEFOLDS WITH AN MCK DECOMPOSITION +5 +Lemma 3.4. Let Xb → P be as in the proposition. Then pullback +Hi(P, Q) → Hi(Xb, Q) +is an isomorphism for i < n, and injective for i = n. +Proof. In case P is smooth, this is a result of Cornalba [6]. The general case is readily deduced +from this: assume P = P ′/G where P ′ is smooth projective and G ⊂ Aut(P ′) is a finite cyclic +group, and consider the fiber square +X′ +b +→ +Xb +↓ +↓ +P ′ +→ +P . +Cornalba’s result applies to the double cover of the left-hand vertical arrow, and so pullback +Hi(P ′, Q) → Hi(X′ +b, Q) +is an isomorphism for i < n, and injective for i = n. The G-action on P ′ lifts to X′ +b, and taking +G-invariants we find that +Hi(P, Q) = Hi(P ′, Q)G → Hi(X′ +b, Q)G = Hi(Xb, Q) +is an isomorphism for i < n, and injective for i = n. +□ +Since H∗(P, Q) is algebraic (this is a general fact for any variety with trivial Chow groups, cf. +[23]), this implies that also Hi(Xb, Q) is algebraic, for all i ̸= n. More precisely, for i ̸= n odd, +one has Hi(Xb, Q) = 0 while for i < n even, one has isomorphisms +Ai/2(P) ∼= Hi(Xb, Q) , +induced by pullback. This implies that for i < n the K¨unneth components πi +Xb are algebraic, and +generically defined. To define the K¨unneth components πi +Xb explicitly, let p: Xb → P denote +the projection morphism, and let πi +P denote the (unique) CK decomposition of P. One can then +define +πi +Xb := 1/2 tΓp ◦ πi +P ◦ Γp if i < n , +πi +Xb := π2n−i +Xb +if i > n , +πn,fix +Xb +:= 1/2 tΓp ◦ πn +P ◦ Γp , +πn,var +Xb +:= ∆Xb − +� +j̸=n +πj +Xb − πn,fix +Xb +, +πn +Xb := πn,fix +Xb ++ πn,var +Xb +∈ An(Xb × Xb) . +(Note that πn +Xb = 0 in case n is odd.) The notation is meant to remind the reader that πn,fix +Xb +and +πn,var +Xb +are projectors on the fixed part resp. the variable part of cohomology in degree n. + +6 +ROBERT LATERVEER +These projectors define a generically defined CK decomposition for each Xb, i.e. all projectors +are in GDAn +B(Xb × Xb). This CK decomposition has the property that +hj(Xb) := (Xb, πj +Xb, 0) = ⊕1(∗) ∀j ̸= n , +hn,fix(Xb) := (Xb, πn,fix +Xb +, 0) = ⊕1(∗) in Mrat . +(1) +Let us now proceed to verify that this CK decomposition is MCK. What we need to check is +the vanishing +πk +Xb ◦ ∆sm +Xb ◦ (πi +Xb × πj +Xb) = 0 in A2n(Xb × Xb × Xb) for all i + j ̸= k . +First, let us assume that at least one of the 3 integers (i, j, k) is different from n, and i+j ̸= k. +In this case, we have that +πk +Xb ◦ ∆sm +Xb ◦ (πi +Xb × πj +Xb) = (tπi +Xb × tπj +Xb × πk +Xb)∗∆sm +Xb += (π2n−i +Xb +× π2n−j +Xb +× πk +Xb)∗∆sm +Xb +֒→ +� +A∗(Xb × Xb) . +Here the first equality is an application of Lieberman’s lemma [41, Lemma 2.1.3], and the in- +clusion follows from property (1). The resulting cycle in � A∗(Xb × Xb) is generically defined +(since the π∗ +Xb and ∆sm +Xb are) and homologically trivial (since i + j ̸= k). By assumption (i), the +resulting cycle in � A∗(Xb × Xb) is rationally trivial, and so +πk +Xb ◦ ∆sm +Xb ◦ (πi +Xb × πj +Xb) = 0 in A2n(Xb × Xb × Xb) , +as desired. +It remains to treat the case i = j = k = n. The decomposition πn +Xb := πn,fix +Xb ++ πn,var +Xb +induces +a decomposition +πn +Xb ◦ ∆sm +Xb ◦ (πn +Xb × πn +Xb) =πn,fix +Xb +◦ ∆sm +Xb ◦ (πn,fix +Xb +× πn,fix +Xb +) ++ πn,fix +Xb +◦ ∆sm +Xb ◦ (πn,fix +Xb +× πn,var +Xb +) ++ · · · · · · ++ πn,var +Xb +◦ ∆sm +Xb ◦ (πn,var +Xb +× πn,var +Xb +) in A2n(Xb × Xb × Xb) . +Using property (1) and the Franchetta property for Xb × Xb, all summands containing πn,fix +Xb +vanish. One is left with the last term. To deal with the last term, we observe that the covering +involution ι ∈ Aut(Xb) of the double cover p: Xb → P induces a splitting of the motive +h(Xb) =h(Xb)+ ⊕ h(Xb)− +:=(Xb, 1/2 (∆Xb + Γι), 0) ⊕ (Xb, 1/2 (∆Xb − Γι), 0) in Mrat , +where Γι denotes the graph of the involution ι. Moreover, there is equality +hn,var(Xb) = h(Xb)− in Mrat . +But the intersection product map +h(Xb)− ⊗ h(Xb)− +∆sm +Xb +−−→ h(Xb) + +SOME MORE FANO THREEFOLDS WITH AN MCK DECOMPOSITION +7 +factors over h(Xb)+, as is readily seen (cf. Lemma 3.5 below), which is saying exactly that +πn,var +Xb +◦ ∆sm +Xb ◦ (πn,var +Xb +× πn,var +Xb +) = 0 in A2n(Xb × Xb × Xb) . +This closes the proof, modulo the following lemma (which is probably well-known, but we +include a proof for completeness): +Lemma 3.5. Let X → P be a double cover, where X and P are quotient varieties, and let +ι ∈ Aut(X) be the covering involution. Let +h(X)+ := (X, 1/2 (∆X + Γι, 0) , +h(X)− := (X, 1/2 (∆X − Γι), 0) in Mrat . +The map of motives +h(X)− ⊗ h(X)− +∆sm +X +−−→ h(X) +factors over h(X)+. +To prove the lemma, let ι ∈ Aut(X) denote the covering involution. The motive h(X)− is +defined by the projector +∆− +X := 1/2 (∆X − Γι) +∈ An(X × X) . +Plugging this in and developing, it follows that +∆− +X ◦ ∆sm +X ◦ (∆− +X × ∆− +X) = 1/8 (∆X − Γι) ◦ ∆sm +X ◦ (∆X×X − ∆X × Γι − Γι × ∆X + Γι × Γι) += 1/8 +� +∆X ◦ ∆sm +X ◦ (∆X × ∆X) + · · · − Γι ◦ ∆sm +X ◦ (Γι × Γι) +� += 1/8 +� +∆sm +X +− (id × id ×ι)∗(∆sm +X ) − (id ×ι × id)∗(∆sm +X ) − (ι × id × id)∗(∆sm +X ) ++ (id ×ι × ι)∗(∆sm +X ) + (ι × id ×ι)∗(∆sm +X ) + (ι × ι × id)∗(∆sm +X ) +− (ι × ι × ι)∗(∆sm +X ) +� +in A2n(X × X × X) . +Here the last equality is by virtue of Lieberman’s lemma [41, Lemma 2.1.3]. However, we have +equality +∆sm +X = {(x, x, x) | x ∈ X} = (ι × ι × ι)∗(∆sm +X ) in A2n(X × X × X) , +and so the sum of the first and last summand vanish. Likewise, we have equality +(id ×ι×ι)∗(∆sm +X ) = (id ×ι×ι)∗(ι×ι×ι)∗(∆sm +X ) = (ι×id × id)∗(∆sm +X ) in A2n(X ×X ×X) , +and so the other summands cancel each other pairwise. This proves the lemma. +□ +As a first application of our general criterion, we now proceed to show the following: +Proposition 3.6. Let X be a smooth projective variety such that X → Pn is a double cover +ramified along a smooth divisor D ⊂ Pn, and assume either dim Hn(X, Q) > 1, or D has +degree d > n. Then X admits an MCK decomposition. + +8 +ROBERT LATERVEER +Proof. Double covers X as in the proposition are exactly the smooth hypersurfaces of degree 2d +in the weighted projective space P := P(1n+1, d), where 2d := deg D. Let +B ⊂ ¯B := PH0(P, OP(2d)) +denote the Zariski open parametrizing smooth hypersurfaces, and let +B × P ⊃ X → B +denote the universal family. In view of Proposition 3.3, it suffices to check that the family +X ×B X → B has the Franchetta property. +To this end, we remark that the line bundle OP(2d) is very ample (cf. Lemma 3.7 below), +which means that the set-up verifies condition (∗2) of [12, Definition 2.5]. An application of the +stratified projective bundle argument [12, Proposition 2.6] then implies that +(2) +GDA∗ +B(Xb × Xb) = +� +(pi)∗(h), ∆Xb +� +, +where we write h ∈ A1(Xb) for the hyperplane class. The excess intersection formula [14, +Theorem 6.3] gives an equality +∆Xb · (pi)∗(h) = 2d +� +j +(p1)∗(hj) · (p2)∗(hn+1−j) in An+1(Xb × Xb) , +and so equality (2) reduces to the equality +GDA∗ +B(Xb × Xb) = +� +(p1)∗(h), (p2)∗(h) +� +⊕ Q[∆Xb] . +The “decomposable part” ⟨(p1)∗(h), (p2)∗(h)⟩ injects into cohomology, because of the K¨unneth +formula for H∗(Xb × Xb, Q). The class of the diagonal in cohomology is linearly independent +from the decomposable part: indeed, if the diagonal were decomposable it would act as zero on +the primitive cohomology +Hn +prim(Xb, Q) := Coker +� +Hn(Pn, Q) → Hn(Xb, Q) +� +. +But the assumption dim Hn(Xb, Q) > 1 is equivalent to having Hn +prim(Xb, Q) ̸= 0. This proves +the Franchetta property for X ×B X → B, and closes the proof. +The case d > n is a special case where Hn +prim(Xb, Q) ̸= 0, because it is known that the +geometric genus of Xb is +pg(Xb) = +�d − 1 +n +� +[9, Section 3.5.4]. +It remains to prove the following, which we have used above: +Lemma 3.7. Let P := P(1n+1, d). The sheaf OP(d) is locally free and very ample. +The assertion about the sheaf being locally free is just because d is a multiple of the weights +of P (cf. [7, Remarques 1.8]). As for the very ampleness, we apply Delorme’s criterion [7, +Proposition 2.3(iii)] (cf. also [4, Theorem 4.B.7]). To prove very ampleness of OP(d), we need +to prove that the integer E as defined in [7] and [4] is equal to 0. +Let us write x0, . . . , xn, y for the weighted homogeneous coefficients of P, where xj and y +have weight 1 resp. d. It is readily seen that every monomial in xj, y of (weighted) degree + +SOME MORE FANO THREEFOLDS WITH AN MCK DECOMPOSITION +9 +m + dk (where m is a positive multiple of d, and k is any positive integer) is divisible by a +monomial of (weighted) degree dk. This means that the integer E defined in loc. cit. is 0, and so +[7, Proposition 2.3(iii)] implies the very ampleness of OP(d). +This proves the lemma, and ends the proof of the proposition. +□ +Here is another sample application of our general criterion: +Proposition 3.8. Let X ⊂ P(1n, 2, 3) be a smooth hypersurface of (weighted) degree 6. Assume +dim Hn(X, Q) > 1. Then X has an MCK decomposition. +Proof. The varieties X as in the proposition are exactly the smooth double covers of P := +P(1n, 2) branched along a (weighted) degree 6 divisor (cf. [26, Remark 2.3] and for n = 3 +also [17, Theorem 4.2]). Let X → B denote the family of such double covers. We are going to +check that the family X ×B X → B has the Franchetta property. Proposition 3.8 is then a special +case of our general criterion Proposition 3.3. +Let ¯ +X → ¯B ∼= Pr denote the universal family of all (possibly singular) hypersurfaces of +weighted degree 6 in P. The line bundle OP(6) is very ample (cf. Lemma 3.9 below), and so the +projection +¯ +X × ¯B ¯ +X → P × P +has the structure of a stratified projective bundle (with strata the diagonal ∆P and its comple- +ment). One can thus use the stratified projective bundle argument [12, Proposition 2.6] to deduce +the identity +GDA∗ +B(X × X) = +� +(pi)∗GDA∗ +B(X), ∆X +� += +� +(pi)∗(h), ∆X +� +(here, h ∈ A1(X) denotes the restriction to X of an ample generator of A1(P) ∼= Q). +Since X ⊂ P is a hypersurface, the excess intersection formula gives +∆X · (pi)∗(h) = ∆P|X +∈ +� +(pi)∗(h) +� +. +The above identification thus simplifies to +GDA∗ +B(X × X) = +� +(pi)∗(h) +� +⊕ Q[∆X] . +The assumption that dim Hn(X, Q) > 1 implies that the diagonal ∆X is linearly independent +in cohomology from the decomposable classes +� +(pi)∗(h) +� +(indeed, the decomposable classes act +as zero on the primitive cohomology of X, while the diagonal acts as the identity). This shows +that GDA∗ +B(X × X) injects into cohomology, as requested. +Lemma 3.9. Let P := P(1n, 2, 3). The sheaf OP(6) is (locally free and) very ample. +The assertion about the sheaf being locally free is just because 6 is a multiple of all the weights +(cf. [7, Remarques 1.8]). As for the very ampleness, we apply Delorme’s criterion [7, Proposition +2.3(iii)] (cf. also [4, Theorem 4.B.7]). To prove very ampleness of OP(6), we need to prove that +the integer E defined in [7] and [4] is equal to 0. +Let us write x1, . . . , y, z for the weighted homogeneous coefficients of P, where y and z have +weight 2 resp. 3. We need to check that every monomial in xj, y, z of (weighted) degree 6 + 6k +is divisible by a monomial of (weighted) degree 6k (if this is the case, then E = 0 and [7, + +10 +ROBERT LATERVEER +Proposition 2.3(iii)] implies the very ampleness of OP(6)). In case the monomial contains z2, it +is divisible by z2 and so the condition is satisfied. Assume now the monomial contains only one +z. In case the monomial contains y3 it is divisible by y3. Next, if the monomial contains y (or +y2) it is divisible by zyxj (for some j) and so the condition is satisfied. A monomial in z and xj +obviously satisfies the condition. Finally, monomials in xj satisfy the condition. +This proves the lemma, and ends the proof of the proposition. +□ +4. MAIN RESULT +Theorem 4.1. The following Fano threefolds admit an MCK decomposition: +(i) hypersurfaces of weighted degree 6 in weighted projective space P(13, 2, 3); +(ii) quartic double solids; +(iii) sextic double solids; +(iv) double covers of a quadric in P4 branched along the intersection with a quartic; +(v) special Gushel–Mukai threefolds. +Proof. The cases (ii) and (iii) are immediate applications of Proposition 3.6. The case (i) is a +special case of Proposition 3.8. +Before proving case (iv), let us first state a preparatory lemma: +Lemma 4.2. Let Z ⊂ P := P(15, 2) be a smooth weighted hypersurface of degree 2. Then +∆Z = 1 +2 +4 +� +j=0 +hj × h4−j +in A4(Z × Z) . +Proof. Z is a quotient of a non-singular quadric in P5 and so Z has trivial Chow groups (i.e. +A∗ +hom(Z) = 0). Using [9, 4.4.2], one can compute the Betti numbers of Z and one finds that +they are the same as those of projective space P4. This means that there is a cohomological +decomposition of the diagonal +∆Z = 1 +2 +4 +� +j=0 +hj × h4−j +in H8(Z × Z, Q) . +Since Z (and hence also Z × Z) has trivial Chow groups, the same decomposition holds modulo +rational equivalence, proving the lemma. +□ +Now, to prove case (iv) of Theorem 4.1, we apply our general criterion Proposition 3.3. Let +P := P(15, 2), and let Y → B be the universal family of smooth dimensionally transverse +complete intersections of OP(2) ⊕ OP(4), where the base B is a Zariski open +B ⊂ ¯B := PH0(P, OP(2) ⊕ OP(4)) . +It follows from Lemma 3.7 that OP(2) and OP(4) are very ample line bundles on P, and so +¯Y × ¯B ¯Y → P × P is a stratified projective bundle with strata ∆P and its complement. The usual +stratified projective bundle argument [12, Proposition 2.6] applies, and we find that +GDA∗ +B(Y × Y ) = +� +(pi)∗GDA∗ +B(Y ), ∆Y +� += +� +(pi)∗(h), ∆Y +� + +SOME MORE FANO THREEFOLDS WITH AN MCK DECOMPOSITION +11 +(here, h ∈ A1(Y ) denotes the restriction to Y of an ample generator of A1(P) ∼= Q). Let +Y = Z ∩ Z′, where Z and Z′ ⊂ P are hypersurfaces of (weighted) degree 2 and 4. Up to +shrinking B, we may assume the hypersurface Z is smooth. Since Y ⊂ Z is a divisor, the excess +intersection formula gives +∆Y · (pi)∗(h) = ∆Z|Y +in A4(Y × Y ) . +Using Lemma 4.2, it follows that +∆Y · (pi)∗(h) ∈ +� +(pi)∗(h) +� +. +The above identification thus simplifies to +GDA∗ +B(Y × Y ) = +� +(pi)∗(h) +� +⊕ Q[∆Y ] . +As before, the fact that the diagonal ∆Y is linearly independent from the decomposable corre- +spondences in cohomology now shows that +GDA∗ +B(Y × Y ) → H∗(Y × Y, Q) +is injective, and so Y verifies the hypotheses of Proposition 3.3. +The argument for case (v) is similar to that of (iv). First, in view of the spread argument +[55, Lemma 3.2], it suffices to establish an MCK decomposition for the generic special Gushel– +Mukai threefold Y . Thus we may assume that there exists P ⊂ Gr(2, 5), a smooth complete +intersection of Pl¨ucker hyperplanes, and a double cover p: Y → P branched along a smooth +Gushel–Mukai surface. We now consider the family Y → B of all double covers of P branched +along smooth Gushel–Mukai surfaces (so B ⊂ ¯B is a Zariski open in the projectivized space of +quadratic sections of the cone over P), and we apply our general criterion Proposition 3.3 to this +family. +Lemma 4.3. Let Y → B be the family of double covers of P branched along smooth Gushel– +Mukai surfaces. The family Y → B has the Franchetta property. +Proof. We consider the family ¯Y → ¯B with the projection to the cone C over P. This is a +projective bundle, and so for any fiber Y = Yb with b ∈ B we have +GDA∗ +B(Y ) = Im +� +A∗(C) → A∗(Y ) +� +. +The condition b ∈ B means exactly that Y avoids the summit of the cone C, and so (writing +C◦ ⊂ C for the complement of the summit of the cone) we have +(3) +GDA∗ +B(Y ) = Im +� +A∗(C◦) → A∗(Y ) +� +. +But C◦ → P is an affine bundle, and +A∗(P) = Im +� +A∗(Gr(2, 5)) → A∗(P) +� += +� +h +� +, +where h denotes the restriction to P of a Pl¨ucker hyperplane (this follows from [35, Theorem +3.17], or alternatively from the fact that the derived category of P has a full exceptional collection +of length 4 [44]). Thus, (3) reduces to +GDA∗ +B(Y ) = +� +h +� +. +This proves the Franchetta property for Y . +□ + +12 +ROBERT LATERVEER +Lemma 4.4. Let Y → B be as in Lemma 4.3. The family Y ×B Y → B has the Franchetta +property. +Proof. Let us consider the family ¯Y × ¯B ¯Y → ¯B with the projection to C × C. This is a stratified +projective bundle, with strata ∆C and its complement. Thus, the stratified projective bundle +argument [12, Proposition 2.6] implies that +GDA∗ +B(Y × Y ) = +� +Im +� +A∗(C◦ × C◦) → A∗(Y × Y ) +� +, ∆Y +� +. +Since A∗(C◦) = Im +� +A∗(Gr(2, 5)) → A∗(C◦), we find that +GDA∗ +B(Y × Y ) = +� +Im +� +A∗(Gr(2, 5) × Gr(2, 5)) → A∗(Y × Y ) +� +, ∆Y +� +. +But A∗(Gr(2, 5) × Gr(2, 5)) = A∗(Gr(2, 5)) ⊗ A∗(Gr(2, 5)) since the Grassmannian has trivial +Chow groups, and so +GDA∗ +B(Y × Y ) = +� +GDB(Y ), ∆Y +� += +� +h, ∆Y +� +(where the last equality follows from Lemma 4.3). +To finish the proof of the lemma, we now claim that for any (ordinary or special) Gushel– +Mukai threefold Y we have +(4) +∆Y · h ∈ +� +Im +� +A∗(Gr(2, 5)) → A∗(Y ) +�� +. +Combined with Lemma 4.3, this means that for a special Gushel–Mukai threefold Y (and Y → B +as above) there is equality +GDA∗ +B(Y × Y ) = +� +h +� +⊕ Q[∆Y ] . +Then, since the diagonal is linearly independent in cohomology of +� +h +� +(since h1,2(Y ) ̸= 0), this +proves the lemma. +It remains to prove the claim (4). Using the spread argument [55, Lemma 3.2], it suffices to +prove equality (4) for the very general Gushel–Mukai threefold. Thus, we may assume that Y is +ordinary, and moreover that +Y = Y ′ ∩ Q , +where Q is a quadric and Y ′ = Gr(2, 5) ∩ H1 ∩ H2 is a smooth fourfold (where H1, H2 are +Pl¨ucker hyperplanes) and Y ′ is such that +A∗(Y ′) = Im +� +A∗(Gr(2, 5)) → A∗(Y ′) +� +. +(Indeed, the smooth fourfold Y ′ has trivial Chow groups [35, Corollary 4.6], and the very general +Y ′ has no primitive cohomology, as follows from [35, Lemma 3.15]). The excess intersection +formula then implies that +∆Y · h = 1 +2 ∆Y ′|Y ×Y , +and the claim (4) follows. +□ +Lemma 4.4 being proven, all conditions of Proposition 3.3 are met with, and so fibers Y of the +family Y → B have an MCK decomposition; this settles (v). +□ + +SOME MORE FANO THREEFOLDS WITH AN MCK DECOMPOSITION +13 +5. THE TAUTOLOGICAL RING +Theorem 5.1. Let Y be a Fano threefold of Picard number 1. Assume that Y has an MCK +decomposition, and Y is member of a family Y → B such that Y ×B Y → B has the Franchetta +property. For m ∈ N, let +R∗(Y m) := +� +(pi)∗(h), (pij)∗(∆Y ) +� +⊂ +A∗(Y m) +be the Q-subalgebra generated by pullbacks of the polarization h ∈ A1(Y ) and pullbacks of the +diagonal ∆Y ∈ A3(Y × Y ). (Here pi and pij denote the various projections from Y m to Y resp. +to Y × Y ). The cycle class map induces injections +R∗(Y m) ֒→ H∗(Y m, Q) for all m ∈ N . +Proof. This is inspired by the analogous result for cubic hypersurfaces [11, Section 2.3]. In its +turn, the result of [11] was inspired by analogous results for hyperelliptic curves [49], [50] (cf. +Remark 5.2 below) and for K3 surfaces [54], [57]. +Let d denote the degree of Y , and let 2b := dim H3(Y, Q). As in [11, Section 2.3], let us write +o := 1 +dh3 ∈ A3(Y ) (the “distinguished zero-cycle”) and +τ := ∆Y − 1 +d +3 +� +j=0 +hj × h3−j +∈ A3(Y × Y ) +(this cycle τ is nothing but the projector on the motive h3(Y ) considered above). Moreover, let +us write +hi := (pi)∗(h) ∈ A1(Y m) , +oi := (pi)∗(o) ∈ A3(Y m) , +τi,j := (pij)∗(τ) ∈ A3(Y m) . +We define the Q-subalgebra +¯R∗(Y m) := ⟨oi, hi, τi,j⟩ +⊂ H∗(Y m, Q) +(where i ranges over 1 ≤ i ≤ m, and 1 ≤ i < j ≤ m). One can prove (just as [11, Lemma 2.11] +and [57, Lemma 2.3]) that the Q-algebra ¯R∗(Y m) is isomorphic to the free graded Q-algebra +generated by oi, hi, τij, modulo the following relations: +(5) +oi · oi = 0, +hi · oi = 0, +h3 +i = d oi ; +(6) +τi,j · oi = 0, +τi,j · hi = 0, +τi,j · τi,j = 2b oi · oj ; +(7) +τi,j · τi,k = τj,k · oi ; +(8) +� +σ∈S2b+2 +b+1 +� +i=1 +τσ(2i−1),σ(2i) = 0 . +To prove Theorem 5.1, we need to check that these relations are also verified modulo ratio- +nal equivalence. The relations (5) take place in R∗(Y ) and so they follow from the Franchetta + +14 +ROBERT LATERVEER +property for Y . The relations (6) take place in R∗(Y 2). The first and the last relations are triv- +ially verified, because Y being Fano one has A6(Y 2) = Q. As for the second relation of (6), +this follows from the Franchetta property for Y × Y . (Alternatively, it is possible to deduce the +second relation from the MCK decomposition: indeed, the product τ · hi lies in A4 +(0)(Y 2), and it +is readily checked that A4 +(0)(Y 2) injects into H8(Y 2, Q).) +Relation (7) takes place in R∗(Y 3) and follows from the MCK relation. Indeed, we have +∆sm +Y +◦ (π3 +Y × π3 +Y ) = π6 +Y ◦ ∆sm +Y +◦ (π3 +Y × π3 +Y ) in A6(Y 3) , +which (using Lieberman’s lemma) translates into +(π3 +Y × π3 +Y × ∆Y )∗∆sm +Y += (π3 +Y × π3 +Y × π6 +Y )∗∆sm +Y +in A6(Y 3) , +which means that +τ1,3 · τ2,3 = τ1,2 · o3 in A6(Y 3) . +It is left to consider relation (8), which takes place in R∗(Y 2b+2). To check that this relation is +also verified modulo rational equivalence, we observe that relation (8) involves a cycle contained +in +A∗� +Sym2b+2(h3(Y ) +� +. +But we have vanishing of the Chow motive +Sym2b+2 h3(Y ) = 0 in Mrat , +because dim H3(Y, Q) = 2b and h3(Y ) is oddly finite-dimensional in the sense of Kimura [22] +(all Fano threefolds are known to have Kimura finite-dimensional motive [51, Theorem 4]). This +establishes relation (8), modulo rational equivalence, and ends the proof. +□ +Remark 5.2. Given a curve C and an integer m ∈ N, one can define the tautological ring +R∗(Cm) := +� +(pi)∗(KC), (pij)∗(∆C) +� +⊂ A∗(Cm) +(where pi, pij denote the various projections from Cm to C resp. C × C). Tavakol has proven +[50, Corollary 6.4] that if C is a hyperelliptic curve, the cycle class map induces injections +R∗(Cm) ֒→ H∗(Cm, Q) for all m ∈ N . +On the other hand, there are many (non hyperelliptic) curves for which the tautological ring +R∗(C3) does not inject into cohomology (this is related to the non-vanishing of the Ceresa cycle, +cf. [50, Remark 4.2] and also [12, Example 2.3 and Remark 2.4]). + +SOME MORE FANO THREEFOLDS WITH AN MCK DECOMPOSITION +15 +6. A TABLE +Table 1 below lists all Fano threefolds with Picard number 1 (the classification of Fano three- +folds is contained in [18]). The last column indicates the existence of an MCK decomposition. +Note that a Fano threefold X with h1,2(X) = 0 has trivial Chow groups (i.e. A∗ +hom(X) = 0), and +so these Fano threefolds have an MCK decomposition for trivial reasons. The asterisks indicate +new cases settled in this paper. Question marks indicate cases I am not able to settle. +Label +Index +Degree +h1,2 +Description +MCK +4 +4 +1 +0 +P3 +trivial +3 +3 +2 +0 +X2 ⊂ P4 +trivial +2.1 +2 +1 +21 +X6 ⊂ P(13, 2, 3) +∗ +2.2 +2 +2 +10 +X4 ⊂ P(14, 2) +∗ +2.3 +2 +3 +5 +X3 ⊂ P4 +[8], [12] +2.4 +2 +4 +2 +X(2,2) ⊂ P5 +[32] +2.5 +2 +5 +0 +Gr(2, 5) ∩ L ⊂ P9 +trivial +1.2 +1 +2 +52 +X6 ⊂ P(14, 3) +∗ +1.4.a +1 +4 +30 +X4 ⊂ P4 +? +1.4.b +1 +4 +30 +X +2:1 +−→ Q with quartic branch locus +∗ +1.6 +1 +6 +20 +X(2,3) ⊂ P5 +[34] +1.8 +1 +8 +14 +X(2,2,2) ⊂ P6 +? +1.10.a +1 +10 +10 +ordinary Gushel–Mukai 3fold +? +1.10.b +1 +10 +10 +special Gushel–Mukai 3fold +∗ +1.12 +1 +12 +7 +OGr+(5, 10) ∩ L ⊂ P15 +? +1.14 +1 +14 +5 +Gr(2, 6) ∩ L ⊂ P14 +[37] +1.16 +1 +16 +3 +LGr(3, 6) ∩ L ⊂ P13 +? +1.18 +1 +18 +2 +G2/P ∩ L ⊂ P13 +[38] +1.22 +1 +22 +0 +V (s) ⊂ Gr(3, 7) +trivial +TABLE 1. All Fano threefolds with Picard number 1. 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Helv. 90 (2015), +503–511. +INSTITUT DE RECHERCHE MATH´EMATIQUE AVANC´EE, CNRS – UNIVERSIT´E DE STRASBOURG, 7 RUE +REN´E DESCARTES, 67084 STRASBOURG CEDEX, FRANCE. +Email address: robert.laterveer@math.unistra.fr + diff --git a/CNA0T4oBgHgl3EQfAP_i/content/tmp_files/load_file.txt b/CNA0T4oBgHgl3EQfAP_i/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c58ef417386f2b3df9a4424253a8bf1cb41e90a5 --- /dev/null +++ b/CNA0T4oBgHgl3EQfAP_i/content/tmp_files/load_file.txt @@ -0,0 +1,715 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf,len=714 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='01961v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='AG] 5 Jan 2023 SOME MORE FANO THREEFOLDS WITH A MULTIPLICATIVE CHOW–K ¨UNNETH DECOMPOSITION ROBERT LATERVEER ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' We exhibit several families of Fano threefolds with a multiplicative Chow–K¨unneth decomposition, in the sense of Shen–Vial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' As a consequence, a certain tautological subring of the Chow ring of powers of these threefolds injects into cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' As a by-product of the argu- ment, we observe that double covers of projective spaces admit a multiplicative Chow–K¨unneth decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' INTRODUCTION Given a smooth projective variety Y over C, let Ai(Y ) := CHi(Y )Q denote the Chow groups of Y (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' the groups of codimension i algebraic cycles on Y with Q-coefficients, modulo rational equivalence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The intersection product defines a ring structure on A∗(Y ) = � i Ai(Y ), the Chow ring of Y [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' In the special case of K3 surfaces, this ring structure has remarkable properties: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='1 (Beauville–Voisin [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let S be a projective K3 surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The Q-subalgebra � A1(S), cj(S) � ⊂ A∗(S) injects into cohomology under the cycle class map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='2 (Voisin [54], Yin [57]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let S be a projective K3 surface, and m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The Q- subalgebra R∗(Sm) := � A1(S), ∆S � ⊂ A∗(Sm) (generated by pullbacks of divisors and pullbacks of the diagonal ∆S ⊂ S × S) injects into cohomology under the cycle class map for all m ≤ 2 dim H2 tr(S, Q)+1 (where H2 tr(S, Q) denotes the transcendental part of cohomology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Moreover, R∗(Sm) injects into cohomology for all m ∈ N if and only if S is Kimura finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The Chow ring of abelian varieties also has an interesting property: there is a multiplicative splitting, defined by the Fourier transform [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Motivated by the particular behaviour of K3 surfaces and abelian varieties, Beauville [2] has conjectured that for certain special varieties, the Chow ring should admit a multiplicative split- ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' In the wake of Beauville’s “splitting property conjecture”, Shen–Vial [47] have introduced the concept of multiplicative Chow–K¨unneth decomposition (we will abbreviate this to “MCK Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Algebraic cycles, Chow group, motive, Beauville’s “splitting property” conjecture, mul- tiplicative Chow–K¨unneth decomposition, Fano threefolds, tautological ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' 2020 Mathematics Subject Classification: 14C15, 14C25, 14C30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Supported by ANR grant ANR-20-CE40-0023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' 1 2 ROBERT LATERVEER decomposition”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' With the concept of MCK decomposition, it is possible to make concrete sense of this elusive “splitting property conjecture” of Beauville.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' It is hard to understand precisely which varieties admit an MCK decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' To give an idea of what is known: hyperelliptic curves have an MCK decomposition [47, Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='16], but the very general curve of genus ≥ 3 does not have an MCK decomposition [12, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' K3 surfaces have an MCK decomposition, but certain high degree surfaces in P3 do not have an MCK decomposition (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' the examples given in [43], cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' also section 2 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' In this note, we will focus on Fano threefolds and ask the following question: Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let X be a Fano threefold with Picard number 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Does X admit an MCK decom- position ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The restriction on the Picard number is necessary to rule out a counterexample of Beauville [2, Examples 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The answer to Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3 is affirmative for cubic threefolds [8], [12], for intersections of 2 quadrics [32], for intersections of a quadric and a cubic [34], and for prime Fano threefolds of genus 8 [37] and of genus 10 [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The main result of this paper answers Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3 for several more families of Fano three- folds: Theorem (=Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The following smooth Fano threefolds have a multiplicative Chow– K¨unneth decomposition: hypersurfaces of weighted degree 6 in weighted projective space P(13, 2, 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' quartic double solids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' sextic double solids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' double covers of a quadric in P4 branched along the intersection with a quartic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' special Gushel–Mukai threefolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' In Table 1 (at the end of this paper), we have listed all Fano threefolds of Picard number 1 and what is known about MCK for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' To prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='1, we provide a general criterion (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3), that may be useful in other situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' For example, using this criterion we also prove the following: Proposition (=Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let X be a smooth projective variety such that X → Pn is a double cover ramified along a smooth divisor D ⊂ Pn of degree d > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Then X admits an MCK decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' As a consequence of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='1, we obtain an injectivity result similar to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='2: Corollary (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let Y be a Fano threefold as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='1, and m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let R∗(Y m) := � h, ∆Y � ⊂ A∗(Y m) be the Q-subalgebra generated by pullbacks of the polarization h ∈ A1(Y ) and pullbacks of the diagonal ∆Y ∈ A3(Y × Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The cycle class map induces injections R∗(Y m) ֒→ H∗(Y m, Q) for all m ∈ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' SOME MORE FANO THREEFOLDS WITH AN MCK DECOMPOSITION 3 Conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' In this paper, the word variety will refer to a reduced irreducible scheme of finite type over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' A subvariety is a (possibly reducible) reduced subscheme which is equidimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' All Chow groups will be with rational coefficients: we will denote by Aj(Y ) the Chow group of j-dimensional cycles on Y with Q-coefficients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' for Y smooth of dimension n the notations Aj(Y ) and An−j(Y ) are used interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The notation Aj hom(Y ) will be used to indicate the subgroup of homologically trivial cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' For a morphism f : X → Y , we will write Γf ∈ A∗(X × Y ) for the graph of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The contravariant category of Chow motives (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=', pure motives with respect to rational equiv- alence as in [46], [41]) will be denoted Mrat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' MCK DECOMPOSITION Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='1 (Murre [40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let X be a smooth projective variety of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' We say that X has a CK decomposition if there exists a decomposition of the diagonal ∆X = π0 X + π1 X + · · · + π2n X in An(X × X) , such that the πi X are mutually orthogonal idempotents and (πi X)∗H∗(X, Q) = Hi(X, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' (NB: “CK decomposition” is shorthand for “Chow–K¨unneth decomposition”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=') Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Murre has conjectured that any smooth projective variety should have a CK de- composition [40], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3 (Shen–Vial [47]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let X be a smooth projective variety of dimension n, and let ∆sm X ∈ A2n(X × X × X) denote the class of the small diagonal ∆sm X := � (x, x, x) | x ∈ X � ⊂ X × X × X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' An MCK decomposition is defined as a CK decomposition {πi X} of X that is multiplicative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' it satisfies πk X ◦ ∆sm X ◦ (πi X × πj X) = 0 in A2n(X × X × X) for all i + j ̸= k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' (NB: “MCK decomposition” is shorthand for “multiplicative Chow–K¨unneth decomposition”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=') Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The small diagonal (when considered as a correspondence from X × X to X) induces the multiplication morphism ∆sm X : h(X) ⊗ h(X) → h(X) in Mrat .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let us assume X has a CK decomposition h(X) = 2n � i=0 hi(X) in Mrat .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' By definition, this decomposition is multiplicative if for any i, j the composition hi(X) ⊗ hj(X) → h(X) ⊗ h(X) ∆sm X −−→ h(X) in Mrat factors through hi+j(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' 4 ROBERT LATERVEER If X has an MCK decomposition, then setting Ai (j)(X) := (π2i−j X )∗Ai(X) , one obtains a bigraded ring structure on the Chow ring: that is, the intersection product sends Ai (j)(X) ⊗ Ai′ (j′)(X) to Ai+i′ (j+j′)(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' It is conjectured that for any X with an MCK decomposition, one has Ai (j)(X) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='= 0 for j < 0 , Ai (0)(X) ∩ Ai hom(X) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='= 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' this is related to Murre’s conjectures B and D, that have been formulated for any CK decompo- sition [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' For more background on the concept of MCK, and for examples of varieties with an MCK decomposition, we refer to [47, Section 8], as well as [53], [48], [13], [28], [39], [29], [30], [31], [12], [33], [34], [36], [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' A GENERAL CRITERION We develop a general criterion for having an MCK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The criterion hinges on the Franchetta property for families of varieties, which is defined as follows: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let X → B be a smooth projective morphism, where X , B are smooth quasi- projective varieties, and let us write Xb for the fiber over b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' We say that X → B has the Franchetta property in codimension j if the following holds: for every Γ ∈ Aj(X ) such that the restriction Γ|Xb is homologically trivial for the very general b ∈ B, the restriction Γ|b is zero in Aj(Xb) for all b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' We say that X → B has the Franchetta property if X → B has the Franchetta property in codimension j for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' This property is studied in [45], [5], [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Given a family X → B as in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='1, we use the shorthand GDAj B(Xb) := Im � Aj(X ) → Aj(Xb) � ⊂ Aj(Xb) (GDA∗() stands for the “generically defined cycles”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The Franchetta property for X → B means that the generically defined cycles inject into cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let X → B be a family of smooth projective varieties of relative dimension n, with fiber Xb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Assume the following: (i) the family X ×B X → B has the Franchetta property;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' (ii) there exists a projective quotient variety P (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' P = P ′/G where P ′ is smooth projective and G ⊂ Aut(P ′) is a finite cyclic group) with trivial Chow groups (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' A∗ hom(P) = 0), such that Xb → P is a double cover with branch locus a smooth ample divisor, for all b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Then Xb admits an MCK decomposition, for all b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' We have the following Lefschetz-type result in cohomology: SOME MORE FANO THREEFOLDS WITH AN MCK DECOMPOSITION 5 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let Xb → P be as in the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Then pullback Hi(P, Q) → Hi(Xb, Q) is an isomorphism for i < n, and injective for i = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' In case P is smooth, this is a result of Cornalba [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The general case is readily deduced from this: assume P = P ′/G where P ′ is smooth projective and G ⊂ Aut(P ′) is a finite cyclic group, and consider the fiber square X′ b → Xb ↓ ↓ P ′ → P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Cornalba’s result applies to the double cover of the left-hand vertical arrow, and so pullback Hi(P ′, Q) → Hi(X′ b, Q) is an isomorphism for i < n, and injective for i = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The G-action on P ′ lifts to X′ b, and taking G-invariants we find that Hi(P, Q) = Hi(P ′, Q)G → Hi(X′ b, Q)G = Hi(Xb, Q) is an isomorphism for i < n, and injective for i = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' □ Since H∗(P, Q) is algebraic (this is a general fact for any variety with trivial Chow groups, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' [23]), this implies that also Hi(Xb, Q) is algebraic, for all i ̸= n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' More precisely, for i ̸= n odd, one has Hi(Xb, Q) = 0 while for i < n even, one has isomorphisms Ai/2(P) ∼= Hi(Xb, Q) , induced by pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' This implies that for i < n the K¨unneth components πi Xb are algebraic, and generically defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' To define the K¨unneth components πi Xb explicitly, let p: Xb → P denote the projection morphism, and let πi P denote the (unique) CK decomposition of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' One can then define πi Xb := 1/2 tΓp ◦ πi P ◦ Γp if i < n , πi Xb := π2n−i Xb if i > n , πn,fix Xb := 1/2 tΓp ◦ πn P ◦ Γp , πn,var Xb := ∆Xb − � j̸=n πj Xb − πn,fix Xb , πn Xb := πn,fix Xb + πn,var Xb ∈ An(Xb × Xb) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' (Note that πn Xb = 0 in case n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=') The notation is meant to remind the reader that πn,fix Xb and πn,var Xb are projectors on the fixed part resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' the variable part of cohomology in degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' 6 ROBERT LATERVEER These projectors define a generically defined CK decomposition for each Xb, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' all projectors are in GDAn B(Xb × Xb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' This CK decomposition has the property that hj(Xb) := (Xb, πj Xb, 0) = ⊕1(∗) ∀j ̸= n , hn,fix(Xb) := (Xb, πn,fix Xb , 0) = ⊕1(∗) in Mrat .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' (1) Let us now proceed to verify that this CK decomposition is MCK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' What we need to check is the vanishing πk Xb ◦ ∆sm Xb ◦ (πi Xb × πj Xb) = 0 in A2n(Xb × Xb × Xb) for all i + j ̸= k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' First, let us assume that at least one of the 3 integers (i, j, k) is different from n, and i+j ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' In this case, we have that πk Xb ◦ ∆sm Xb ◦ (πi Xb × πj Xb) = (tπi Xb × tπj Xb × πk Xb)∗∆sm Xb = (π2n−i Xb × π2n−j Xb × πk Xb)∗∆sm Xb ֒→ � A∗(Xb × Xb) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Here the first equality is an application of Lieberman’s lemma [41, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3], and the in- clusion follows from property (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The resulting cycle in � A∗(Xb × Xb) is generically defined (since the π∗ Xb and ∆sm Xb are) and homologically trivial (since i + j ̸= k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' By assumption (i), the resulting cycle in � A∗(Xb × Xb) is rationally trivial, and so πk Xb ◦ ∆sm Xb ◦ (πi Xb × πj Xb) = 0 in A2n(Xb × Xb × Xb) , as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' It remains to treat the case i = j = k = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The decomposition πn Xb := πn,fix Xb + πn,var Xb induces a decomposition πn Xb ◦ ∆sm Xb ◦ (πn Xb × πn Xb) =πn,fix Xb ∆sm Xb ◦ (πn,fix Xb × πn,fix Xb ) + πn,fix Xb ∆sm Xb ◦ (πn,fix Xb × πn,var Xb ) + · · · · · · + πn,var Xb ∆sm Xb ◦ (πn,var Xb × πn,var Xb ) in A2n(Xb × Xb × Xb) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Using property (1) and the Franchetta property for Xb × Xb, all summands containing πn,fix Xb vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' One is left with the last term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' To deal with the last term, we observe that the covering involution ι ∈ Aut(Xb) of the double cover p: Xb → P induces a splitting of the motive h(Xb) =h(Xb)+ ⊕ h(Xb)− :=(Xb, 1/2 (∆Xb + Γι), 0) ⊕ (Xb, 1/2 (∆Xb − Γι), 0) in Mrat , where Γι denotes the graph of the involution ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Moreover, there is equality hn,var(Xb) = h(Xb)− in Mrat .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' But the intersection product map h(Xb)− ⊗ h(Xb)− ∆sm Xb −−→ h(Xb) SOME MORE FANO THREEFOLDS WITH AN MCK DECOMPOSITION 7 factors over h(Xb)+, as is readily seen (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='5 below), which is saying exactly that πn,var Xb ∆sm Xb ◦ (πn,var Xb × πn,var Xb ) = 0 in A2n(Xb × Xb × Xb) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' This closes the proof, modulo the following lemma (which is probably well-known, but we include a proof for completeness): Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let X → P be a double cover, where X and P are quotient varieties, and let ι ∈ Aut(X) be the covering involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let h(X)+ := (X, 1/2 (∆X + Γι, 0) , h(X)− := (X, 1/2 (∆X − Γι), 0) in Mrat .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The map of motives h(X)− ⊗ h(X)− ∆sm X −−→ h(X) factors over h(X)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' To prove the lemma, let ι ∈ Aut(X) denote the covering involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The motive h(X)− is defined by the projector ∆− X := 1/2 (∆X − Γι) ∈ An(X × X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Plugging this in and developing, it follows that ∆− X ◦ ∆sm X ◦ (∆− X × ∆− X) = 1/8 (∆X − Γι) ◦ ∆sm X ◦ (∆X×X − ∆X × Γι − Γι × ∆X + Γι × Γι) = 1/8 � ∆X ◦ ∆sm X ◦ (∆X × ∆X) + · · · − Γι ◦ ∆sm X ◦ (Γι × Γι) � = 1/8 � ∆sm X − (id × id ×ι)∗(∆sm X ) − (id ×ι × id)∗(∆sm X ) − (ι × id × id)∗(∆sm X ) + (id ×ι × ι)∗(∆sm X ) + (ι × id ×ι)∗(∆sm X ) + (ι × ι × id)∗(∆sm X ) − (ι × ι × ι)∗(∆sm X ) � in A2n(X × X × X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Here the last equality is by virtue of Lieberman’s lemma [41, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' However, we have equality ∆sm X = {(x, x, x) | x ∈ X} = (ι × ι × ι)∗(∆sm X ) in A2n(X × X × X) , and so the sum of the first and last summand vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Likewise, we have equality (id ×ι×ι)∗(∆sm X ) = (id ×ι×ι)∗(ι×ι×ι)∗(∆sm X ) = (ι×id × id)∗(∆sm X ) in A2n(X ×X ×X) , and so the other summands cancel each other pairwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' This proves the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' □ As a first application of our general criterion, we now proceed to show the following: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let X be a smooth projective variety such that X → Pn is a double cover ramified along a smooth divisor D ⊂ Pn, and assume either dim Hn(X, Q) > 1, or D has degree d > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Then X admits an MCK decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' 8 ROBERT LATERVEER Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Double covers X as in the proposition are exactly the smooth hypersurfaces of degree 2d in the weighted projective space P := P(1n+1, d), where 2d := deg D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let B ⊂ ¯B := PH0(P, OP(2d)) denote the Zariski open parametrizing smooth hypersurfaces, and let B × P ⊃ X → B denote the universal family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' In view of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3, it suffices to check that the family X ×B X → B has the Franchetta property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' To this end, we remark that the line bundle OP(2d) is very ample (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='7 below), which means that the set-up verifies condition (∗2) of [12, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' An application of the stratified projective bundle argument [12, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='6] then implies that (2) GDA∗ B(Xb × Xb) = � (pi)∗(h), ∆Xb � , where we write h ∈ A1(Xb) for the hyperplane class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The excess intersection formula [14, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3] gives an equality ∆Xb · (pi)∗(h) = 2d � j (p1)∗(hj) · (p2)∗(hn+1−j) in An+1(Xb × Xb) , and so equality (2) reduces to the equality GDA∗ B(Xb × Xb) = � (p1)∗(h), (p2)∗(h) � ⊕ Q[∆Xb] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The “decomposable part” ⟨(p1)∗(h), (p2)∗(h)⟩ injects into cohomology, because of the K¨unneth formula for H∗(Xb × Xb, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The class of the diagonal in cohomology is linearly independent from the decomposable part: indeed, if the diagonal were decomposable it would act as zero on the primitive cohomology Hn prim(Xb, Q) := Coker � Hn(Pn, Q) → Hn(Xb, Q) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' But the assumption dim Hn(Xb, Q) > 1 is equivalent to having Hn prim(Xb, Q) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' This proves the Franchetta property for X ×B X → B, and closes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The case d > n is a special case where Hn prim(Xb, Q) ̸= 0, because it is known that the geometric genus of Xb is pg(Xb) = �d − 1 n � [9, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' It remains to prove the following, which we have used above: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let P := P(1n+1, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The sheaf OP(d) is locally free and very ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The assertion about the sheaf being locally free is just because d is a multiple of the weights of P (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' [7, Remarques 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' As for the very ampleness, we apply Delorme’s criterion [7, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3(iii)] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' also [4, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' To prove very ampleness of OP(d), we need to prove that the integer E as defined in [7] and [4] is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let us write x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' , xn, y for the weighted homogeneous coefficients of P, where xj and y have weight 1 resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' It is readily seen that every monomial in xj, y of (weighted) degree SOME MORE FANO THREEFOLDS WITH AN MCK DECOMPOSITION 9 m + dk (where m is a positive multiple of d, and k is any positive integer) is divisible by a monomial of (weighted) degree dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' This means that the integer E defined in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' is 0, and so [7, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3(iii)] implies the very ampleness of OP(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' This proves the lemma, and ends the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' □ Here is another sample application of our general criterion: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let X ⊂ P(1n, 2, 3) be a smooth hypersurface of (weighted) degree 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Assume dim Hn(X, Q) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Then X has an MCK decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The varieties X as in the proposition are exactly the smooth double covers of P := P(1n, 2) branched along a (weighted) degree 6 divisor (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' [26, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3] and for n = 3 also [17, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let X → B denote the family of such double covers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' We are going to check that the family X ×B X → B has the Franchetta property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='8 is then a special case of our general criterion Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let ¯ X → ¯B ∼= Pr denote the universal family of all (possibly singular) hypersurfaces of weighted degree 6 in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The line bundle OP(6) is very ample (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='9 below), and so the projection ¯ X × ¯B ¯ X → P × P has the structure of a stratified projective bundle (with strata the diagonal ∆P and its comple- ment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' One can thus use the stratified projective bundle argument [12, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='6] to deduce the identity GDA∗ B(X × X) = � (pi)∗GDA∗ B(X), ∆X � = � (pi)∗(h), ∆X � (here, h ∈ A1(X) denotes the restriction to X of an ample generator of A1(P) ∼= Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Since X ⊂ P is a hypersurface, the excess intersection formula gives ∆X · (pi)∗(h) = ∆P|X ∈ � (pi)∗(h) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The above identification thus simplifies to GDA∗ B(X × X) = � (pi)∗(h) � ⊕ Q[∆X] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The assumption that dim Hn(X, Q) > 1 implies that the diagonal ∆X is linearly independent in cohomology from the decomposable classes � (pi)∗(h) � (indeed, the decomposable classes act as zero on the primitive cohomology of X, while the diagonal acts as the identity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' This shows that GDA∗ B(X × X) injects into cohomology, as requested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let P := P(1n, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The sheaf OP(6) is (locally free and) very ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The assertion about the sheaf being locally free is just because 6 is a multiple of all the weights (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' [7, Remarques 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' As for the very ampleness, we apply Delorme’s criterion [7, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3(iii)] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' also [4, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' To prove very ampleness of OP(6), we need to prove that the integer E defined in [7] and [4] is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let us write x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' , y, z for the weighted homogeneous coefficients of P, where y and z have weight 2 resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' We need to check that every monomial in xj, y, z of (weighted) degree 6 + 6k is divisible by a monomial of (weighted) degree 6k (if this is the case, then E = 0 and [7, 10 ROBERT LATERVEER Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3(iii)] implies the very ampleness of OP(6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' In case the monomial contains z2, it is divisible by z2 and so the condition is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Assume now the monomial contains only one z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' In case the monomial contains y3 it is divisible by y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Next, if the monomial contains y (or y2) it is divisible by zyxj (for some j) and so the condition is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' A monomial in z and xj obviously satisfies the condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Finally, monomials in xj satisfy the condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' This proves the lemma, and ends the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' MAIN RESULT Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The following Fano threefolds admit an MCK decomposition: (i) hypersurfaces of weighted degree 6 in weighted projective space P(13, 2, 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' (ii) quartic double solids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' (iii) sextic double solids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' (iv) double covers of a quadric in P4 branched along the intersection with a quartic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' (v) special Gushel–Mukai threefolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The cases (ii) and (iii) are immediate applications of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The case (i) is a special case of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Before proving case (iv), let us first state a preparatory lemma: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let Z ⊂ P := P(15, 2) be a smooth weighted hypersurface of degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Then ∆Z = 1 2 4 � j=0 hj × h4−j in A4(Z × Z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Z is a quotient of a non-singular quadric in P5 and so Z has trivial Chow groups (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' A∗ hom(Z) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Using [9, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='2], one can compute the Betti numbers of Z and one finds that they are the same as those of projective space P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' This means that there is a cohomological decomposition of the diagonal ∆Z = 1 2 4 � j=0 hj × h4−j in H8(Z × Z, Q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Since Z (and hence also Z × Z) has trivial Chow groups, the same decomposition holds modulo rational equivalence, proving the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' □ Now, to prove case (iv) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='1, we apply our general criterion Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let P := P(15, 2), and let Y → B be the universal family of smooth dimensionally transverse complete intersections of OP(2) ⊕ OP(4), where the base B is a Zariski open B ⊂ ¯B := PH0(P, OP(2) ⊕ OP(4)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' It follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='7 that OP(2) and OP(4) are very ample line bundles on P, and so ¯Y × ¯B ¯Y → P × P is a stratified projective bundle with strata ∆P and its complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The usual stratified projective bundle argument [12, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='6] applies, and we find that GDA∗ B(Y × Y ) = � (pi)∗GDA∗ B(Y ), ∆Y � = � (pi)∗(h), ∆Y � SOME MORE FANO THREEFOLDS WITH AN MCK DECOMPOSITION 11 (here, h ∈ A1(Y ) denotes the restriction to Y of an ample generator of A1(P) ∼= Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let Y = Z ∩ Z′, where Z and Z′ ⊂ P are hypersurfaces of (weighted) degree 2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Up to shrinking B, we may assume the hypersurface Z is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Since Y ⊂ Z is a divisor, the excess intersection formula gives ∆Y · (pi)∗(h) = ∆Z|Y in A4(Y × Y ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='2, it follows that ∆Y · (pi)∗(h) ∈ � (pi)∗(h) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The above identification thus simplifies to GDA∗ B(Y × Y ) = � (pi)∗(h) � ⊕ Q[∆Y ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' As before, the fact that the diagonal ∆Y is linearly independent from the decomposable corre- spondences in cohomology now shows that GDA∗ B(Y × Y ) → H∗(Y × Y, Q) is injective, and so Y verifies the hypotheses of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The argument for case (v) is similar to that of (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' First, in view of the spread argument [55, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='2], it suffices to establish an MCK decomposition for the generic special Gushel– Mukai threefold Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Thus we may assume that there exists P ⊂ Gr(2, 5), a smooth complete intersection of Pl¨ucker hyperplanes, and a double cover p: Y → P branched along a smooth Gushel–Mukai surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' We now consider the family Y → B of all double covers of P branched along smooth Gushel–Mukai surfaces (so B ⊂ ¯B is a Zariski open in the projectivized space of quadratic sections of the cone over P), and we apply our general criterion Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3 to this family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let Y → B be the family of double covers of P branched along smooth Gushel– Mukai surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The family Y → B has the Franchetta property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' We consider the family ¯Y → ¯B with the projection to the cone C over P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' This is a projective bundle, and so for any fiber Y = Yb with b ∈ B we have GDA∗ B(Y ) = Im � A∗(C) → A∗(Y ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The condition b ∈ B means exactly that Y avoids the summit of the cone C, and so (writing C◦ ⊂ C for the complement of the summit of the cone) we have (3) GDA∗ B(Y ) = Im � A∗(C◦) → A∗(Y ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' But C◦ → P is an affine bundle, and A∗(P) = Im � A∗(Gr(2, 5)) → A∗(P) � = � h � , where h denotes the restriction to P of a Pl¨ucker hyperplane (this follows from [35, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='17], or alternatively from the fact that the derived category of P has a full exceptional collection of length 4 [44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Thus, (3) reduces to GDA∗ B(Y ) = � h � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' This proves the Franchetta property for Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' □ 12 ROBERT LATERVEER Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let Y → B be as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The family Y ×B Y → B has the Franchetta property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let us consider the family ¯Y × ¯B ¯Y → ¯B with the projection to C × C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' This is a stratified projective bundle, with strata ∆C and its complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Thus, the stratified projective bundle argument [12, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='6] implies that GDA∗ B(Y × Y ) = � Im � A∗(C◦ × C◦) → A∗(Y × Y ) � , ∆Y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Since A∗(C◦) = Im � A∗(Gr(2, 5)) → A∗(C◦), we find that GDA∗ B(Y × Y ) = � Im � A∗(Gr(2, 5) × Gr(2, 5)) → A∗(Y × Y ) � , ∆Y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' But A∗(Gr(2, 5) × Gr(2, 5)) = A∗(Gr(2, 5)) ⊗ A∗(Gr(2, 5)) since the Grassmannian has trivial Chow groups, and so GDA∗ B(Y × Y ) = � GDB(Y ), ∆Y � = � h, ∆Y � (where the last equality follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' To finish the proof of the lemma, we now claim that for any (ordinary or special) Gushel– Mukai threefold Y we have (4) ∆Y · h ∈ � Im � A∗(Gr(2, 5)) → A∗(Y ) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Combined with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3, this means that for a special Gushel–Mukai threefold Y (and Y → B as above) there is equality GDA∗ B(Y × Y ) = � h � ⊕ Q[∆Y ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Then, since the diagonal is linearly independent in cohomology of � h � (since h1,2(Y ) ̸= 0), this proves the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' It remains to prove the claim (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Using the spread argument [55, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='2], it suffices to prove equality (4) for the very general Gushel–Mukai threefold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Thus, we may assume that Y is ordinary, and moreover that Y = Y ′ ∩ Q , where Q is a quadric and Y ′ = Gr(2, 5) ∩ H1 ∩ H2 is a smooth fourfold (where H1, H2 are Pl¨ucker hyperplanes) and Y ′ is such that A∗(Y ′) = Im � A∗(Gr(2, 5)) → A∗(Y ′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' (Indeed, the smooth fourfold Y ′ has trivial Chow groups [35, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='6], and the very general Y ′ has no primitive cohomology, as follows from [35, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The excess intersection formula then implies that ∆Y · h = 1 2 ∆Y ′|Y ×Y , and the claim (4) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='4 being proven, all conditions of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3 are met with, and so fibers Y of the family Y → B have an MCK decomposition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' this settles (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' □ SOME MORE FANO THREEFOLDS WITH AN MCK DECOMPOSITION 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' THE TAUTOLOGICAL RING Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let Y be a Fano threefold of Picard number 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Assume that Y has an MCK decomposition, and Y is member of a family Y → B such that Y ×B Y → B has the Franchetta property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' For m ∈ N, let R∗(Y m) := � (pi)∗(h), (pij)∗(∆Y ) � ⊂ A∗(Y m) be the Q-subalgebra generated by pullbacks of the polarization h ∈ A1(Y ) and pullbacks of the diagonal ∆Y ∈ A3(Y × Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' (Here pi and pij denote the various projections from Y m to Y resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' to Y × Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The cycle class map induces injections R∗(Y m) ֒→ H∗(Y m, Q) for all m ∈ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' This is inspired by the analogous result for cubic hypersurfaces [11, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' In its turn, the result of [11] was inspired by analogous results for hyperelliptic curves [49], [50] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='2 below) and for K3 surfaces [54], [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Let d denote the degree of Y , and let 2b := dim H3(Y, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' As in [11, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3], let us write o := 1 dh3 ∈ A3(Y ) (the “distinguished zero-cycle”) and τ := ∆Y − 1 d 3 � j=0 hj × h3−j ∈ A3(Y × Y ) (this cycle τ is nothing but the projector on the motive h3(Y ) considered above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Moreover, let us write hi := (pi)∗(h) ∈ A1(Y m) , oi := (pi)∗(o) ∈ A3(Y m) , τi,j := (pij)∗(τ) ∈ A3(Y m) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' We define the Q-subalgebra ¯R∗(Y m) := ⟨oi, hi, τi,j⟩ ⊂ H∗(Y m, Q) (where i ranges over 1 ≤ i ≤ m, and 1 ≤ i < j ≤ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' One can prove (just as [11, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='11] and [57, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3]) that the Q-algebra ¯R∗(Y m) is isomorphic to the free graded Q-algebra generated by oi, hi, τij, modulo the following relations: (5) oi · oi = 0, hi · oi = 0, h3 i = d oi ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' (6) τi,j · oi = 0, τi,j · hi = 0, τi,j · τi,j = 2b oi · oj ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' (7) τi,j · τi,k = τj,k · oi ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' (8) � σ∈S2b+2 b+1 � i=1 τσ(2i−1),σ(2i) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' To prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='1, we need to check that these relations are also verified modulo ratio- nal equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The relations (5) take place in R∗(Y ) and so they follow from the Franchetta 14 ROBERT LATERVEER property for Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The relations (6) take place in R∗(Y 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The first and the last relations are triv- ially verified, because Y being Fano one has A6(Y 2) = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' As for the second relation of (6), this follows from the Franchetta property for Y × Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' (Alternatively, it is possible to deduce the second relation from the MCK decomposition: indeed, the product τ · hi lies in A4 (0)(Y 2), and it is readily checked that A4 (0)(Y 2) injects into H8(Y 2, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=') Relation (7) takes place in R∗(Y 3) and follows from the MCK relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Indeed, we have ∆sm Y (π3 Y × π3 Y ) = π6 Y ◦ ∆sm Y (π3 Y × π3 Y ) in A6(Y 3) , which (using Lieberman’s lemma) translates into (π3 Y × π3 Y × ∆Y )∗∆sm Y = (π3 Y × π3 Y × π6 Y )∗∆sm Y in A6(Y 3) , which means that τ1,3 · τ2,3 = τ1,2 · o3 in A6(Y 3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' It is left to consider relation (8), which takes place in R∗(Y 2b+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' To check that this relation is also verified modulo rational equivalence, we observe that relation (8) involves a cycle contained in A∗� Sym2b+2(h3(Y ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' But we have vanishing of the Chow motive Sym2b+2 h3(Y ) = 0 in Mrat , because dim H3(Y, Q) = 2b and h3(Y ) is oddly finite-dimensional in the sense of Kimura [22] (all Fano threefolds are known to have Kimura finite-dimensional motive [51, Theorem 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' This establishes relation (8), modulo rational equivalence, and ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Given a curve C and an integer m ∈ N, one can define the tautological ring R∗(Cm) := � (pi)∗(KC), (pij)∗(∆C) � ⊂ A∗(Cm) (where pi, pij denote the various projections from Cm to C resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' C × C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Tavakol has proven [50, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='4] that if C is a hyperelliptic curve, the cycle class map induces injections R∗(Cm) ֒→ H∗(Cm, Q) for all m ∈ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' On the other hand, there are many (non hyperelliptic) curves for which the tautological ring R∗(C3) does not inject into cohomology (this is related to the non-vanishing of the Ceresa cycle, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' [50, Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='2] and also [12, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3 and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' SOME MORE FANO THREEFOLDS WITH AN MCK DECOMPOSITION 15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' A TABLE Table 1 below lists all Fano threefolds with Picard number 1 (the classification of Fano three- folds is contained in [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The last column indicates the existence of an MCK decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Note that a Fano threefold X with h1,2(X) = 0 has trivial Chow groups (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' A∗ hom(X) = 0), and so these Fano threefolds have an MCK decomposition for trivial reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The asterisks indicate new cases settled in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Question marks indicate cases I am not able to settle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Label Index Degree h1,2 Description MCK 4 4 1 0 P3 trivial 3 3 2 0 X2 ⊂ P4 trivial 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='1 2 1 21 X6 ⊂ P(13, 2, 3) ∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='2 2 2 10 X4 ⊂ P(14, 2) ∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='3 2 3 5 X3 ⊂ P4 [8], [12] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='4 2 4 2 X(2,2) ⊂ P5 [32] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='5 2 5 0 Gr(2, 5) ∩ L ⊂ P9 trivial 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='2 1 2 52 X6 ⊂ P(14, 3) ∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='a 1 4 30 X4 ⊂ P4 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='b 1 4 30 X 2:1 −→ Q with quartic branch locus ∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='6 1 6 20 X(2,3) ⊂ P5 [34] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='8 1 8 14 X(2,2,2) ⊂ P6 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='a 1 10 10 ordinary Gushel–Mukai 3fold ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='b 1 10 10 special Gushel–Mukai 3fold ∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='12 1 12 7 OGr+(5, 10) ∩ L ⊂ P15 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='14 1 14 5 Gr(2, 6) ∩ L ⊂ P14 [37] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='16 1 16 3 LGr(3, 6) ∩ L ⊂ P13 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='18 1 18 2 G2/P ∩ L ⊂ P13 [38] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='22 1 22 0 V (s) ⊂ Gr(3, 7) trivial TABLE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' All Fano threefolds with Picard number 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Here, X(d1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=',dr) denotes a complete intersection of multidegree (d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' , dr), Q is a quadric, and L ⊂ Pr is a linear subspace of the appropriate dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' The notations LGr(3, 6) and OGr+(5, 10) indicate the Lagrangian Grassmannian, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' a connected compo- nent of the orthogonal Grassmannian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' In 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='22, V (s) denotes the zero locus of a section of some vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Thanks to Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Kai Laterveer of the Lego University of Schiltigheim who provided inspiration for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' 16 ROBERT LATERVEER REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Beauville, Sur l’anneau de Chow d’une vari´et´e ab´elienne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} 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UNIVERSIT´E DE STRASBOURG, 7 RUE REN´E DESCARTES, 67084 STRASBOURG CEDEX, FRANCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content=' Email address: robert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='laterveer@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} +page_content='fr' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNA0T4oBgHgl3EQfAP_i/content/2301.01961v1.pdf'} diff --git a/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf b/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..84dbca2f5f82c2342a2c2315ee3e99ffb8419d41 --- /dev/null +++ b/CNAzT4oBgHgl3EQfTvwq/content/2301.01253v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:fc2afdd2090143536748eeb9b7768b7a4db132711abb4f702191ae6b42a0882c +size 8730594 diff --git a/CdE4T4oBgHgl3EQfFgzX/content/tmp_files/2301.04887v1.pdf.txt b/CdE4T4oBgHgl3EQfFgzX/content/tmp_files/2301.04887v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..add3d7b4d5c3dc6c953ab4308a98314f3205aa9f --- /dev/null +++ b/CdE4T4oBgHgl3EQfFgzX/content/tmp_files/2301.04887v1.pdf.txt @@ -0,0 +1,1466 @@ +Learning Partial Differential Equations by Spectral Approximates of General +Sobolev Spaces +Juan Esteban Suarez Cardona 1 Michael Hecht 1 +Abstract +We introduce a novel spectral, finite-dimensional approximation of general Sobolev spaces in terms of Chebyshev +polynomials. Based on this polynomial surrogate model (PSM), we realise a variational formulation, solving +a vast class of linear and non-linear partial differential equations (PDEs). The PSMs are as flexible as the +physics-informed neural nets (PINNs) and provide an alternative for addressing inverse PDE problems, such as +PDE-parameter inference. In contrast to PINNs, the PSMs result in a convex optimisation problem for a vast +class of PDEs, including all linear ones, in which case the PSM-approximate is efficiently computable due to the +exponential convergence rate of the underlying variational gradient descent. +As a practical consequence prominent PDE problems were resolved by the PSMs without High Performance +Computing (HPC) on a local machine. This gain in efficiency is complemented by an increase of approximation +power, outperforming PINN alternatives in both accuracy and runtime. +Beyond the empirical evidence we give here, the translation of classic PDE theory in terms of the Sobolev +space approximates suggests the PSMs to be universally applicable to well-posed, regular forward and inverse +PDE problems. +1. Introduction +Partial differential equations (PDEs) are omnipresent mathematical models governing the dynamics and (physical) +laws of complex systems (Jost, 2002; Brezis, 2011). However, analytic PDE solutions are rarely known for most of the +systems being the centre of current research. Therefore, there is a strong demand on efficient and accurate numerical solvers +and simulations. +Main classic numerical solvers divide into: Finite Elements (Ern & Guermond, 2004); Finite Differences (LeVeque, 2007); +Finite Volumes(Eymard et al., 2000); Spectral Methods (Bernardi & Maday, 1997; Canuto et al., 2007) and Particle Methods +(Li & Liu, 2007). +Machine learning methods such as: Physics-Informed GAN (Arjovsky et al., 2017), Deep Galerkin Method (Sirignano +& Spiliopoulos, 2018), and Physics Informed Neural Networks (PINNs) (Raissi et al., 2019), gain big traction in the +scientific computing community. In contrast to classic solvers, PINNs provide a neural net (NN) surrogate model e.g., +ˆu : (−1, 1)m −→ R, m ∈ N, parametrising the solution space of the PDEs and enabling to solve inverse problems like +inference of PDE parameters or initial condition detection. PINN-learning is given by minimising a variational problem, +which is typically formulated in L2-loss terms +� +Ω +��ˆu(x) − u(x) +��2dΩ ≈ +1 +|P| +� +p∈P +��ˆu(p) − u(p) +��2 +(1) +being approximated by the mean square error (MSE) in random (data) nodes P, (Yang et al., 2020),(Long et al., 2018). +The applications of PINNs range from fluid mechanics (Jin et al., 2020) to biology (Lagergren et al., 2020) or medicine +(Sahli Costabal et al., 2020), physics (Ellis et al., 2021) and beyond. +1CASUS - Center for Advanced System Understanding, Helmholtz-Zentrum Dresden-Rossendorf e.V. (HZDR), G¨orlitz, Germany. +Correspondence to: Juan Esteban Suarez Cardona , Michael Hecht . +This work was partially funded by the Center of Advanced Systems Understanding (CASUS), financed by Germany’s Federal Ministry of +Education and Research (BMBF) and by the Saxon Ministry for Science, Culture and Tourism (SMWK) with tax funds on the basis of the +budget approved by the Saxon State Parliament. +arXiv:2301.04887v1 [math.NA] 12 Jan 2023 + +Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces +1.1. Related work – Physics Informed Neural Nets (PINNs) +We identify the essential approaches addressing stability and accuracy of PINNs below. +1.1.1. VARIATIONAL PINNS (VPINNS) +VPINNs were introduced in (Kharazmi et al., 2019; 2020) resting on variational Sobolev losses for PINN-training. +The approach exploits analytic integration and differentiation formulas of shallow neural networks with specified activation +functions. The method is extended by using quadrature rules and automatic differentiation for computing the losses and is +complemented by a domain decomposition approach. The drawback of VPINNs, we identify and demonstrate here, is their +highly consuming runtime performance, preventing the approach to be applicable for multi-dimensional PDE problems. +1.1.2. INVERSE DIRICHLET LOSS BALANCING +The Inverse Dirichlet method (Maddu et al., 2021) was shown to increase the numerical stability of PINNS by +dynamically balancing the occurring variational gradient amplitudes, which if unbalanced cause numerical stiffness +phenomena (Wang et al., 2021). However, the PINN formulation rests on classic MSE losses, limiting the approach to +consider only strong PDE problem formulations. +1.1.3. SOBOLEV CUBATURES PINNS (SC-PINN) +In our prior work (Cardona & Hecht, 2022) we gave a PINN formulation, by replacing the MSE loss by Sobolev +Cubatures. In contrast to ID-PINNs approximating Sobolev losses enables the approach to consider PDE problems in the +weak and strong sense. As a consequence, the automatic differentiation (A.D.) is replaced by polynomial differentiation +implicitly realised in the Sobolev cubatures. As we demonstrated this results in an increase of accuracy and runtime +efficiency by several orders of magnitude compared to PINNs relying on A.D. +1.2. Related Work - Classic spectral methods +Spectral methods are well established techniques solving PDEs and ODEs. Hereby, one aims to approximate the PDE +solution by an expansion u = � +α∈A cαϕα, A ⊆ Nm with respect to a specific finite dimensional space Π = span{ϕα}α∈A +generated by a chosen basis, e.g., Fourier basis for periodic PDEs or Jacobi-Chebyshev polynomials for general, non-periodic +problems. The coefficients of the expansion are constrained by the PDE and its corresponding boundary conditions. For +example: Consider a (non-linear) differential operator L and the equation +Lu = f +in Ω, +with homogeneous Dirichlet boundary conditions. By sampling the function f = f(pα)α∈A ∈ R|A|, A ⊆ Nm in some +node set P = {pα}α∈A determination of the coefficients C := (cα)α∈A ⊆ R|A| demands solving the truncated (non-linear) +system: +L[C] − f +!= 0 , +where L = L|Π denotes the truncated operator. This system of equations is typically formulated as the solution of the +weighted residual: +⟨ϕi, L[C] − f⟩ +!= 0 , +∀α ∈ A. +Depending on the choice of the test functions ϕi we obtain pseudo-spectral methods or Galerkin spectral methods (Kang & +Suh, 2008; Canuto et al., 2007; Bernardi & Maday, 1997). If the operator L is linear, the problem is reduced to solving a +linear system. In the non-linear case, least square methods with Newton-Raphson minimiser are commonly used (Hessari +& Shin, 2013; Kim & Shin, 2006). Extending this formulation to inverse problems (inferring parameters) with general +boundary conditions and/or additional constraints without causing ill-conditioned problems is a unresolved challenge for +classic spectral methods. Our contribution relies on providing the demanded extensions, enabling to addresses general +forward and inverse PDE problems in a numerically stable, efficient and accurate fashion. +1.3. Contribution +We present a generalised soft-constrained spectral method that results in a λ-convex variational optimisation problem +for linear and a class of non-linear PDEs. We theoretically guarantee exponentially fast convergence of the resulting +variational gradient descent. While established PINN alternatives result in non-convex variational problems, already for +linear PDEs, the spectral polynomial surrogate models (PSMs) provide approximates of the PDE solutions outperforming +PINNs in runtime and accuracy, as demonstrated in Section 4. + +Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces +Our approach rests on using Chebyshev Polynomial Surrogate Models (PSMs): +ˆu(x, Θ) = +� +α∈Am,n +θαTα(x) , +Θ = (θα)α∈Am,n ∈ R|Am,n|, x ∈ Rm , +(2) +where Am,n denotes a multi-index set, see Section 2.1, and Tα denotes the Cheybshev polynomial basis of first kind given +by the relation: +Tα(cos(x)) = Tα(cos(x1), . . . , cos(xm)) = +m +� +i=1 +cos(αixi) = cos(αx) +(3) +for all α ∈ Am,n. The Chebyshev polynomials are widely used due to their excellent approximation properties extensively +discussed in (Trefethen, 2019). In our recent work (Cardona & Hecht, 2022), we already formulated (weak) PDE losses by +generalising classic Gauss-Legendre cubature rules, we termed Sobolev cubatures. As aforementioned, for linear and a +class of non-linear PDEs the induced variational λ-convex gradient flows possess an exponential rate of convergence. The +resulting PSMs deliver an increase of accuracy up to 10 orders of magnitude, by reducing the runtime costs up to 3 orders of +magnitude compared to PINN alternatives. Moreover, we demonstrate the PSMs to be as flexible as PINNs for addressing +inverse PDE problems, such as PDE-parameter inference. +In contrast to PINNs, the prominent PDE problems considered in Section 4 were solved by our PSM-method without +High Performance Computing (HPC) on a local machine. We consequently expect the approach to deeply impact current +methodology addressing computational challenges arising across all scientific disciplines and believe that even currently +non-reachable (high-dimensional, strongly varying) PDE problems can be successfully resolved due to our contribution. +2. PDE theory +In this section we introduce the mathematical concepts on which our approach rest. This includes the formulation of +Sobolev cubatures (Cardona & Hecht, 2022), approximating general Sobolev norms. To start with we fix the notation used +throughout this article. +2.1. Notation and basic concepts +We denote with Ω = (−1, 1)m the open m-dimensional standard hypercube, with ¯Ω = [−1, 1]m its closure, and with +∂Ω its boundary. ∥x∥p = (�m +i=1 |xi|p)1/p, x = (x1, . . . , xm) ∈ Rm, 1 ≤ p < ∞, ∥x∥∞ = max1≤i≤m |xi| denotes the +lp-norm, and ⟨x, y⟩, ∥x∥, x, y ∈ Rm the standard Euclidean inner product and norm on Rm. +Moreover, Πm,n = span{xα}∥α∥∞≤n denotes the R-vector space of all real polynomials in m variables spanned by +all monomials xα = �m +i=1 xαi +i +of maximum degree n ∈ N, whereas Πm,n(∂Ω) = {Q|Ω : Q ∈ Πm,n} denotes the space of +restricted polynomials with support Ω. +We consider the multi-index set Am,n = {α ∈ Nm : ∥α∥∞ ≤ n} with |Am,n| = (n + 1)m and order Am,n with +respect to the lexicographic order ⪯ on Nm starting from last entry to the 1st, e.g., (5, 3, 1) ⪯ (1, 0, 3) ⪯ (1, 1, 3). Let +D ∈ R|Am,n|×|Am,n| be a matrix we slightly abuse notation by writing +D = (dα,β)α,β∈Am,n , +(4) +where dα,β ∈ R is the α-th, β-th entry of D. +2.2. Sobolev space theory +We recommend (Adams & Fournier, 2003; Neuberger, 2008; Brezis, 2011) for an excellent overview on functional +analysis and Sobolev space theory including the concepts we shortly summarise: We denote with Ck(Ω, R), k ∈ N∪{∞} the +Banach spaces of all k-times continuously differentiable functions with norm ∥f∥Ck(Ω) = �k +i=0 supx∈Ω,∥α∥1=i |Dαf(x)|. +The Sobolev spaces +Hk(Ω, R) = +� +f ∈ L2(Ω, R) : Dαf ∈ L2(Ω, R) +� +, +∥α∥1 = �m +i=1 αi ≤ k, k ∈ N are given by all L2-integrable functions f : Ω −→ R with existing L2-integrable weak +derivatives Dαf = ∂α1 +x1 . . . ∂αm +xm f up to order k. In fact, Hk(Ω, R) is a Hilbert space with inner product +⟨f, g⟩Hk(Ω) = +� +0≤∥α∥1≤k +⟨Dαf, Dαg⟩L2(Ω) + +Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces +and norm ∥f∥2 +Hk(Ω) = ⟨f, f⟩Hk(Ω). Thus, the embeddings j : Hk(Ω, R) �→ Hk′(Ω) are well defined and continuous for +all k′ ≤ k due to ∥ · ∥Hk′(Ω ≤ ∥ · ∥Hk(Ω,R), whereas H0(Ω, R) = L2(Ω, R), with ⟨f, g⟩L2(Ω) = +� +Ω +f · g dΩ. +For k ≥ 1 the trace operator +tr : Hk(Ω, R) −→ L2(∂Ω, R) +(5) +is defined as usual as the Hk-extension of the classic continuous trace tr(u) = u|∂Ω with domain dom(tr) = C0(¯Ω, R). +The Sobolev spaces with zero trace are denoted as usual with Hk +0 (Ω, R) = {u ∈ Hk(Ω, R) : tr(u) = 0}, k ≥ 1 and can be +alternatively defined as completion of the space of smooth functions that vanish on the boundary ∂Ω of Ω, i.e., +Hk +0 (Ω, R) = C∞ +0 (Ω, R) +∥·∥Hk(Ω) , +C∞ +0 (Ω, R) = {f ∈ C∞(Ω, R) : f|∂Ω = 0} . +We further consider the space of all distributions D′(Ω) = {F : C∞ +0 (¯Ω) −→ R} also known as generalised functions +(being the dual space of all test functions C∞ +0 (¯Ω) = {f ∈ C∞(Ω) : f|∂Ω = 0} with respect to the canonical LF topology). +We associate the negative order Sobolev space as the completion of D′(Ω) with respect to the following norm +H−k(Ω, R) := D′(Ω) +∥·∥H−k(Ω) , +∥F∥H−k(Ω,R) = +sup +u∈Hk(Ω,R) +|Fu| +∥u∥Hk(Ω,R) +, +(6) +yielding a separable, reflexive Hilbert space (Lax, 1955). +The weak PDE formulations and their underlying Hilbert space choice we will propose later on require the notion of +adjoint (differential) operators. We recall the definition. +Definition 1 (Adjoint operators). Let (K, ∥ · ∥K), (H, ∥ · ∥H) be Hilbert spaces and T : dom(T) ⊆ K −→ H, T ∗ : +dom(T ∗) ⊆ H −→ K be linear operators with dense domains. Then T ∗ is called an adjoint operator of T if and only if +⟨Tx, y⟩H = ⟨x, T ∗y⟩K +for all x ∈ dom(T) and y ∈ dom(T ∗). +Example 2. Consider ∂xi : L2(Ω, R) −→ L2(Ω, R) as the differential operator in the weak sense. Then its domain is given +by dom(∂xi) = H1(Ω, R) ⊆ L2(Ω, R), which is a dense subset. Following Definition 1, and applying integration by parts, +an adjoint operator ∂∗ +xi : L2(Ω, R) −→ L2(Ω, R), with domain dom(∂∗ +xi) = H1 +0(Ω, R) is given by ∂∗ +xi = −∂xi. +We link the spaces H−k(Ω, R) and Hk(Ω, R) due to the following fact. +Proposition 3. Let j : Hk(Ω, R) �→ L2(Ω, R), k ∈ N be the embedding with adjoint operator j∗ : L2(Ω, R) −→ +Hk(Ω, R). Let f, g ∈ L2(Ω, R) and the distributions F = ⟨f, ·⟩L2(Ω,R), G = ⟨g, ·⟩L2(Ω,R) ∈ H−k(Ω, R), with f ∈ +L2(Ω, R). Then +∥F∥H−k(Ω,R) = ∥j∗f∥Hk(Ω) , +⟨F, G⟩H−k(Ω) = ⟨j∗f, j∗g⟩Hk(Ω) . +Proof. The proof is derived directly from the definition of the H−k(Ω, R)-norm in Eq. (6): +∥j∗f∥Hk(Ω) = +∥j∗f∥2 +Hk(Ω) +∥j∗f∥Hk(Ω) += |⟨jf, j∗f⟩L2(Ω)| +∥j∗f∥Hk(Ω) += |⟨f, j∗f⟩L2(Ω)| +∥j∗f∥Hk(Ω) +≤ +sup +u∈Hk(Ω,R) +|⟨f, u⟩L2(Ω)| +∥u∥Hk(Ω) += ∥F∥H−k(Ω) . +Vice versa, applying the Cauchy-Schwarz inequality yields +∥F∥H−k(Ω,R) = +sup +u∈Hk(Ω,R) +|⟨f, ju⟩L2(Ω)| +∥u∥Hk(Ω) += +sup +u∈Hk(Ω,R) +|⟨j∗f, u⟩Hk(Ω)| +∥u∥Hk(Ω) +≤ +sup +u∈Hk(Ω,R) +∥j∗f∥Hk(Ω)∥u∥Hk(Ω) +∥u∥Hk(Ω) += ∥j∗f∥Hk(Ω) , +implying the claimed equality. The statement for the inner product follows analogously. +A main ingredient of all further considerations are the truncated L2- or Hk-inner products that rest on adaptions of +classic Gauss-Legendre cubatures, which we provide next. + +Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces +2.3. Orthogonal polynomials and Gauss-Legendre cubatures +Here, we recapture the underlying concept of orthogonal polynomials: Let m, n ∈ N and Pm,n = ⊕m +i=1Legn ⊆ Ω be +the we the m-dimensional Legendre grids, where Legn = {p0, . . . , pn} are the n + 1 Legendre nodes given by the roots of +the Legendre polynomials of degree n + 2 We denote pα = (pα1, . . . , pαm) ∈ Pm,n, α ∈ Am,n. It is a classic fact (Stroud, +1971; 2011; Trefethen, 2017; 2019), that the Lagrange polynomials Lα ∈ Πm,n, α ∈ Am,n given by +Lα = +m +� +i=1 +lαi,i , +lj,i = +m +� +j̸=i,j=0 +xi − pj +pi − pj +, +(7) +satisfy Lα(pβ) = δα,β, ∀ α, β ∈ Am,n and form an orthogonal L2-basis of Πm,n, i.e., +⟨Lα, Lβ⟩L2(Ω) = +� +Ω +Lα(x)Lβ(x)dΩ = wαδα,β , +∀ α, β ∈ Am,n, where δ·,· denotes the Kronecker delta and +wα = ∥Lα∥2 +L2(Ω) +(8) +the efficiently computable Gauss-Legendre cubature weight (Stroud, 1971; 2011; Trefethen, 2017; 2019). Consequently, for +any polynomial Q ∈ Πm,2n+1 of degree 2n + 1 the following cubature rule applies: +� +Ω +Q(x)dΩ = +� +α∈Am,n +wαQ(pα) . +(9) +Summarising: Polynomials of degree 2n + 1 can be (numerically) integrated exactly when sampled on the Legendre grid +Pm,n of order n + 1. Thanks to |Pm,n| = (n + 1)m ≪ (2n + 1)m this makes Gauss-Legendre integration a very powerful +scheme yielding +⟨Q1, Q2⟩L2(Ω) = +� +Ωm +Q1(x)Q2(x)dΩm = +� +α∈Am,n +Q1(pα)Q2(pα)wα , +(10) +for all Q1, Q2 ∈ Πm,n. In light of this fact, we propose the following definition. +Definition 4 (Legendre interpolation and L2-projection ). Let m, n ∈ N, Pm,n be the Legendre grid and Lα, α ∈ Am,n be +the corresponding Lagrange polynomials from Eq.(7). For continuous functions f : ¯Ω −→ R we denote with +Im,n : C0(Ω, R) −→ Πm,n , +Im,n(f) = +� +α∈Am,n +f(pα)Lα ∈ Πm,n +(11) +the interpolation operator. Moreover, we denote with +πm,n : L2(Ω, R) −→ Πm,n , +πm,n(f) = +� +α∈Am,n +1 +wα +⟨f, Lα⟩L2(Ω)Lα ∈ Πm,n +(12) +the L2-projection. +Remark 5. It is important to note that Im,n(f) ̸= πm,n(f) in general. However, both operators are projections that due to +Eq. (10) satisfy +πm,n(πm,n(f)) = πm,n(f) , +Im,n(Im,n(f)) = Im,n(f) , +Im,n(πm,n(f)) = Im,n(f) , +πm,n(Im,n(f)) = Im,n(f) . +In fact, both concepts can deliver exponential fast approximation rates (truncation errors) in case the considered function f +is analytic (Trefethen, 2019). +How differential operators acting on polynomial spaces can be understood due to these concepts is proposed in the +next section. + +Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces +2.4. Truncated differential and adjoint operators +Based on Eq. (7) we derive exact matrix representations of differential operators acting on the polynomial spaces +Πm,n. This allows to extend Eq. (10) and deliver approximates of the Sobolev norms for general functions f ∈ Hk(Ω, R), +k ∈ N. +For Lα ∈ Πm,n from Eq. (7) and 1 ≤ i ≤ m the computation of the values dα,β = ∂xiLα(pβ), pβ ∈ Pm,n, +∀ β ∈ Am,n yield the Lagrange expansion +∂xiLα(x) = +� +β∈Am,n +dα,βLβ(x) . +(13) +Consequently, the matrix +Di = (dα,β)α,β∈Am,n ∈ R|Am,n|×|Am,n| , +(14) +represents the finite dimensional truncation of the differential operator ∂xi : C1(Ω, R) −→ C0(Ω, R) to the polynomial +space Πm,n and for β ∈ Nm we set +Dβ = +m +� +j=1 +Dβi , +with D0 = I , +(15) +to be the approximation of the differential operator ∂β := ∂β1 +x1 . . . ∂βm +xm . +For representing the truncation of general adjoint operators we we consider the Legendre grid Pm,n = {pα : α ∈ +Am,n}, m, n, ∈ N the positive, symmetric Gauss-Legendre cubature weight matrix Wm,n = diag(wα)α∈Am,n, and the +evaluation vector f = (f(Pα))α∈Am,n ∈ R|Am,n| for a given function f : Ω −→ R. With these ingredients we state: +Proposition 6. Let Dβ : L2(Ω, R) −→ L2(Ω, R), β ∈ Nm be a differential operator and Dβ : Πm,n(Ω) −→ Πm,n(Ω) be +its truncation to the polynomial space. Then the matrix representation of the truncated adjoint operator D∗ +β : Πm,n(Ω) −→ +Πm,n(Ω) is given by: +D∗ +β = W−1 +m,nD⊤ +β Wm,n . +(16) +Proof. We derive Eq. (16) due to the Gauss-cubature in terms of Eq. (10). Let Q1, Q2 ∈ Πm,n, and denote with q1 = +(Q1(pα))α∈Am,n, q2 = (Q2(pα))α∈Am,n ∈ R|Am,n| the corresponding evaluation vectors. Then we compute +⟨DβQ1, Q2⟩L2(Ω,R) = ⟨Dq1, Wm,nq2⟩ = q⊤ +1 D⊤ +β Wm,nq2 = q⊤ +1 Wm,nW−1 +m,nD⊤ +β Wm,nq2 += ⟨W⊤ +m,nq1, D∗ +βq2⟩ = ⟨q1, Wm,nD∗ +βq2⟩ = ⟨Q1, D∗ +βQ2⟩L2(Ω,R) , +proving the statement. +We provide a matrix representation of the truncation of the adjoint operator j∗ : Hk(Ω, R) −→ L2(Ω, R) of the +embedding j : Hk(Ω, R) −→ L2(Ω, R). +Theorem 7. Let j∗ : L2(Ω, R) −→ Hk(Ω, R) be the adjoint operator of the embedding j : Hk(Ω, R) −→ L2(Ω, R). +Denote with Dβ the representations of the derivatives from Eq. (15) then its truncation J∗ : Πm,n(Ω) ⊆ L2(Ω, R) −→ +Πm,n(Ω) ⊆ Hk(Ω, R) can be represented by the matrix J∗ ∈ R|Am,n|×|Am,n| given by +J∗ = +� � +|β|≤k +D∗ +βDβ +�−1 +. +(17) +Proof. Let Q1, Q2 ∈ Πm,n, Pm,n the Legendre grid and q1 = (Q1(pα))α∈Am,n, q2 = (Q2(pα))α∈Am,n ∈ R|Am,n| the +evaluation vectors,respectively. Then we compute +⟨Q1, Q2⟩Hk(Ω) = +� +|β|≤k +⟨DβQ1, DβQ2⟩L2(Ω,R) = +� +|β|≤k +⟨D∗ +βDβQ1, Q2⟩L2(Ω,R) += ⟨ +� � +|β|≤k +D∗ +βDβ +� +Q1, Q2⟩L2(Ω,R) . + +Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces +Thus, setting J∗−1 := � +|β|≤k D∗ +βDβ yields that due to the identity above J∗−1 is a symmetric and positive definite linear +operator on a finite dimensional space implying its invertibility. Due to +⟨ +� � +|β|≤k +D∗ +βDβ +� +Q1, Q2⟩L2(Ω,R) = ⟨ +� � +|β|≤k +D∗ +βDβ +� +q1, q2⟩ +we realise that J∗−1 := � +|β|≤k D∗ +βDβ represents J∗−1. +As introduced, the PSMs rely on the Chebyshev polynomials {Tα}α∈Am,n, m, n ∈ N, Eq. (3). For later purpose we +provide the basis transformation between the Tα and the Lagrange basis Lα in the Legendre grid Pm,n. That is to consider +the matrix +T = (Tβ(pα))α,β∈Am,n ∈ R|Am,n|×|Am,n| +and its inverse +T−1 ∈ R|Am,n|×|Am,n| . +(18) +Given Lagrange coefficients C = (cα)α∈Am,n of a polynomial Q = � +α∈Am,n cαLα, Θ = (θα)α∈Am,n = T−1C yields the +coefficients of its Chebyshev representation Q = � +α∈Am,n θαTα. Vice versa D = (dα)α∈Am,n = TΘ yields the Lagrange +coefficients of its Chebyshev expansion. We close this section, by deriving a matrix representation of the trace operator, +Eq. (5): +Definition 8 (Truncated trace operator). Let tr : Hk(Ω, R) −→ L2(∂Ω, R) be the trace operator, Eq. (5). Denote with +P ± +m−1,n,j ⊆ ∂Ω± +j the m-1-dimensional Legendre grids for each of the faces ∂Ω± +j = {x ∈ Ω : xj = ±1} of the hypercube +Ω. Then the matrix S± +m,n,j ∈∈ R|Am−1,n|×|Am,n| with +S± +m,n,j = (Tα(pγ))(γ,α)∈Am−1,n×Am,n , +pγ ∈ P ± +m−1,n,j , j = 1, . . . , m . +(19) +represents the truncated trace operator tr : Πm,n −→ Πm−1,n(∂Ω± +j ) for each of the faces ∂Ω± +j . +The derived representations of the truncated differential and adjoint operators enable to derive cubature rules for the +truncated Sobolev spaces. +2.5. Sobolev cubatures +Based on the classic Gauss-Legendre cubature Eq. (10) we, here, derive general Sobolev cubatures. We start by +defining: +Definition 9 (Truncated (dual) inner product and norm). For β ∈ Nm, ∥β∥1 ≤ k, m, n ∈ N we consider the truncated +differential operator Dβ and its adjoint Dβ : Πm,n(Ω) −→ Πm,n(Ω), D∗ +β : Πm,n(Ω) −→ Πm,n(Ω) satisfying +⟨DβQ1, Q2⟩L2(Ω) = ⟨Q1, D∗ +βQ2⟩L2(Ω) , +∀Q1, Q2 ∈ Πm,n +Given the matrix representations Dβ, D∗ +β = W −1 +m,nDT +β Wm,n from Proposition 6, J∗ from Eq. (17) and its formal dual +J∗ = +� � +|β|≤k +D∗ +βDβ +�−1 +, +J∗ = +� � +|β|≤k +DβD∗ +β +�−1 +, +we introduce +Wm,n,k = Wm,nJ∗−1 , Wm,n,−k = Wm,nJ∗ , +Wm,n,k = Wm,nJ∗−1 , Wm,n,−k = Wm,nJ∗ , +and for f, g ∈ Πm,n and their dual distributions F = ⟨f, ·⟩L2(Ω), G = ⟨g, ·⟩L2(Ω) we set +⟨f, g⟩Hk(Ω) += +� +β∈Nm,∥β∥1≤k +⟨Dβf, Dβg⟩L2(Ω) +=⟨f, Wm,n,kg⟩ +⟨f, g⟩Hk(Ω),∗ += +� +β∈Nm,∥β∥1≤k +⟨D∗ +βf, D∗ +βg⟩L2(Ω) +=⟨f, Wm,n,kg⟩ +⟨F, G⟩H−k(Ω) = +� +β∈Nm,∥β∥1≤k +⟨DβJ∗f, DβJ∗g⟩L2(Ω)=⟨f, Wm,n,−kg⟩ +⟨F, G⟩H−k(Ω),∗= +� +β∈Nm,∥β∥1≤k +⟨D∗ +βJ∗f, D∗ +βJ∗g⟩L2(Ω)=⟨f, Wm,n,−kg⟩ , +(20) + +Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces +where f = (f(pα))α∈Am,n ∈ R|Am,n|, g = (g(pα))α∈Am,n ∈ R|Am,n| are the evaluation vectors of f, g in the Legendre +nodes pα ∈ Pm,n, respectively. The corresponding norms are given by +∥f∥Hk(Ω) = ⟨f, f⟩1/2 +Hk(Ω) , +∥f∥Hk(Ω),∗ = ⟨f, f⟩1/2 +Hk(Ω),∗ +∥F∥H−k(Ω) = ⟨F, F⟩1/2 +H−k(Ω) , +∥F∥H−k(Ω),∗ = ⟨F, F⟩1/2 +H−k(Ω),∗ . +(21) +In fact, while including the L2-inner product for β = 0, the expressions above define inner products and norms. We +deduce the exactness of the equations. +Theorem 10 (Sobolev cubatures). Let f, g ∈ Hk(Ω, R) and F = ⟨f, ·⟩, G = ⟨g, ·⟩ ∈ H−k(Ω, R). Then the approximations +given by Definition 9, Eq. (20), are exact for all f, g ∈ Πm,n. +Proof. By combining Proposition 3, Theorem 7 and Im,n(πm,n(f)) = πm,n(f) the proof follows. +The following observation is helpful for computing the Sobolev cubatures. +Corollary 11. Let f ∈ Πm,n and the assumptions of Definition 9 be fulfilled. Then the following identities hold: +⟨Dβf, Dβf⟩L2(Ω,R) = +� +α∈Am,n +1 +wα +⟨Dβf, Lβ⟩2 +L2(Ω,R) +⟨D∗ +βf, D∗ +βf⟩L2(Ω,R) = +� +α∈Am,n +1 +wα +⟨f, DβLα⟩2 +L2(Ω,R) +(22) +Proof. We use Proposition 6 in terms of D∗ +β = W−1 +m,nDT +β Wm,n and due to Theorem 10 compute +⟨D∗ +βf, D∗ +βf⟩L2(Ω,R) = ⟨D∗ +βf, Wm,nD∗ +βf⟩ = ⟨W−1 +m,nD⊤ +β Wm,nf, D⊤ +β Wm,nf⟩ += +� +α∈Am,n +1 +wα +⟨f, D⊤ +β Wm,neα⟩2 = +� +α∈Am,n +1 +wα +⟨f, DβLα⟩2 +L2(Ω,R) , +where eα is the α-th standard basis vector of R|Am,n|. The analog computation applies for Dβ. +In fact, when considering the truncated (dual) norms (∥·∥H−k(Ω),∗, ∥·∥Hk(Ω),∗), ∥·∥H−k(Ω), ∥·∥Hk(Ω), computations +based on Eq. (22) are straightforwardly achieved and documented in (ABC, 2021). We provide the formal setup next. +3. PDE formulations +In light of the provided perspectives, we follow (Jost, 2002; Brezis, 2011) to propose the following formalization of +classic PDE problems. For the sake of simplicity, we focus on classic Poisson type equations. Extensions to more general +PDE problems can be derived once the notion is given, see Section 4. +3.1. Poisson equation +Let us consider the Poisson equation, for f ∈ C0(Ω, R). The strong Poisson problem with Dirichlet boundary +condition g ∈ C0(∂Ω, R) seeks for solutions u ∈ C2(Ω, R) fulfilling: +� −∆u(x) − f(x) += 0 +, ∀x ∈ Ω +u(x) − g(x) += 0 +, ∀x ∈ ∂Ω . +(23) +By using the notion of weak derivatives we can formulate a weaker version of the Poisson equation. That is, finding +u ∈ H2(Ω, R) ⊆ C0(Ω, R) fulfilling +� +Ω +(−∆u − f)φ dx, ∀φ ∈ C∞(Ω, R), +(24) +subjected to the same Dirichlet boundary conditions as in equation (23). The notions give rise to the following optimisation +problems. + +Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces +3.2. PDE loss +We use the Sobolev space setting Hk(Ω, R), Hl(∂Ω, R), k, l ∈ Z for introducing soft-constrained PDE-losses that +impose the Poisson-PDE-solution with general boundary condition as one global variational optimisation problem. +Definition 12. Given the setup of Eq. (23) the strong PDE-loss Lstrong : Hk+2(Ω, R) ∩ Hl(∂Ω, R) −→ R, k, l ∈ N is +defined by +Lstrong(u) = rstrong(u) + sstrong(u) = ∥ − ∆u − f∥2 +Hk(Ω) + ∥u|∂Ω − g∥2 +L2(Ω) . +(25) +The weak PDE-loss Lweak : Hk+2(Ω, R) ∩ Hl(∂Ω, R) −→ R, reflecting the weak formulation in Eq. (24), is given by +Lweak(u) = rweak(u) + sweak(u) += +sup +φ∈C∞(Ω,R) +⟨−∆u − f, φ⟩2 +Hk(Ω) + +sup +φ∈C∞(∂Ω,R) +⟨u − g, φ⟩2 +L2(Ω) . +(26) +Truncations of the the strong loss Lstrong : Πm,n −→ R+ can be derived by applying the Sobolev cubatures from +Definition 9. A truncation Lweak : Πm,n −→ R+ of the weak PDE-loss, Eq. (26) is given by requiring Eq. (24) to be +fulfilled only for all polynomial test functions ϕ ∈ Πm,n = span(Lα)α∈Am,n spanned by the Lagrange polynomials. Hence, +we consider +rweak(u) ≈ +� +α∈Am,n +⟨−∆u − f, Lα⟩2 +Hk(Ω) , +sweak(u) ≈ +� +α∈Am,n +⟨u − g, Lα⟩2 +Hl(Ω) . +(27) +While Definition 12 includes the case k, l < 0 the corresponding losses occur when replacing ∥ · ∥Hk(Ω), ∥ · ∥H−k(Ω) +with ∥·∥Hk(Ω), ∥·∥H−k(Ω),∗, yielding well-defined notions due to Proposition 3. Next, we derive the corresponding gradient +flows of the given losses. +3.3. Variational gradient flows +Given a polynomial QC0 += +� +α∈Am,n cαLα in Lagrange expansion with respect to the Legendre +grid Pm,n +⊆ +Ω with coefficients C0 += +(cα)α∈Am,n +∈ +R|Am,n|. +We consider the truncated loss +L : R|Am,n| −→ R+, L = L[C] acting on the coefficients and the gradient flow ODE +∂tC(t) = −∇L(QC(t)) +, C(0) = C0 . +(28) +Combining the identity QC(pα) = cα, with Definition 9 for the evaluation vector f = (f(pα))α∈Am,n we derive the +following expression for the L2-gradient in case for the strong loss L = Lstrong from Eq. (25),i.e, +∇C(rstrong) = ∇C⟨ +� +(D2 +x1 + · · · + D2 +xm)C + f +� +, Wm,n +� +(D2 +x1 + · · · + D2 +xm)C + f +� +⟩ , +where according to Eq. (15), D2 +xi = D2ei with ei ∈ Rm being the standard basis, i = 1, . . . , m. Thus, +∇C(rstrong) = −2(D2 +x1 + · · · + D2 +xm)T Wm,n +� +(D2 +x1 + · · · + D2 +xm)C + f +� +, +(29) +∇C(sstrong)± +j = 2Wm−1,n(S± +m,n,jC − g± +j ) , +j = 1, . . . , m , +where g± +j is the evaluation vector of g in the m-1-dimensional Legendre grid P ± +m−1,n,j ⊆ ∂Ω± +j contained in each face ∂Ω± +j +of Ω, and S± +m,n,j denotes the truncated trace operator, Definition 8. +Analogously, in case of the weak loss L = Lweak from Eq. (26) we derive +∇C(rweak) = −2(D2 +x1 + · · · D2 +xm)T W2 +m,n +� +(D2 +x1 + · · · + D2 +xm)C + f +� +(30) +∇C(sweak)± +j = 2W2 +m−1,n(S± +m,n,jC − g± +j ) . +Formulas for choosing truncated dual norms ∥ · ∥Hk(Ω), ∥ · ∥Hk(Ω),∗, 0 < k < ∞ as in Definition 9 result when replacing +Wm,n with the corresponding cubature matrix, e.g. Wm,nJ∗−1, from Definition 9 in Eq. (29), while in Eq. (30) W2 +m,nJ∗−1 +occurs. +For all cases, Corollary 11 provides the baseline for numerical stable implementations, which are realised and +documented in (ABC, 2021). + +Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces +3.3.1. ANALYTIC VARIATION OF LINEAR PDES +Given the analytic expressions of the variational gradients in Eq. (29),(30) we derive the analytic solution of the +gradient descent, Eq. (28): To do so, we shorten D := (D2 +x1 + · · · + D2 +xm), D∗ := DT Wm,n, S := �m +j=1 S± +m,n,j, +S∗g := Wm−1,n +�m +j=1 g± +j and realise that Eq. (28) becomes: +d +dtC(t) = −2(D∗D + S∗S)C(t) + 2(S∗g − D∗f) . +By applying the variation of parameters we derive the solution of the ODE as: +C(t) = exp(−t · K∗K)C0 + 2(I − exp(−t · K∗K))(K∗K)+(S∗g − D∗f) , +where K∗K := 2(D∗D+S∗S), and (K∗K)+ denotes the Moore–Penrose pseudo-left-inverse, see e.g., (Ben-Israel & Greville, +2003; Trefethen & Bau III, 1997). In case, where K∗K is a positive definite matrix that imples +C∞ := lim +t→∞ C(t) = (K∗K)−1(S∗g − D∗f) . +(31) +While we expect that K∗K is positive definite, and thus invertible, whenever the underlying PDE problem is well posed and +posses a unique solution a formal proof of this implication requires a deeper theoretical study that is out of scope of this +article. Empirical demonstrations in Section 4, however, suggest this expectation to be genuine. +Whatsoever, non-linear PDEs or inverse PDE problems can not be solved due to Eq. (31) and require gradient descent +methods, realising Eq. (28). A deeper investigation of such approaches is given in the next section. +3.4. Exponential convergence of λ-convex gradient flows +In practice more general problems than linear (forward) PDE problems occur. We motivate this section by considering +an inverse problem for the Poisson equation (23). That is to consider a function f : Ω −→ R and an unknown parameter +µ ∈ R and pose the PDE problem +� −∆u(x) − µf(x) += 0 +, ∀x ∈ Ω +u(x) − g(x) += 0 +, ∀x ∈ Ω +(32) +where g is one specific Poisson solution, i.e., ∆g = µf on Ω. For inferring the parameter µ ∈ R and the PDE solutions +simultaneously we assume that g can be sampled at the Legendre grid Pm,n and formulate the truncated (polynomial) loss +by: +L[C, µ] = ∥ − ∆QC − µf∥2 +Hk(Ω) + ∥QC − g∥2 +Hl(Ω) , +k, l ∈ N . +(33) +While the PDE solution depends on µ itself, we cannot compute the analytic solution directly. Instead, we apply an iterative +gradient descent for deriving the solution based on Eq. (33). We prove that the proposed approach converges exponentially +fast for even more general problems. +Definition 13. A differentiable functional F : R|Am,n| → R is called λ-convex if there is a λ > 0 such that: +F[x] ≥ F[y] + ∇F[y]T (x − y) + λ +2 ∥x − y∥2, ∀x, y ∈ R|A| +(34) +Theorem 14. Given a truncated loss L : R|Am,n| −→ R+, m, n ∈ N, as in Section 3.2, that is λ-convex and differentiable +and assume that the optimal solution C∞ := argminC∈R|Am,n|L[C] minimizing the variational problem exists and is unique. +Then both the loss and the gradient descent +∂tC(t) = −∇L(QC(t)) +, C(0) = C0 . +converge exponentially fast as t → ∞: +λ +2 ∥C(t) − C∞∥2 ≤ L[C(t)] − L[C∞] ≤ e−2λt(L[C0] − L[C∞]). +(35) +Proof. The proof of the statement is given in the appendix. +We give some insights to assert in which situations Theorem 14 applies: + +Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces +Proposition 15. Let A ∈ Rr×s, r ≥ s ∈ N be a positive definite matrix, λ > 0 be the smallest eigenvalue of A then the +affine loss +L(C) = ∥AC + b∥2 , +b ∈ Rr +(36) +is λ-convex. +Proof. We start by observing that any norm is 1−convex, in particular it holds: +∥x∥2 = ∥y∥2 + (∇∥y∥2)T (x − y) + ∥x − y∥2 , +(37) +where (∇∥y∥2)T (x − y) = 2⟨y, x − y⟩. +By replacing the roles of x, y with Ax + b, Ay + b, respectively, we compute: +∥Ax + b∥2 = ∥Ay + b∥2 + 2⟨Ay + b, A(x − y)⟩ + ∥A(x − y)∥2 += ∥Ay + b∥2 + 2⟨AT (Ay + b), x − y⟩ + ∥A(x − y)∥2 +≥ ∥Ay + b∥2 + 2(∇(∥Ay + b∥2), x − y) + λ∥x − y∥2 , +where ∇(∥Ay + b∥2) = 2(AT (Ay + b)). +We want to note that the assumption on A in Proposition 15 can be relaxed: +Remark 16 (Exponential convergence of non-unique solutions). Given that ker A ̸= 0, but b ∈ Rr in Eq. (36) satisfies +b ∈ cokerAT = {x ∈ Rs : AT x ̸= 0} we observe that solving AC = b is equivalent to minimising +L(C) = ∥AT AC + AT b∥2 = ∥A′C + b′∥2 , +(38) +with b′ = AT b, A′ = AT A. Let λ > 0 be the smallest non-vanishing eigenvalue of A′ = AT A. While cokerAT ∼= imA, +L is λ-convex on (ker A)⊥. Due to Theorem 14 and Proposition 15 this implies that the gradient descent of well-posed +problems, Eq. (38), converges exponentially fast to a solution as long as the initial coefficients C0 = C(0) ̸∈ ker A were +proper chosen. +The practical relevance of the observation above is part of the empirical demonstrations of our proposed concepts +given in the next section. +4. Numerical experiments +We designed several numerical experiments for validating our theoretical results. The computations of the PSMs were +executed on a standard Linux laptop (Intel(R) Core(TM) i7-1065G7 CPU @ 1.30GHz, 32 GB RAM). Precomputation of the +Sobolev cubature matrices is realised as a feature of the open source package (Hernandez Acosta et al., 2021). The PSMs are +realised by Chebyshev polynomials, Eq. (3), constrained on Legendre grids as asserted in Eq. (18). All PINN experiments +were executed on the NVIDIA V100 cluster at HZDR. Complete code and benchmark sets is available at (ABC, 2021). We +intensively compared several PINN approaches in our previous work (Cardona & Hecht, 2022). That is why, apart from +classic PINNs, here, we focus on comparing our approach with the PINN-methods that turned out to be most reliable: +i) Classic PINNs with the strong L2-MSE loss based on (Raissi et al., 2019), as described in the introduction. +ii) Inverse Dirichlet Balancing (ID-PINNs) with the L2-MSE loss (Maddu et al., 2021), as described in the introduction. +iii) Sobolev Cubature PINNs (SC-PINNs) (Cardona & Hecht, 2022), with the weak L2-loss for all the experiments unless +specified otherwise. +iv) Gradient flow optimised PSMs (GF-PSM), using the LBFGS-optimiser (Byrd et al., 1995) for the forward problem +with the H−1 +⋆ -norm for the PDE loss and the strong L2−loss for the other terms (unless further specified). Poisson and +QHO Inverse problems are solved by an Implicit-Euler time integration (Butcher, 2001) with the strong L2 loss and +Newton-Raphson (Chong & Zak, 1996) for the Navier Stokes inverse problem, with the H−1 +⋆ +loss. +iv) Analytic Descent (AD-PSM), deriving the PSM by the analytic descent given in Eq. (31) by choosing the dual H−1 +⋆ -loss, +Eq. (20), for the PDE-loss and the strong L2-loss for the remaining terms. + +Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces +For measuring the approximation errors of a ground truth function g : Ω −→ R by a surrogate model u we evaluate +both on equidistant grids g = (g(pi))i=1,...,N ∈ RN u(u(pi))i=1,...,N ∈ RN of size N and compute the l1, l∞-errors +ϵ1 := ∥g − u∥1/N, ϵ∞ := ∥g − u∥∞. We used N = 1002 points for the 2D problems and N = 204 points for the 4D +problem. The parameter inference error is denoted with ϵµ := |µ − µgt|. +All models are trained with the same number of training points T. For the PINN and ID-PINN methods, the training +points are given by randomly sampling from an equidistant grid G of size |G| ≫ N. For the SC-PINN and the PSM methods +the training points are given by the Legendre grids. CPU-training-runtimes are reported in seconds. +4.1. 2D and 4D Poisson equations +We start by considering the Poisson problem in dimension m = 2 in the strong formulation with Dirichlet boundary +conditions, Eq. (23). +Figure 1. Solution for 2D Poisson problem +Approximation error +Runtime (s) +dim = 2 +ϵ1 +ϵ∞ +PINN +4.43 · 10−3 +5.2 · 10−2 +t = 886 +ID-PINN +5.23 · 10−3 +1.9 · 10−2 +t = 1356 +SC-PINN +2.52 · 10−3 +3.33 · 10−2 +t = 79.2 +GF-PSM +5.37 · 10−5 +2.94 · 10−3 +t = 12.84 +AD-PSM +8.79 · 10−10 +1.25 · 10−8 +t = 1.21 +Approximation error +Runtime (s) +dim = 4 +ϵ1 +ϵ∞ +GF-PSM +1.33 · 10−6 +1.0 · 10−3 +t = 173.59s +AD-PSM +5.42 · 10−8 +6.37 · 10−7 +t = 7.66s +Table 1. Errors for 2D and 4D Poisson forward problem +Experiment 4.1 (Non-periodic 2D-Poisson forward problem with hard transitions). We consider the Poisson equation with +right hand side function f given by +f(x, y) =C(A sin(ωy) + tanh(βy))(−Aω2 sin(ωx) − 2β2 tanh(βx)sech2(βx)) ++ C(A sin(ωx) + tanh(βx))(−Aω2 sin(ωy) − 2β2 tanh(βy)sech2(βy)), +with C = 0.1, A = 0.1, β = 5, ω = 10π. All the experiments where conducted with the same number of training points, as +required for the Sobolev cubatures of degree n = 50 in the domain and n = 100 for the boundary. For the SC-PINN the +weak L2 -loss was used for the PDE loss and for the boundary. +Table 1 (top) reports the results and shows that the PSM methods outperform all PINN approaches, both, in accuracy +and runtime. AD-PSM reaches seven orders of magnitude smaller ϵ1-error and requires up to three orders of magnitude +less runtime. The GF-PSM performance is non-compatible to AD-PSM, but still far better than the PINN alternatives. The +results clearly demonstrate the PSM method to be capable of finding solutions to non-trivial linear PDEs with general +non-periodic boundary conditions. +The following experiment indicates that this observation maintains true even for higher dimensional problems. +Experiment 4.2 (4D Poisson equation forward problem). We seek for a solution of a Poisson problem in dimension m = 4. +We choose +f(x) := −4ω2g(x), +with ω = 1 and periodic boundary condition g(x) := sin(ωx1) cos(ωx2) sin(ωx3) cos(ωx4) yielding u(x) = g(x) to be +the analytic solution. We choose Sobolev cubatures of degree n = 8 for both, the domain and the boundary loss. +In Table 1 (bottom) the approximation errors are reported. While all PINN approaches failed to provide any reasonable +solution, the PINN-results were skipped. In contrast, the PSMs can recover the solution accurately. We want to stress that +the PSM runtimes are still smaller than the training runtimes of ID-PINN or the standard PINNs occuring for the analogue +2D Poisson problem, validating again its superior efficiency. + +1.0 +0.10 +0.5 +0.05 +> 0.0 +0.00 +-0.05 +-0.5 +-0.10 +.0 +-0.5 +0.0 +0.5 +1.0 +XLearning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces +Figure 2. Solution for 2D inverse Poisson +problem with ωgt = π. +Approximation error +Runtime (s) +ϵµ +ϵ1 +ϵ∞ +PINN +4.63 · 10−1 +1.13 · 10−2 +1.24 · 10−1 +t ≈ 1592 +ID-PINN +2.14 · 10−2 +8.09 · 10−4 +1.52 · 10−2 +t ≈ 2184 +SC-PINN +3.0 · 10−4 +5.49 · 10−4 +1.01 · 10−2 +t ≈ 103 +GF-PSM +5.8 · 10−8 +6.0 · 10−10 +3.47 · 10−9 +t ≈ 0.49 +Table 2. Errors for 2D Poisson inverse problem +Figure 4. Solution for 2D QHO with µ = 31 on Ω′ = 5.3Ω due to AD-PSM. +Experiment 4.3 (2D Poisson inverse problem). We consider the inverse 2D-Poisson problem, as introduced in Section 3.4, +Eq. (32): We are seeking for inferring the parameter µ in the right hand side f(x) = µ cos(ωx) sin(ωy), for the unknown +ground truth µgt = 2ω2 +gt, ωgt = π and the corresponding PDE solution simultaneously, with the L2-loss (k = l = 0) given +in equation (33). The GF-PSM is applied for a Sobolev cubature with degree n = 100 for the boundary and n = 30 for the +PDE loss. Benchmarks for the standard PINN and the ID-PINN are executed with the same number of training points. +Table 2 reports the reached accuracy and the required runtimes. The GF-PSM outperforms all other methods by +several orders of magnitude in accuracy for both the solution of the PDE, as well as the inferred parameter µ. As discussed in +Section 3.4 the analytic variation, Eq. (31), does not directly apply for this task and is, thus, omitted here. The exponentially +fast convergence of the GF-PSM, Section 3.4, is reflected in the required runtime being 4 orders of magnitude less than the +PINN alternatives. +4.2. Quantum Harmonic Oscillator in 2D +We consider eigenvalue problem for the time-independent Quantum Harmonic Oscillator in dimension m = 2, which +is a special case of the Schr¨odinger equation with linear potential V (u(x)) := (x2 +1 + x2 +2)u(x), u ∈ C2(Ω, R), see e.g., +(Liboff, 1980; Griffiths & Schroeter, 2018): +� +−∆u(x) + V (u(x)) += µu(x) +, ∀x ∈ Ω +u(x) − g(x) += 0 +, ∀x ∈ ∂Ω , +It is a classic fact, that the the eigenvalues are given by µ = n1 + n2 + 1, n1, n2 ∈ N with corresponding eigenfunctions +g(x1, x2) = +π−1/4 +√2n1+n2n1!n2!e− +(x2 +1+x2 +2) +2 +Hn1(x1)Hn2(x2) , +whereas Hn denotes the n-th Hermite polynomial. +Experiment 4.4 (QHO forward problem). For solving the QHO forward problem with eigenvalue µ = 21 and extended +domain Ω′ = [−5.3, 5.3], GF-PSM and the AD-PSM use Sobolev cubatures of degree n = 100 for the boundary and + +1.0 +1.00 +0.75 +0.5 +0.50 +0.25 +V0.0 +0.00 +0.25 +-0.5 +0.50 +0.75 +-1.0 - +1.00 +-1.0 +-0.5 +0.0 +0.5 +1.0 +XGround Truth +Prediction +Point-wise Error le-8 +5.3 +0.4 +5.3 +0.4 +5.3 +2.5 +2.6 +2.6 +2.6 +2.0 +0.2 +0.2 +1.5 +0.0 +y +0.0 +y 0.0 +0.0 +0.0 +1.0 +-2.6 +-2.6 +-2.6 +0.5 +-0.2 +-0.2 +-5.3 +-5.3 +-5.3 +5.3 +-2.6 +0.0 +2.6 +5.3 +-5.3 +-2.6 +0.0 +2.6 +5.3 +-5.3 +-2.6 +0.0 +2.6 +5.3 +x +xLearning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces +Figure 3. Solution of 2D QHO +forward problem with µ = 21. +Approximation error +Runtime (s) +µ = 21 +ϵ1 +ϵ∞ +PINN +6.97 · 10−2 +1. · 10−3 +t ≈ 776 +ID-PINN +4.29 · 10−2 +1.30 · 10−1 +t ≈ 948 +SC-PINN +8.16 · 10−4 +7.27 · 10−3 +t ≈ 167 +GF-PSM +1.6 · 10−8 +5.4 · 10−8 +t ≈ 0.16 +AD-PSM +7.61 · 10−13 +2.37 · 10−12 +t ≈ 0.07 +µ = 31 +ϵ1 +ϵ∞ +GF-PSM +1.09 · 10−9 +1.45 · 10−8 +t ≈ 2.39 +AD-PSM +2.25 · 10−9 +9.82 · 10−9 +t ≈ 1.07 +Table 3. Errors for 2D QHO forward problem with µ = 21, 31. +Figure 5. Solution for 2D QHO with +µgt = 9 on Ω′ = 5.3Ω. +Approximation error +Runtime (s) +ϵµ +ϵ1 +ϵ∞ +PINN +6.01 +7.32 · 10−2 +4.37 · 10−1 +t ≈ 1414 +ID-PINN +6.21 · 10−2 +7.51 · 10−3 +9.40 · 10−2 +t ≈ 1346 +SC-PINN +2.18 · 10−4 +5.68 · 10−4 +1.39 · 10−2 +t ≈ 192 +GF-PSM +9.50 · 10−11 +1.49 · 10−12 +5.13 · 10−10 +t ≈ 5 +Table 4. Errors for 2D QHO inverse problem with µgt = 9 +n = 30 for the PDE loss, whereas we choose n = 200 and n = 50 for eigenvalue µ = 31 on the standard hypercube Ω, +respectively. The AD-PSM uses the by default chosen H−1(Ω), ∗ norm, while the GF-PSM was applied with weak L2-loss, +as in Eq. (26). +Results are reported in Table 3. SC-PINN was the only PINN method that gains reasonable results for µ = 31 +and Ω = [−1, 1]2. However, as in Section 4.1 the PSMs-methods outperform SC-PINN in both runtime and accuracy +performance. In the second scenario, µ = 21, Ω′ = 5.3Ω, none of PINN approaches was able to reach close approximations, +while AD-PSM and GF-PSM do. AD-PSM performs best and its solution is visualised in Fig. 4. +Experiment 4.5 (QHO inverse problem). Similar to Exp. 4.3 we seek for inferring the unknown eigenvalue µ, set to µgt = 9, +and the corresponding continuous approximation of the PDE solution simultaneously, with given data u ∈ R|Am,n| sampled +on the Legendre grid by optimising the loss: +L[C, µ] = ∥∆Qc + V (Qc) − µQC∥2 +L2 + ∥QC − u∥2 +L2 +(39) +We choose a n = 50 degree Sobolev cubature for the domain and n = 200 on the boundary and compare it with the PINN +and the ID-PINN for the same number of training points. +As shown in Table 4 the GF-PSM outperforms the ID-PINN by several orders of magnitude in both accuracy and +runtime. This reflects the strength and flexibility of the method when addressing linear inverse problems. While na¨ıve, +unconditioned Implicit-Euler implementations are inherently unstable the insights of Section 3.4 enable us to exploit the +structure of the gradient flow to realize stable numerical integrators. Applying the PSM method to non-linear forward +problems is our next demonstration task. + +1.0 +0.15 +0.10 +0.5 +0.05 +y 0.0 +0.00 +-0.05 +-0.5 +0.10 +-1.0 +-0.15 +-1.0 +-0.5 +0.0 +0.5 +1.0 +X5.3 +0.4 +0.3 +2.6 +0.2 +0.1 +> 0.0 +0.0 +-0.1 +-2.6 +-0.2 +-0.3 +-5.35.3 +-0.4 +-2.6 +0.0 +2.6 +5.3 +XLearning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces +4.3. 2D Incompressible Navier Stokes equation +We consider the incompressible 2D Navier Stokes equation as an example of a non-linear PDE problem: Let +u = (u1, u2), u ∈ C2(Ω, R2) be the vector velocity field and p ∈ C1(Ω; R) the scalar pressure field the equation becomes: +� +� +� +−ν∆u(x, y) + (u(x, y) · ∇)u(x, y) + ∇p(x, y) += f(x, y) +, ∀(x, y) ∈ Ω +∇ · u(x, y) += 0 +, ∀(x, y) ∈ Ω +u(x, y) − g(x, y) += 0 +, ∀(x, y) ∈ ∂Ω , +where +f(x, y) = 2νπ2(u1(x, y), u2(x, y)) + π cos(πx) cos(πy)(−u1(x, y), u2(x, y)) ++ π sin(πx) sin(πy)(u2, −u1) + exp(πy)(1, πx) , +g(x, y) = [− sin(πx) cos(πy), cos(πx) sin(πy)]T +Experiment 4.6 (Navier-Stokes Forward and Inverse Problem). We solve the Navier-Stokes forward problem by applying +GF-PSM with n = 100 and n = 30 degree Sobolev cubature for the boundary and the domain respectively. We set the +viscosity to ν = 0.05 and use the analytic pressure field p = x exp(πy) with Dirichlet boundary conditions. +The inverse problem seeks for inferring ν and the scalar pressure field p for the ground truth viscosity νgt = 0.05 +and u1 = − sin(πx) cos(πy), u2 = cos(πx) sin(πy). The errors ϵ1 and ϵ∞ reported for this experiment, correspond to the +predicted pressure against the ground truth one. +Figure 6. Solution u1. +Approximation error +Runtime (s) +Forward Problem +ϵ1 +ϵ∞ +GF-PSM +u1 +3.31 · 10−10 +2.35 · 10−9 +t ≈ 405.22 +GF-PSM +u2 +3.28 · 10−10 +2.35 · 10−9 +t ≈ 405.22 +Table 5. Approximation errors of the forward problem. +Approximation error +Runtime (s) +Inverse Problem +ϵν +ϵ1 +ϵ∞ +GF-PSM +2.91 · 10−16 +2.63 · 10−14 +1.21 · 10−11 +t ≈ 0.79 +Table 6. Approximation errors of the inverse problem. +While none of the PINN approaches was able to address the problem reasonably the PSM methods reach similar +accuracy as in the prior (linear) experiments, as reported in Tables 5,6. +We summarise the experimental and theoretical findings in the concluding thoughts below. +5. Conclusion +We introduced a novel variational spectral method solving linear, non-linear, forward and inverse PDE problems. +In contrast to neural network - PINN approaches Chebyshev polynomials surve as a polynomial surrogate model - PSM, +maintainig the same flexibility as PINNs. +Based on our prior work (Cardona & Hecht, 2022), we gave weak PDE formulations, resting on the novel Sobolev +cubatures approximating general Sobolev norms. Allowing us to formulate and compute the resulting finite-dimensional +gradient flow for finding the optimal coefficients for the PSMs, in the case of linear PDEs, we could even derive the analytical +solution of the gradient flow. In particular, the resulting efficient computation of the negative order dual Sobolev norm + +1.0 +1.0 +0.5 +0.5 +0.0 +0.0 +-0.5 +-0.5 +-1.0 +-1.0 +-1.0 +-0.5 +0.0 +0.5 +1.0Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces +∥ · ∥H−k(Ω),∗ was demonstrated to perform best compared to the alternative formulations. While we meanwhile deepened +the theoretical insights, presented here, to deliver the optimal choice of the Sobolev norm beforehand these subjects are +part of a follow-up study. This includes a relaxation of the Sobolev cubatures, resisting the curse of dimensionality when +addressing higher dimensional problems. +In summary, the PSMs methods outperformed all other benchmark methods by far, showing the superiority in runtime +and accuracy performance of the PSMs formulation on the whole spectrum of the considered problems. 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SIAM, 1997. +Wang, S., Teng, Y., and Perdikaris, P. Understanding and mitigating gradient flow pathologies in physics-informed neural +networks. SIAM Journal on Scientific Computing, 43(5):A3055–A3081, 2021. +Yang, L., Zhang, D., and Karniadakis, G. E. Physics-informed generative adversarial networks for stochastic differential +equations. ArXiv, abs/1811.02033, 2020. +Appendix +The result provided in Theorem 14 is a known fact and could be also found for example in (Karimi et al., 2016) in +a more general setting. We prove it by combining the following lemmas. Given a differentiable λ-convex truncated loss +L : R|Am,n| −→ R+, m, n ∈ N, as in Theorem 14, inducing the gradient descent ODE +∂tC(t) = −∇L(QC(t)) +, C(0) = C0 , +where C0 ∈ R|Am,n| is some initial guess of the coefficients. The Implicit Euler discretisation of the ODE is given by +Cn+1 = Cn − τ∇L[Cn+1] , +(40) +where τ ∈ R is the learning rate. We will use the following two definitions: +Definition 17. A functional F : R|Am,n| → R is convex if: +F[tx + (1 − t)y] ≤ tF[x] + (1 − t)F[y], +(41) +it is called strictly convex, if the inequality is strict. +Definition 18. A functional F : R|Am,n| → R is coercive if: +lim +||u||→∞ F[u] = ∞ +(42) +Lemma 19. Let the assumptions of Theorem 14 be fulfilled then the following estimate applies: +λ +2 ∥Cn − C∞∥2 ≤ L[Cn] − L[C∞] ≤ 1 +2λ∥∇L[Cn]∥2 . +Proof. We prove the first inequality by rephrasing the λ- convexity property,Eq. (34). Let γt := tx + (1 − t)y, then +L = L(x) is λ-convex if +L[γt] ≤ tL[x] + (1 − t)L[y] − λ +2 t(1 − t)∥x − y∥2 . +By replacing x and y with Cn and C∞, respectively, and re-arranging, we obtain: +λ +2 t(1 − t)∥Cn − C∞∥2 ≤ t(L[Cn] − L[C∞]) + L[C∞] − L[γt] ≤ t(L[Cn] − L[C∞]) , +where we used the minimality of C∞ for the last inequality. Dividing by t and taking the limit for t → 0 yields the first +inequality of Lemma 19. The second inequality follows directly from the λ-convexity, Eq. (34), implying +L[Cn] − L[C∞] ≤ −∇L[Cn]T (C∞ − Cn) − λ +2 ∥C∞ − Cn∥2, + +Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces +We set F[C∞] := ∇L[Cn]T (C∞ − Cn) + λ +2 ∥C∞ − Cn∥2 +2 and realise that F is a coercive, strictly convex functional with +respect to C∞. Hence, the uniquely determined minimum C∗ +∞ is given by: +∇F +!= 0 ⇐⇒ (C∗ +∞ − Cn) = − 1 +λ∇L[Cn]. +In light of this fact, we can bound −∇F by +L[Cn] − L[C∞] ≤ ( 1 +λ − 1 +2λ)∥∇L[Cn]∥2 , +yielding the desired result. +The following lemma provides the monotonicity property of the gradient flow, being a necessary ingredient for proving +the exponential convergence. +Lemma 20. Let the assumptions of Theorem 14 be fulfilled the the following estimate holds: +L[Cn−1] − L[C∞] ≥ (1 + λτ)2(L[Cn] − L[C∞]) +Proof. Due to the λ-convexity and the Implicit Euler update, Eq. (40), we realise that: +L[Cn−1] ≥ L[Cn] + ∇L[Cn](Cn−1 − Cn) + λ +2 ∥Cn−1 − Cn∥2 += L[Cn] + τ(τλ +2 + 1)∥∇L[Cn]∥2 . +Due to Lemma 19 we further conclude +L[Cn−1] ≥ L[Cn] + 2λτ(τλ +2 + 1)(L[Cn] − L[C∞]) . +(43) +Adding −L[C∞] at both sides provides the claim. +Lemma 21. Let the assumptions of Theorem 14 be fulfilled and define ˆλ := 1 +τ log(1 + λτ). Then the sequence: +∆nL := L[Cn] − L[C∞], +decreases monotonically with an exponential rate of e−2ˆλτn, i.e. +∆nL ≤ e−2ˆλτn(L[C0] − L[C∞]) +(44) +Proof. Due to Lemma (20) we compute +e2ˆλτn(L[Cn] − L[C∞]) = (1 + λτ)2n(L[Cn] − L[C∞]) +≤ (1 + λτ)2(n−1)(L[Cn−1] − L[C∞]) +· · · +≤ L[C0] − L[C∞] . +Proof of Theorem 14. Theorem (14) now follows by combing Lemma (19) and (21) yielding: +1 +λ∥Cn − C∞∥2 +2 ≤ L[Cn] − L[C∞] ≤ e−2ˆλτn(L[C0] − L[C∞]) . +(45) +Thus, for τ → 0, it follows by the definition of ˆλ that ˆλ → λ and Cn → C(t), with t = nτ due to the continuity of +C = C(t) inherited from the differentiability of F. Hence, the continuity of the norm implies the statement. +Remark 22. Lemma 20 implies that also the Implicit Euler discretised gradient flow, converges exponentially fast. + diff --git a/CdE4T4oBgHgl3EQfFgzX/content/tmp_files/load_file.txt b/CdE4T4oBgHgl3EQfFgzX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..81aa944c706ceafaba5ac8d5e0518609b3218b44 --- /dev/null +++ b/CdE4T4oBgHgl3EQfFgzX/content/tmp_files/load_file.txt @@ -0,0 +1,1163 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf,len=1162 +page_content='Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces Juan Esteban Suarez Cardona 1 Michael Hecht 1 Abstract We introduce a novel spectral, finite-dimensional approximation of general Sobolev spaces in terms of Chebyshev polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=' Based on this polynomial surrogate model (PSM), we realise a variational formulation, solving a vast class of linear and non-linear partial differential equations (PDEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=' The PSMs are as flexible as the physics-informed neural nets (PINNs) and provide an alternative for addressing inverse PDE problems, such as PDE-parameter inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=' In contrast to PINNs, the PSMs result in a convex optimisation problem for a vast class of PDEs, including all linear ones, in which case the PSM-approximate is efficiently computable due to the exponential convergence rate of the underlying variational gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=' As a practical consequence prominent PDE problems were resolved by the PSMs without High Performance Computing (HPC) on a local machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=' This gain in efficiency is complemented by an increase of approximation power, outperforming PINN alternatives in both accuracy and runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=' Beyond the empirical evidence we give here, the translation of classic PDE theory in terms of the Sobolev space approximates suggests the PSMs to be universally applicable to well-posed, regular forward and inverse PDE problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=' Introduction Partial differential equations (PDEs) are omnipresent mathematical models governing the dynamics and (physical) laws of complex systems (Jost, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=' Brezis, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=' However, analytic PDE solutions are rarely known for most of the systems being the centre of current research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=' Therefore, there is a strong demand on efficient and accurate numerical solvers and simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=' Main classic numerical solvers divide into: Finite Elements (Ern & Guermond, 2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=' Finite Differences (LeVeque, 2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=' Finite Volumes(Eymard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=', 2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=' Spectral Methods (Bernardi & Maday, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=' Canuto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=', 2007) and Particle Methods (Li & Liu, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=' Machine learning methods such as: Physics-Informed GAN (Arjovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=', 2017), Deep Galerkin Method (Sirignano & Spiliopoulos, 2018), and Physics Informed Neural Networks (PINNs) (Raissi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=', 2019), gain big traction in the scientific computing community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=' In contrast to classic solvers, PINNs provide a neural net (NN) surrogate model e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=', ˆu : (−1, 1)m −→ R, m ∈ N, parametrising the solution space of the PDEs and enabling to solve inverse problems like inference of PDE parameters or initial condition detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=' PINN-learning is given by minimising a variational problem, which is typically formulated in L2-loss terms � Ω ��ˆu(x) − u(x) ��2dΩ ≈ 1 |P| � p∈P ��ˆu(p) − u(p) ��2 (1) being approximated by the mean square error (MSE) in random (data) nodes P, (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=', 2020),(Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=' The applications of PINNs range from fluid mechanics (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=', 2020) to biology (Lagergren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=', 2020) or medicine (Sahli Costabal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=', 2020), physics (Ellis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=', 2021) and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=' 1CASUS - Center for Advanced System Understanding, Helmholtz-Zentrum Dresden-Rossendorf e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=' (HZDR), G¨orlitz, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfFgzX/content/2301.04887v1.pdf'} +page_content=' Correspondence to: Juan Esteban Suarez Cardona 1), yielding +measurements: +y[n] = xhi(t) ⋆ h(t)|t=nT lo = +K +� +k=1 +ckh(nTlo − nkThi) +(1) +The goal of super-resolution is to recover the spike locations nk +and amplitudes ck, k = 1, 2, · · · , K from a limited number (M) +of low-rate samples {y[n]}M−1 +n=0 . The problem is typically ill- +posed due to systematic attenuation of high-frequency contents +of the spikes by the low-pass filter h(t). In order to make the +problem well-posed, it becomes necessary to exploit priors such +as sparsity [6]–[9] and/or non-negativity [10], [11]. In recent +times, there has been a substantial progress towards developing +efficient algorithms for provably solving the super-resolution +problem [7]–[19]. +In this paper, we investigate the problem of binary super- +resolution, where the amplitudes of the spikes are known +apriori to be ck = A, but their number (K) and locations +(nk) are unknown. Motivated by the problem of neural spike +deconvolution in two-photon calcium imaging [2], [20], we +will focus on a blurring kernel that can be represented as a +stable first order auto-regressive (AR(1)) filter. Each neural +spike results in a sharp rise in Ca2+ concentration followed by +a slow exponential decay (modeled as the impulse response of +an AR(1) filter), which results in an overlap of the responses +from nearby spiking events, leading to poor temporal resolution +[2], [21]. +A. Related Works +Early +works +on +super-resolution +date +back +to +algebraic/subspace-based +techniques +such +as +Prony’s +method, MUSIC [12], [22], ESPRIT [8], [23] and matrix +pencil [9], [24]. Following the seminal work in [6], substantial +progress has been made in understanding the role of sparsity +as a prior for super-resolution [7], [25], [26]. In recent times, +convex optimization-based techniques have been developed +that employ Total Variational (TV) norm and atomic norm +regularizers, in order to promote sparsity [7], [18], [19], [25], +[26] and/or non-negativity [10], [11], [27]. These techniques +primarily employ sampling in the Fourier/frequency domain by +assuming the kernel h(t) to be (approximately) bandlimited. +However, selecting the appropriate cut-off frequency is crucial +for super-resolution and needs careful consideration [25], +[28]. Unlike subspace-based methods, theoretical guarantees +for these convex algorithms rely on a minimum separation +between the spikes, which is also shown to be necessary even +in absence of noise [29]. The finite rate of innovation (FRI) +framework [30]–[34] also considers the recovery of spikes +from measurements acquired using an exponentially decaying +kernel, which includes the AR(1) filter considered in this +paper. In the absence of noise, FRI enables the exact recovery +of K spikes with arbitrary amplitudes from M = Ω(K)1 +measurements, without any separation condition [32]. It is +to be noted that all of the above methods require M > K +measurements for resolving K spikes. In contrast, we will +show that it is possible to recover K spikes from M ≪ K +1This notation essentially means that there exists a positive constant c such +that M ≥ cK. +arXiv:2301.01724v1 [eess.SP] 4 Jan 2023 + +2 +measurements by exploiting the binary nature of the spiking +signal. The above algorithms are designed to handle arbitrary +real-valued amplitudes and as such, they are oblivious to +binary priors. Therefore, they cannot successfully recover +spikes in the regime M < K, which is henceforth referred to +as the extreme compression regime. +The problem of recovering binary signals from underde- +termined linear measurements (with more unknowns than +equations/measurements) has been recently studied under the +parlance of Binary Compressed Sensing (BCS) [35]–[42]. +In BCS, the undersampling operation employs random (and +typically dense) sampling matrices, whereas we consider a +deterministic and structured measurement matrix derived from +a filter, followed by uniform downsampling. Moreover, existing +theoretical guarantees for BCS crucially rely on sparsity +assumptions that will be shown to be inadequate for our +problem (discussed in Section II-C). Most importantly, in order +to achieve computational tractability, BCS relaxes the binary +constraints and solves continuous-valued optimization problems. +Consequently, their theoretical guarantees do not apply in the +extreme compression regime M < K. +As mentioned earlier, our study is motivated by the problem +of neural spike deconvolution arising in calcium imaging [3], +[4], [20], [32], [43]–[45]. A majority of the existing spike +deconvolution techniques [4], [43], [44] infer the spiking +activity at the same (low) rate at which the fluorescence signal +is sampled, and a single estimate such as spike counts or +rates are obtained over a temporal bin equal to the resolution +of the imaging rate. Although sequential Monte-Carlo based +techniques have been proposed that generate spikes at a rate +higher than the calcium frame rate [3], no theoretical guarantees +are available that prove that these methods can indeed uniquely +identify the high-rate spiking activity. Algorithms that rely +on sparsity and non-negativity [43], [44] alone are ineffective +for inferring the neural spiking activity that occurs at a much +higher rate than the calcium sampling rate. On the other hand, +at the high-rate, the spiking activity is often assumed to be +binary since the probability of two or more spikes occurring +within two time instants on the fine temporal grid is negligible +[2], [46]. Therefore, we propose to exploit the inherent binary +nature of the neural spikes and provide the first theoretical +guarantees that it is indeed possible to resolve the high-rate +binary neural spikes from calcium fluorescence signal acquired +at a much lower rate. +B. Our Contributions +We make both theoretical and algorithmic contributions to +the problem of binary super-resolution in the setting when +the spikes lie on a fine grid. We theoretically establish that +at very low sampling rates, sparsity and non-negativity are +inadequate for the exact reconstruction of binary spikes (Lemma +2). However, by exploiting the binary nature of the spiking +activity, much stronger identifiability results can be obtained +compared to classical sparsity-based results (Theorem 1). In the +absence of noise, we show that it is possible to uniquely recover +K binary spikes from only M = Ω(1) low-rate measurements. +The analysis also provides interesting insights into the interplay +between binary priors and the “infinite memory" of the AR(1) +filter. +Although it is possible to uniquely identify binary spikes in +the extreme compression regime (M ≪ K), the combinatorial +nature of binary constraints introduce computational hurdles in +exactly enforcing them. Our second contribution is to leverage +the special structure of the AR(1) measurements to overcome +this computational challenge in the extreme compression +regime M < K (Section III-A). Our formulation reveals +an interesting and novel connection between binary super- +resolution, and finding the generalized radix representation of +real numbers, known as β-expansion [47]–[49] (Section III). In +order to circumvent the problem of exhaustive search, we pre- +construct and store (in memory) a binary tree that is completely +determined by the model parameters (filter and undersampling +factor). When the low-rate measurements are acquired, we can +efficiently perform a binary search to traverse the tree and find +the desired binary solution. This ability to trade-off memory +for computational efficiency is made possible by the unique +structure of the measurement model governed by the AR(1) +filter. The algorithm guarantees exact super-resolution even +when the measurements are corrupted by a small bounded +(adversarial) noise, the strength of which depends on the +AR filter parameter and the undersampling factor. When the +measurements are corrupted by additive Gaussian noise, we +characterize the probability of erroneous decoding (Theorem +3) in the extreme compression regime M < K and indicate +the trade-off among the filter parameter, SNR and the extent +of compression. Finally, we also demonstrate how binary +priors can improve the performance of a popularly used spike +deconvolution algorithm (OASIS [43]) on real calcium imaging +datasets. +II. FUNDAMENTAL SAMPLE COMPLEXITY OF BINARY +SUPER-RESOLUTION +Let yhi[n] be the output of a stable first-order Autoregressive +AR(1) filter with parameter α, 0 < α < 1, driven by an +unknown binary-valued input signal xhi[n] ∈ {0, A}, A > 0: +yhi[n] = αyhi[n − 1] + xhi[n] +(2) +In this paper, we consider a super-resolution setting where +we do not directly observe yhi[n], and instead acquire M +measurements {ylo[n]}M−1 +n=0 +at a lower-rate by uniformly +subsampling yhi[n] by a factor of D: +ylo[n] = yhi[Dn], +n = 0, 1, · · · , M − 1, +(3) +The signal ylo[n] corresponds to a filtered and downsampled +version of the signal xhi[n] where the filter is an infinite impulse +response (IIR) filter with a single pole at α. Let ylo ∈ RM +be a vector obtained by stacking the low-rate measurements +{ylo[n]}M−1 +n=0 : +ylo = [ylo[0], ylo[1], · · · , ylo[M − 1]]⊤ +Since (2) represents a causal filtering operation, the low rate +signal ylo only depends on the present and past high-rate +binary signal. Denote L := (M − 1)D + 1. The M low-rate +measurements in ylo are a function of L samples of the high + +3 +rate binary input signal {xhi[n]}L−1 +n=0. These L samples are +given by the following vector xhi ∈ {0, A}L: +xhi := [xhi[0], xhi[1], · · · , xhi[L − 1]]⊤. +Assuming the system to be initially at rest, i.e., yhi[n] = 0, n < +0, we can represent the M samples from (3) in a compact +matrix-vector form as: +ylo := SDyhi = SDGαxhi +(4) +where Gα ∈ RL×L is a Toeplitz matrix given by: +Gα = +� +���� +1 +0 +· · · +0 +α +1 +· · · +0 +... +... +... +... +αL−1 +αL−2 +· · · +1 +� +���� +(5) +and SD ∈ RM×L is defined as: +[SD]i,j = +� +1, +j = (i − 1)D + 1 +0, else +. +The matrix SD represents the D−fold downsampling operation. +Our goal is to infer the unknown high-rate binary input signal +xhi[n] from the low-rate measurements ylo[n]. This is essentially +a “super-resolution" problem because the AR(1) filter first +attenuates the high-frequency components of xhi[n], and +the uniform downsampling operation systematically discards +measurements. As a result, it may seem that the spiking activity +{xhi[(n − 1)D + k]}D +k=1 occurring “in-between" two low-rate +measurements ylo[n − 1] and ylo[n] is apparently lost. One can +potentially interpolate arbitrarily, making the problem hopeless. +In the next section, we will show that surprisingly, xhi still +remains identifiable from ylo in the absence of noise, due to +the binary nature of xhi and “infinite memory" of the AR(1) +filter. +A. Identifiability Conditions for Binary super-resolution +Consider the following partition of xhi into M disjoint blocks, +where the first block is a scalar and the remaining M −1 blocks +are of length D, xhi = [xhi(0), xhi(1)⊤, . . . , xhi(M−1)⊤]⊤. Here, +xhi(0) = xhi[0] and xhi(n) ∈ {0, A}D is given by: +[xhi +(n)]k = xhi[(n − 1)D + k], +1 ≤ n ≤ M − 1 +(6) +The sub-vectors xhi(n), and xhi(n−1) (n ≥ 1) represent consec- +utive and disjoint blocks (of length D) of the high-rate binary +spike signal. In order to study the identifiability of xhi from ylo, +we first introduce an alternative (but equivalent) representation +for (4), by constructing a sequence c[n] as follows c[0] = ylo[0], +c[n] = ylo[n] − αDylo[n − 1], 1 ≤ n ≤ M − 1 +(7) +Given the high rate AR(1) model defined in (2), it is possible +to recursively represent yhi[Dn] in terms of yhi[Dn − 1], which +in turn, can be represented in terms of yhi[Dn − 2], and so +on. By this recursive relation, we can represent yhi[Dn − 1] in +terms of yhi[Dn−D] and {xhi[Dn−i]}D−1 +i=0 and re-write ylo[n] +as +ylo[n] = yhi[Dn] = αyhi[Dn − 1] + xhi[Dn] += αDyhi[Dn − D] + αD−1xhi[D(n − 1) + 1] + · · · ++ αxhi[Dn − 1] + xhi[Dn], +ylo[n] − αDylo[n − 1] = αD−1xhi[D(n − 1) + 1] + · · · ++ αxhi[Dn − 1] + xhi[Dn] +(8) +The last equality holds due to the fact that ylo[n−1] = yhi[Dn− +D]. Combining (7) and (8), the sequence c[n] can be re-written +as c[0] = ylo[0] = xhi(0), and for 1 ≤ n ≤ M − 1 +c[n] = +D +� +i=1 +αD−ixhi[(n − 1)D + i] = hT +αxhi +(n) +(9) +where hα = [αD−1, αD−2, . . . , α, 1]T ∈ RD. This implies +that c[n] depends only on the block xhi(n). Denote c := +[c[0], c[1], . . . , c[M − 1]]⊤ ∈ RM. For any D, (9) can be +compactly represented as: +c = HD(α)xhi +(10) +where HD(α) ∈ RM×L is given by: +HD(α) = +� +������ +1 +0⊤ +0⊤ +· · · +0⊤ +0 +h⊤ +α +0⊤ +· · · +0⊤ +0 +0⊤ +h⊤ +α +· · · +0⊤ +... +... +... +... +... +0 +0⊤ +0⊤ +· · · +h⊤ +α +� +������ +The following Lemma establishes the equivalence between (4) +and (10). +Lemma 1. Given ylo, construct c following (7). Then, there +is a unique binary xhi ∈ {0, A}L satisfying (4) if and only if +xhi is a unique binary vector satisfying (10). +Proof. First suppose that there is a unique binary xhi ∈ {0, A}L +satisfying (4) but (10) has a non-unique binary solution, i.e., +there exists xhi′ ∈ {0, A}L, xhi′ ̸= xhi, such that +c = HD(α)xhi = HD(α)xhi +′ +(11) +Define yhi′ := Gαxhi′ whose entries are given by: +yhi +′[n] = +n +� +k=0 +αn−kxhi +′[k], +0 ≤ n ≤ L − 1 +(12) +Notice that (7) can be re-written as +ylo[0] = c[0] = xhi[0], ylo[1] = c[1] + αDylo[0] = c[1] + αDc[0] +ylo[2] = c[2] + αDylo[1] = c[2] + αDc[1] + α2Dc[0] +... +Following this recursive relation, and using (9) and (11), we +can further re-write ylo[n] as: +ylo[n] = +n +� +i=0 +α(n−i)Dc[i] = αnDx′ +hi +(0) + +n +� +i=1 +α(n−i)Dh⊤ +α xhi +′(i) += αnDx′ +hi +(0) + +n +� +i=1 +D +� +j=1 +αnD−(i−1)D−jx′ +hi[(i − 1)D + j] +(a) += +nD +� +k=0 +αnD−kx′ +hi[k] +(b) += y′ +hi[nD] +(13) + +4 +The equality (a) follows by a re-indexing of the summation +into a single sum, and (b) follows from (12). By arranging +(13) in a matrix form we obtain the following relation: +ylo = SDGαxhi +′ +However from (4), we have ylo = SDGαxhi. This contradicts +the supposition that (4) has a unique binary solution. +Next, suppose that (10) has a unique binary solution but the +binary solution to (4) is non-unique, i.e., there exists xhi′ ∈ +{0, A}L, xhi′ ̸= xhi such that +ylo = SDGαxhi +′ = SDGαxhi +By following (7) and (10), we also have c = HD(α)xhi′ = +HD(α)xhi which contradicts the assumption that solution of +(10) is unique. +Lemma 1 assures that a binary xhi is uniquely identifiable +from measurements ylo if and only if there is a unique binary +solution xhi ∈ {0, A}L to (10). From (9), it can be seen that +c[n] and c[n − 1] have contributions from only disjoint blocks +of high rate spikes xhi(n), and xhi(n−1). Hence effectively, +we only have a single scalar measurement c[n] to decode an +entire block xhi(n) of length D, regardless of how sparse it +is. The task of decoding xhi(n) from a single measurement +seems like a hopelessly “ill-posed" problem, caused by the +uniform downsampling operation. But this is precisely where +the binary nature of xhi can be used as a powerful prior to +make the problem well-posed. Theorem 1 specifies conditions +under which it is possible to do so. +Theorem 1. (Identifiability) For any α ∈ (0, 1), with the +possible exception of α belonging to a set of Lebesgue measure +zero, there is a unique xhi ∈ {0, A}L that satisfies (10) for +every D ≥ 1. +Proof. In Appendix A. +Using Lemma 1 and Theorem 1, we can conclude that xhi +is uniquely identifiable from ylo for almost all α ∈ (0, 1). It +can be verified that for α = 1 the mapping is non-injective. +Theorem 1 establishes that it is fundamentally possible to +decode each block xhi(n) of length D, from effectively a single +measurement c[n]. Since xhi(n) can take 2D possible values, in +principle, one can always perform an exhaustive search over +these 2D possible binary sequences and by Theorem 1, only +one of them will satisfy c[n] = h⊤ +α xhi(n). Since exhaustive +search is computationally prohibitive, this leads to the natural +question regarding alternative solutions. In Section III, we will +develop an alternative algorithm that leverages the trade-off +between memory and computation to achieve a significantly +lower run-time decoding complexity. +B. Comparison with Finite Rate of Innovation Approach +In a related line of work [30]–[32], [34], the FRI framework +has been developed to reconstruct spikes from the measurement +model considered here. However, in the general FRI framework, +there is no assumption on the amplitude of the spikes, and there +are a total of 2D real valued unknowns corresponding to the +locations and amplitudes of D spikes. In [32], it was shown that +by leveraging the property of exponentially reproducing kernels, +it is possible to recover arbitrary amplitudes and spike locations +using Prony-type algorithms, provided at least 2D+1(> D) low- +rate measurements are available. However, since we exploit +the binary nature of spiking activity, we can operate at a +much smaller sample complexity than FRI. In fact, Theorem +1 shows that when we exploit the fact that the spikes occur +on a high-resolution grid with binary amplitudes, M = Ω(1) +measurements suffice to identify D spikes regardless of how +large D is. A direct application of the FRI approach cannot +succeed in this regime, since the number of spikes is larger than +the number of measurements. That being said, with enough +measurements, FRI techniques are powerful, and they can also +identify off-grid spikes. In future, it would be interesting to +combine the two approaches by incorporating binary priors to +FRI based techniques and remove the grid assumptions. +C. Curse of Uniform Downsampling: Inadequacy of sparsity +and non-negativity +By virtue of being a binary signal, xhi is naturally sparse and +non-negative. Therefore, one may ask if sparsity and/or non- +negativity are sufficient to uniquely identify xhi from c, without +the need for imposing any binary constraints. In particular, we +would like to understand if the solution to the following problem +that seeks the sparsest non-negative vector in RL satisfying +(10) indeed coincides with the true xhi ∈ {0, A}L +min +x∈RL +∥x∥0 +subject to c = HD(α)x, +x ≥ 0 +(P0) +Lemma 2. For every xhi ∈ {0, A}L (except xhi = Ae1), +and c ∈ RM satisfying (10), the following are true +(i) There exists a solution x⋆ ̸= xhi to (P0) satisfying +∥x⋆∥0 ≤ ∥xhi∥0 +(14) +(ii) The inequality in (14) is strict as long as there exists an +integer n0 ≥ 1 such that the block x(n0) +hi +of xhi (defined +in (6)) satisfies ∥x(n0) +hi +∥0 ≥ 2. +Proof. The proof is in Appendix B. +Lemma 2 shows there exist other non-binary solution(s) to +(10) (different from xhi) that have the same or smaller sparsity +as the binary signal xhi ∈ {0, A}L. Furthermore, there exist +problem instances where the sparsest solution to (P0) is strictly +sparser than xhi. Hence, sparsity and/or non-negativity are +inadequate to identify the ground truth xhi uniquely. +Implicit Bias of Relaxation: The optimization problem (P0) +is non-convex and the binary constraints are not enforced. In +binary compressed sensing [35], [36], it is common to relax the +binary constraints using box-constraint and l0 norm is relaxed +to l1 norm in the following manner: +min +x∈RL ∥x∥1 +subject to c = HD(α)x, 0 ≤ x ≤ A1 (P1-B) +In the following Lemma, we show that there is an implicit bias +introduced to the solution of (P1-B). +Lemma 3. For every xhi ∈ {0, A}L, and c ∈ RM satisfying +(10). There exists a solution x⋆ to (P1-B) satisfying +∥x⋆∥1 ≤ ∥xhi∥1. +(15) + +5 +Moreover, for all n ≥ 1, the blocks x(n)⋆ ∈RD of x⋆ satisfy: +supp(x(n)⋆) = {D, D − 1, · · · , D − jn}, if c[n] ̸= 0 +(16) +for some 0 ≤ jn ≤ D − 1 and x(n)⋆ = 0 if c[n] = 0, +irrespective of the support of xhi. +Proof. The proof is in Appendix B. +Lemma 3 shows that even in the noiseless setting, introducing +the box-constraint as a means of relaxing the binary constraint +introduces a bias in the support of the recovered spikes. +The optimal solution always results in spikes with support +clustered towards the end of each block of length D, irrespective +of the ground truth spiking pattern xhi that generated the +measurements. This bias is a consequence of the nature of +relaxation, as well as the specific structure of the measurement +matrix HD(α) arising in the problem. +D. Role of Memory in Super-resolution: IIR vs. FIR filters +The ability to identify the high-rate binary signal xhi ∈ +{0, A}L from D−fold undersampled measurements ylo (for +arbitrarily large D) in the absence of noise, is in parts also due to +the “infinite memory" or infinite impulse response of the AR(1) +filter. Indeed, for an Finite Impulse Response (FIR) filter, there +is a limit to downsampling without losing identifiability. This +was recently studied in our earlier work [40] where we showed +that the undersampling limit is determined by the length of +the FIR filter. To see this, consider the convolution of a binary +valued signal xhi with a FIR filter u = [u[0], u[1], · · · , u[r − +1]]T ∈ Rr of length r: zf[n] = �r−1 +i=0 u[r − 1 − i]xhi[n + i]. +These samples are represented in the vector form as zf := +u⋆xhi ∈ RL (by suitable zero padding). Suppose, as before, we +only observe a D−fold downsampling of the output zD[n] = +zf[Dn]. Two consecutive samples zD[p], zD[p + 1] of the low- +rate observation are given by: +zD[p] = +r−1 +� +i=0 +u[r − 1 − i]xhi[Dp + i], +zD[p + 1] = +r−1 +� +i=0 +u[r − 1 − i]xhi[D(p + 1) + i] +If D > r, notice that none of the measurements is a function of +the samples xhi[Dp+r], xhi[Dp+r +1], · · · , xhi[D(p+1)−1]. +Hence, it is possible to assign them arbitrary binary values and +yet be consistent with the low-rate measurements zD[n]. This +makes it impossible to exactly recover xhi (even if it is known +to be binary valued) if the decimation is larger than the filter +length (D > r). The following lemma summarizes this result. +Lemma 4. For every FIR filter u ∈ Rr, if the undersampling +factor exceeds the filter length, i.e. D > r, there exist x0, x1 ∈ +{0, A}L, x0 ̸= x1 such that SD(u ⋆ x0) = SD(u ⋆ x1). +This shows that the identifiability result presented in Theorem +1 is not merely a consequence of binary priors but the infinite +memory of the autoregressive process is also critical in allowing +arbitrary undersampling D > 1 in absence of noise. For such +IIR filters, the memory of all past (binary) spiking activity +is encoded (with suitable weighting) into every measurement +captured after the spike, which would not be the case for a +finite impulse response filter. +III. EFFICIENT BINARY SUPER-RESOLUTION USING +BINARY SEARCH WITH STRUCTURED MEASUREMENTS +By Theorem 1, we already know that it is possible to uniquely +identify xhi from c (or equivalently, each block xhi(n) from +a single measurement c[n]) by exhaustive search. We now +demonstrate how this exhaustive search can be avoided by +formulating the decoding problem in terms of “binary search" +over an appropriate set, and thereby attaining computational +efficiency. We begin by introducing some notations and +definitions. Given a non-negative integer k, 0 ≤ k ≤ 2D − 1, +let (b1(k), b2(k), · · · , bD(k)) be the unique D-bit binary repre- +sentation of k: k = �D +d=1 2D−dbd(k), +bd(k) ∈ {0, 1} ∀ 1 ≤ +d ≤ D. Here b1(k) is the most significant bit and bD(k) is +the least significant bit. Using this notation, we define the +following set: +Sall := {v0, v1, v2, · · · , v2D−1}, +(17) +where each vk ∈ {0, A}D is a binary vector given by +[vk]d = Abd(k). +1 ≤ d ≤ D +(18) +In other words, the binary vector +1 +Avk is the D-bit binary +representation of its index k. Using this convention, v0 = 0 +(i.e., a binary sequence of all 0′s) and v2D−1 = A1 (i.e., a +binary sequence of all A′s). Recall the partition of xhi defined +in (6), where each block xhi(n) (n ≥ 1) is a binary vector of +length D and xhi(0) ∈ {0, A} is a scalar. It is easy to see that +(17) comprises of all possible values that each block xhi(n) can +assume. According to (9) each scalar measurement c[n] can be +written as: c[0] = x(0), +c[n] = hα⊤xhi(n), 1 ≤ n ≤ M − 1. +For every α, we define the following set: +Θα := {θ0, θ1, · · · , θ2D−1}, where θk := h⊤ +α vk +(19) +Observe that every measurement c[n] = �D +i=1 αD−ixhi[(n − +1)D+i] takes values from this set Θα, depending on the value +taken by the underlying block of spiking pattern from Sall. Our +goal is to recover the spikes {xhi[(n − 1)D + i]}D +i=1 from c[n]. +In the following, we show that this problem is equivalent to +finding the representation of a real number over an arbitrary +radix, which is known as “β-expansion" [49]. Given a real +(potentially non-integer) number β > 1, the representation of +another real number p ≥ 0 of the form: +p = +∞ +� +n=1 +anβ−n, where 0 ≤ an < ⌊β⌋ +(20) +is referred to as a β-expansion of p. The coefficients 0 ≤ an < +⌊β⌋ are integers. This is a generalization of the representation +of numbers beyond integer-radix to a system where the radix +can be chosen as an arbitrary real number. This notion of +representation over arbitrary radix was first introduced by Renyi +in [49], and since then has been extensively studied [47], [48], +[50]. There is a direct connection between β-expansion and +the binary super-resolution problem considered here. In the +problem at hand, any element θk ∈ Θα can be written as: +θk = h⊤ +α vk = +D +� +i=1 +αD−i[vk]i +When 1/2 < α < 1, by letting β = 1/α, we see that the +coefficients in (20) must satisfy 0 ≤ an < ⌊1/α⌋ < 2, i.e., + +6 +they are restricted to be binary valued an ∈ {0, 1}. Therefore, +decoding the spikes vk from the observation θk is equivalent +to finding a D−bit representation for the number θk/A over +the non-integer radix β = 1/α. Questions regarding the +existence of β-expansion, and finding the coefficients of a finite +β−expansion (whenever it exists) has been an active topic of +research [47], [48], [50], [51]. When β ≥ 2 (equivalently, +0 < α ≤ 1/2), it is possible to find the coefficients using +a greedy algorithm which proceeds in a fashion similar to +finding the D-bit binary representation of an integer [47], [51]. +However, the regime β ∈ (1, 2) (equivalently 1/2 < α < 1), +is significantly more complicated and is of continued research +interest [47], [48], [50]. To the best of our knowledge, there +are no known computationally efficient ways to find the finite +β-expansion when 1/2 < α < 1 (if it exists) [N. Sidorov, +personal communication, May 24, 2022]. In practice, we +encounter filter values α (= 1/β) that are much closer to +1, and hence, we need an alternative approach to find this +finite β-radix representation for θk. In the next section, we +show that by performing a suitable preprocessing, finite β-radix +representation can be formulated as a binary search problem +which is guaranteed to succeed for all values of β that permit +unique finite β−expansions. +A. Formulation as a Binary Search Problem +Before describing the algorithm, we first introduce the notion +of a collision-free set. +Definition 1 (Collision Free set). Given an undersampling +factor D, define a class of “collision free" AR(1) filters as: +GD = {α ∈ (0, 1) s.t. h⊤ +α vi ̸= h⊤ +α vj ∀ i ̸= j, vi, vj ∈ Sall} +The set GD denotes permissible values of the AR(1) filter +parameter α such that each of the 2D binary sequences in +Sall maps to a unique element in the set Θα. In other words, +every θk ∈ Θα has a unique D−bit expansion for all α ∈ GD. +This naturally raises the question “How large is the set GD?". +Theorem 1 already provided the answer to this question, where +the identifiability result implies that for every D, almost all +α ∈ (0, 1) belong to this set GD (with the possible exception +of a measure zero set). Hence, Theorem 1 ensures that there +are infinite choices for collision-free filter parameters. +Lemma 5. For every α ∈ GD, the mapping Φα(.) : Sall → Θα, +Φα(v) = h⊤ +α v forms a bijection between Sall and Θα. +Proof. Since α ∈ GD, from the definition of the set GD, it is +clear that for any vi, vj ∈ Sall, vi ̸= vj we have hα⊤vi ̸= +hα⊤vj. Therefore, the mapping is injective. Furthermore, from +(19) we also have |Θα| ≤ |Sall| = 2D. Since Φα(·) is injective, +we must also have |Θα| = 2D and hence the mapping Φα(.) +forms a bijection between Sall and Θα. +When α ∈ GD, Lemma 5 states that the finite beta expansion +for every θk ∈ Θα is unique. Lemma 5 provides a way to avoid +exhaustive search over Sall, and yet identify xhi(n) from c[n] in +a computationally efficient way. From Lemma 5, we know that +each of the 2D spiking patterns in Sall maps to a unique element +in Θα, and each element in Θα has a corresponding spiking +pattern. Hence instead of searching Sall, we can equivalently +search the set Θα in order to determine the unknown spiking +pattern. Since Θα permits “ordering", searching Θα has a +distinct computational advantage over searching Sall. This +ordering enables us to employ binary search over (an ordered) +Θα and find the desired element in a computationally efficient +manner. To do this, we first sort the set Θα (in ascending order) +and arrange the corresponding elements of Sall in the same +order. Given Θα as an input, the function SORT(·) returns +a sorted list Θsort +α , and an index set I = {i0, i1, · · · , i2D−1} +containing the indices of the sorted elements in the list Θα. +Θsort +α , I ← SORT(Θα) +Let us denote the elements of the sorted lists as Θsort +α += +{�θ0, · · · , �θ2D−1}, and Ssort +all = {�v0, · · · , �v2D−1} where: +�θ0 < �θ1 < · · · < �θ2D−1 +and �θj = θij, +�vj = vij +∀j. +It is important to note that this sorting step does not depend +on the measurements c, and can therefore be part of a pre- +processing pipeline that can be performed offline. However, +it does require memory to store the sorted lists. In the +Algorithm 1 Noiseless Spike Recovery +1: Input: Measurement c[n], Sorted list Θsort +α +and the corre- +sponding (ordered) spike patterns Ssort +all +2: Output: Decoded spike block �xhi(n) +3: i⋆ ← BINSEARCH(Θsort +α , c[n]) +4: Return �xhi(n) ← �vi⋆ +noiseless setting, we know that every scalar measurement +c[n] = h⊤ +α xhi(n) belongs to the set Θsort +α . Therefore, if we +identify its index, say i⋆, then we can successfully recover +xhi(n) by returning the corresponding binary vector �vi⋆ from +Ssort +all . Therefore, we can formulate the decoding problem as +searching for the input c[n] in the sorted list Θsort +α . This can be +efficiently done by using “Binary Search". The noiseless spike +decoding procedure is summarized as Algorithm 1. Since the +complexity of performing a binary search over an ordered list +of N elements is O(log N), the complexity of Algorithm 1 +is logarithmic in the cardinality of Θsort +α , which results in a +complexity of O(log(2D)) = O(D). We summarize this result +in the following Lemma. +Lemma 6. Assume α ∈ GD. Given the ordered set Θsort +α +, and +an input c[n] = h⊤ +α xhi(n), Algorithm 1 terminates in O(D) +steps and its output �xhi(n) satisfies �xhi(n) = xhi(n). +B. Noisy Measurements and 1 D Nearest Neighbor Search +We demonstrate how binary search can still be useful in +presence of noise by formulating noisy spike detection as a +one dimensional nearest neighbor search problem. Suppose +{zlo[n]}M−1 +n=0 denote noisy D-fold decimated filter output +zlo[n] = ylo[n] + w[n], +0 ≤ n ≤ M − 1 +(21) + +7 +Here w[n] represents the additive noise term that corrupts the +(noiseless) low-rate measurements ylo[n]. Similar to (7), we +compute ce[n] from zlo[n] as follows: +ce[n] = zlo[n] − αDzlo[n − 1] +(22) += +D +� +i=1 +αD−ixhi[(n − 1)D + i] + e[n]= c[n] + e[n] (23) +where c[n] = h⊤ +α xhi(n) ∈ Θsort +α , and e[n] = w[n] − αDw[n − +1]. We can interpret ce[n] as a noisy/perturbed version of an +element c[n] ∈ Θsort +α , with e[n] representing the noise. This +perturbed signal may no longer belong to Θsort +α +(i.e. ce[n] ̸∈ +Θsort +α ) and hence, we cannot find an exact match in the set +Θsort +α . Instead, we aim to find the closest element in Θsort +α +(the +nearest neighbor of ce[n]) by solving the following problem: +�xhi +(n) = arg min +v∈Ssort +all +|ce[n] − h⊤ +α v| +(24) +Solving (24) is equivalent to finding the spike sequence +�v ∈ Ssort +all +that maps to the nearest neighbor of ce[n] in the +set Θsort +α . By leveraging the sorted list Θsort +α , it is no longer +necessary to parse the list sequentially (which would incur +O(2D) complexity), instead we can perform a modified binary +search as summarized in Algorithm 2, that keeps track of +additional indices compared to the vanilla binary search. Finally, +we return the unique spiking pattern from Ssort +α +that gets +mapped to the nearest neighbor of the noisy measurement +ce[n]. It is well-known that the nearest neighbor for any query +could be found in O(log(2D)) = O(D) steps, instead of the +linear complexity of O(2D). This guarantees a computationally +efficient decoding of spikes by solving (24). +Next, we characterize the error events that lead to erroneous +detection of a block of spikes. Recall that the set Θsort +α +is sorted, +and its elements satisfy the ordering: +0 = �θ0 < �θ1 < · · · < �θlD = 1 + α + · · · + αD−1 +where lD := 2D−1. We also have �θk = h⊤ +α �vk, where �vk ∈ Ssort +all +is a binary spiking sequence of length D. +For each �vk and each n, we will determine the error event +�xhi(n) ̸= xhi(n), when xhi(n) = �vk. First, consider the scenario +when xhi(n) = �vk for some 0 < k < lD (excluding �v0, �vlD). +The corresponding noiseless measurement is c[n] = �θk = +h⊤ +α �vk which satisfies �θk−1 < c[n] = �θk < �θk+1. Since Θsort +α +is +sorted, it can be easily verified that the nearest neighbor of +ce[n] will be �θk, if and only if ce[n] satisfies the following +condition: +(�θk−1 + �θk)/2 ≤ ce[n] ≤ (�θk+1 + �θk)/2 +(25) +Since �θk = h⊤ +α �vk, the solution to (24) is attained at �vk ∈ Ssort +all , +and the decoding is successful. Therefore Algorithm 2 produces +an erroneous estimate of �vk if and only if ce[n] violates (25). +The event ce[n] ̸∈ [ +�θk−1+�θk +2 +, +�θk+1+�θk +2 +] is equivalent to e[n] ∈ +Ek (e[n] is defined earlier in (23)), where +Ek = {e[n] < − +�θk − �θk−1 +2 +, or e[n] > +�θk+1 − �θk +2 +} +(26) +Finally, we characterize the error events for k = 0, lD. The +error events for c[n] = θ0 = 0 or c[n] = θlD are given by: +E0 = {e[n] ≥ �θ1/2}, ElD = {e[n] ≤ −(�θlD − �θlD−1)/2} (27) +Define the “minimum distance" between points in Θsort +α : +∆θmin(α, D) = +min +1≤k≤lD |�θk − �θk−1|. +This minimum distance depends on A, α and D. From (26), +(27) it can be verified that if 2|w[n]| < ∆θmin(α, D)/2 (which +would imply |e[n]| < ∆θmin(α, D)/2) for all n, then �xhi(n) = +xhi(n). As summarized in Theorem 2, Algorithm 2 can exactly +recover the ground truth spikes from measurements corrupted +by bounded adversarial noise, the extent of the robustness is +determined by the parameters A, α, D. +Algorithm 2 Noisy Spike Recovery +1: Input: Measurement ce[n], Sorted list Θsort +α +and the +corresponding (ordered) spike patterns Ssort +all +2: Output: Decoded spike block �xhi(n) +3: Set l ← 0, u ← 2D − 1 +4: +while u − l > 1 +5: +Set m ← l + ⌊(u − l)/2⌋ +6: +if �θm > ce[n] then +7: +u ← m +8: +else +9: +l ← m +10: +end if +11: +end while +12: Find the nearest neighbor i⋆ = arg mini∈{l,u}(ce[n]− �θi)2 +13: Return �xhi(n) ← �vi⋆ +Theorem 2. Assume α ∈ GD. Given the ordered set Θsort +α , the +output of Algorithm 2 with input ce[n] exactly coincides with +the solution of the optimization problem (24) in at most O(D) +steps. Furthermore, if for all n, |w[n]| < ∆θmin(α, D)/4, then +the output of Algorithm 2 satisfies �xhi(n) = xhi(n). +From Theorem 2, it is evident that ∆θmin(α, D) plays an +important role in characterizing the upper bound on noise. +We attempt to gain insight into how ∆θmin(α, D) varies as a +function of α when D is held fixed. +Lemma 7. Given D, ∆θmin(α, D) = αD−1 for α ∈ (0, 0.5]. +Proof. The proof is in Appendix C. +When α ∈ (0, 0.5], ∆θmin(α, D) is monotonically increasing +with α. However, for α > 0.5 the trend fluctuates with α +differently for different D, and becomes quite challenging to +predict. This is also confirmed by the empirical plot in Fig. 1. +A refined analysis of ∆θmin(α, D) to gain insight into desirable +filter parameters α is an interesting direction for future work. +C. Trade-off between memory and computational complexity +A crucial aspect of Algorithms 1 and 2 is that they +achieve efficient run-time complexity by leveraging the off- +line construction of the sorted list Θsort +α +and Ssort +all . These lists, +each with 2D elements, need to be stored in memory and +made available during run-time. Since there is no free lunch, +the resulting computational efficiency of O(D) at run-time +is attained at the expense of the additional memory that is +required to store the sorted lists Θsort +α , Ssort +all . + +8 +D. Parallelizable Implementation +Algorithm 2 (also Algo. 1) only takes ce[n](c[n]) as input +and returns �xhi(n), and is completely de-coupled from any +other �xhi(n′), n′ ̸= n. Recall that in reality, we are provided +with measurements zlo[n](ylo[n]), and ce[n](respectively c[n]) +needs to be computed. Due to this de-coupling, we can compute +ce[n]′s in parallel using two consecutive low-rate samples +zlo[n], zlo[n−1] and perform a nearest neighbor search without +waiting for any previously decoded spikes. Therefore, the total +decoding complexity can be further improved depending on +the available parallel computing resources. +IV. ERROR ANALYSIS FOR GAUSSIAN NOISE +Algorithm 2 solves (24) without requiring any knowledge +of the noise statistics. However, in order to analyze its per- +formance, we will make the following (standard) assumptions +on the statistics of the high-rate spiking signal xhi and the +measurement noise w[n] as follows: +• (A1) The entries of the binary vector xhi ∈ {0, A}L are +i.i.d random variables distributed as xhi[n] ∼ ABern(p). +• (A2) The additive noise w[n], 0 ≤ n ≤ M − 1 is +independent of xhi[n], and distributed as w[n] ∼ N(0, σ2) +A. Probability of Erroneous Decoding +Under assumption (A2), the ML estimate of xhi is given by +the solution to the following problem: +�xML = arg +min +v∈{0,A}L ∥zlo − SDGαv∥2 +(PNN) +The proposed Algorithm 2 does not attempt to solve +(PNN), which is computationally intractable. Instead, it solves +a set of M − 1 one dimensional nearest neighbor search +problems, by finding the nearest neighbor of ce[n] for each +n = 1, 2, · · · , M − 1. This scalar nearest neighbor search is +implemented in a computationally efficient manner by using +parallel binary search on a pre-sorted list. Notice that by the +operation (22), the variance of the equivalent noise term e[n] +gets amplified by a factor of at most (1+α2D) < 2. This can be +thought of as a price paid to achieve computational efficiency +and parallelizability. The following theorem characterizes the +dependence of certain key quantities of interest, such as the +signal-to-noise ratio (SNR), undersampling factor D, and filter’s +frequency response (controlled by α) on the performance of +Algorithm 2. +Theorem 3. Suppose α ∈ GD and assumptions (A1-A2) hold. +Given δ > 0, if the following condition is satisfied: +∆θ2 +min(α, D)/σ2 ≥ 4 ln (2M/δ) +(28) +then Algorithm 2 can exactly recover the binary signal xhi +with probability at least 1 − δ. +Proof. The proof follows standard arguments for computing +the probability of error for symbol detection in Gaussian noise, +followed by certain simplifications and is included in Appendix +D for completeness. +In Fig. 1, we plot ∆θmin(α, D) as a function of D for +different values of α. As expected, ∆θmin(α, D) decays as the +D increases. Understandably, for a fixed α, as D increases, +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +Minimum distance +For D=4 +For D=5 +Min. Dist (D=4) +Min. Dist (D=5) +Cluster Min. Dist (D=4) +Cluster Min. Dist (D=5) +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +Undersampling factor (D) +0 +0.2 +0.4 +0.6 +0.8 +1 +Minimum distance +=0.2 +=0.5 +=0.9 +Fig. 1: Variation of ∆θmin(α, D) as a function of undersampling factor +D and α. The cluster-distance ∆c +min(α, D) vs. α is also overlaid. Each +dotted line denotes the start of the interval FD. +it becomes harder to recover the spikes exactly, and higher +SNR is needed to compensate for the lower sampling rate. +This can be interpreted as the price paid for super-resolution +in presence of noise. This phenomenon is also reminiscent of +the noise amplification effect in super-resolution, where the +ability to super-resolve point sources becomes more severely +hindered by noise as the target resolution grid becomes finer +[6]. In Fig. 1, we plot ∆θmin(α, D) as a function of α and as +predicted by Lemma 7, it monotonically increases upto 0.5, +but for α > 0.5, the behavior becomes much more erratic +and a precise characterization becomes challenging. It is to +be noted that in Theorem 3, we aim to exactly recover xhi. +The SNR requirement can be relaxed if our goal is to recover +only spike counts instead of the true spikes as discussed in the +next subsection. One can define other notions of approximate +recovery, the analysis of which will be a topic of future research. +B. Relaxed Spike reconstruction: Count Estimation +As shown in Theorem 2, exact recovery of spikes is possible +under somewhat restrictive condition on the noise in terms +of ∆θmin(α, D), which becomes quite small as D increases. +This naturally calls for other relaxed notions of recovery +which can handle larger noise levels. In neuroscience, it is +believed that information is encoded as either the spike timing +(temporal code) or the firing rates (rate coding) of individual +neurons in the brain. Therefore, the spike counts over an +interval can be informative to understand neural functions, even +when it is impossible to temporally localize the neural spikes. +For example, neurons in the visual cortex encode stimulus +orientations as their firing rates [52]. We will therefore focus +on spike count as an approximate recovery metric, which +concerns estimating the number of spikes occurring between +two consecutive low-rate measurements instead of resolving +the individual spiking activity at a higher resolution. +Let γ[n] denote the total number of spikes occurring between +two consecutive low-rate samples zlo[n] and zlo[n − 1]. Since +xhi and its estimate �xhi are both binary valued (amplitude A), +the true spike count (γ[n]) and estimated count (�γ[n]) are given +by: γ[n] = ∥xhi(n)∥0, +�γ[n] = ∥�x(n) +hi ∥0, n = 1, · · · , M − 1, + +9 +γ[0] = xhi[0]/A and �γ[0] = �xhi[0]/A since the first block is of +size 1 as described in (6). Define a set CD +k as: +CD +k := {v ∈ {0, A}D, ∥v∥0 = k}, +0 ≤ k ≤ D +It is a collection of all binary vectors (of length D) with spike +count k. The ground truth spike block belongs to CD +γ[n]. Any +element from CD +γ[n] will give the true spike count. Hence, exact +recovery of count can be possible even when spikes cannot be +recovered. +For a fixed D, we define a set of α denoted by FD: +FD := {α ∈ (0, 1)|αD − αD−k0−1 − αk0 + 1 < 0} +(29) +where k0 = ⌊D/2⌋. We will obtain a sufficient condition for +robust spike count estimation when α ∈ FD. It can be shown +that for any D, FD will always be non-empty. Define +θk +min := min +u∈CD +k +h⊤ +α u +θk +max := max +u∈CD +k +h⊤ +α u +(30) +Observe that if +θk+1 +min > θk +max, k = 0, 1, · · · , D − 1 +(31) +then all spike patterns ui ∈ CD +k (with the same spike count k) +are clustered together when mapped on to the real line by the +transformation h⊤ +α u as shown in Figure 2. When (31) holds, +we can define a “cluster-restricted minimum distance" as: +∆c +min(α, D) := +min +0≤k≤D−1 θk+1 +min − θk +max +(32) +Given a noisy observation ce[n] = h⊤ +α xhi(n)+e[n], the solution +to the nearest neighbor problem (24) may return an incorrect +neighbor θj ̸= h⊤ +α xhi(n). However, when (31) holds and if +the noisy observation satisfies the following conditions: +(θγ[n] +min + θγ[n]−1 +max +)/2 < ce[n] < (θγ[n]+1 +min ++ θγ[n] +max)/2 +(33) +then the nearest-neighbor decision rule in Algorithm 2 will still +ensure that θj ∈ CD +γ[n]. This has also been visualized in Fig. 2 +where each colored band represents the “safe-zone" for each +count and the black dotted-line denotes the boundary. This will +result in correct identification of the spike count but will incur +error in terms of spiking pattern. We formally summarize this +in the following Theorem that provides robustness guarantee +for exact count recovery from measurements corrupted by +adversarial noise (similar to Theorem 2 for spike recovery). +Theorem 4. Assume α ∈ FD. Given the ordered set Θsort +α , let +�γ[n] be the estimated spike count obtained from Algorithm 2 +with input ce[n]. If for all n, |w[n]| < ∆c +min(α, D)/4, then the +count can be exactly recovered, i.e., �γ[n] = γ[n]. +Proof. Proof is in Appendix E. +It is clear that when (31) holds, ∆c +min(α, D) is no smaller +than ∆θmin(α, D), since the former is computed over neigh- +boring elements of the cluster whereas ∆θmin(D, α) computes +the minimum distance over all consecutive elements (both +inter-cluster as well as intra-cluster) in Θsort +α . This essentially +suggests that estimation of counts (for this range of α and +D) can be more robust compared to inferring the individual +spiking patterns. We also illustrate this numerically in Figure +1 (top), where we plot both ∆c +min and ∆θmin as a function of +α and the start of the interval FD (computed numerically) is +C0 +C1 +C2 +C3 +000 +100 +010 +001 +110 +101 +011 +111 +Fig. 2: Visualization of the sets CD +k for D = 3. In this scenario, the +spiking patterns corresponding to the same count are clustered together +and hence, are favorable for robust count estimation. +denoted using dotted lines. For both values of D, we can see +that ∆c +min > ∆θmin and the gap grows as α increases. +V. NUMERICAL EXPERIMENTS +We conduct numerical experiments to evaluate the per- +formance of the proposed super-resolution spike decoding +algorithm on both synthetic and real calcium imaging datasets. +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Undersampling Factor (D) +0 +0.2 +0.4 +0.6 +0.8 +1 +F-score +p=0.35, s=350 +Algo 2 ( =0.5) +l1 Box ( =0.5) +Algo 2 ( =0.9) +l1 Box ( =0.9) +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Undersampling factor (D) +0.6 +0.7 +0.8 +0.9 +1 +F1-score +p=0.5, s=500 +D=3 +D=5 +D=7 +AR(1), =0.5 +FIR (r=3) +FIR (r=5) +FIR (r=7) +Fig. 3: (Top) Quantitative comparison of Algorithm 2 against box- +constrained l1 minimization method with noiseless measurements +(with tolerance t0 = 0). (Bottom) (Role of Filter Memory): Average +F-score vs. D for FIR and IIR (AR(1)) filters. Each dotted line indicates +the corresponding theoretical transition point (D = r). +A. Synthetic Data Generation and Evaluation Metrics +We create a synthetic dataset by generating high-rate binary +spike sequence xhi ∈ {0, 1}L (A = 1 and L = 1000) that +satisfies assumption (A1). The spiking probability p controls +the average sparsity level given by s := E[∥xhi∥0] = Lp. We +aim to reconstruct xhi from M ≈ L/D low-rate measurements +zlo[n] defined in (21). Notice that we operate in a regime where +the expected sparsity is greater than the total number of low- +rate measurements, i.e., s > M. We employ the widely-used +F-score metric to evaluate the accuracy of spike detection [4], +[10]. The F-score is computed by first matching the estimated +and ground truth spikes. An estimated spike is considered a +“match" to a ground truth spike if it is within a distance of t0 +of the ground truth (many-to-one matching is not allowed) [4], +[10]. Let K and K′ be the total number of ground truth and +estimated spikes, respectively. The number of spikes declared as +true positives is denoted by Tp. After the matching procedure, +we compute the recall (R = +Tp +K ) which is defined as the +ratio of true positives (Tp) and the total number of ground +truth spikes (K). Precision (P = Tp +K′ ) measures the fraction +of the total detected spikes which were correct. Finally, the +F-score is given by the harmonic mean of recall and precision +F-score = 2PR/(P + R). + +10 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +24 +26 +28 +30 +32 +34 +36 +38 +40 +42 +44 +46 +48 +50 +0 +4.99 +yhi[n] +D = 5 +(Top) +(Bottom) +0 +4.99 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +ylo[n] +0 +1.02 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +xhi[n] +0 +1.02 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +�xhi[n] +0 +1.02 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +�xl1[n] +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +24 +26 +28 +30 +32 +34 +36 +38 +40 +42 +44 +46 +48 +50 +0 +4.47 +yhi[n] +D = 10 +0 +4.47 +0 +10 +20 +30 +40 +50 +ylo[n] +0 +1.02 +0 +10 +20 +30 +40 +50 +xhi[n] +0 +1.02 +0 +10 +20 +30 +40 +50 +�xhi[n] +0 +1.02 +0 +10 +20 +30 +40 +50 +�xl1[n] +0 +4.99 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +ylo[n] +0 +1.02 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +xhi[n] +0 +1.02 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +�xhi[n] +0 +1.02 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +�xl1[n] +0 +3.45 +0 +10 +20 +30 +40 +50 +ylo[n] +0 +1.02 +0 +10 +20 +30 +40 +50 +xhi[n] +0 +1.02 +0 +10 +20 +30 +40 +50 +�xhi[n] +0 +1.02 +0 +10 +20 +30 +40 +50 +�xl1[n] +xhi[n]: Ground Truth Spikes, �xhi[n]: Output of Algorithm 2, �xl1[n]: Output of l1 minimization, +yhi[n]: High rate waveform, ylo[n]: Low rate samples +Fig. 4: Qualitative comparison of Algorithm 2 and box-constrained l1 minimization on simulated data. For each simulation noisy measurements +are generated with α = 0.9 such that the noise realization (Top) obeys the bound |w[n]| ≤ ∆θmin (from Theorem 2) and (Bottom) violates +the bound. For larger noise (Bottom), the spike recovery is imperfect but the spike count can still be exactly recovered using Algorithm 2. +B. Noiseless Recovery: Role of Binary priors and memory +We first consider the noiseless setting (w[n] = 0 in (21)). +We compare the performance of Algorithm 2 against box- +constrained l1 minimization method [35], [36], where we solve: +min +x∈RL ∥x∥1 s.t. ∥ylo − SDGαx∥2 ≤ ϵ, 0 ≤ x ≤ A1 +(P1) +For synthetic data, ϵ is chosen using the norm of the noise term +∥w∥2. This oracle choice ensures most favorable parameter +tuning for the (P1), although a more realistic choice would +be to set ϵ = +√ +Mσ according to the noise power (σ). In the +noiseless setting, we choose ϵ = 0. The problem (P1) is a +standard convex relaxation of (P0) which promotes sparsity +as well as tries to impose the binary constraint via the box- +relaxation (introduced in Section II-C). In Fig. 3 (Top), we plot +the F-score (t0 = 0) as a function of D. As can be observed, +Algorithm 2 consistently achieves an F-score of 1, whereas the +F-score of l1 minimization shows a decay as D increases. This +confirms Lemma 3 that for D > 1, using box-constraints with l1 +norm minimization is not enough to enable exact recovery from +low rate measurements. In absence of noise, the performance +of Algorithm 2 is not affected by the filter parameter α as +shown in Fig. 3 (Top). +Next, we compare the reconstruction from the decimated +output of (i) an AR(1) filter and (ii) an FIR filter of length +r driven by the same input xhi ∈ {0, 1}1000. We choose the +FIR filter h = [1, α, · · · , αr−1]⊤ (truncation of the IIR filter) +with α = 0.5. Algorithm 2 is applied to the low-rate AR(1) +measurements, whereas the algorithm proposed in [40] is used +for the FIR case. The algorithm applied for the FIR case can +provably operate with the optimal number of measurements +when α = 0.5 and hence, we chose this specific value for +the filter parameter. In Figure 3 (Bottom), we again compare +the average F-score as a function of D, averaged over 10000 +Monte Carlo runs, for p = 0.5. As predicted by Lemma 4, +despite utilizing binary priors, the error for the FIR filter shows +a phase transition when D > r. This demonstrates the critical +role played by the infinite memory of the AR(1) filter in +achieving exact recovery with arbitrary D. +C. Performance of noisy spike decoding +We generate noisy measurements of the form (21), where +w[n] and xhi[n] satisfy assumptions (A1-A2). We illustrate +some representative examples of recovered spikes on synthetic +data. In Fig. (4), we display the recovered super-resolution +estimates on synthetically generated measurements for two +undersampling factors D = 5 (left), 10 (right). For each D, the +top plots show the spikes recovered using Algorithm 2 and l1 +minimization with box-constraint where the noise realization +obeys the bound in Theorem 2, while the bottom plots show +the same for noise realization violating the bound. The output +of l1 minimization with box-constraint is inaccurate, and the +spikes are clustered towards the end of each block of length +D. This bias is consistent with the prediction made by our +theoretical results in Lemma 3. When the noise is small enough +(top), Algorithm 2 exactly decodes the spikes, including the +ones occurring between two consecutive low-rate samples as +predicted by Theorem 2. In presence of larger noise (violating +the bound), the spikes estimated using l1 minimization continue +to be biased to be clustered towards the end of the block. +Although the spikes recovered using Algorithm 2 are not exact, +most of the detected spikes are within a tolerance window of +ground truth spikes. In fact, the spike count estimation is perfect +as predicted by Theorem 4. We next quantitatively evaluate +the performance in presence of noise, where the metrics are +computed with t0 = 2. In Fig. 5 (Top), we plot the F-score +as a function of D for different values of α. For a fixed α, +the F-score of both methods decays with increasing D, but +Algorithm 2 consistently attains a higher F-score compared to + +11 +3 +4 +5 +6 +7 +8 +9 +10 +Undersampling Factor (D) +0.2 +0.4 +0.6 +0.8 +1 +1.2 +F-score +p=0.35, s=350>M +Algo 2 (alpha=0.9) +l1 Box (alpha=0.9) +Algo 2 (alpha=0.5) +l1 Box (alpha=0.5) +Fig. 5: Spike detection performance with noisy measurements. (Top) +F-score vs. D for different filter parameters α (σ = 0.01). Here, +L = 1000 and expected sparsity s = 350 where we operate in the +regime s > M. The F-score is computed with a tolerance of t0 = 2. +l1 minimization. We observe that α = 0.5 leads to a higher F- +score potentially due to having a larger ∆θmin(α, D) compared +to α = 0.9. Next, in Fig. 7, we study the behavior of spike +detection as a function of the spiking probability p, while +keeping D fixed at D = 5. When σ is fixed, the performance +trend is not significantly affected by the spiking probability. +At first, this may seem surprising as the expected sparsity +is growing while the number of measurements is unchanged. +However, since our algorithm exploits the binary nature of +the spikes (and not just sparsity), it can handle larger sparsity +levels. The spikes reconstructed using l1 minimization achieve +a much lower F-score than Algorithm 2 since the former fails +to succeed when the sparsity is large. As expected, smaller σ +leads to higher F-scores. +In Fig. 8, we study the probability of erroneous spike +detection as a function of D and validate the upper bound +derived in Theorem 3. Recall that the decoding is considered +successful if “every" spike is detected correctly. Therefore, it +becomes more challenging to “exactly super-resolve" all the +spikes in presence of noise as the desired resolution becomes +finer. We calculate the empirical probability of error and overlay +the corresponding theoretical bound. As shown in Fig. 8, the +empirical probability of error is indeed upper bounded by the +bound computed by our analysis. The empirical probability of +error increases as a function of undersampling factor D. +10-5 +10-4 +10-3 +10-2 +10-1 +100 +Noise Level +0.2 +0.4 +0.6 +0.8 +1 +F-score +D=5, M=200, s/M>1 +Algo 2 ( =0.5) +l1 Box ( =0.5) +Algo 2 ( =0.9) +l1 Box ( =0.9) +10-5 +10-4 +10-3 +10-2 +10-1 +100 +Noise Level +10-10 +10-5 +100 +105 +Count Estimation Error +D=5, M=200, s/M>1 +Algo 2 ( =0.5) +l1 Box ( =0.5) +Algo 2 ( =0.9) +l1 Box ( =0.9) +Fig. 6: Spike detection performance with noisy measurements for +different filter parameters α. (Top) F-score vs. noise level (σ) (Bottom) +Count estimation error vs. noise level. Here, L = 1000 and expected +sparsity is fixed at s = 350 where we operate in the regime s > M. +The F-score is computed with a tolerance of t0 = 2. +Finally, we evaluate the noise tolerance of the proposed +methodology by comparing the average F-score as a function +of the noise level σ, while keeping the spiking rate and +undersampling factor fixed at p = 0.35 and D = 5, respectively. +As seen in Fig. 6 (Top), the performance of both algorithms +degrades with increasing noise level and this is also consistent +with the intuition that it becomes harder to super-resolve spikes +with more noise. However, for both filter parameters considered +in this experiment Algorithm 2 has a higher F-score compared +to box-constrained l1 minimization. For large noise levels +(comparable to spike amplitude A = 1), the performance gap +decreases for α = 0.9 but Algorithm 2 achieves a much higher +F-score for α = 0.5 at all noise levels. +As discussed in Section IV-B, we next study a relaxed +notion of spike recovery which focuses on the spike counts +occurring between two consecutive low-rate samples. Let Γ = +[γ[0], · · · , γ[M − 1]]⊤ be the vector of counts and �Γ be its +estimate. In Fig. 6 (Bottom) we plot the average l1 distance +∥Γ − �Γ∥1 as a function of the noise level. We observe that for +α = 0.9 (it can be verified from Fig. 1 (Top) that 0.9 ∈ F5), it +is possible to exactly recover the spike counts at higher noise +even though the F-score (for timing recovery) has dropped +below 1. However, this is not the case for α = 0.5, since +0.5 ̸∈ F5. This is consistent with the conclusion of Theorem 4 +which states that when α ∈ FD, the noise tolerance for exact +count recovery can be much larger than exact spike recovery +since ∆c +min(α, D) > ∆θmin(α, D). +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +Spiking Probability (p) +0.2 +0.4 +0.6 +0.8 +1 +F-score +D=5, M=200, s/M>1 +Algo 2 (sigma=0.001) +l1 Box (sigma=0.001) +Algo 2 (sigma=0.01) +l1 Box (sigma=0.01) +Fig. 7: Spike detection performance with noisy measurements. F-score +vs. spiking probability (p) for different noise levels σ (fix α = 0.9, +D = 5, L = 1000) in the extreme compression regime s > M. +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Undersampling factor (D) +0 +0.2 +0.4 +0.6 +0.8 +1 +Probability of Error +s=30, L=100 +Algo 2 ( =0.9) +Theoretical Bound ( =0.9) +Algo 2 ( =0.95) +Theoretical Bound ( =0.95) +Fig. 8: Probability of erroneous detection of high-rate spikes xhi ∈ +{0, 1}L as a function of the undersampling factor D. Theoretical +upper bounds are overlaid using dotted lines. Here, L = 100. +D. Spike Deconvolution from Real Calcium Imaging Datasets +We now discuss how the mathematical framework developed +in this paper can be used for super-resolution spike deconvo- +lution in calcium imaging. Two-photon calcium imaging is a +widely used imaging technique for large scale recording of +neural activity with high spatial but poor temporal resolution. In +calcium imaging, the signal xhi corresponds to the underlying +neural spikes which is modeled to be binary valued on a finer +temporal scale [2], [46]. Each neural spike results in a sharp +rise in Ca2+ concentration followed by a slow exponential +decay, leading to superposition of the responses from nearby + +12 +spiking events [2]–[4]. This calcium transient can be modeled +by the first order autoregressive model introduced in Section +II. The decay time constant depends on the calcium indicator +and essentially determines the filter parameter α. The signal +yhi[n] is an unobserved signal corresponding to sampling the +calcium fluorescence at a high sampling rate (at the same rate +as the underlying spikes). The observed calcium signal ylo[n] +corresponds to downsampling yhi[n] at an interval determined +by the frame rate of the microscope. The frame rate of a +typical scanning microscopy system (that captures the changes +in the calcium fluorescence) is determined by the amount of +time required to spatially scan the desired field of view, which +makes it significantly slower compared to the temporal scale +of the neural spiking activity. We model this discrepancy by +the downsampling operation (by a factor D). Therefore, the +mathematical framework developed in this paper can be directly +applied to reconstruct the underlying spiking activity at a +temporal scale finer than the sampling rate of the calcium signal. +Using real calcium imaging data, we demonstrate a way to fuse +our algorithm with a popular spike deconvolution algorithm +called OASIS [43]. OASIS solves an l1 minimization problem +similar to (P1) with only the non-negativity constraint, in order +to exploit the sparse nature of the spiking activity. Unlike our +approach where we wish to obtain spikes representation on a +finer temporal scale, OASIS returns the spike estimates on the +low-resolution grid. This is typically used to infer the spiking +rate over a temporal bin equal to the sampling interval. We +demonstrate that our proposed framework can be integrated with +OASIS and improve its performance. As we saw in the synthetic +experiments, the noise level is an important consideration. By +augmenting Algorithm 2 with OASIS, referred as “B-OASIS", +the denoising power of l1 minimization can be leveraged.Let +�xl1 ∈ RM be the estimate obtained on a low-resolution grid +by solving the l1 minimization problem such as the one +implemented in OASIS. We can obtain an estimate of the +denoised calcium signal as �ylo[n] = αD�ylo[n] + �xl1[n], n ≥ 1 +and �ylo[0] = �xl1[0]. We can now utilize the denoised calcium +signal �ylo[n] generated by OASIS to obtain the estimate ce[n] +indirectly. Due to the non-linear processing done by OASIS, it +is difficult to obtain the resulting noise statistics. An important +advantage of Algorithm 2 is that it does not rely on the +knowledge of the noise statistics. Hence, we can directly apply +Algorithm 2 on �ce[n] = �ylo[n]−αD�ylo[n−1] (instead of ce[n]) +to obtain a binary “fused super-resolution spike estimate". +B-OASIS +OASIS +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Recall +F-score +B-OASIS +OASIS +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Recall +F-score +Fig. 9: Spike detection performance of OASIS and B-OASIS on +GCaMP6f dataset sampled at (Left) 60 Hz and (Right) 30 Hz. We +compare the average F-score of data points where the F-score of +OASIS is < 0.5. Standard deviation is depicted using the error bars. +20 +20.2 +20.4 +20.6 +20.8 +21 +21.2 +21.4 +21.6 +21.8 +22 +ylo[n] +20 +20.2 +20.4 +20.6 +20.8 +21 +21.2 +21.4 +21.6 +21.8 +22 +xhi[n] +20 +20.2 +20.4 +20.6 +20.8 +21 +21.2 +21.4 +21.6 +21.8 +22 +�xhi +B-OA[n] +20 +20.2 +20.4 +20.6 +20.8 +21 +21.2 +21.4 +21.6 +21.8 +22 +�xhi +OA[n] +Fig. 10: Example of spike reconstruction on GENIE dataset (GCaMP6f +indicator) using OASIS and B-OASIS (binary augmented) with +calcium signal sampled at 30Hz. +E. Results +We evaluate the algorithms on the publicly available GENIE +dataset [53], [54] which consists of simultaneous calcium imag- +ing and in vivo cell-attached recording from the mouse visual +cortex using genetically encoded GCaMP6f calcium indicator +GCaMP6f [53], [54]. The calcium images were acquired at a +frame rate of 60 Hz and the ground truth electrophysiology +signal was digitized at 10 KHz and synchronized with the +calcium frames. In addition to using the original data, we also +synthetically downsample it to emulate the effect of a lower +frame rate of 30 Hz, and evaluate how the performance changes +by this downsampling operation. +In Fig. 10, we extract an interval of ∼ 2 sec (from the neuron +1 of the GCaMP6f indicator dataset) and qualitatively compare +the detected spikes with the ground truth. We downsample +the data by a factor of 2 to emulate frame rate of 30 Hz, +the low-rate grid becomes coarser. As a result of which, we +observe an offset between ground truth spikes and estimate +produced by OASIS. However, with the help of binary priors +(B-OASIS), we can output spikes that are not restricted to be +on the coarser scale, and this mitigates the offset observed in +the raw estimates obtained by OASIS. +We quantify the improvement in the performance by com- +paring the F-scores of OASIS and B-OASIS at both sampling +rates (60 and 30 Hz). Since the output of OASIS is non- +binary, the estimated spikes are binarized by thresholding. +To ensure a fair comparison, we select the threshold by a +80 − 20 cross-validation scheme that maximizes the average +F-score on a held-out validation set (averaged over 3-random +selections of the validation set). The tolerance for the F-score +was set at 100 ms. The dataset consisted of 34 traces of +length ∼ 234 s. The OASIS algorithm has an automated +routine to estimate the parameter α, which we utilize for +our experiments. The amplitude A is estimated using the +procedure described in Appendix F. We use D = 12 to obtain +the spike representation for B-OASIS. In order to quantify +the performance boost achieved by augmentation, we isolate +the traces where the F−score of OASIS drops below 0.5 +and compare the average F-score and recall for these data +points. As shown in Fig. 9, at both sampling rates, we see a +significant improvement in the average F-score of B-OASIS +over OASIS, attributed to an increase in recall while keeping the +precision unchanged. Additionally, despite downsampling, the +spike detection performance is not significantly degraded with +binary priors, although the detection criteria were unchanged. + +13 +VI. CONCLUSION +We theoretically established the benefits of binary priors in +super-resolution, and showed that it is possible to achieve +significant reduction in sample complexity over sparsity- +based techniques. Using an AR(1) model, we developed +and analyzed an efficient algorithm that can operate in the +extreme compression regime ( M ≪ K) by exploiting the +special structure of measurements and trading memory for +computational efficiency at run-time. We also demonstrated that +binary priors can be used to boost the performance of existing +neural spike deconvolution algorithms. In the future, we will +develop algorithmic frameworks for incorporating binary priors +into different neural spike deconvolution pipelines and evaluate +the performance gain on diverse datasets. The extension of +this binary framework for higher-order AR filters is another +exciting future direction. +APPENDIX +APPENDIX A: PROOF OF THEOREM 1 +Proof. We show that for any α in 0 < α < 1, except possibly +for a set consisting of only a finite number of points, (10) +always has a unique binary solution. Consider all possible +D−dimensional ternary vectors with their entries chosen from +{−1, 0, 1}, and denote them as v(i) = [v(i) +1 , v(i) +2 , · · · , v(i) +D ]T ∈ +{−1, 0, 1}D, 0 ≤ i ≤ 3D − 1. We use the convention that +v(0) = 0. For every i > 0, we define a set Zv(i) determined +by v(i) as Zv(i) := +� +x ∈ (0, 1) +�� �D +k=1 v(i) +k xD−k = 0 +� +. Notice +that pi(x) := �D +k=1 v(i) +k xD−k denotes a polynomial (in x) of +degree at most D−1, whose coefficients are given by the ternary +vector v(i). The set Zv(i) denotes the set of zeros of pi(x) that +are contained in (0, 1). Since the degree of pi(x) is at most +D−1, Zv(i) is a finite set with cardinality at most D−1. +Now suppose that the binary solution of (10) is non-unique, +i.e., there exist u, w ∈ {0, A}L, u ̸= w, such that +HD(α)u = HD(α)w ⇒ HD(α)u − HD(α)w = 0 +(34) +By partitioning u, w into blocks u(n), w(n) in the same way +as in (6), we can re-write (34) as u(0) = w(0) and +D +� +i=1 +1 +A([u(j)]i − [w(j)]i)αD−i = 0, +1 ≤ j ≤ M − 1 +(35) +Since u ̸= w, they differ at least at one block, i.e., there exists +some j0, 1 ≤ j0 ≤ M − 1 such that u(j0) ̸= w(j0). Define +b := 1 +A(u(j0) − w(j0)). Then, b is a non-zero ternary vector, +i.e., b ∈ {−1, 0, 1}D. Now from (35), we have +D +� +i=1 +[b]iαD−i = 0, +(36) +which implies that α ∈ Zb. Since b can be any one of the 3D−1 +ternary vectors {v(i)}3D−1 +i=1 , (36) holds if and only if α ∈ S := +�3D−1 +i=1 Zv(i), i.e., α is a root of at least one of the polynomials +pi(x) defined by the vectors v(i) as their coefficients. For each +v(i), since the cardinality of Zv(i) is at most D−1, S is a finite +set (of cardinality at most (D − 1)(3D − 1)), and therefore its +Lebesgue measure is 0. This implies that (10) has a non-unique +binary solution only if α belongs to the measure zero set S, +thereby proving the theorem. +APPENDIX B: PROOF OF LEMMA 2 AND LEMMA 3 +Proof. (i) Let sn denote the sparsity (number of non-zero +elements) of the nth block xhi(n) of xhi. Then, the total +sparsity is ∥xhi∥0 = �M−1 +n=0 sn. We will construct a vec- +tor v ∈ RL, v ̸= xhi that satisfies c = HD(α)v and +∥xhi∥0 ≥ ∥v∥0. Following (6), consider the partition of v +v = [v(0), v(1)⊤, · · · , v(M−1)⊤]⊤. Firstly, we assign v(0) = +c[0] = xhi(0). We construct v(n) as follows. For each n ≥ 1, +there are three cases: +Case I: sn = 0. In this case, xhi(n) = 0 and hence c[n] = 0. +Therefore, we assign v(n) = xhi(n) = 0. +Case II: sn = 1. First suppose that [xhi(n)]D = 0. We +construct v(n) as follows: +[v(n)]k = +� +c[n], +if k = D +0, +else +. +(37) +Next suppose that [xhi(n)]D ̸= 0. Since sn = 1, this implies +that [xhi(n)]k = 0, k = 1, · · · , D−1. In this case, we construct +v(n) as follows: +[v(n)]k = +� +c[n]/α, +if k = D − 1 +0, +else +. +(38) +Notice that both (37) and (38) ensure that v(n) ̸= xhi(n) and +c[n] = hT +αv(n). Moreover, ∥v(n)∥0 = sn. +Case III: sn ≥ 2. In this case, we follow the same +construction as (37). As before v(n) satisfies c[n] = h⊤ +α v(n). +Since ∥xhi(n)∥0 ≥ 2 and ∥v(n)∥0 = 1, we automatically have +v(n) ̸= xhi(n), and ∥v(n)∥0 < sn. Therefore, combining the +three cases, we can construct the desired vector v that satisfies +v ̸= xhi, c = HD(α)v, and ∥v∥0 ≤ �M−1 +n=0 sn = ∥xhi(n)∥0. +Therefore, the solution x⋆ to (P0) satisfies ∥x⋆∥0 ≤ ∥v∥0 ≤ +∥xhi(n)∥0. +(ii) In this case, we construct v(n0) according to Case III. +Since ∥v(n0)∥0 < sn0, and ∥v(n)∥0 ≤ sn, n ̸= n0, we have +∥v∥0 < ∥xhi∥0, implying ∥x⋆∥0 ≤ ∥v∥0 < ∥xhi∥0. +A. Proof of Lemma 3 +Proof. We will construct a vector v ∈ RL whose support is of +the form (16), that is feasible for (P1-B), and we will prove +that it has the smallest l1 norm. Using the block structure given +by (6), we choose v(0) = c[0]. For each n ≥ 1, we construct +v(n) based on the following two cases: +Case I: c[n] ≥ A. Let kn be the largest integer such that the +following holds: µ[n] := A(1 + α + · · · + αkn−1) ≤ c[n], +where 1 ≤ kn ≤ D. Note that kn = 1 always produces a valid +lower bound. However, we are interested in the largest lower +bound on c[n] of the above form. We choose +[v(n)]k = +� +� +� +� +� +A, +if D − kn + 1 ≤ k ≤ D +(c[n] − µ[n])/αkn, if k = D − kn +0, else +It is easy to verify that h⊤ +α v(n) = c[n]. From the definition +of kn, it follows that µ[n] ≤ c[n] < µ[n] + Aαkn and hence, +0 ≤ (c[n] − µ[n])/αkn < A, which ensures that v obeys the +box-constraints in (P1-B). Now, let vf ∈ RL be any feasible +point of (P1-B) which must be of the form v(0) +f += c[0], v(n) +f += +v(n) + r(n), where r(n) ∈ N(h⊤ +α ) is a vector in the null-space + +14 +of h⊤ +α . It can be verified that the following vectors {wt}D−1 +t=1 +form a basis for N(h⊤ +α ): +[wt]k = +� +� +� +� +� +1, +k = t +−α, +k = t + 1 +0, +else +, +Therefore, ∃ {β(n) +t +}D−1 +t=1 such that r(n) = �D−1 +t=1 β(n) +t +wt. We +further consider two scenarios: (i) 1 ≤ kn ≤ D − 2. In this +case [v(n)]1 = 0, and for k = 1, 2, · · · D, [v(n) +f ]k satisfies 2 +[v(n) +f ]k = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +β(n) +k , if k = 1 +β(n) +k +− αβ(n) +k−1, if 2 ≤ k ≤ D − kn − 1 +[v(n)]k + β(n) +k +− αβ(n) +k−1, if k = D − kn +A + β(n) +k +− αβ(n) +k−1, if D − kn + 1 ≤ k ≤ D − 1 +A − αβ(n) +k−1, if k = D +To ensure v(n) +f +is a feasible point for (P1-B), the following must +hold: 0 ≤ β(n) +D−1 ≤ A/α and 0 ≤ β(n) +1 +≤ A. For 2 ≤ k ≤ D − +kn−1, the constraint [v(n) +f ]k ≥ 0 implies β(n) +k +≥ αβ(n) +k−1. Since +β(n) +1 +≥ 0, it follows that β(n) +k +≥ 0 for all 2 ≤ k ≤ D − kn − 1. +For D−kn+1 ≤ k ≤ D−1, the constraint [v(n) +f ]k ≤ A implies +β(n) +k−1 ≥ β(n) +k /α. Since β(n) +D−1 ≥ 0, it follows that β(n) +k +≥ 0 for +all D − kn ≤ k ≤ D − 1. (ii) kn ∈ {D − 1, D}. In this case, +for k = 1, 2, · · · , D, [v(n) +f ]k satisfies +[v(n) +f ]k = +� +� +� +� +� +[v(n)]1 + β(n) +1 +, if k = 1 +A + β(n) +k +− αβ(n) +k−1, if 2 ≤ k ≤ D − 1 +A − αβ(n) +k−1, if k = D +For 2 ≤ k ≤ D − 1, the box-constraint [v(n) +f ]k ≤ A implies +β(n) +k−1 ≥ β(n) +k /α. Since β(n) +D−1 ≥ 0, it follows that β(n) +k +≥ 0 for +all 1 ≤ k ≤ D − 1. Summarizing, we have established that +β(n) +i +≥ 0, ∀i. +Case II: c[n] < A. In this case, v(n) is constructed following +(37), and hence v(n) +f +has the following structure: +[v(n) +f ]k = +� +� +� +� +� +β(n) +k , if k = 1 +−αβ(n) +k−1 + β(n) +k , if 2 ≤ k ≤ D − 1 +c[n] − αβ(n) +k−1, if k = D +To ensure v(n) +f +is a feasible point, it must hold that β(n) +1 +≥ +0, β(n) +k +≥ αβ(n) +k−1 ≥ 0 for 2 ≤ k ≤ D − 1. Hence, in both +Cases I and II, we established that β(n) +k +≥ 0. For each case, +since v(n) +f +is a non-negative vector ∀n, it can be verified that +∥vf∥1 = +M−1 +� +n=0 +∥v(n) +f ∥1 = v(0) +f ++ +M−1 +� +n=1 +D +� +k=1 +[v(n) +f ]k += c[0] + +M−1 +� +n=1 +D +� +k=1 +[v(n)]k +� +�� +� +∥v∥1 ++ +M−1 +� +n=1 +D−1 +� +k=1 +(1 − α)β(n) +k +2In the definition of v(n) +f +, an assignment will be ignored if the specified +interval for k is empty. +We used the fact that �D +k=1 +�D−1 +t=1 β(n) +t +[wt]k = �D−1 +t=1 (1 − +α)β(n) +t +. If vf ̸= v, we must have β(n) +k +̸= 0 for some k and +n > 0. This implies that ∥vf∥1 > ∥v∥1. It is easy to see +that the support of the constructed vector is of the form (16). +Moreover, based on the above argument, v is the only vector +that has the minimum l1 norm among all possible feasible +points of (P1-B). +APPENDIX C: PROOF OF LEMMA 7 +Proof. For any 0 < α ≤ 0.5, we begin by showing that for an +integer p ≥ 1 the following inequality holds: +p +� +k=1 +αD−k = αD−p−1 +� 1 − αp +1/α − 1 +� +< αD−p−1 +(39) +since 1/α − 1 ≥ 1 and 1 − αp < 1 in the regime 0 < α ≤ 0.5. +Let S1 = {0, αD−1, αD−2, αD−1 + αD−2}. Notice that the +elements of S1 are sorted in ascending order for any α and D. +Now, we recursively define the sets Si as follows: +Si := {Si−1, Si−1 + αD−1−i}, 2 ≤ i ≤ D − 1 +(40) +Our hypothesis is that for every 2 ≤ i ≤ D − 1 α ∈ (0, 0.5] +and D, the set Si as defined in (40), is automatically sorted in +ascending order. We prove this via induction. For i = 2, the +sets S1 and S1 + αD−3 are individually sorted. Moreover from +(39), we can show that: maxa∈S1 a = αD−1+αD−2 < αD−3 = +minb∈S1+αD−3 b. This shows that S2 is ordered, establishing the +the base case of our induction. Now, assume Si is ordered for +some 2 ≤ i ≤ D−2. We need to show that Si+1 is also ordered. +As a result of the induction hypothesis, both Si and Si+αD−2−i +are ordered. Using the ordering of Si, we have: maxa∈Si a = +�i+1 +j=1 αD−j, minb∈Si+αD−2−i b = αD−(i+1)−1. From (39), we +can conclude that maxa∈Si a < minb∈Si+αD−2−i b and hence, +Si+1 is also ordered. This completes the induction proof. Also, +note that for α ∈ (0, 0.5], we have Θsort +α += SD−1. +Let ∆min(Si) be the min. distance between the elements of the +set Si. It is easy to see that ∆min(Si) = ∆min(Si + αD−2−i). +Since Si is sorted for α ∈ (0, 0.5], ∆min(Si) is given by: +∆min(Si) = min(∆min(Si−1), +min +x∈Si−1+αD−1−i x − max +y∈Si−1 y) += min{∆min(Si−1), αD−i−1 − +i +� +j=1 +αD−j}. +(41) +Now, we use induction to establish the following conjecture: +∆min(Si) = αD−1, 1 ≤ i ≤ D − 1 +(42) +For the base case i = 1, ∆min(S1) = min(αD−1, αD−2 − +αD−1) = αD−1, where the last equality holds since α ∈ +(0, 0.5] ⇒ αD−1(1/α − 1) ≥ αD−1. Suppose (42) holds for +some 1 ≤ i ≤ D − 2. From the definition of ∆min(Si+1) and +the induction hypothesis that ∆min(Si) = αD−1, it follows that +∆min(Si+1) = min{αD−1, αD−(i+1)−1 −�i+1 +j=1 αD−j}. Again, +from the definition of ∆min(Si) in (41), and the induction +hypothesis we also have αD−i−1 −�i +j=1 αD−j ≥ ∆min(Si) = +αD−1. Using this and the fact that α ≤ 0.5, we can show: +αD−i−2 −αD−i−1 − �i +j=1 αD−j ≥ αD−i−2 − 2αD−i−1 + αD−1 +≥ αD−1 + αD−i−1(1/α − 2) ≥ αD−1 + +15 +Therefore ∆min(Si+1)=min{αD−1, αD−i−2−�i+1 +j=1 αD−j} = +αD−1. +Thus, +we +can +conclude +that +∆min(α, D) += +∆min(SD−1)=αD−1. +APPENDIX D: PROOF OF THEOREM 3 +Proof. The probability of incorrectly identifying xhi(n) from a +single measurement ce[n] is given by +pe := P(�xhi +(n) ̸= xhi +(n)) += +lD +� +k=0 +P(�xhi +(n) ̸= xhi +(n)|xhi +(n) = �vk)P(xhi +(n) = �vk) +Given a binary vector z ∈ {0, 1}D, define the function ψ(z) := +�D +k=1 zk, which denotes the count of ones in z. Since the +noisy observations are given by ce[n] = c[n] + e[n], where +e[n] = w[n] − αDw[n − 1], it follows from assumption (A2) +that e[n] ∼ N(0, σ2 +1) where σ2 +1 = (1 + α2D)σ2. From (27), +we obtain P(�xhi(n) ̸= xhi(n)|xhi(n) = �v0) = P(e[n] ∈ E0) = +Q(αD−1/(2σ1)). Similarly, P(�xhi(n) ̸= xhi(n)|xhi(n) = �vlD) = +P(e[n] ∈ ElD) = Q((�θlD − �θlD−1)/(2σ1)) = Q(αD−1/(2σ1)). +The last equality follows from the fact that �θlD − �θlD−1 = αD−1. +Finally, when conditioned on xhi(n) = �vk for 0 < k < lD, +from (26), we obtain P(�x(n) ̸= xhi(n)|xhi(n) = �vk) = P(e[n] ∈ +Ek) = Q( +�θk−�θk−1 +2σ1 +) + Q( +�θk+1−�θk +2σ1 +). Due to Assumption (A1) +on xhi, we have P(xhi(n) = �vk) = pψ(�vk)(1 − p)D−ψ(�vk). +Therefore, pe is given by +pe = Q(αD−1/(2σ1))(1 − p)D + Q(αD−1/(2σ1))pD+ +lD−1 +� +k=1 +� +Q( +�θk − �θk−1 +2σ1 +) + Q( +�θk+1 − �θk +2σ1 +) +� +pψ(vk)(1 − p)D−ψ(vk) +(43) +The spike train xhi is incorrectly decoded if at least one of the +blocks are decoded incorrectly, hence, the total probability of +error is given by: +P( +M−1 +� +n=0 +�x(n) ̸= xhi +(n)) ≤ +M−1 +� +n=0 +P(�x(n) ̸= xhi +(n)) = Mpe +(a) +≤ 2MQ(∆θmin(α, D)/(2σ1)) +D +� +j=0 +pj(1 − p)D−j +�D +j +� +(b) +≤ 2M exp(−∆θ2 +min(α, D)/(4σ2 +1)) +(44) +where the first inequality follows from union bound and second +equality is a consequence of (43). The inequality (a) follows +from the monotonically decreasing property of Q(.) function +and the sum can be re-written by grouping all terms with the +same count, i.e., ψ(vk) = j. The inequality (b) follows from +the inequality Q(x) ≤ exp(−x2/2) for x > 0. If the SNR +condition (28) holds then from (44) the total probability of +error is bounded by δ. +APPENDIX E: PROOF OF THEOREM 4 +Proof. We first begin by showing that α ∈ FD implies that (31) +holds and hence the mapping of spikes with the same counts are +clustered. Notice that for k = 0, θk +max = θk +min = 0. For k ≥ 1, +it is easy to verify that θk +max and θk +min are attained by the spiking +patterns 00...1111 (with k consecutive spikes at the indices +D − k + 1 to D) and 111...000 (with consecutive spikes at the +indices 1 to k), which allows us to simplify (31) as αD−1 > 0 +for k = 0 and �k+1 +i=1 αD−i > �k−1 +j=0 αj, k = 1, · · · , D − 1. +The values of α that satisfy each of these relations can be +described by the following sets: +G0 = {α ∈ (0, 1)|αD−1 > 0}, Gk = {α ∈ (0, 1)|rk(α) < 0}, +where rk(α) = αD − αD−k−1 − αk + 1 for 1 ≤ k ≤ D − 1. It +is easy to see that FD = Gk0. Observe that the relations are +symmetric, i.e., Gk = GD−k−1. Furthermore, for 1 ≤ k ≤ D/2, +we show that Gk ⊆ Gk−1 as follows. Trivially, G1 ⊂ G0. +For 2 ≤ k ≤ D/2, observe that +rk(α) − rk−1(α) = +αD−k(1 − 1/α) − αk(1 − 1/α) = (1/α − 1)(αk − αD−k) ≥ 0. +Therefore, α ∈ Gk ⇒ α ∈ Gk−1, k = 1, 2 · · · , k0. Moreover, +since Gk = GD−k−1, it follows that FD = Gk0 = ∩D−1 +k=0Gk. +Hence, α ∈ FD ⇒ α ∈ Gi for all 0 ≤ i ≤ D − 1, which +implies that (31) holds. If the noise perturbation satisfies +|w[n]| < ∆c +min(α, D)/4, it implies |e[n]| < ∆c +min(α, D)/2. +For any block xhi(n) ∈ CD +k , θk +min ≤ h⊤ +α xhi(n) ≤ θk +max. If +|e[n]| < ∆c +min(α, D)/2, we have +h⊤ +α xhi +(n) + e[n] < θk +max + ∆c +min(α, D) +2 +< θk +max + θk+1 +min − θk +max +2 +h⊤ +α xhi +(n) + e[n] > θk +min − ∆c +min(α, D) +2 +> θk +min − θk +min − θk−1 +max +2 +This shows that +whenever α ∈ FD, the condition |e[n]| < +∆c +min(α, D)/2 is sufficient for (33) to hold ∀ γ[n] and hence +the spike count can be exactly recovered. +APPENDIX F: AMPLITUDE ESTIMATION +We suggest a procedure to estimate the binary amplitude A, if +it is unknown. We first evaluate the signal c[n] from different +time instants n = 1, 2, · · · , M − 1. For some 1 ≤ n0 ≤ +M − 1, we estimate a set A = {Ak} of candidate amplitudes: +Ak := c[n0]/hT +αvk where vk ∈ Sall. Only a certain amplitudes +can generate c[n0] from a valid binary spiking pattern vk ∈ Sall. +Our goal is to prune A by sequentially eliminating certain +candidate amplitudes from the set based on a consistency +test across the remaining measurements c[n]. At the tth stage +(t = 2, 3, · · · ), for every remaining candidate amplitude Ak ∈ +A, we perform the following consistency test with c[n], to +identify if a candidate amplitude can potentially generate the +corresponding measurement c[n]. Suppose there exists a spiking +pattern vl ∈ Sall such that +c[n] = AkhT +αvl +(45) +then Ak remains a valid candidate. If we cannot find a +corresponding vl ∈ Sall for an amplitude Ak, we remove +it, A = A \ Ak. In presence of noise, (45) can be modified +to allow a tolerance γ as we may not find an exact match. +The tolerance γ is chosen to be 0.5 in the experiments on +the GENIE dataset. 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GENIE Project, Janelia Farm Campus, “Simultaneous imaging +and loose-seal cell-attached electrical recordings from neurons expressing +a variety of genetically encoded calcium indicators,” CRCNS. org, 2015. + diff --git a/DNAzT4oBgHgl3EQfwf6z/content/tmp_files/load_file.txt b/DNAzT4oBgHgl3EQfwf6z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4bd965939e6b09821f8ec91e6886a352f001b29b --- /dev/null +++ b/DNAzT4oBgHgl3EQfwf6z/content/tmp_files/load_file.txt @@ -0,0 +1,1305 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf,len=1304 +page_content='1 Super-resolution with Binary Priors: Theory and Algorithms Pulak Sarangi, Ryoma Hattori, Takaki Komiyama and Piya Pal Abstract—The problem of super-resolution is concerned with the reconstruction of temporally/spatially localized events (or spikes) from samples of their convolution with a low-pass filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Distinct from prior works which exploit sparsity in appropriate domains in order to solve the resulting ill-posed problem, this paper explores the role of binary priors in super-resolution, where the spike (or source) amplitudes are assumed to be binary-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Our study is inspired by the problem of neural spike deconvolution, but also applies to other applications such as symbol detection in hybrid millimeter wave communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This paper makes several theoretical and algorithmic contributions to enable binary super-resolution with very few measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Our results show that binary constraints offer much stronger identifiability guarantees than sparsity, allowing us to operate in “extreme compression" regimes, where the num- ber of measurements can be significantly smaller than the sparsity level of the spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' To ensure exact recovery in this "extreme compression" regime, it becomes necessary to design algorithms that exactly enforce binary constraints without relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In order to overcome the ensuing computational challenges, we consider a first order auto-regressive filter (which appears in neural spike deconvolution), and exploit its special structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This results in a novel formulation of the super-resolution binary spike recovery in terms of binary search in one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We perform numerical experiments that validate our theory and also show the benefits of binary constraints in neural spike deconvolution from real calcium imaging datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Index Terms—Binary compressed sensing, super-resolution, spike deconvolution, sparsity, binary search, beta-expansions I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' INTRODUCTION The problem of recovering localized events (spikes) from their convolution with a blurring kernel, arises in a wide range of scientific and engineering applications such as fluorescence microscopy [1], neural spike deconvolution [2]–[4], hybrid millimeter wave (mmWave) communication [5], to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Consider K temporal spikes, which can be represented as: xhi(t) = K � k=1 ckδ(t − nkThi) Here, the high-rate spikes are supported on a fine temporal grid with spacing Thi, nk ∈ Z is an integer corresponding to the time index of the kth spike and ck denotes its amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The convolution of spikes with a filter h(t) is typically uniformly (down)sampled at a (low) rate Tlo = DThi (D > 1), yielding measurements: y[n] = xhi(t) ⋆ h(t)|t=nT lo = K � k=1 ckh(nTlo − nkThi) (1) The goal of super-resolution is to recover the spike locations nk and amplitudes ck, k = 1, 2, · · · , K from a limited number (M) of low-rate samples {y[n]}M−1 n=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The problem is typically ill- posed due to systematic attenuation of high-frequency contents of the spikes by the low-pass filter h(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In order to make the problem well-posed, it becomes necessary to exploit priors such as sparsity [6]–[9] and/or non-negativity [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In recent times, there has been a substantial progress towards developing efficient algorithms for provably solving the super-resolution problem [7]–[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In this paper, we investigate the problem of binary super- resolution, where the amplitudes of the spikes are known apriori to be ck = A, but their number (K) and locations (nk) are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Motivated by the problem of neural spike deconvolution in two-photon calcium imaging [2], [20], we will focus on a blurring kernel that can be represented as a stable first order auto-regressive (AR(1)) filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Each neural spike results in a sharp rise in Ca2+ concentration followed by a slow exponential decay (modeled as the impulse response of an AR(1) filter), which results in an overlap of the responses from nearby spiking events, leading to poor temporal resolution [2], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Related Works Early works on super-resolution date back to algebraic/subspace-based techniques such as Prony’s method, MUSIC [12], [22], ESPRIT [8], [23] and matrix pencil [9], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Following the seminal work in [6], substantial progress has been made in understanding the role of sparsity as a prior for super-resolution [7], [25], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In recent times, convex optimization-based techniques have been developed that employ Total Variational (TV) norm and atomic norm regularizers, in order to promote sparsity [7], [18], [19], [25], [26] and/or non-negativity [10], [11], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' These techniques primarily employ sampling in the Fourier/frequency domain by assuming the kernel h(t) to be (approximately) bandlimited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' However, selecting the appropriate cut-off frequency is crucial for super-resolution and needs careful consideration [25], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Unlike subspace-based methods, theoretical guarantees for these convex algorithms rely on a minimum separation between the spikes, which is also shown to be necessary even in absence of noise [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The finite rate of innovation (FRI) framework [30]–[34] also considers the recovery of spikes from measurements acquired using an exponentially decaying kernel, which includes the AR(1) filter considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In the absence of noise, FRI enables the exact recovery of K spikes with arbitrary amplitudes from M = Ω(K)1 measurements, without any separation condition [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' It is to be noted that all of the above methods require M > K measurements for resolving K spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In contrast, we will show that it is possible to recover K spikes from M ≪ K 1This notation essentially means that there exists a positive constant c such that M ≥ cK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='01724v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='SP] 4 Jan 2023 2 measurements by exploiting the binary nature of the spiking signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The above algorithms are designed to handle arbitrary real-valued amplitudes and as such, they are oblivious to binary priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Therefore, they cannot successfully recover spikes in the regime M < K, which is henceforth referred to as the extreme compression regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The problem of recovering binary signals from underde- termined linear measurements (with more unknowns than equations/measurements) has been recently studied under the parlance of Binary Compressed Sensing (BCS) [35]–[42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In BCS, the undersampling operation employs random (and typically dense) sampling matrices, whereas we consider a deterministic and structured measurement matrix derived from a filter, followed by uniform downsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Moreover, existing theoretical guarantees for BCS crucially rely on sparsity assumptions that will be shown to be inadequate for our problem (discussed in Section II-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Most importantly, in order to achieve computational tractability, BCS relaxes the binary constraints and solves continuous-valued optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Consequently, their theoretical guarantees do not apply in the extreme compression regime M < K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' As mentioned earlier, our study is motivated by the problem of neural spike deconvolution arising in calcium imaging [3], [4], [20], [32], [43]–[45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' A majority of the existing spike deconvolution techniques [4], [43], [44] infer the spiking activity at the same (low) rate at which the fluorescence signal is sampled, and a single estimate such as spike counts or rates are obtained over a temporal bin equal to the resolution of the imaging rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Although sequential Monte-Carlo based techniques have been proposed that generate spikes at a rate higher than the calcium frame rate [3], no theoretical guarantees are available that prove that these methods can indeed uniquely identify the high-rate spiking activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Algorithms that rely on sparsity and non-negativity [43], [44] alone are ineffective for inferring the neural spiking activity that occurs at a much higher rate than the calcium sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' On the other hand, at the high-rate, the spiking activity is often assumed to be binary since the probability of two or more spikes occurring within two time instants on the fine temporal grid is negligible [2], [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Therefore, we propose to exploit the inherent binary nature of the neural spikes and provide the first theoretical guarantees that it is indeed possible to resolve the high-rate binary neural spikes from calcium fluorescence signal acquired at a much lower rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Our Contributions We make both theoretical and algorithmic contributions to the problem of binary super-resolution in the setting when the spikes lie on a fine grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We theoretically establish that at very low sampling rates, sparsity and non-negativity are inadequate for the exact reconstruction of binary spikes (Lemma 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' However, by exploiting the binary nature of the spiking activity, much stronger identifiability results can be obtained compared to classical sparsity-based results (Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In the absence of noise, we show that it is possible to uniquely recover K binary spikes from only M = Ω(1) low-rate measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The analysis also provides interesting insights into the interplay between binary priors and the “infinite memory" of the AR(1) filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Although it is possible to uniquely identify binary spikes in the extreme compression regime (M ≪ K), the combinatorial nature of binary constraints introduce computational hurdles in exactly enforcing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Our second contribution is to leverage the special structure of the AR(1) measurements to overcome this computational challenge in the extreme compression regime M < K (Section III-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Our formulation reveals an interesting and novel connection between binary super- resolution, and finding the generalized radix representation of real numbers, known as β-expansion [47]–[49] (Section III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In order to circumvent the problem of exhaustive search, we pre- construct and store (in memory) a binary tree that is completely determined by the model parameters (filter and undersampling factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' When the low-rate measurements are acquired, we can efficiently perform a binary search to traverse the tree and find the desired binary solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This ability to trade-off memory for computational efficiency is made possible by the unique structure of the measurement model governed by the AR(1) filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The algorithm guarantees exact super-resolution even when the measurements are corrupted by a small bounded (adversarial) noise, the strength of which depends on the AR filter parameter and the undersampling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' When the measurements are corrupted by additive Gaussian noise, we characterize the probability of erroneous decoding (Theorem 3) in the extreme compression regime M < K and indicate the trade-off among the filter parameter, SNR and the extent of compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Finally, we also demonstrate how binary priors can improve the performance of a popularly used spike deconvolution algorithm (OASIS [43]) on real calcium imaging datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' FUNDAMENTAL SAMPLE COMPLEXITY OF BINARY SUPER-RESOLUTION Let yhi[n] be the output of a stable first-order Autoregressive AR(1) filter with parameter α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 0 < α < 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' driven by an unknown binary-valued input signal xhi[n] ∈ {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' A},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' A > 0: yhi[n] = αyhi[n − 1] + xhi[n] (2) In this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' we consider a super-resolution setting where we do not directly observe yhi[n],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' and instead acquire M measurements {ylo[n]}M−1 n=0 at a lower-rate by uniformly subsampling yhi[n] by a factor of D: ylo[n] = yhi[Dn],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' n = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' M − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' (3) The signal ylo[n] corresponds to a filtered and downsampled version of the signal xhi[n] where the filter is an infinite impulse response (IIR) filter with a single pole at α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Let ylo ∈ RM be a vector obtained by stacking the low-rate measurements {ylo[n]}M−1 n=0 : ylo = [ylo[0], ylo[1], · · · , ylo[M − 1]]⊤ Since (2) represents a causal filtering operation, the low rate signal ylo only depends on the present and past high-rate binary signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Denote L := (M − 1)D + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The M low-rate measurements in ylo are a function of L samples of the high 3 rate binary input signal {xhi[n]}L−1 n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' These L samples are given by the following vector xhi ∈ {0, A}L: xhi := [xhi[0], xhi[1], · · · , xhi[L − 1]]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Assuming the system to be initially at rest, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=', yhi[n] = 0, n < 0, we can represent the M samples from (3) in a compact matrix-vector form as: ylo := SDyhi = SDGαxhi (4) where Gα ∈ RL×L is a Toeplitz matrix given by: Gα = � ���� 1 0 · · 0 α 1 · · 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' αL−1 αL−2 · · 1 � ���� (5) and SD ∈ RM×L is defined as: [SD]i,j = � 1, j = (i − 1)D + 1 0, else .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The matrix SD represents the D−fold downsampling operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Our goal is to infer the unknown high-rate binary input signal xhi[n] from the low-rate measurements ylo[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This is essentially a “super-resolution" problem because the AR(1) filter first attenuates the high-frequency components of xhi[n], and the uniform downsampling operation systematically discards measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' As a result, it may seem that the spiking activity {xhi[(n − 1)D + k]}D k=1 occurring “in-between" two low-rate measurements ylo[n − 1] and ylo[n] is apparently lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' One can potentially interpolate arbitrarily, making the problem hopeless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In the next section, we will show that surprisingly, xhi still remains identifiable from ylo in the absence of noise, due to the binary nature of xhi and “infinite memory" of the AR(1) filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Identifiability Conditions for Binary super-resolution Consider the following partition of xhi into M disjoint blocks, where the first block is a scalar and the remaining M −1 blocks are of length D, xhi = [xhi(0), xhi(1)⊤, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' , xhi(M−1)⊤]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Here, xhi(0) = xhi[0] and xhi(n) ∈ {0, A}D is given by: [xhi (n)]k = xhi[(n − 1)D + k], 1 ≤ n ≤ M − 1 (6) The sub-vectors xhi(n), and xhi(n−1) (n ≥ 1) represent consec- utive and disjoint blocks (of length D) of the high-rate binary spike signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In order to study the identifiability of xhi from ylo, we first introduce an alternative (but equivalent) representation for (4), by constructing a sequence c[n] as follows c[0] = ylo[0], c[n] = ylo[n] − αDylo[n − 1], 1 ≤ n ≤ M − 1 (7) Given the high rate AR(1) model defined in (2), it is possible to recursively represent yhi[Dn] in terms of yhi[Dn − 1], which in turn, can be represented in terms of yhi[Dn − 2], and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' By this recursive relation, we can represent yhi[Dn − 1] in terms of yhi[Dn−D] and {xhi[Dn−i]}D−1 i=0 and re-write ylo[n] as ylo[n] = yhi[Dn] = αyhi[Dn − 1] + xhi[Dn] = αDyhi[Dn − D] + αD−1xhi[D(n − 1) + 1] + · · · + αxhi[Dn − 1] + xhi[Dn], ylo[n] − αDylo[n − 1] = αD−1xhi[D(n − 1) + 1] + · · · + αxhi[Dn − 1] + xhi[Dn] (8) The last equality holds due to the fact that ylo[n−1] = yhi[Dn− D].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Combining (7) and (8), the sequence c[n] can be re-written as c[0] = ylo[0] = xhi(0), and for 1 ≤ n ≤ M − 1 c[n] = D � i=1 αD−ixhi[(n − 1)D + i] = hT αxhi (n) (9) where hα = [αD−1, αD−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' , α, 1]T ∈ RD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This implies that c[n] depends only on the block xhi(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Denote c := [c[0], c[1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' , c[M − 1]]⊤ ∈ RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For any D, (9) can be compactly represented as: c = HD(α)xhi (10) where HD(α) ∈ RM×L is given by: HD(α) = � ������ 1 0⊤ 0⊤ · · 0⊤ 0 h⊤ α 0⊤ · · 0⊤ 0 0⊤ h⊤ α · · 0⊤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 0 0⊤ 0⊤ · · h⊤ α � ������ The following Lemma establishes the equivalence between (4) and (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Given ylo, construct c following (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Then, there is a unique binary xhi ∈ {0, A}L satisfying (4) if and only if xhi is a unique binary vector satisfying (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' First suppose that there is a unique binary xhi ∈ {0, A}L satisfying (4) but (10) has a non-unique binary solution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=', there exists xhi′ ∈ {0, A}L, xhi′ ̸= xhi, such that c = HD(α)xhi = HD(α)xhi ′ (11) Define yhi′ := Gαxhi′ whose entries are given by: yhi ′[n] = n � k=0 αn−kxhi ′[k], 0 ≤ n ≤ L − 1 (12) Notice that (7) can be re-written as ylo[0] = c[0] = xhi[0], ylo[1] = c[1] + αDylo[0] = c[1] + αDc[0] ylo[2] = c[2] + αDylo[1] = c[2] + αDc[1] + α2Dc[0] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Following this recursive relation, and using (9) and (11), we can further re-write ylo[n] as: ylo[n] = n � i=0 α(n−i)Dc[i] = αnDx′ hi (0) + n � i=1 α(n−i)Dh⊤ α xhi ′(i) = αnDx′ hi (0) + n � i=1 D � j=1 αnD−(i−1)D−jx′ hi[(i − 1)D + j] (a) = nD � k=0 αnD−kx′ hi[k] (b) = y′ hi[nD] (13) 4 The equality (a) follows by a re-indexing of the summation into a single sum, and (b) follows from (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' By arranging (13) in a matrix form we obtain the following relation: ylo = SDGαxhi ′ However from (4), we have ylo = SDGαxhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This contradicts the supposition that (4) has a unique binary solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Next, suppose that (10) has a unique binary solution but the binary solution to (4) is non-unique, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=', there exists xhi′ ∈ {0, A}L, xhi′ ̸= xhi such that ylo = SDGαxhi ′ = SDGαxhi By following (7) and (10), we also have c = HD(α)xhi′ = HD(α)xhi which contradicts the assumption that solution of (10) is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Lemma 1 assures that a binary xhi is uniquely identifiable from measurements ylo if and only if there is a unique binary solution xhi ∈ {0, A}L to (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' From (9), it can be seen that c[n] and c[n − 1] have contributions from only disjoint blocks of high rate spikes xhi(n), and xhi(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Hence effectively, we only have a single scalar measurement c[n] to decode an entire block xhi(n) of length D, regardless of how sparse it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The task of decoding xhi(n) from a single measurement seems like a hopelessly “ill-posed" problem, caused by the uniform downsampling operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' But this is precisely where the binary nature of xhi can be used as a powerful prior to make the problem well-posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Theorem 1 specifies conditions under which it is possible to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' (Identifiability) For any α ∈ (0, 1), with the possible exception of α belonging to a set of Lebesgue measure zero, there is a unique xhi ∈ {0, A}L that satisfies (10) for every D ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Using Lemma 1 and Theorem 1, we can conclude that xhi is uniquely identifiable from ylo for almost all α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' It can be verified that for α = 1 the mapping is non-injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Theorem 1 establishes that it is fundamentally possible to decode each block xhi(n) of length D, from effectively a single measurement c[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Since xhi(n) can take 2D possible values, in principle, one can always perform an exhaustive search over these 2D possible binary sequences and by Theorem 1, only one of them will satisfy c[n] = h⊤ α xhi(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Since exhaustive search is computationally prohibitive, this leads to the natural question regarding alternative solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In Section III, we will develop an alternative algorithm that leverages the trade-off between memory and computation to achieve a significantly lower run-time decoding complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Comparison with Finite Rate of Innovation Approach In a related line of work [30]–[32], [34], the FRI framework has been developed to reconstruct spikes from the measurement model considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' However, in the general FRI framework, there is no assumption on the amplitude of the spikes, and there are a total of 2D real valued unknowns corresponding to the locations and amplitudes of D spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In [32], it was shown that by leveraging the property of exponentially reproducing kernels, it is possible to recover arbitrary amplitudes and spike locations using Prony-type algorithms, provided at least 2D+1(> D) low- rate measurements are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' However, since we exploit the binary nature of spiking activity, we can operate at a much smaller sample complexity than FRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In fact, Theorem 1 shows that when we exploit the fact that the spikes occur on a high-resolution grid with binary amplitudes, M = Ω(1) measurements suffice to identify D spikes regardless of how large D is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' A direct application of the FRI approach cannot succeed in this regime, since the number of spikes is larger than the number of measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' That being said, with enough measurements, FRI techniques are powerful, and they can also identify off-grid spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In future, it would be interesting to combine the two approaches by incorporating binary priors to FRI based techniques and remove the grid assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Curse of Uniform Downsampling: Inadequacy of sparsity and non-negativity By virtue of being a binary signal, xhi is naturally sparse and non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Therefore, one may ask if sparsity and/or non- negativity are sufficient to uniquely identify xhi from c, without the need for imposing any binary constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In particular, we would like to understand if the solution to the following problem that seeks the sparsest non-negative vector in RL satisfying (10) indeed coincides with the true xhi ∈ {0, A}L min x∈RL ∥x∥0 subject to c = HD(α)x, x ≥ 0 (P0) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For every xhi ∈ {0, A}L (except xhi = Ae1), and c ∈ RM satisfying (10), the following are true (i) There exists a solution x⋆ ̸= xhi to (P0) satisfying ∥x⋆∥0 ≤ ∥xhi∥0 (14) (ii) The inequality in (14) is strict as long as there exists an integer n0 ≥ 1 such that the block x(n0) hi of xhi (defined in (6)) satisfies ∥x(n0) hi ∥0 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The proof is in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Lemma 2 shows there exist other non-binary solution(s) to (10) (different from xhi) that have the same or smaller sparsity as the binary signal xhi ∈ {0, A}L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Furthermore, there exist problem instances where the sparsest solution to (P0) is strictly sparser than xhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Hence, sparsity and/or non-negativity are inadequate to identify the ground truth xhi uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Implicit Bias of Relaxation: The optimization problem (P0) is non-convex and the binary constraints are not enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In binary compressed sensing [35], [36], it is common to relax the binary constraints using box-constraint and l0 norm is relaxed to l1 norm in the following manner: min x∈RL ∥x∥1 subject to c = HD(α)x, 0 ≤ x ≤ A1 (P1-B) In the following Lemma, we show that there is an implicit bias introduced to the solution of (P1-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For every xhi ∈ {0, A}L, and c ∈ RM satisfying (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' There exists a solution x⋆ to (P1-B) satisfying ∥x⋆∥1 ≤ ∥xhi∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' (15) 5 Moreover, for all n ≥ 1, the blocks x(n)⋆ ∈RD of x⋆ satisfy: supp(x(n)⋆) = {D, D − 1, · · · , D − jn}, if c[n] ̸= 0 (16) for some 0 ≤ jn ≤ D − 1 and x(n)⋆ = 0 if c[n] = 0, irrespective of the support of xhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The proof is in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Lemma 3 shows that even in the noiseless setting, introducing the box-constraint as a means of relaxing the binary constraint introduces a bias in the support of the recovered spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The optimal solution always results in spikes with support clustered towards the end of each block of length D, irrespective of the ground truth spiking pattern xhi that generated the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This bias is a consequence of the nature of relaxation, as well as the specific structure of the measurement matrix HD(α) arising in the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Role of Memory in Super-resolution: IIR vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' FIR filters The ability to identify the high-rate binary signal xhi ∈ {0, A}L from D−fold undersampled measurements ylo (for arbitrarily large D) in the absence of noise, is in parts also due to the “infinite memory" or infinite impulse response of the AR(1) filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Indeed, for an Finite Impulse Response (FIR) filter, there is a limit to downsampling without losing identifiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This was recently studied in our earlier work [40] where we showed that the undersampling limit is determined by the length of the FIR filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' To see this, consider the convolution of a binary valued signal xhi with a FIR filter u = [u[0], u[1], · · · , u[r − 1]]T ∈ Rr of length r: zf[n] = �r−1 i=0 u[r − 1 − i]xhi[n + i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' These samples are represented in the vector form as zf := u⋆xhi ∈ RL (by suitable zero padding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Suppose, as before, we only observe a D−fold downsampling of the output zD[n] = zf[Dn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Two consecutive samples zD[p], zD[p + 1] of the low- rate observation are given by: zD[p] = r−1 � i=0 u[r − 1 − i]xhi[Dp + i], zD[p + 1] = r−1 � i=0 u[r − 1 − i]xhi[D(p + 1) + i] If D > r, notice that none of the measurements is a function of the samples xhi[Dp+r], xhi[Dp+r +1], · · · , xhi[D(p+1)−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Hence, it is possible to assign them arbitrary binary values and yet be consistent with the low-rate measurements zD[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This makes it impossible to exactly recover xhi (even if it is known to be binary valued) if the decimation is larger than the filter length (D > r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The following lemma summarizes this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For every FIR filter u ∈ Rr, if the undersampling factor exceeds the filter length, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' D > r, there exist x0, x1 ∈ {0, A}L, x0 ̸= x1 such that SD(u ⋆ x0) = SD(u ⋆ x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This shows that the identifiability result presented in Theorem 1 is not merely a consequence of binary priors but the infinite memory of the autoregressive process is also critical in allowing arbitrary undersampling D > 1 in absence of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For such IIR filters, the memory of all past (binary) spiking activity is encoded (with suitable weighting) into every measurement captured after the spike, which would not be the case for a finite impulse response filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' EFFICIENT BINARY SUPER-RESOLUTION USING BINARY SEARCH WITH STRUCTURED MEASUREMENTS By Theorem 1, we already know that it is possible to uniquely identify xhi from c (or equivalently, each block xhi(n) from a single measurement c[n]) by exhaustive search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We now demonstrate how this exhaustive search can be avoided by formulating the decoding problem in terms of “binary search" over an appropriate set, and thereby attaining computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We begin by introducing some notations and definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Given a non-negative integer k, 0 ≤ k ≤ 2D − 1, let (b1(k), b2(k), · · · , bD(k)) be the unique D-bit binary repre- sentation of k: k = �D d=1 2D−dbd(k), bd(k) ∈ {0, 1} ∀ 1 ≤ d ≤ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Here b1(k) is the most significant bit and bD(k) is the least significant bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Using this notation, we define the following set: Sall := {v0, v1, v2, · · · , v2D−1}, (17) where each vk ∈ {0, A}D is a binary vector given by [vk]d = Abd(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 1 ≤ d ≤ D (18) In other words, the binary vector 1 Avk is the D-bit binary representation of its index k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Using this convention, v0 = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=', a binary sequence of all 0′s) and v2D−1 = A1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=', a binary sequence of all A′s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Recall the partition of xhi defined in (6), where each block xhi(n) (n ≥ 1) is a binary vector of length D and xhi(0) ∈ {0, A} is a scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' It is easy to see that (17) comprises of all possible values that each block xhi(n) can assume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' According to (9) each scalar measurement c[n] can be written as: c[0] = x(0), c[n] = hα⊤xhi(n), 1 ≤ n ≤ M − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For every α, we define the following set: Θα := {θ0, θ1, · · · , θ2D−1}, where θk := h⊤ α vk (19) Observe that every measurement c[n] = �D i=1 αD−ixhi[(n − 1)D+i] takes values from this set Θα, depending on the value taken by the underlying block of spiking pattern from Sall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Our goal is to recover the spikes {xhi[(n − 1)D + i]}D i=1 from c[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In the following, we show that this problem is equivalent to finding the representation of a real number over an arbitrary radix, which is known as “β-expansion" [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Given a real (potentially non-integer) number β > 1, the representation of another real number p ≥ 0 of the form: p = ∞ � n=1 anβ−n, where 0 ≤ an < ⌊β⌋ (20) is referred to as a β-expansion of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The coefficients 0 ≤ an < ⌊β⌋ are integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This is a generalization of the representation of numbers beyond integer-radix to a system where the radix can be chosen as an arbitrary real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This notion of representation over arbitrary radix was first introduced by Renyi in [49], and since then has been extensively studied [47], [48], [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' There is a direct connection between β-expansion and the binary super-resolution problem considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In the problem at hand, any element θk ∈ Θα can be written as: θk = h⊤ α vk = D � i=1 αD−i[vk]i When 1/2 < α < 1, by letting β = 1/α, we see that the coefficients in (20) must satisfy 0 ≤ an < ⌊1/α⌋ < 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=', 6 they are restricted to be binary valued an ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Therefore, decoding the spikes vk from the observation θk is equivalent to finding a D−bit representation for the number θk/A over the non-integer radix β = 1/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Questions regarding the existence of β-expansion, and finding the coefficients of a finite β−expansion (whenever it exists) has been an active topic of research [47], [48], [50], [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' When β ≥ 2 (equivalently, 0 < α ≤ 1/2), it is possible to find the coefficients using a greedy algorithm which proceeds in a fashion similar to finding the D-bit binary representation of an integer [47], [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' However, the regime β ∈ (1, 2) (equivalently 1/2 < α < 1), is significantly more complicated and is of continued research interest [47], [48], [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' To the best of our knowledge, there are no known computationally efficient ways to find the finite β-expansion when 1/2 < α < 1 (if it exists) [N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Sidorov, personal communication, May 24, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In practice, we encounter filter values α (= 1/β) that are much closer to 1, and hence, we need an alternative approach to find this finite β-radix representation for θk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In the next section, we show that by performing a suitable preprocessing, finite β-radix representation can be formulated as a binary search problem which is guaranteed to succeed for all values of β that permit unique finite β−expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Formulation as a Binary Search Problem Before describing the algorithm, we first introduce the notion of a collision-free set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Definition 1 (Collision Free set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Given an undersampling factor D, define a class of “collision free" AR(1) filters as: GD = {α ∈ (0, 1) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' h⊤ α vi ̸= h⊤ α vj ∀ i ̸= j, vi, vj ∈ Sall} The set GD denotes permissible values of the AR(1) filter parameter α such that each of the 2D binary sequences in Sall maps to a unique element in the set Θα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In other words, every θk ∈ Θα has a unique D−bit expansion for all α ∈ GD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This naturally raises the question “How large is the set GD?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' ".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Theorem 1 already provided the answer to this question, where the identifiability result implies that for every D, almost all α ∈ (0, 1) belong to this set GD (with the possible exception of a measure zero set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Hence, Theorem 1 ensures that there are infinite choices for collision-free filter parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For every α ∈ GD, the mapping Φα(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=') : Sall → Θα, Φα(v) = h⊤ α v forms a bijection between Sall and Θα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Since α ∈ GD, from the definition of the set GD, it is clear that for any vi, vj ∈ Sall, vi ̸= vj we have hα⊤vi ̸= hα⊤vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Therefore, the mapping is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Furthermore, from (19) we also have |Θα| ≤ |Sall| = 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Since Φα(·) is injective, we must also have |Θα| = 2D and hence the mapping Φα(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=') forms a bijection between Sall and Θα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' When α ∈ GD, Lemma 5 states that the finite beta expansion for every θk ∈ Θα is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Lemma 5 provides a way to avoid exhaustive search over Sall, and yet identify xhi(n) from c[n] in a computationally efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' From Lemma 5, we know that each of the 2D spiking patterns in Sall maps to a unique element in Θα, and each element in Θα has a corresponding spiking pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Hence instead of searching Sall, we can equivalently search the set Θα in order to determine the unknown spiking pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Since Θα permits “ordering", searching Θα has a distinct computational advantage over searching Sall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This ordering enables us to employ binary search over (an ordered) Θα and find the desired element in a computationally efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' To do this, we first sort the set Θα (in ascending order) and arrange the corresponding elements of Sall in the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Given Θα as an input, the function SORT(·) returns a sorted list Θsort α , and an index set I = {i0, i1, · · · , i2D−1} containing the indices of the sorted elements in the list Θα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Θsort α , I ← SORT(Θα) Let us denote the elements of the sorted lists as Θsort α = {�θ0, · · · , �θ2D−1}, and Ssort all = {�v0, · · · , �v2D−1} where: �θ0 < �θ1 < · · · < �θ2D−1 and �θj = θij, �vj = vij ∀j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' It is important to note that this sorting step does not depend on the measurements c, and can therefore be part of a pre- processing pipeline that can be performed offline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' However, it does require memory to store the sorted lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In the Algorithm 1 Noiseless Spike Recovery 1: Input: Measurement c[n], Sorted list Θsort α and the corre- sponding (ordered) spike patterns Ssort all 2: Output: Decoded spike block �xhi(n) 3: i⋆ ← BINSEARCH(Θsort α , c[n]) 4: Return �xhi(n) ← �vi⋆ noiseless setting, we know that every scalar measurement c[n] = h⊤ α xhi(n) belongs to the set Θsort α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Therefore, if we identify its index, say i⋆, then we can successfully recover xhi(n) by returning the corresponding binary vector �vi⋆ from Ssort all .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Therefore, we can formulate the decoding problem as searching for the input c[n] in the sorted list Θsort α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This can be efficiently done by using “Binary Search".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The noiseless spike decoding procedure is summarized as Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Since the complexity of performing a binary search over an ordered list of N elements is O(log N), the complexity of Algorithm 1 is logarithmic in the cardinality of Θsort α , which results in a complexity of O(log(2D)) = O(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We summarize this result in the following Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Assume α ∈ GD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Given the ordered set Θsort α , and an input c[n] = h⊤ α xhi(n), Algorithm 1 terminates in O(D) steps and its output �xhi(n) satisfies �xhi(n) = xhi(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Noisy Measurements and 1 D Nearest Neighbor Search We demonstrate how binary search can still be useful in presence of noise by formulating noisy spike detection as a one dimensional nearest neighbor search problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Suppose {zlo[n]}M−1 n=0 denote noisy D-fold decimated filter output zlo[n] = ylo[n] + w[n], 0 ≤ n ≤ M − 1 (21) 7 Here w[n] represents the additive noise term that corrupts the (noiseless) low-rate measurements ylo[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Similar to (7), we compute ce[n] from zlo[n] as follows: ce[n] = zlo[n] − αDzlo[n − 1] (22) = D � i=1 αD−ixhi[(n − 1)D + i] + e[n]= c[n] + e[n] (23) where c[n] = h⊤ α xhi(n) ∈ Θsort α , and e[n] = w[n] − αDw[n − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We can interpret ce[n] as a noisy/perturbed version of an element c[n] ∈ Θsort α , with e[n] representing the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This perturbed signal may no longer belong to Θsort α (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' ce[n] ̸∈ Θsort α ) and hence, we cannot find an exact match in the set Θsort α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Instead, we aim to find the closest element in Θsort α (the nearest neighbor of ce[n]) by solving the following problem: �xhi (n) = arg min v∈Ssort all |ce[n] − h⊤ α v| (24) Solving (24) is equivalent to finding the spike sequence �v ∈ Ssort all that maps to the nearest neighbor of ce[n] in the set Θsort α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' By leveraging the sorted list Θsort α , it is no longer necessary to parse the list sequentially (which would incur O(2D) complexity), instead we can perform a modified binary search as summarized in Algorithm 2, that keeps track of additional indices compared to the vanilla binary search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Finally, we return the unique spiking pattern from Ssort α that gets mapped to the nearest neighbor of the noisy measurement ce[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' It is well-known that the nearest neighbor for any query could be found in O(log(2D)) = O(D) steps, instead of the linear complexity of O(2D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This guarantees a computationally efficient decoding of spikes by solving (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Next, we characterize the error events that lead to erroneous detection of a block of spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Recall that the set Θsort α is sorted, and its elements satisfy the ordering: 0 = �θ0 < �θ1 < · · · < �θlD = 1 + α + · · · + αD−1 where lD := 2D−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We also have �θk = h⊤ α �vk, where �vk ∈ Ssort all is a binary spiking sequence of length D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For each �vk and each n, we will determine the error event �xhi(n) ̸= xhi(n), when xhi(n) = �vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' First, consider the scenario when xhi(n) = �vk for some 0 < k < lD (excluding �v0, �vlD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The corresponding noiseless measurement is c[n] = �θk = h⊤ α �vk which satisfies �θk−1 < c[n] = �θk < �θk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Since Θsort α is sorted, it can be easily verified that the nearest neighbor of ce[n] will be �θk, if and only if ce[n] satisfies the following condition: (�θk−1 + �θk)/2 ≤ ce[n] ≤ (�θk+1 + �θk)/2 (25) Since �θk = h⊤ α �vk, the solution to (24) is attained at �vk ∈ Ssort all , and the decoding is successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Therefore Algorithm 2 produces an erroneous estimate of �vk if and only if ce[n] violates (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The event ce[n] ̸∈ [ �θk−1+�θk 2 , �θk+1+�θk 2 ] is equivalent to e[n] ∈ Ek (e[n] is defined earlier in (23)), where Ek = {e[n] < − �θk − �θk−1 2 , or e[n] > �θk+1 − �θk 2 } (26) Finally, we characterize the error events for k = 0, lD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The error events for c[n] = θ0 = 0 or c[n] = θlD are given by: E0 = {e[n] ≥ �θ1/2}, ElD = {e[n] ≤ −(�θlD − �θlD−1)/2} (27) Define the “minimum distance" between points in Θsort α : ∆θmin(α, D) = min 1≤k≤lD |�θk − �θk−1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This minimum distance depends on A, α and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' From (26), (27) it can be verified that if 2|w[n]| < ∆θmin(α, D)/2 (which would imply |e[n]| < ∆θmin(α, D)/2) for all n, then �xhi(n) = xhi(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' As summarized in Theorem 2, Algorithm 2 can exactly recover the ground truth spikes from measurements corrupted by bounded adversarial noise, the extent of the robustness is determined by the parameters A, α, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Algorithm 2 Noisy Spike Recovery 1: Input: Measurement ce[n], Sorted list Θsort α and the corresponding (ordered) spike patterns Ssort all 2: Output: Decoded spike block �xhi(n) 3: Set l ← 0, u ← 2D − 1 4: while u − l > 1 5: Set m ← l + ⌊(u − l)/2⌋ 6: if �θm > ce[n] then 7: u ← m 8: else 9: l ← m 10: end if 11: end while 12: Find the nearest neighbor i⋆ = arg mini∈{l,u}(ce[n]− �θi)2 13: Return �xhi(n) ← �vi⋆ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Assume α ∈ GD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Given the ordered set Θsort α , the output of Algorithm 2 with input ce[n] exactly coincides with the solution of the optimization problem (24) in at most O(D) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Furthermore, if for all n, |w[n]| < ∆θmin(α, D)/4, then the output of Algorithm 2 satisfies �xhi(n) = xhi(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' From Theorem 2, it is evident that ∆θmin(α, D) plays an important role in characterizing the upper bound on noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We attempt to gain insight into how ∆θmin(α, D) varies as a function of α when D is held fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Given D, ∆θmin(α, D) = αD−1 for α ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The proof is in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' When α ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5], ∆θmin(α, D) is monotonically increasing with α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' However, for α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5 the trend fluctuates with α differently for different D, and becomes quite challenging to predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This is also confirmed by the empirical plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' A refined analysis of ∆θmin(α, D) to gain insight into desirable filter parameters α is an interesting direction for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Trade-off between memory and computational complexity A crucial aspect of Algorithms 1 and 2 is that they achieve efficient run-time complexity by leveraging the off- line construction of the sorted list Θsort α and Ssort all .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' These lists, each with 2D elements, need to be stored in memory and made available during run-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Since there is no free lunch, the resulting computational efficiency of O(D) at run-time is attained at the expense of the additional memory that is required to store the sorted lists Θsort α , Ssort all .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 8 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Parallelizable Implementation Algorithm 2 (also Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 1) only takes ce[n](c[n]) as input and returns �xhi(n), and is completely de-coupled from any other �xhi(n′), n′ ̸= n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Recall that in reality, we are provided with measurements zlo[n](ylo[n]), and ce[n](respectively c[n]) needs to be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Due to this de-coupling, we can compute ce[n]′s in parallel using two consecutive low-rate samples zlo[n], zlo[n−1] and perform a nearest neighbor search without waiting for any previously decoded spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Therefore, the total decoding complexity can be further improved depending on the available parallel computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' ERROR ANALYSIS FOR GAUSSIAN NOISE Algorithm 2 solves (24) without requiring any knowledge of the noise statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' However, in order to analyze its per- formance, we will make the following (standard) assumptions on the statistics of the high-rate spiking signal xhi and the measurement noise w[n] as follows: (A1) The entries of the binary vector xhi ∈ {0, A}L are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='d random variables distributed as xhi[n] ∼ ABern(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' (A2) The additive noise w[n], 0 ≤ n ≤ M − 1 is independent of xhi[n], and distributed as w[n] ∼ N(0, σ2) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Probability of Erroneous Decoding Under assumption (A2), the ML estimate of xhi is given by the solution to the following problem: �xML = arg min v∈{0,A}L ∥zlo − SDGαv∥2 (PNN) The proposed Algorithm 2 does not attempt to solve (PNN), which is computationally intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Instead, it solves a set of M − 1 one dimensional nearest neighbor search problems, by finding the nearest neighbor of ce[n] for each n = 1, 2, · · · , M − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This scalar nearest neighbor search is implemented in a computationally efficient manner by using parallel binary search on a pre-sorted list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Notice that by the operation (22), the variance of the equivalent noise term e[n] gets amplified by a factor of at most (1+α2D) < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This can be thought of as a price paid to achieve computational efficiency and parallelizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The following theorem characterizes the dependence of certain key quantities of interest, such as the signal-to-noise ratio (SNR), undersampling factor D, and filter’s frequency response (controlled by α) on the performance of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Suppose α ∈ GD and assumptions (A1-A2) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Given δ > 0, if the following condition is satisfied: ∆θ2 min(α, D)/σ2 ≥ 4 ln (2M/δ) (28) then Algorithm 2 can exactly recover the binary signal xhi with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The proof follows standard arguments for computing the probability of error for symbol detection in Gaussian noise, followed by certain simplifications and is included in Appendix D for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 1, we plot ∆θmin(α, D) as a function of D for different values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' As expected, ∆θmin(α, D) decays as the D increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Understandably, for a fixed α, as D increases, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='9 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='8 1 Minimum distance For D=4 For D=5 Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Dist (D=4) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Dist (D=5) Cluster Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Dist (D=4) Cluster Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Dist (D=5) 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5 5 Undersampling factor (D) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='8 1 Minimum distance =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='2 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 1: Variation of ∆θmin(α, D) as a function of undersampling factor D and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The cluster-distance ∆c min(α, D) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' α is also overlaid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Each dotted line denotes the start of the interval FD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' it becomes harder to recover the spikes exactly, and higher SNR is needed to compensate for the lower sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This can be interpreted as the price paid for super-resolution in presence of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This phenomenon is also reminiscent of the noise amplification effect in super-resolution, where the ability to super-resolve point sources becomes more severely hindered by noise as the target resolution grid becomes finer [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 1, we plot ∆θmin(α, D) as a function of α and as predicted by Lemma 7, it monotonically increases upto 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5, but for α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5, the behavior becomes much more erratic and a precise characterization becomes challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' It is to be noted that in Theorem 3, we aim to exactly recover xhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The SNR requirement can be relaxed if our goal is to recover only spike counts instead of the true spikes as discussed in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' One can define other notions of approximate recovery, the analysis of which will be a topic of future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Relaxed Spike reconstruction: Count Estimation As shown in Theorem 2, exact recovery of spikes is possible under somewhat restrictive condition on the noise in terms of ∆θmin(α, D), which becomes quite small as D increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This naturally calls for other relaxed notions of recovery which can handle larger noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In neuroscience, it is believed that information is encoded as either the spike timing (temporal code) or the firing rates (rate coding) of individual neurons in the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Therefore, the spike counts over an interval can be informative to understand neural functions, even when it is impossible to temporally localize the neural spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For example, neurons in the visual cortex encode stimulus orientations as their firing rates [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We will therefore focus on spike count as an approximate recovery metric, which concerns estimating the number of spikes occurring between two consecutive low-rate measurements instead of resolving the individual spiking activity at a higher resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Let γ[n] denote the total number of spikes occurring between two consecutive low-rate samples zlo[n] and zlo[n − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Since xhi and its estimate �xhi are both binary valued (amplitude A), the true spike count (γ[n]) and estimated count (�γ[n]) are given by: γ[n] = ∥xhi(n)∥0, �γ[n] = ∥�x(n) hi ∥0, n = 1, · · · , M − 1, 9 γ[0] = xhi[0]/A and �γ[0] = �xhi[0]/A since the first block is of size 1 as described in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Define a set CD k as: CD k := {v ∈ {0, A}D, ∥v∥0 = k}, 0 ≤ k ≤ D It is a collection of all binary vectors (of length D) with spike count k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The ground truth spike block belongs to CD γ[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Any element from CD γ[n] will give the true spike count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Hence, exact recovery of count can be possible even when spikes cannot be recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For a fixed D, we define a set of α denoted by FD: FD := {α ∈ (0, 1)|αD − αD−k0−1 − αk0 + 1 < 0} (29) where k0 = ⌊D/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We will obtain a sufficient condition for robust spike count estimation when α ∈ FD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' It can be shown that for any D, FD will always be non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Define θk min := min u∈CD k h⊤ α u θk max := max u∈CD k h⊤ α u (30) Observe that if θk+1 min > θk max, k = 0, 1, · · · , D − 1 (31) then all spike patterns ui ∈ CD k (with the same spike count k) are clustered together when mapped on to the real line by the transformation h⊤ α u as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' When (31) holds, we can define a “cluster-restricted minimum distance" as: ∆c min(α, D) := min 0≤k≤D−1 θk+1 min − θk max (32) Given a noisy observation ce[n] = h⊤ α xhi(n)+e[n], the solution to the nearest neighbor problem (24) may return an incorrect neighbor θj ̸= h⊤ α xhi(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' However, when (31) holds and if the noisy observation satisfies the following conditions: (θγ[n] min + θγ[n]−1 max )/2 < ce[n] < (θγ[n]+1 min + θγ[n] max)/2 (33) then the nearest-neighbor decision rule in Algorithm 2 will still ensure that θj ∈ CD γ[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This has also been visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 2 where each colored band represents the “safe-zone" for each count and the black dotted-line denotes the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This will result in correct identification of the spike count but will incur error in terms of spiking pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We formally summarize this in the following Theorem that provides robustness guarantee for exact count recovery from measurements corrupted by adversarial noise (similar to Theorem 2 for spike recovery).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Assume α ∈ FD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Given the ordered set Θsort α , let �γ[n] be the estimated spike count obtained from Algorithm 2 with input ce[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' If for all n, |w[n]| < ∆c min(α, D)/4, then the count can be exactly recovered, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=', �γ[n] = γ[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Proof is in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' It is clear that when (31) holds, ∆c min(α, D) is no smaller than ∆θmin(α, D), since the former is computed over neigh- boring elements of the cluster whereas ∆θmin(D, α) computes the minimum distance over all consecutive elements (both inter-cluster as well as intra-cluster) in Θsort α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This essentially suggests that estimation of counts (for this range of α and D) can be more robust compared to inferring the individual spiking patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We also illustrate this numerically in Figure 1 (top), where we plot both ∆c min and ∆θmin as a function of α and the start of the interval FD (computed numerically) is C0 C1 C2 C3 000 100 010 001 110 101 011 111 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 2: Visualization of the sets CD k for D = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In this scenario, the spiking patterns corresponding to the same count are clustered together and hence, are favorable for robust count estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' denoted using dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For both values of D, we can see that ∆c min > ∆θmin and the gap grows as α increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' NUMERICAL EXPERIMENTS We conduct numerical experiments to evaluate the per- formance of the proposed super-resolution spike decoding algorithm on both synthetic and real calcium imaging datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 Undersampling Factor (D) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='8 1 F-score p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='35, s=350 Algo 2 ( =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5) l1 Box ( =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5) Algo 2 ( =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='9) l1 Box ( =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='9) 1 2 3 4 5 6 7 8 9 10 Undersampling factor (D) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='9 1 F1-score p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5, s=500 D=3 D=5 D=7 AR(1), =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5 FIR (r=3) FIR (r=5) FIR (r=7) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 3: (Top) Quantitative comparison of Algorithm 2 against box- constrained l1 minimization method with noiseless measurements (with tolerance t0 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' (Bottom) (Role of Filter Memory): Average F-score vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' D for FIR and IIR (AR(1)) filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Each dotted line indicates the corresponding theoretical transition point (D = r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Synthetic Data Generation and Evaluation Metrics We create a synthetic dataset by generating high-rate binary spike sequence xhi ∈ {0, 1}L (A = 1 and L = 1000) that satisfies assumption (A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The spiking probability p controls the average sparsity level given by s := E[∥xhi∥0] = Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We aim to reconstruct xhi from M ≈ L/D low-rate measurements zlo[n] defined in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Notice that we operate in a regime where the expected sparsity is greater than the total number of low- rate measurements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=', s > M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We employ the widely-used F-score metric to evaluate the accuracy of spike detection [4], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The F-score is computed by first matching the estimated and ground truth spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' An estimated spike is considered a “match" to a ground truth spike if it is within a distance of t0 of the ground truth (many-to-one matching is not allowed) [4], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Let K and K′ be the total number of ground truth and estimated spikes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The number of spikes declared as true positives is denoted by Tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' After the matching procedure, we compute the recall (R = Tp K ) which is defined as the ratio of true positives (Tp) and the total number of ground truth spikes (K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Precision (P = Tp K′ ) measures the fraction of the total detected spikes which were correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Finally, the F-score is given by the harmonic mean of recall and precision F-score = 2PR/(P + R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 10 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='99 yhi[n] D = 5 (Top) (Bottom) 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='99 0 5 10 15 20 25 30 35 40 45 50 ylo[n] 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='02 0 5 10 15 20 25 30 35 40 45 50 xhi[n] 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='02 0 5 10 15 20 25 30 35 40 45 50 �xhi[n] 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='02 0 5 10 15 20 25 30 35 40 45 50 �xl1[n] 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='47 yhi[n] D = 10 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='47 0 10 20 30 40 50 ylo[n] 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='02 0 10 20 30 40 50 xhi[n] 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='02 0 10 20 30 40 50 �xhi[n] 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='02 0 10 20 30 40 50 �xl1[n] 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='99 0 5 10 15 20 25 30 35 40 45 50 ylo[n] 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='02 0 5 10 15 20 25 30 35 40 45 50 xhi[n] 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='02 0 5 10 15 20 25 30 35 40 45 50 �xhi[n] 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='02 0 5 10 15 20 25 30 35 40 45 50 �xl1[n] 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='45 0 10 20 30 40 50 ylo[n] 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='02 0 10 20 30 40 50 xhi[n] 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='02 0 10 20 30 40 50 �xhi[n] 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='02 0 10 20 30 40 50 �xl1[n] xhi[n]: Ground Truth Spikes, �xhi[n]: Output of Algorithm 2, �xl1[n]: Output of l1 minimization, yhi[n]: High rate waveform, ylo[n]: Low rate samples Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 4: Qualitative comparison of Algorithm 2 and box-constrained l1 minimization on simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For each simulation noisy measurements are generated with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='9 such that the noise realization (Top) obeys the bound |w[n]| ≤ ∆θmin (from Theorem 2) and (Bottom) violates the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For larger noise (Bottom), the spike recovery is imperfect but the spike count can still be exactly recovered using Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Noiseless Recovery: Role of Binary priors and memory We first consider the noiseless setting (w[n] = 0 in (21)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We compare the performance of Algorithm 2 against box- constrained l1 minimization method [35], [36], where we solve: min x∈RL ∥x∥1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' ∥ylo − SDGαx∥2 ≤ ϵ, 0 ≤ x ≤ A1 (P1) For synthetic data, ϵ is chosen using the norm of the noise term ∥w∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This oracle choice ensures most favorable parameter tuning for the (P1), although a more realistic choice would be to set ϵ = √ Mσ according to the noise power (σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In the noiseless setting, we choose ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The problem (P1) is a standard convex relaxation of (P0) which promotes sparsity as well as tries to impose the binary constraint via the box- relaxation (introduced in Section II-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 3 (Top), we plot the F-score (t0 = 0) as a function of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' As can be observed, Algorithm 2 consistently achieves an F-score of 1, whereas the F-score of l1 minimization shows a decay as D increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This confirms Lemma 3 that for D > 1, using box-constraints with l1 norm minimization is not enough to enable exact recovery from low rate measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In absence of noise, the performance of Algorithm 2 is not affected by the filter parameter α as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 3 (Top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Next, we compare the reconstruction from the decimated output of (i) an AR(1) filter and (ii) an FIR filter of length r driven by the same input xhi ∈ {0, 1}1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We choose the FIR filter h = [1, α, · · · , αr−1]⊤ (truncation of the IIR filter) with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Algorithm 2 is applied to the low-rate AR(1) measurements, whereas the algorithm proposed in [40] is used for the FIR case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The algorithm applied for the FIR case can provably operate with the optimal number of measurements when α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5 and hence, we chose this specific value for the filter parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In Figure 3 (Bottom), we again compare the average F-score as a function of D, averaged over 10000 Monte Carlo runs, for p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' As predicted by Lemma 4, despite utilizing binary priors, the error for the FIR filter shows a phase transition when D > r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This demonstrates the critical role played by the infinite memory of the AR(1) filter in achieving exact recovery with arbitrary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Performance of noisy spike decoding We generate noisy measurements of the form (21), where w[n] and xhi[n] satisfy assumptions (A1-A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We illustrate some representative examples of recovered spikes on synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' (4), we display the recovered super-resolution estimates on synthetically generated measurements for two undersampling factors D = 5 (left), 10 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For each D, the top plots show the spikes recovered using Algorithm 2 and l1 minimization with box-constraint where the noise realization obeys the bound in Theorem 2, while the bottom plots show the same for noise realization violating the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The output of l1 minimization with box-constraint is inaccurate, and the spikes are clustered towards the end of each block of length D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This bias is consistent with the prediction made by our theoretical results in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' When the noise is small enough (top), Algorithm 2 exactly decodes the spikes, including the ones occurring between two consecutive low-rate samples as predicted by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In presence of larger noise (violating the bound), the spikes estimated using l1 minimization continue to be biased to be clustered towards the end of the block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Although the spikes recovered using Algorithm 2 are not exact, most of the detected spikes are within a tolerance window of ground truth spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In fact, the spike count estimation is perfect as predicted by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We next quantitatively evaluate the performance in presence of noise, where the metrics are computed with t0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 5 (Top), we plot the F-score as a function of D for different values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For a fixed α, the F-score of both methods decays with increasing D, but Algorithm 2 consistently attains a higher F-score compared to 11 3 4 5 6 7 8 9 10 Undersampling Factor (D) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='2 F-score p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='35, s=350>M Algo 2 (alpha=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='9) l1 Box (alpha=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='9) Algo 2 (alpha=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5) l1 Box (alpha=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 5: Spike detection performance with noisy measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' (Top) F-score vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' D for different filter parameters α (σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Here, L = 1000 and expected sparsity s = 350 where we operate in the regime s > M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The F-score is computed with a tolerance of t0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' l1 minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We observe that α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5 leads to a higher F- score potentially due to having a larger ∆θmin(α, D) compared to α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Next, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 7, we study the behavior of spike detection as a function of the spiking probability p, while keeping D fixed at D = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' When σ is fixed, the performance trend is not significantly affected by the spiking probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' At first, this may seem surprising as the expected sparsity is growing while the number of measurements is unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' However, since our algorithm exploits the binary nature of the spikes (and not just sparsity), it can handle larger sparsity levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The spikes reconstructed using l1 minimization achieve a much lower F-score than Algorithm 2 since the former fails to succeed when the sparsity is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' As expected, smaller σ leads to higher F-scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 8, we study the probability of erroneous spike detection as a function of D and validate the upper bound derived in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Recall that the decoding is considered successful if “every" spike is detected correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Therefore, it becomes more challenging to “exactly super-resolve" all the spikes in presence of noise as the desired resolution becomes finer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We calculate the empirical probability of error and overlay the corresponding theoretical bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 8, the empirical probability of error is indeed upper bounded by the bound computed by our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The empirical probability of error increases as a function of undersampling factor D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 10-5 10-4 10-3 10-2 10-1 100 Noise Level 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='8 1 F-score D=5, M=200, s/M>1 Algo 2 ( =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5) l1 Box ( =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5) Algo 2 ( =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='9) l1 Box ( =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='9) 10-5 10-4 10-3 10-2 10-1 100 Noise Level 10-10 10-5 100 105 Count Estimation Error D=5, M=200, s/M>1 Algo 2 ( =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5) l1 Box ( =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5) Algo 2 ( =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='9) l1 Box ( =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='9) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 6: Spike detection performance with noisy measurements for different filter parameters α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' (Top) F-score vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' noise level (σ) (Bottom) Count estimation error vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Here, L = 1000 and expected sparsity is fixed at s = 350 where we operate in the regime s > M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The F-score is computed with a tolerance of t0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Finally, we evaluate the noise tolerance of the proposed methodology by comparing the average F-score as a function of the noise level σ, while keeping the spiking rate and undersampling factor fixed at p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='35 and D = 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 6 (Top), the performance of both algorithms degrades with increasing noise level and this is also consistent with the intuition that it becomes harder to super-resolve spikes with more noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' However, for both filter parameters considered in this experiment Algorithm 2 has a higher F-score compared to box-constrained l1 minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For large noise levels (comparable to spike amplitude A = 1), the performance gap decreases for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='9 but Algorithm 2 achieves a much higher F-score for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5 at all noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' As discussed in Section IV-B, we next study a relaxed notion of spike recovery which focuses on the spike counts occurring between two consecutive low-rate samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Let Γ = [γ[0], · · · , γ[M − 1]]⊤ be the vector of counts and �Γ be its estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 6 (Bottom) we plot the average l1 distance ∥Γ − �Γ∥1 as a function of the noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We observe that for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='9 (it can be verified from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 1 (Top) that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='9 ∈ F5), it is possible to exactly recover the spike counts at higher noise even though the F-score (for timing recovery) has dropped below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' However, this is not the case for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5, since 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5 ̸∈ F5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This is consistent with the conclusion of Theorem 4 which states that when α ∈ FD, the noise tolerance for exact count recovery can be much larger than exact spike recovery since ∆c min(α, D) > ∆θmin(α, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5 Spiking Probability (p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='8 1 F-score D=5, M=200, s/M>1 Algo 2 (sigma=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='001) l1 Box (sigma=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='001) Algo 2 (sigma=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='01) l1 Box (sigma=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='01) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 7: Spike detection performance with noisy measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' F-score vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' spiking probability (p) for different noise levels σ (fix α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='9, D = 5, L = 1000) in the extreme compression regime s > M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 Undersampling factor (D) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='8 1 Probability of Error s=30, L=100 Algo 2 ( =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='9) Theoretical Bound ( =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='9) Algo 2 ( =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='95) Theoretical Bound ( =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='95) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 8: Probability of erroneous detection of high-rate spikes xhi ∈ {0, 1}L as a function of the undersampling factor D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Theoretical upper bounds are overlaid using dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Here, L = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Spike Deconvolution from Real Calcium Imaging Datasets We now discuss how the mathematical framework developed in this paper can be used for super-resolution spike deconvo- lution in calcium imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Two-photon calcium imaging is a widely used imaging technique for large scale recording of neural activity with high spatial but poor temporal resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In calcium imaging, the signal xhi corresponds to the underlying neural spikes which is modeled to be binary valued on a finer temporal scale [2], [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Each neural spike results in a sharp rise in Ca2+ concentration followed by a slow exponential decay, leading to superposition of the responses from nearby 12 spiking events [2]–[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This calcium transient can be modeled by the first order autoregressive model introduced in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The decay time constant depends on the calcium indicator and essentially determines the filter parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The signal yhi[n] is an unobserved signal corresponding to sampling the calcium fluorescence at a high sampling rate (at the same rate as the underlying spikes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The observed calcium signal ylo[n] corresponds to downsampling yhi[n] at an interval determined by the frame rate of the microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The frame rate of a typical scanning microscopy system (that captures the changes in the calcium fluorescence) is determined by the amount of time required to spatially scan the desired field of view, which makes it significantly slower compared to the temporal scale of the neural spiking activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We model this discrepancy by the downsampling operation (by a factor D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Therefore, the mathematical framework developed in this paper can be directly applied to reconstruct the underlying spiking activity at a temporal scale finer than the sampling rate of the calcium signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Using real calcium imaging data, we demonstrate a way to fuse our algorithm with a popular spike deconvolution algorithm called OASIS [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' OASIS solves an l1 minimization problem similar to (P1) with only the non-negativity constraint, in order to exploit the sparse nature of the spiking activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Unlike our approach where we wish to obtain spikes representation on a finer temporal scale, OASIS returns the spike estimates on the low-resolution grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This is typically used to infer the spiking rate over a temporal bin equal to the sampling interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We demonstrate that our proposed framework can be integrated with OASIS and improve its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' As we saw in the synthetic experiments, the noise level is an important consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' By augmenting Algorithm 2 with OASIS, referred as “B-OASIS", the denoising power of l1 minimization can be leveraged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='Let �xl1 ∈ RM be the estimate obtained on a low-resolution grid by solving the l1 minimization problem such as the one implemented in OASIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We can obtain an estimate of the denoised calcium signal as �ylo[n] = αD�ylo[n] + �xl1[n], n ≥ 1 and �ylo[0] = �xl1[0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We can now utilize the denoised calcium signal �ylo[n] generated by OASIS to obtain the estimate ce[n] indirectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Due to the non-linear processing done by OASIS, it is difficult to obtain the resulting noise statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' An important advantage of Algorithm 2 is that it does not rely on the knowledge of the noise statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Hence, we can directly apply Algorithm 2 on �ce[n] = �ylo[n]−αD�ylo[n−1] (instead of ce[n]) to obtain a binary “fused super-resolution spike estimate".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' B-OASIS OASIS 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='9 Recall F-score B-OASIS OASIS 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='9 Recall F-score Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 9: Spike detection performance of OASIS and B-OASIS on GCaMP6f dataset sampled at (Left) 60 Hz and (Right) 30 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We compare the average F-score of data points where the F-score of OASIS is < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Standard deviation is depicted using the error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 20 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='4 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='8 21 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='2 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='8 22 ylo[n] 20 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='4 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='8 21 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='2 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='8 22 xhi[n] 20 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='4 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='8 21 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='2 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='8 22 �xhi B-OA[n] 20 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='4 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='8 21 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='2 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='8 22 �xhi OA[n] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 10: Example of spike reconstruction on GENIE dataset (GCaMP6f indicator) using OASIS and B-OASIS (binary augmented) with calcium signal sampled at 30Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Results We evaluate the algorithms on the publicly available GENIE dataset [53], [54] which consists of simultaneous calcium imag- ing and in vivo cell-attached recording from the mouse visual cortex using genetically encoded GCaMP6f calcium indicator GCaMP6f [53], [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The calcium images were acquired at a frame rate of 60 Hz and the ground truth electrophysiology signal was digitized at 10 KHz and synchronized with the calcium frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In addition to using the original data, we also synthetically downsample it to emulate the effect of a lower frame rate of 30 Hz, and evaluate how the performance changes by this downsampling operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 10, we extract an interval of ∼ 2 sec (from the neuron 1 of the GCaMP6f indicator dataset) and qualitatively compare the detected spikes with the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We downsample the data by a factor of 2 to emulate frame rate of 30 Hz, the low-rate grid becomes coarser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' As a result of which, we observe an offset between ground truth spikes and estimate produced by OASIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' However, with the help of binary priors (B-OASIS), we can output spikes that are not restricted to be on the coarser scale, and this mitigates the offset observed in the raw estimates obtained by OASIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We quantify the improvement in the performance by com- paring the F-scores of OASIS and B-OASIS at both sampling rates (60 and 30 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Since the output of OASIS is non- binary, the estimated spikes are binarized by thresholding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' To ensure a fair comparison, we select the threshold by a 80 − 20 cross-validation scheme that maximizes the average F-score on a held-out validation set (averaged over 3-random selections of the validation set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The tolerance for the F-score was set at 100 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The dataset consisted of 34 traces of length ∼ 234 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The OASIS algorithm has an automated routine to estimate the parameter α, which we utilize for our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The amplitude A is estimated using the procedure described in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We use D = 12 to obtain the spike representation for B-OASIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In order to quantify the performance boost achieved by augmentation, we isolate the traces where the F−score of OASIS drops below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5 and compare the average F-score and recall for these data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 9, at both sampling rates, we see a significant improvement in the average F-score of B-OASIS over OASIS, attributed to an increase in recall while keeping the precision unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Additionally, despite downsampling, the spike detection performance is not significantly degraded with binary priors, although the detection criteria were unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 13 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' CONCLUSION We theoretically established the benefits of binary priors in super-resolution, and showed that it is possible to achieve significant reduction in sample complexity over sparsity- based techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Using an AR(1) model, we developed and analyzed an efficient algorithm that can operate in the extreme compression regime ( M ≪ K) by exploiting the special structure of measurements and trading memory for computational efficiency at run-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We also demonstrated that binary priors can be used to boost the performance of existing neural spike deconvolution algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In the future, we will develop algorithmic frameworks for incorporating binary priors into different neural spike deconvolution pipelines and evaluate the performance gain on diverse datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The extension of this binary framework for higher-order AR filters is another exciting future direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' APPENDIX APPENDIX A: PROOF OF THEOREM 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We show that for any α in 0 < α < 1, except possibly for a set consisting of only a finite number of points, (10) always has a unique binary solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Consider all possible D−dimensional ternary vectors with their entries chosen from {−1, 0, 1}, and denote them as v(i) = [v(i) 1 , v(i) 2 , · · · , v(i) D ]T ∈ {−1, 0, 1}D, 0 ≤ i ≤ 3D − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We use the convention that v(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For every i > 0, we define a set Zv(i) determined by v(i) as Zv(i) := � x ∈ (0, 1) �� �D k=1 v(i) k xD−k = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Notice that pi(x) := �D k=1 v(i) k xD−k denotes a polynomial (in x) of degree at most D−1, whose coefficients are given by the ternary vector v(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The set Zv(i) denotes the set of zeros of pi(x) that are contained in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Since the degree of pi(x) is at most D−1, Zv(i) is a finite set with cardinality at most D−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Now suppose that the binary solution of (10) is non-unique, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=', there exist u, w ∈ {0, A}L, u ̸= w, such that HD(α)u = HD(α)w ⇒ HD(α)u − HD(α)w = 0 (34) By partitioning u, w into blocks u(n), w(n) in the same way as in (6), we can re-write (34) as u(0) = w(0) and D � i=1 1 A([u(j)]i − [w(j)]i)αD−i = 0, 1 ≤ j ≤ M − 1 (35) Since u ̸= w, they differ at least at one block, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=', there exists some j0, 1 ≤ j0 ≤ M − 1 such that u(j0) ̸= w(j0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Define b := 1 A(u(j0) − w(j0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Then, b is a non-zero ternary vector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=', b ∈ {−1, 0, 1}D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Now from (35), we have D � i=1 [b]iαD−i = 0, (36) which implies that α ∈ Zb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Since b can be any one of the 3D−1 ternary vectors {v(i)}3D−1 i=1 , (36) holds if and only if α ∈ S := �3D−1 i=1 Zv(i), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=', α is a root of at least one of the polynomials pi(x) defined by the vectors v(i) as their coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For each v(i), since the cardinality of Zv(i) is at most D−1, S is a finite set (of cardinality at most (D − 1)(3D − 1)), and therefore its Lebesgue measure is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This implies that (10) has a non-unique binary solution only if α belongs to the measure zero set S, thereby proving the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' APPENDIX B: PROOF OF LEMMA 2 AND LEMMA 3 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' (i) Let sn denote the sparsity (number of non-zero elements) of the nth block xhi(n) of xhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Then, the total sparsity is ∥xhi∥0 = �M−1 n=0 sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We will construct a vec- tor v ∈ RL, v ̸= xhi that satisfies c = HD(α)v and ∥xhi∥0 ≥ ∥v∥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Following (6), consider the partition of v v = [v(0), v(1)⊤, · · · , v(M−1)⊤]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Firstly, we assign v(0) = c[0] = xhi(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We construct v(n) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For each n ≥ 1, there are three cases: Case I: sn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In this case, xhi(n) = 0 and hence c[n] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Therefore, we assign v(n) = xhi(n) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Case II: sn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' First suppose that [xhi(n)]D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We construct v(n) as follows: [v(n)]k = � c[n], if k = D 0, else .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' (37) Next suppose that [xhi(n)]D ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Since sn = 1, this implies that [xhi(n)]k = 0, k = 1, · · · , D−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In this case, we construct v(n) as follows: [v(n)]k = � c[n]/α, if k = D − 1 0, else .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' (38) Notice that both (37) and (38) ensure that v(n) ̸= xhi(n) and c[n] = hT αv(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Moreover, ∥v(n)∥0 = sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Case III: sn ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In this case, we follow the same construction as (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' As before v(n) satisfies c[n] = h⊤ α v(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Since ∥xhi(n)∥0 ≥ 2 and ∥v(n)∥0 = 1, we automatically have v(n) ̸= xhi(n), and ∥v(n)∥0 < sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Therefore, combining the three cases, we can construct the desired vector v that satisfies v ̸= xhi, c = HD(α)v, and ∥v∥0 ≤ �M−1 n=0 sn = ∥xhi(n)∥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Therefore, the solution x⋆ to (P0) satisfies ∥x⋆∥0 ≤ ∥v∥0 ≤ ∥xhi(n)∥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' (ii) In this case, we construct v(n0) according to Case III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Since ∥v(n0)∥0 < sn0, and ∥v(n)∥0 ≤ sn, n ̸= n0, we have ∥v∥0 < ∥xhi∥0, implying ∥x⋆∥0 ≤ ∥v∥0 < ∥xhi∥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Proof of Lemma 3 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We will construct a vector v ∈ RL whose support is of the form (16), that is feasible for (P1-B), and we will prove that it has the smallest l1 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Using the block structure given by (6), we choose v(0) = c[0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For each n ≥ 1, we construct v(n) based on the following two cases: Case I: c[n] ≥ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Let kn be the largest integer such that the following holds: µ[n] := A(1 + α + · · · + αkn−1) ≤ c[n], where 1 ≤ kn ≤ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Note that kn = 1 always produces a valid lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' However, we are interested in the largest lower bound on c[n] of the above form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We choose [v(n)]k = � � � � � A, if D − kn + 1 ≤ k ≤ D (c[n] − µ[n])/αkn, if k = D − kn 0, else It is easy to verify that h⊤ α v(n) = c[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' From the definition of kn, it follows that µ[n] ≤ c[n] < µ[n] + Aαkn and hence, 0 ≤ (c[n] − µ[n])/αkn < A, which ensures that v obeys the box-constraints in (P1-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Now, let vf ∈ RL be any feasible point of (P1-B) which must be of the form v(0) f = c[0], v(n) f = v(n) + r(n), where r(n) ∈ N(h⊤ α ) is a vector in the null-space 14 of h⊤ α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' It can be verified that the following vectors {wt}D−1 t=1 form a basis for N(h⊤ α ): [wt]k = � � � � � 1, k = t −α, k = t + 1 0, else , Therefore, ∃ {β(n) t }D−1 t=1 such that r(n) = �D−1 t=1 β(n) t wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We further consider two scenarios: (i) 1 ≤ kn ≤ D − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In this case [v(n)]1 = 0, and for k = 1, 2, · · · D, [v(n) f ]k satisfies 2 [v(n) f ]k = � � � � � � � � � � � � � � � β(n) k , if k = 1 β(n) k − αβ(n) k−1, if 2 ≤ k ≤ D − kn − 1 [v(n)]k + β(n) k − αβ(n) k−1, if k = D − kn A + β(n) k − αβ(n) k−1, if D − kn + 1 ≤ k ≤ D − 1 A − αβ(n) k−1, if k = D To ensure v(n) f is a feasible point for (P1-B), the following must hold: 0 ≤ β(n) D−1 ≤ A/α and 0 ≤ β(n) 1 ≤ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For 2 ≤ k ≤ D − kn−1, the constraint [v(n) f ]k ≥ 0 implies β(n) k ≥ αβ(n) k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Since β(n) 1 ≥ 0, it follows that β(n) k ≥ 0 for all 2 ≤ k ≤ D − kn − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For D−kn+1 ≤ k ≤ D−1, the constraint [v(n) f ]k ≤ A implies β(n) k−1 ≥ β(n) k /α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Since β(n) D−1 ≥ 0, it follows that β(n) k ≥ 0 for all D − kn ≤ k ≤ D − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' (ii) kn ∈ {D − 1, D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In this case, for k = 1, 2, · · · , D, [v(n) f ]k satisfies [v(n) f ]k = � � � � � [v(n)]1 + β(n) 1 , if k = 1 A + β(n) k − αβ(n) k−1, if 2 ≤ k ≤ D − 1 A − αβ(n) k−1, if k = D For 2 ≤ k ≤ D − 1, the box-constraint [v(n) f ]k ≤ A implies β(n) k−1 ≥ β(n) k /α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Since β(n) D−1 ≥ 0, it follows that β(n) k ≥ 0 for all 1 ≤ k ≤ D − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Summarizing, we have established that β(n) i ≥ 0, ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Case II: c[n] < A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In this case, v(n) is constructed following (37), and hence v(n) f has the following structure: [v(n) f ]k = � � � � � β(n) k , if k = 1 −αβ(n) k−1 + β(n) k , if 2 ≤ k ≤ D − 1 c[n] − αβ(n) k−1, if k = D To ensure v(n) f is a feasible point, it must hold that β(n) 1 ≥ 0, β(n) k ≥ αβ(n) k−1 ≥ 0 for 2 ≤ k ≤ D − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Hence, in both Cases I and II, we established that β(n) k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For each case, since v(n) f is a non-negative vector ∀n, it can be verified that ∥vf∥1 = M−1 � n=0 ∥v(n) f ∥1 = v(0) f + M−1 � n=1 D � k=1 [v(n) f ]k = c[0] + M−1 � n=1 D � k=1 [v(n)]k � �� � ∥v∥1 + M−1 � n=1 D−1 � k=1 (1 − α)β(n) k 2In the definition of v(n) f , an assignment will be ignored if the specified interval for k is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We used the fact that �D k=1 �D−1 t=1 β(n) t [wt]k = �D−1 t=1 (1 − α)β(n) t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' If vf ̸= v, we must have β(n) k ̸= 0 for some k and n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This implies that ∥vf∥1 > ∥v∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' It is easy to see that the support of the constructed vector is of the form (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Moreover, based on the above argument, v is the only vector that has the minimum l1 norm among all possible feasible points of (P1-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' APPENDIX C: PROOF OF LEMMA 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For any 0 < α ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5, we begin by showing that for an integer p ≥ 1 the following inequality holds: p � k=1 αD−k = αD−p−1 � 1 − αp 1/α − 1 � < αD−p−1 (39) since 1/α − 1 ≥ 1 and 1 − αp < 1 in the regime 0 < α ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Let S1 = {0, αD−1, αD−2, αD−1 + αD−2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Notice that the elements of S1 are sorted in ascending order for any α and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Now, we recursively define the sets Si as follows: Si := {Si−1, Si−1 + αD−1−i}, 2 ≤ i ≤ D − 1 (40) Our hypothesis is that for every 2 ≤ i ≤ D − 1 α ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5] and D, the set Si as defined in (40), is automatically sorted in ascending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We prove this via induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For i = 2, the sets S1 and S1 + αD−3 are individually sorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Moreover from (39), we can show that: maxa∈S1 a = αD−1+αD−2 < αD−3 = minb∈S1+αD−3 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This shows that S2 is ordered, establishing the the base case of our induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Now, assume Si is ordered for some 2 ≤ i ≤ D−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We need to show that Si+1 is also ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' As a result of the induction hypothesis, both Si and Si+αD−2−i are ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Using the ordering of Si, we have: maxa∈Si a = �i+1 j=1 αD−j, minb∈Si+αD−2−i b = αD−(i+1)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' From (39), we can conclude that maxa∈Si a < minb∈Si+αD−2−i b and hence, Si+1 is also ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This completes the induction proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Also, note that for α ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5], we have Θsort α = SD−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Let ∆min(Si) be the min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' distance between the elements of the set Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' It is easy to see that ∆min(Si) = ∆min(Si + αD−2−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Since Si is sorted for α ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5], ∆min(Si) is given by: ∆min(Si) = min(∆min(Si−1), min x∈Si−1+αD−1−i x − max y∈Si−1 y) = min{∆min(Si−1), αD−i−1 − i � j=1 αD−j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' (41) Now, we use induction to establish the following conjecture: ∆min(Si) = αD−1, 1 ≤ i ≤ D − 1 (42) For the base case i = 1, ∆min(S1) = min(αD−1, αD−2 − αD−1) = αD−1, where the last equality holds since α ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5] ⇒ αD−1(1/α − 1) ≥ αD−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Suppose (42) holds for some 1 ≤ i ≤ D − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' From the definition of ∆min(Si+1) and the induction hypothesis that ∆min(Si) = αD−1, it follows that ∆min(Si+1) = min{αD−1, αD−(i+1)−1 −�i+1 j=1 αD−j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Again, from the definition of ∆min(Si) in (41), and the induction hypothesis we also have αD−i−1 −�i j=1 αD−j ≥ ∆min(Si) = αD−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Using this and the fact that α ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5, we can show: αD−i−2 −αD−i−1 − �i j=1 αD−j ≥ αD−i−2 − 2αD−i−1 + αD−1 ≥ αD−1 + αD−i−1(1/α − 2) ≥ αD−1 15 Therefore ∆min(Si+1)=min{αD−1, αD−i−2−�i+1 j=1 αD−j} = αD−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Thus, we can conclude that ∆min(α, D) = ∆min(SD−1)=αD−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' APPENDIX D: PROOF OF THEOREM 3 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The probability of incorrectly identifying xhi(n) from a single measurement ce[n] is given by pe := P(�xhi (n) ̸= xhi (n)) = lD � k=0 P(�xhi (n) ̸= xhi (n)|xhi (n) = �vk)P(xhi (n) = �vk) Given a binary vector z ∈ {0, 1}D, define the function ψ(z) := �D k=1 zk, which denotes the count of ones in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Since the noisy observations are given by ce[n] = c[n] + e[n], where e[n] = w[n] − αDw[n − 1], it follows from assumption (A2) that e[n] ∼ N(0, σ2 1) where σ2 1 = (1 + α2D)σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' From (27), we obtain P(�xhi(n) ̸= xhi(n)|xhi(n) = �v0) = P(e[n] ∈ E0) = Q(αD−1/(2σ1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Similarly, P(�xhi(n) ̸= xhi(n)|xhi(n) = �vlD) = P(e[n] ∈ ElD) = Q((�θlD − �θlD−1)/(2σ1)) = Q(αD−1/(2σ1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The last equality follows from the fact that �θlD − �θlD−1 = αD−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Finally, when conditioned on xhi(n) = �vk for 0 < k < lD, from (26), we obtain P(�x(n) ̸= xhi(n)|xhi(n) = �vk) = P(e[n] ∈ Ek) = Q( �θk−�θk−1 2σ1 ) + Q( �θk+1−�θk 2σ1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Due to Assumption (A1) on xhi, we have P(xhi(n) = �vk) = pψ(�vk)(1 − p)D−ψ(�vk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' pe is given by pe = Q(αD−1/(2σ1))(1 − p)D + Q(αD−1/(2σ1))pD+ lD−1 � k=1 � Q( �θk − �θk−1 2σ1 ) + Q( �θk+1 − �θk 2σ1 ) � pψ(vk)(1 − p)D−ψ(vk) (43) The spike train xhi is incorrectly decoded if at least one of the blocks are decoded incorrectly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' the total probability of error is given by: P( M−1 � n=0 �x(n) ̸= xhi (n)) ≤ M−1 � n=0 P(�x(n) ̸= xhi (n)) = Mpe (a) ≤ 2MQ(∆θmin(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' D)/(2σ1)) D � j=0 pj(1 − p)D−j �D j � (b) ≤ 2M exp(−∆θ2 min(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' D)/(4σ2 1)) (44) where the first inequality follows from union bound and second equality is a consequence of (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The inequality (a) follows from the monotonically decreasing property of Q(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=') function and the sum can be re-written by grouping all terms with the same count, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=', ψ(vk) = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The inequality (b) follows from the inequality Q(x) ≤ exp(−x2/2) for x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' If the SNR condition (28) holds then from (44) the total probability of error is bounded by δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' APPENDIX E: PROOF OF THEOREM 4 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We first begin by showing that α ∈ FD implies that (31) holds and hence the mapping of spikes with the same counts are clustered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Notice that for k = 0, θk max = θk min = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For k ≥ 1, it is easy to verify that θk max and θk min are attained by the spiking patterns 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='1111 (with k consecutive spikes at the indices D − k + 1 to D) and 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='000 (with consecutive spikes at the indices 1 to k), which allows us to simplify (31) as αD−1 > 0 for k = 0 and �k+1 i=1 αD−i > �k−1 j=0 αj, k = 1, · · · , D − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The values of α that satisfy each of these relations can be described by the following sets: G0 = {α ∈ (0, 1)|αD−1 > 0}, Gk = {α ∈ (0, 1)|rk(α) < 0}, where rk(α) = αD − αD−k−1 − αk + 1 for 1 ≤ k ≤ D − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' It is easy to see that FD = Gk0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Observe that the relations are symmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=', Gk = GD−k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Furthermore, for 1 ≤ k ≤ D/2, we show that Gk ⊆ Gk−1 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Trivially, G1 ⊂ G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For 2 ≤ k ≤ D/2, observe that rk(α) − rk−1(α) = αD−k(1 − 1/α) − αk(1 − 1/α) = (1/α − 1)(αk − αD−k) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Therefore, α ∈ Gk ⇒ α ∈ Gk−1, k = 1, 2 · · · , k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Moreover, since Gk = GD−k−1, it follows that FD = Gk0 = ∩D−1 k=0Gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Hence, α ∈ FD ⇒ α ∈ Gi for all 0 ≤ i ≤ D − 1, which implies that (31) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' If the noise perturbation satisfies |w[n]| < ∆c min(α, D)/4, it implies |e[n]| < ∆c min(α, D)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For any block xhi(n) ∈ CD k , θk min ≤ h⊤ α xhi(n) ≤ θk max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' If |e[n]| < ∆c min(α, D)/2, we have h⊤ α xhi (n) + e[n] < θk max + ∆c min(α, D) 2 < θk max + θk+1 min − θk max 2 h⊤ α xhi (n) + e[n] > θk min − ∆c min(α, D) 2 > θk min − θk min − θk−1 max 2 This shows that whenever α ∈ FD, the condition |e[n]| < ∆c min(α, D)/2 is sufficient for (33) to hold ∀ γ[n] and hence the spike count can be exactly recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' APPENDIX F: AMPLITUDE ESTIMATION We suggest a procedure to estimate the binary amplitude A, if it is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' We first evaluate the signal c[n] from different time instants n = 1, 2, · · · , M − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' For some 1 ≤ n0 ≤ M − 1, we estimate a set A = {Ak} of candidate amplitudes: Ak := c[n0]/hT αvk where vk ∈ Sall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Only a certain amplitudes can generate c[n0] from a valid binary spiking pattern vk ∈ Sall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Our goal is to prune A by sequentially eliminating certain candidate amplitudes from the set based on a consistency test across the remaining measurements c[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' At the tth stage (t = 2, 3, · · · ), for every remaining candidate amplitude Ak ∈ A, we perform the following consistency test with c[n], to identify if a candidate amplitude can potentially generate the corresponding measurement c[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Suppose there exists a spiking pattern vl ∈ Sall such that c[n] = AkhT αvl (45) then Ak remains a valid candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' If we cannot find a corresponding vl ∈ Sall for an amplitude Ak, we remove it, A = A \\ Ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' In presence of noise, (45) can be modified to allow a tolerance γ as we may not find an exact match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' The tolerance γ is chosen to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content='5 in the experiments on the GENIE dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This procedure prunes out possible values for the amplitude by leveraging the shared amplitude across multiple measurements c[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' ACKNOWLEDGEMENT The authors would like to thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Nikita Sidorov, Department of Mathematics at the University of Manchester, for helpful discussions regarding computational challenges in finding finite β-expansion in the range β ∈ (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' This work was supported by Grants ONR N00014-19-1-2256, DE- SC0022165, NSF 2124929, and NSF CAREER ECCS 1700506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 16 REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Small and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' Stahlheber, “Fluorophore localization algorithms for super-resolution microscopy,” Nature methods, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfwf6z/content/2301.01724v1.pdf'} +page_content=' 11, no.' metadata={'source': 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Romero7, Jahna +Otterbacher10, Carsten Schwemmer11, Kenneth Joseph12, David Garcia13, Fred Morstatter14 +1School of Social Sciences and Technology, Technical University of Munich, 2Centre for Trusted Internet and Community, +National University of Singapore, 3Sabanci University, 4University of Washington (Bothell), 5Graz University of Technology, +6GESIS - Leibniz Institute for the Social Sciences, 7University of Michigan, 8Stanford University, 9Karlstad University, +10Open University of Cyprus & CYENS CoE, 11Ludwig Maximilian University of Munich, 12University at Buffalo, +13University of Konstanz, 14Information Sciences Institute, University of Southern California 15Complexity Science Hub +Vienna +Abstract +At the end of October 2022, Elon Musk concluded his acqui- +sition of Twitter. In the weeks and months before that, sev- +eral questions were publicly discussed that were not only of +interest to the platform’s future buyers, but also of high rele- +vance to the Computational Social Science research commu- +nity. For example, how many active users does the platform +have? What percentage of accounts on the site are bots? And, +what are the dominating topics and sub-topical spheres on the +platform? In a globally coordinated effort of 80 scholars to +shed light on these questions, and to offer a dataset that will +equip other researchers to do the same, we have collected all +375 million tweets published within a 24-hour time period +starting on September 21, 2022. To the best of our knowl- +edge, this is the first complete 24-hour Twitter dataset that +is available for the research community. With it, the present +work aims to accomplish two goals. First, we seek to an- +swer the aforementioned questions and provide descriptive +metrics about Twitter that can serve as references for other +researchers. Second, we create a baseline dataset for future +research that can be used to study the potential impact of the +platform’s ownership change. +Introduction +On March 21, 2006, Twitter’s first CEO Jack Dorsey sent +the first message on the platform. In the subsequent 16 years, +close to 3 trillion tweets have been sent.1 Roughly two-thirds +of these have been either removed from the platform be- +cause the senders deleted them or because the accounts (and +all their tweets) have been banned from the platform, have +been made private by the users, or are otherwise inaccessi- +ble via the historic search with the v2 API endpoints. We +estimate that about 900 billion public tweets were on the +platform when Elon Musk acquired Twitter in October 2022 +for $44B., i.e., he paid about 5 cents per tweet. +Besides its possible economic value, Twitter has been +instrumental in studying human behavior with social me- +dia data and the entire field of Computational Social Sci- +ence (CSS) has heavily relied on data from Twitter. At the +Copyright © 2022, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +1While we do not have an official source for this number, it rep- +resents an educated guess from a collaboration of dozens of schol- +ars of Twitter. +AAAI International Conference on Web and Social Media +(ICWSM), in the past two years alone (2021-2022), over +30 scientific papers analyzed a subset of Twitter for a wide +range of topics ranging from public and mental health anal- +yses to politics and partisanship. Indeed, since its emer- +gence, Twitter has been described as a digital socioscope +(i.e., social telescope) by researchers in fields of social sci- +ence (Mejova, Weber, and Macy 2015), “a massive antenna +for social science that makes visible both the very large (e.g., +global patterns of communications) and the very small (e.g., +hourly changes in emotions)”. Beyond CSS, there is increas- +ing use of Twitter data for training large pre-trained language +models in the field of natural language processing and ma- +chine learning, such as Bernice (DeLucia et al. 2022), where +2.5 billion tweets are used to develop representations for +Twitter-specific languages, and TwHIN-BERT (Zhang et al. +2022) that leverages 7 billion tweets covering over 100 dis- +tinct languages to model short, noisy, and user-generated +text. +Although Twitter data has fostered interdisciplinary re- +search across many fields and has become a “model organ- +ism” of big data, scholarship using Twitter data has also +been criticized for various forms of bias that can emerge +during analyses (Tufekci 2014). One major challenge giv- +ing rise to these biases is getting access to data and knowing +about data quality and possible data biases (Ruths and Pfef- +fer 2014; Gonz´alez-Bail´on et al. 2014; Olteanu et al. 2019). +While Twitter has long served as one of the most collabora- +tive big social media platforms in the context of data-sharing +with academic researchers, there nonetheless exists a lack +of transparency in sampling procedures and possible biases +created from technical artifacts (Morstatter et al. 2013; Pf- +effer, Mayer, and Morstatter 2018). These unknown biases +may jeopardize research quality. At the same time, access to +unfiltered/unsampled Twitter data is nearly impossible to ac- +cess, and thus the above mentioned studies, as well as thou- +sands of others, still retain unknown and potentially signifi- +cant biases in their use of sampled data. +Contributions. +The data collection efforts presented in +this paper were driven by a desire to address these concerns +about sampling bias that exist because of the lack of a com- +plete sample of Twitter data. Consequently, the main contri- +arXiv:2301.11429v1 [cs.SI] 26 Jan 2023 + +bution of this article is to create the first complete dataset of +24 hours on Twitter and make these Tweets available via fu- +ture collaborations with the authors and contributors of this +article. The dataset collected and described here can be used +by the research community to: +• Promote a better understanding of the communication +dynamics on the platform. For example, it can be used +to answer questions like, how many active (posting) ac- +counts are on the platform? And, what are the dominating +languages and topics? +• Create a set of descriptive metrics that can serve as refer- +ences for the research community and provide context to +past and present research papers on Twitter. +• Provide a baseline for the situation before the recent +sale of Twitter. With the new ownership of Twitter, plat- +form policies as well as the company structures are un- +der significant change, which will create questions about +whether previous Twitter studies will be still valuable ref- +erences for future studies. +In the following sections, we describe the data collection +process and provide some descriptive analyses of the dataset. +We also discuss ethical considerations and data availability. +Data +Data Collection. +We have collected 24 hours of Twitter +data from September 20, 15:00:00 UTC to September 21 +14:59:59 UTC. The data collection was accomplished by +utilizing the Academic API (Pfeffer et al. 2022) that is free +and openly available for researchers. The technical setup of +the data collection pipeline was dominated by two major +challenges: First, how can we avoid—at least to a satisfying +extent—a temporal bias in data collection? Second, how can +we get a good representation of Twitter? In the following, +these two aspects are discussed in more detail. +What is a complete dataset? +What does complete mean +when we want to collect a day’s worth of Twitter data? It has +been shown previously that the availability of tweets fluctu- +ates, especially in the first couple of minutes (Pfeffer et al. +2022)—people might delete their tweets because of typos, +tweets might be removed because of violations of terms of +service, etc. To reduce this initial uncertainty, we have de- +cided to collect the data 10 minutes after the tweets were +sent. Consequently, this dataset does not include all tweets +that were sent on the collection day but instead tries to create +a somewhat stable representation of Twitter. +Avoiding temporal collection bias. +We wanted to collect +a set of tweets close to the time when they were created. +However, collecting data takes time, which can introduce +possible temporal bias, e.g., if we want to collect data from +the previous hour and the data collection job takes three +hours, then the data that is collected at the end of the col- +lection job will be much older (with potentially more tweet +removals) than the data that is collected at the beginning. To +tackle this challenge, we have split the day into 86,400 col- +lection tasks, each consisting of 1 second of Twitter activ- +ity. The collection of every second of data started exactly 10 +Time +Tweets per minute +200,000 +300,000 +400,000 +15 +18 +21 +00 +03 +06 +09 +12 +Figure 1: Tweets per minute over the 24-hour collection pe- +riod, time in UTC. +minutes after the data creation time. Because the data collec- +tion of a second took more than a minute during peak times, +we have distributed the workload to 80 collection processes, +i.e., Academic API tokens, in order to avoid backlogs. +Number of tweets. +With the above-described process, we +have collected 374,937,971 tweets within the 24 hours time +span. On average, this amounts to 4,340 [2,989 – 8,955] +tweets per second. Fig. 1 plots the number of tweets per +Minute (avg=260,374, min=192,322, max=435,721). The +data collection started at 15:00 UTC, when almost the en- +tire Twitter world is awake. Then, we can see from Japan to +Europe time zone after time zone getting off the platform. +While Europe and the Americas are sleeping, Asia keeps +the number of tweets at around 200,000. Starting at 7:00 +UTC, Europe is getting active again, followed by the Amer- +icas from East to West. Another astonishing observation of +this time series is that the first minute of every hour has on +average 15.5% more tweets than the minute before—most +likely due to bot activities and other timed tweet releases, +e.g., news. +Descriptive Analyses +Active Users +The 375 million tweets in our dataset were sent by +40,199,195 accounts. While the publicly communicated +numbers of users of a platform are often based on the num- +ber of active and passive visitors, we can state that Twitter +has (or at least had on our observed day) 40 million active +contributors who have sent at least one tweet. Less than 100 +accounts have created about 1% (=3.5M) tweets. ∼175, 000 +accounts (0.44%) created 50% of all tweets. +These numbers are not surprising when we consider that +> 95% of active accounts have sent one or two tweets. How- +ever, these numbers lend more nuance to recent reports from +the Pew Research Center, which reported that while the ma- +jority of Americans use social media, approximately 97% of +all tweets were posted by 25% of the users (McClain 2021). + +hi +fr +qme +in +zxx +fa +ko +th +pt +und +ar +tr +es +ja +en +Proportion of Tweets +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.01 +0.015 +0.017 +0.022 +0.023 +0.024 +0.03 +0.04 +0.044 +0.049 +0.05 +0.053 +0.073 +0.165 +0.31 +Figure 2: All languages occurring in at least 1% of the +tweets. +In fact, our dataset suggests that worldwide, the numbers +may be more skewed than previously suggested. +User metrics +Followers. +The active accounts on our day of Twitter +data have a mean of 2,123 followers (median=99). We can +find six accounts with more than 100 million followers +(max=133,301,854), and 427/8,635 accounts with more than +10/1 million followers. Exactly 50% of accounts that were +active on our collection day have less than 100 followers. +Following. +These accounts follow much fewer other ac- +counts: mean=547, median=197, range: 0–4,103,801. Inter- +estingly, there are 2,377 accounts that follow more than +100,000 other accounts. One-third of accounts follow less +than 100 accounts, but only 1.7% of accounts follow zero +other accounts. +Listed. +Lists are a Twitter feature for users to organize +accounts around topics and filter tweets. While there is lit- +tle evidence that lists are used widely on the platform, this +feature might be useful for getting an impression about the +number of interesting content creators on the platform. The +40 million active accounts in our dataset are listed (i.e., +number of lists that include a user) in 0 to 3,086,443 lists +(mean=10.1, median=0). 1,692/46,139 accounts are in lists +of at least 10,000/1,000 accounts. +Tweets sent. +The user information of the tweet metadata +also includes the number of tweets that a user has sent—or +at least how many of those tweets are still available on Twit- +ter. The sum of the sent tweets variable of all 40 million ac- +Table 1: Distribution of user activity +% Total Tweets +% Total Users +Min. no. of Tweets +1% +0.00023% +2,267 +10% +0.01199% +465 +25% +0.07284% +152 +50% +0.43526% +39 +75% +1.70955% +11 +90% +4.18836% +3 +counts is ∼404 billion (mean=9,704, median=1,522). If we +assume that our initial estimate of having 900 billion tweets +on the platform at the time of data collection is somewhat +correct, the accounts active in our dataset have contributed +∼45% of all of the available tweets over the entire lifetime +of Twitter. +Verified accounts. +At the time of our data collection, we +can identify 221,246 verified accounts among the 40 million +active users. +Tweets and retweets +79.2% of all tweets refer to other tweets, i.e. they are +retweets or quotes of or replies to other tweets. Conse- +quently, 20.8% of the tweets in our dataset are original +tweets. The tweets with references are of the following +types: 50.7% retweets, 4.3% quotes, 24.2% replies, i.e. half +of all tweets are retweets and a fourth are replies. +Retweeted and liked. +Studying the retweet and like num- +bers from the tweets’ metadata has created little insight since +the top retweeted tweets are very old tweets that have been +retweeted by chance on our collection day. Furthermore, we +can see the number of likes only for tweets that have been +tweeted and retweeted. In any case, the retweeted number +is interesting—the 374 million tweets have been retweeted +401 billion times. In other words, significant parts of historic +Twitter get retweeted on a daily basis. +Languages +Twitter annotates a language variable for every tweet. Fig. 2 +shows those languages that were annotated on at least 1% of +our dataset. Together, these 15 languages make up 92.5% of +all tweets. Besides the most common languages on Twitter, +we can also find interesting language codes in this list: und +stands for undefined and represents tweets for which Twitter +was not able to identify a language; qme and zxx seem to +be used by Twitter for tweets consisting of only media or a +Twitter card. +Media +There are 112,779,266 media attachments in our data collec- +tion (76.9% photos, 20.7% videos, 2.4% animated GIFs), of +which 37,803,473 have unique media keys (83.8% photos, +10.0% videos, 6.2% animated GIFs). +Geo-tags +We found only 0.5% of tweets to be geo-tagged. This is +not surprising as previous works have shown that the per- +centage of geo-tagging in Twitter has been declining (Ajao, +Hong, and Liu 2015). Fig. 3 shows the distribution of the +geo-tagged tweets across the world, with USA (20%), Brazil +(11%), Japan (8%), Saudi Arabia (6%) and India (4%) being +the top five countries. +Estimating prevalence of bot accounts +Twitter has a pivotal role in public discourse and entities +that are after power and influence often utilize this platform + +Figure 3: Choropleth map of the geo-tagged tweets across +the world. +through social bots and other means of automated activi- +ties. Since the early days of Twitter, researchers have been +studying bot behavior, and it has become an active research +area (Ferrara et al. 2016; Cresci 2020). The first estimation +of bot prevalence on Twitter indicates that 9-15% of Twit- +ter accounts exhibit automated behavior (Varol et al. 2017), +while others have observed significantly higher percentages +of tweets produced by bot-likely accounts on specific dis- +courses (Uyheng and Carley 2021; Antenore, Camacho Ro- +driguez, and Panizzi 2022). One major challenge in estimat- +ing bot prevalence is the variety of definitions, datasets, and +models used for detection (Varol 2022). +In this study, we employed BotometerLite (Yang et al. +2020), a scalable and light-weight version of the Botome- +ter (Sayyadiharikandeh et al. 2020), for computing bot +scores for unique accounts in our collection. In Fig. 4a, we +present the distribution of bot scores and nearly 20% of the +40 million active accounts have scores above 0.5 suggesting +bot-likely behavior. +While identification of bots is a complex and possi- +bly controversial challenge, plotting the distributions of +BotometerLite scores grouped by account age in Fig. 4b sug- +gests the proportions of accounts that show bot-like behavior +has increased dramatically in recent years. This result may +also suggest that the longevity of simpler bot accounts is +limited and they are no longer active on the platform. In Fig. +4c, we also present the distribution of bot scores for differ- +ent rates of activities in our dataset. Accounts that have over +1,000 posts exhibit higher rates of bot-like behaviors. +It is important to mention that accounts studied in this pa- +per were identified due to their content creation activities. +Our collection cannot capture passive accounts that are sim- +ply used to boost follower counts without visible activity on +tweet streams. Fair assessment of bot prevalence is only pos- +sible with complete access to Twitter’s internal database; +since activity streams, network data, and historical tweet +archives can capture different sets of accounts (Varol 2022). +Content on Twitter +The top 500 hashtags occurred 81,468,508 times in the +tweets. Via manual inspection, we were able to identify the +meaning of 95% of these top hashtags. They can be aggre- +gated into ten the categories. +Table 2 suggests that a large proportion of tweets referred +to entertainment, which together comprised about 30% of +tweets. These included mentions of celebrities (25.5%) and +other entertainment-related tweets (5.4%) such as mentions +of South Korean boy band members, and other references +to music, movies, and TV shows. Our data collection time +window occurred during Fall/Winter 2022, when the world +was discussing the protests in Iran after the death of Mahsa +Amini. Therefore, the Iranian protests also comprised a large +proportion of the hashtag volume at 16.6%. +Finally, and perhaps surprisingly, the category sex com- +prised over a quarter of all content covered by the top hash- +tags, and was almost completely related to escorts. “Other” +topics reflect that on “regular” Twitter days, sports, tech, and +art may take up only about 3.3% of Twitter volume. +Fig. 5 is a hashtag visualization that attempts to provide an +overview of the entire content on Twitter. We first removed +all tweets from accounts with more than 240 tweets to re- +duce the noise from bots using random trending hashtags. +From the remaining tweets, we extracted the 10,000 most +often used hashtags in our dataset and created a hashtag sim- +ilarity matrix with the number of accounts that have used a +pair of two hashtags on the day of data collection. Every el- +ement in Fig. 5 represents a hashtag. The position is the re- +sult of Multidimensional Scaling (MDS) and the color shows +the dominant language that was used in the tweets with the +particular hashtag. In this figure, we can see how languages +separate the Twitter universe but that there are also topical +sub-communities within languages. +Discussion and Potential Applications +Twitter is a social media platform with a worldwide user- +base. Open access to its data also makes it attractive to a +large community of researchers, journalists, technologists, +and policymakers who are interested in examining social +and civic behavior online. Early studies of Twitter explored +who says what to whom on Twitter (Wu et al. 2011), char- +acterizing its primary use as a communication tool. Other +early work mapped follower communities through ego net- +works (Gruzd, Wellman, and Takhteyev 2011). However, +Twitter has since expanded into its own universe, with a +plethora of users, uses, modalities, communities, and real- +life implications. Twitter is increasingly the source of break- +Table 2: The categories of the top 500 hashtags in the dataset +Category +Hashtags +Occurrence +Celebrities +159 +20,809,742 +25.5% +Sex +104 +20,529,196 +25.2% +Iranian Protests +15 +13,488,295 +16.6% +Entertainment +45 +4,392,227 +5.4% +Advertisement +32 +4,644,540 +5.7% +Politics +38 +3,858,550 +4.7% +Finance +30 +3,549,107 +4.4% +Games +21 +3,348,128 +4.1% +Other +31 +2,672,291 +3.3% +Unknown +25 +4,176,432 +5.1% +Sum +500 +81,468,508 +100.0% + +GeotaggedTweets +500k +400k +300k +200k +100k0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +BotometerLite score +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Histogram of botscores +1e6 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +CDF +(a) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +BotometerLite score +0 +1 +2 +3 +4 +5 +6 +Density +2007 +2008 +2009 +2010 +2011 +2012 +2013 +2014 +2015 +2016 +2017 +2018 +2019 +2020 +2021 +2022 +0 +2 +4 +6 +8 +# of accounts +1e6 +(b) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +BotometerLite score +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Density +Nt = Tweet count +Nt < 101 +101 +Nt < 102 +102 +Nt < 103 +103 +Nt +(c) +Figure 4: BotometerLite scores distribution: (a) histogram and cumulative distribution, (b) by account age, (c) by tweet counts +in our dataset. +ing news, and many studies from the U.S. and Europe have +reported that Twitter is one of the primary sources of news +for their citizens. Twitter has been used for political engage- +ment and citizen activism worldwide. During the COVID- +19 pandemic, Twitter even assumed the role of the official +mouthpiece and crisis communication tool for many gov- +ernments to contact their citizens, and from which citizens +could seek help and information. +Fig. 3 confirms prior reports that geotagging practices are +limited in many low- and middle-income countries (Malik +et al. 2015); however, this should not deter scholars from ex- +ploring alternative methods of triangulating the location of +users (Schwartz et al. 2013), and creating post-stratified es- +timates of regional language use (Jaidka et al. 2020; Giorgi +et al. 2022). In prior studies, the difficulties in widespread +data collection and analyses have so far implied that most +answers are based on smaller samples (usually constrained +by geography, for convenience) of a burgeoning Twitter pop- +ulation. Fig. 5 and Table 2 also impressively illustrate that +Twitter is about so much more than US politics. +We hope that our dataset is the first step in creating al- +ternatives for conducting a representative and truly inclusive +analysis of the Twitterverse. Temporal snapshots are invalu- +able to map the national and international migration patterns +that increasingly blur geopolitical boundaries (Zagheni et al. +2014). +The increasing popularity of Twitter has led it into issues +of scale, where its moderation can no longer check the large +proportion of bots on the platform. Our findings in Fig. 4 +indicate that the infestation of bots may be more pernicious +than previously imagined. We are especially concerned that +the escalation of the war on Ukraine by Russia may reflect a +spike (in our dataset) in the online activity of bots from Rus- +sia operated either by the Russian government or its allied +intelligence agencies (Badawy, Ferrara, and Lerman 2018). +These and other bots serve to amplify trending topics and +facilitate the spread of misinformation (though, perhaps, at +a rate less than humans do (Vosoughi, Roy, and Aral 2018)). +They may also misuse hashtags to divert attention away from +social or political topics (Earl, Maher, and Pan 2022; Broni- +atowski et al. 2018) or strategically target influential users +(Shao et al. 2018; Varol and Uluturk 2020). We hope that +our work will spur more studies on these topics, and we wel- +come researchers to explore our data. +By observing bursts of discussions around politically +charged events and characterizing the temporal spikes in +Twitter topics, we can better rationalize how our experience +of Twitter as a political hotbed differs from the simplified +understanding of the American Twitter landscape reported +in Mukerjee, Jaidka, and Lelkes (2022), which suggested +that politics is largely a sideshow on Twitter. It is worth con- +sidering that these politically active users may not be rep- +resentative of social media users at large (McClain 2021; +Wojcieszak et al. 2022). +Twitter is also under scrutiny for how its platform gover- +nance may conflict with users’ interests and rights (Van Di- +jck, Poell, and De Waal 2018). Concerns have been raised +about alleged biases in the algorithmic amplification (and +deamplification) of content, with evidence from France, +Germany, Turkey, and the United States, among other coun- +tries (Maj´o-V´azquez et al. 2021; Tanash et al. 2015; Jaidka, +Mukerjee, and Lelkes 2023). Other scholars have also criti- +cized Twitter’s use as a censorship weapon by governments +and political propagandists worldwide (Varol 2016; Elmas, +Overdorf, and Aberer 2021; Jakesch et al. 2021). They, and +others, may be interested in examining the trends in the en- +forcement of content moderation policies by Twitter. +Besides answering questions of data, representativeness, +access, and censorship, we anticipate that our dataset +is suited to explore the temporal dynamics of online +(mis)information in the following directions: +• Content characteristics: We have provided a high-level +exploration of the topics on Twitter. However, more can + +Figure 5: MDS of top 10,000 hashtags based on co-usage by same accounts; colors represent dominant language in tweets using +a hashtag. +be done with regard to understanding users’ concerns and +priorities. While hashtags act as signposts for the broader +Twitter community to find and engage in topics of mu- +tual interest (Cunha et al. 2011), tweets without hashtags +may offer a different understanding of Twitter discourse, +where users may engage in more interpersonal discus- +sions of news, politics, and sports than the numbers sug- +gest (Rajadesingan, Budak, and Resnick 2021). +• Patterns of information dissemination: Informational +exchanges occurring on Twitter can overcome spatio- +temporal limitations as they essentially reconfigure user +connections to create newly emergent communities. +However, these communities may vanish as quickly as +they are created, as the lifecycle of a tweet determines +how long it continues to circulate on Twitter timelines. +To the best of our knowledge, no prior research has re- +ported on the average “age” of a tweet, and we hope that +a 24-hour snapshot will enable us to answer this question +empirically. +• Content moderation and fake news: Prior research +suggests that 0.1% of Twitter users accounted for 80% +of all fake news sources shared in the lead-up to a +US election (Grinberg et al. 2019). However, we ex- +pect there to be cross-lingual differences in this distri- +bution, especially for low- or under-resourced languages +with fewer open tools for fact-checking. Similarly, we +expect that the quality of moderation and hate speech +will vary by geography and language, and recommend +the use of multilingual large language models to explore +these trends (with attention to persisting representative- + +fa +hi +ko +th +tr +un +ar +en +it +de +es +ja +pt +und +zh +frness caveats (Wu and Dredze 2020)). +• Mass mobilization: Twitter is increasingly the hotbed of +protest, which has led to some activists donning the role +of “movement spilloverers” (Zhou and Yang 2021) or se- +rial activists (Bastos and Mercea 2016) who broker infor- +mation across different online movements, thereby acting +as key coordinators, itinerants, or gatekeepers in the ex- +change of information. Such users, as well as the constant +communities in which they presumably reside (Chowd- +hury et al. 2022), may be easier to study through tempo- +ral snapshots, as facilitated by this dataset. +• Echo chambers and filter bubbles: On Twitter, algo- +rithms can affect the information diets of users in over +200 countries, with an estimated 396.5 million monthly +users (Kemp 2022). Recent surveys of the literature have +considered the evidence on how platforms’ designs and +affordances influence users behaviors, attitudes, and be- +liefs (Gonz´alez-Bail´on and Lelkes 2022). Studies of the +structural and informational networks based on snapshots +of Twitter can offer clues to solving these puzzles with- +out the constraints of data selection. +Ethics Statement and Data Availability +Ethics statement. +We acknowledge that privacy and ethi- +cal concerns are associated with collecting and using social +media data for research. However, we took several steps to +avoid risks to human subjects since participants no longer +opt into being part of our study, in a traditional sense (Zim- +mer 2020). In our analysis, we only studied and reported +population level, and aggregated observations of our dataset. +We share publicly only the tweet IDs with the research com- +munity to account for privacy issues and Twitter’s TOS. For +this purpose, we use a data sharing and long-term archiving +service provided by GESIS - Leibniz Institute for the Social +Sciences, a German infrastructure institute for the social sci- +ences 2. +With regards to data availability, this repository adheres +to the FAIR principles (Wilkinson et al. 2016) as follows: +• Findability: In compliance with Twitter’s terms of ser- +vice, only tweet IDs are made publicly available at DOI: +https://doi.org/10.7802/2516. A unique Document Ob- +ject Identifier (DOI) is associated with the dataset. Its +metadata and licenses are also readily available. +• Accessibility: The dataset can be downloaded using stan- +dard APIs and communications protocol (the REST API +and OAI-PMH). +• Interoperability: The data is provided in raw text for- +mat. +• Reusability: The CC BY 4.0 license implies that re- +searchers are free to use the data with proper attribution. +Furthermore, we want to invite the broader research com- +munity to approach one or more of the authors and collab- +orators (see Acknowledgments) of this paper with research +ideas about what can be done with this dataset. We will be +very happy to collaborate with you on your ideas! +2https://www.gesis.org/en/data-services/share-data +Acknowledgments +The data collection effort described in this paper could +not have been possible without the great collaboration of +a large number of scholars, here are some of them (in +random order): Chris Schoenherr, Leonard Husmann, Diyi +Liu, Benedict Witzenberger, Joan Rodriguez-Amat, Flo- +rian Angermeir, Stefanie Walter, Laura Mahrenbach, Isaac +Bravo, Anahit Sargsyan, Luca Maria Aiello, Sophie Brandt, +Wienke Strathern, Bilal C¸ akir, David Schoch, Yuliia Holu- +bosh, Savvas Zannettou, Kyriaki Kalimeri. +References +Ajao, O.; Hong, J.; and Liu, W. 2015. A survey of loca- +tion inference techniques on Twitter. Journal of Information +Science, 41(6): 855–864. +Antenore, M.; Camacho Rodriguez, J. M.; and Panizzi, E. +2022. A Comparative Study of Bot Detection Techniques +With an Application in Twitter Covid-19 Discourse. 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Routledge. + diff --git a/EtFJT4oBgHgl3EQfCSzh/content/tmp_files/load_file.txt b/EtFJT4oBgHgl3EQfCSzh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..385840665b6833fc841c5ce14d3f42cb6cd2ccf1 --- /dev/null +++ b/EtFJT4oBgHgl3EQfCSzh/content/tmp_files/load_file.txt @@ -0,0 +1,986 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf,len=985 +page_content='Just Another Day on Twitter: A Complete 24 Hours of Twitter Data J¨urgen Pfeffer1, Daniel Matter1, Kokil Jaidka2, Onur Varol3, Afra Mashhadi4, Jana Lasser5, 15, Dennis Assenmacher6, Siqi Wu7, Diyi Yang8, Cornelia Brantner9, Daniel M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Romero7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Jahna Otterbacher10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Carsten Schwemmer11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Kenneth Joseph12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' David Garcia13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Fred Morstatter14 1School of Social Sciences and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Technical University of Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2Centre for Trusted Internet and Community,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' National University of Singapore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 3Sabanci University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 4University of Washington (Bothell),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 5Graz University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 6GESIS - Leibniz Institute for the Social Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 7University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 8Stanford University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 9Karlstad University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 10Open University of Cyprus & CYENS CoE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 11Ludwig Maximilian University of Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 12University at Buffalo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 13University of Konstanz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 14Information Sciences Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' University of Southern California 15Complexity Science Hub Vienna Abstract At the end of October 2022,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Elon Musk concluded his acqui- sition of Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' In the weeks and months before that, sev- eral questions were publicly discussed that were not only of interest to the platform’s future buyers, but also of high rele- vance to the Computational Social Science research commu- nity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' For example, how many active users does the platform have?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' What percentage of accounts on the site are bots?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' And, what are the dominating topics and sub-topical spheres on the platform?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' In a globally coordinated effort of 80 scholars to shed light on these questions, and to offer a dataset that will equip other researchers to do the same, we have collected all 375 million tweets published within a 24-hour time period starting on September 21, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' To the best of our knowl- edge, this is the first complete 24-hour Twitter dataset that is available for the research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' With it, the present work aims to accomplish two goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' First, we seek to an- swer the aforementioned questions and provide descriptive metrics about Twitter that can serve as references for other researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Second, we create a baseline dataset for future research that can be used to study the potential impact of the platform’s ownership change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Introduction On March 21, 2006, Twitter’s first CEO Jack Dorsey sent the first message on the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' In the subsequent 16 years, close to 3 trillion tweets have been sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='1 Roughly two-thirds of these have been either removed from the platform be- cause the senders deleted them or because the accounts (and all their tweets) have been banned from the platform, have been made private by the users, or are otherwise inaccessi- ble via the historic search with the v2 API endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' We estimate that about 900 billion public tweets were on the platform when Elon Musk acquired Twitter in October 2022 for $44B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=', i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=', he paid about 5 cents per tweet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Besides its possible economic value, Twitter has been instrumental in studying human behavior with social me- dia data and the entire field of Computational Social Sci- ence (CSS) has heavily relied on data from Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' At the Copyright © 2022, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 1While we do not have an official source for this number, it rep- resents an educated guess from a collaboration of dozens of schol- ars of Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' AAAI International Conference on Web and Social Media (ICWSM), in the past two years alone (2021-2022), over 30 scientific papers analyzed a subset of Twitter for a wide range of topics ranging from public and mental health anal- yses to politics and partisanship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Indeed, since its emer- gence, Twitter has been described as a digital socioscope (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=', social telescope) by researchers in fields of social sci- ence (Mejova, Weber, and Macy 2015), “a massive antenna for social science that makes visible both the very large (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=', global patterns of communications) and the very small (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=', hourly changes in emotions)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Beyond CSS, there is increas- ing use of Twitter data for training large pre-trained language models in the field of natural language processing and ma- chine learning, such as Bernice (DeLucia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2022), where 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='5 billion tweets are used to develop representations for Twitter-specific languages, and TwHIN-BERT (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2022) that leverages 7 billion tweets covering over 100 dis- tinct languages to model short, noisy, and user-generated text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Although Twitter data has fostered interdisciplinary re- search across many fields and has become a “model organ- ism” of big data, scholarship using Twitter data has also been criticized for various forms of bias that can emerge during analyses (Tufekci 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' One major challenge giv- ing rise to these biases is getting access to data and knowing about data quality and possible data biases (Ruths and Pfef- fer 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Gonz´alez-Bail´on et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Olteanu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' While Twitter has long served as one of the most collabora- tive big social media platforms in the context of data-sharing with academic researchers, there nonetheless exists a lack of transparency in sampling procedures and possible biases created from technical artifacts (Morstatter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Pf- effer, Mayer, and Morstatter 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' These unknown biases may jeopardize research quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' At the same time, access to unfiltered/unsampled Twitter data is nearly impossible to ac- cess, and thus the above mentioned studies, as well as thou- sands of others, still retain unknown and potentially signifi- cant biases in their use of sampled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' The data collection efforts presented in this paper were driven by a desire to address these concerns about sampling bias that exist because of the lack of a com- plete sample of Twitter data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Consequently, the main contri- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='11429v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='SI] 26 Jan 2023 bution of this article is to create the first complete dataset of 24 hours on Twitter and make these Tweets available via fu- ture collaborations with the authors and contributors of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' The dataset collected and described here can be used by the research community to: Promote a better understanding of the communication dynamics on the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' For example, it can be used to answer questions like, how many active (posting) ac- counts are on the platform?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' And, what are the dominating languages and topics?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Create a set of descriptive metrics that can serve as refer- ences for the research community and provide context to past and present research papers on Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Provide a baseline for the situation before the recent sale of Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' With the new ownership of Twitter, plat- form policies as well as the company structures are un- der significant change, which will create questions about whether previous Twitter studies will be still valuable ref- erences for future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' In the following sections, we describe the data collection process and provide some descriptive analyses of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' We also discuss ethical considerations and data availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Data Data Collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' We have collected 24 hours of Twitter data from September 20, 15:00:00 UTC to September 21 14:59:59 UTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' The data collection was accomplished by utilizing the Academic API (Pfeffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2022) that is free and openly available for researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' The technical setup of the data collection pipeline was dominated by two major challenges: First, how can we avoid—at least to a satisfying extent—a temporal bias in data collection?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Second, how can we get a good representation of Twitter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' In the following, these two aspects are discussed in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' What is a complete dataset?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' What does complete mean when we want to collect a day’s worth of Twitter data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' It has been shown previously that the availability of tweets fluctu- ates, especially in the first couple of minutes (Pfeffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2022)—people might delete their tweets because of typos, tweets might be removed because of violations of terms of service, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' To reduce this initial uncertainty, we have de- cided to collect the data 10 minutes after the tweets were sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Consequently, this dataset does not include all tweets that were sent on the collection day but instead tries to create a somewhat stable representation of Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Avoiding temporal collection bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' We wanted to collect a set of tweets close to the time when they were created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' However, collecting data takes time, which can introduce possible temporal bias, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=', if we want to collect data from the previous hour and the data collection job takes three hours, then the data that is collected at the end of the col- lection job will be much older (with potentially more tweet removals) than the data that is collected at the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' To tackle this challenge, we have split the day into 86,400 col- lection tasks, each consisting of 1 second of Twitter activ- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' The collection of every second of data started exactly 10 Time Tweets per minute 200,000 300,000 400,000 15 18 21 00 03 06 09 12 Figure 1: Tweets per minute over the 24-hour collection pe- riod, time in UTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' minutes after the data creation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Because the data collec- tion of a second took more than a minute during peak times, we have distributed the workload to 80 collection processes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=', Academic API tokens, in order to avoid backlogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Number of tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' With the above-described process, we have collected 374,937,971 tweets within the 24 hours time span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' On average, this amounts to 4,340 [2,989 – 8,955] tweets per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 1 plots the number of tweets per Minute (avg=260,374, min=192,322, max=435,721).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' The data collection started at 15:00 UTC, when almost the en- tire Twitter world is awake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Then, we can see from Japan to Europe time zone after time zone getting off the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' While Europe and the Americas are sleeping, Asia keeps the number of tweets at around 200,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Starting at 7:00 UTC, Europe is getting active again, followed by the Amer- icas from East to West.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Another astonishing observation of this time series is that the first minute of every hour has on average 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='5% more tweets than the minute before—most likely due to bot activities and other timed tweet releases, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=', news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Descriptive Analyses Active Users The 375 million tweets in our dataset were sent by 40,199,195 accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' While the publicly communicated numbers of users of a platform are often based on the num- ber of active and passive visitors, we can state that Twitter has (or at least had on our observed day) 40 million active contributors who have sent at least one tweet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Less than 100 accounts have created about 1% (=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='5M) tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' ∼175, 000 accounts (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='44%) created 50% of all tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' These numbers are not surprising when we consider that > 95% of active accounts have sent one or two tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' How- ever, these numbers lend more nuance to recent reports from the Pew Research Center, which reported that while the ma- jority of Americans use social media, approximately 97% of all tweets were posted by 25% of the users (McClain 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' hi fr qme in zxx fa ko th pt und ar tr es ja en Proportion of Tweets 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='049 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='073 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='165 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='31 Figure 2: All languages occurring in at least 1% of the tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' In fact, our dataset suggests that worldwide, the numbers may be more skewed than previously suggested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' User metrics Followers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' The active accounts on our day of Twitter data have a mean of 2,123 followers (median=99).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' We can find six accounts with more than 100 million followers (max=133,301,854), and 427/8,635 accounts with more than 10/1 million followers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Exactly 50% of accounts that were active on our collection day have less than 100 followers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' These accounts follow much fewer other ac- counts: mean=547, median=197, range: 0–4,103,801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Inter- estingly, there are 2,377 accounts that follow more than 100,000 other accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' One-third of accounts follow less than 100 accounts, but only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='7% of accounts follow zero other accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Lists are a Twitter feature for users to organize accounts around topics and filter tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' While there is lit- tle evidence that lists are used widely on the platform, this feature might be useful for getting an impression about the number of interesting content creators on the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' The 40 million active accounts in our dataset are listed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=', number of lists that include a user) in 0 to 3,086,443 lists (mean=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='1, median=0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 1,692/46,139 accounts are in lists of at least 10,000/1,000 accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Tweets sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' The user information of the tweet metadata also includes the number of tweets that a user has sent—or at least how many of those tweets are still available on Twit- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' The sum of the sent tweets variable of all 40 million ac- Table 1: Distribution of user activity % Total Tweets % Total Users Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' of Tweets 1% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='00023% 2,267 10% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='01199% 465 25% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='07284% 152 50% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='43526% 39 75% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='70955% 11 90% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='18836% 3 counts is ∼404 billion (mean=9,704, median=1,522).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' If we assume that our initial estimate of having 900 billion tweets on the platform at the time of data collection is somewhat correct, the accounts active in our dataset have contributed ∼45% of all of the available tweets over the entire lifetime of Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Verified accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' At the time of our data collection, we can identify 221,246 verified accounts among the 40 million active users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Tweets and retweets 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='2% of all tweets refer to other tweets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' they are retweets or quotes of or replies to other tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Conse- quently, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='8% of the tweets in our dataset are original tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' The tweets with references are of the following types: 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='7% retweets, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='3% quotes, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='2% replies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' half of all tweets are retweets and a fourth are replies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Retweeted and liked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Studying the retweet and like num- bers from the tweets’ metadata has created little insight since the top retweeted tweets are very old tweets that have been retweeted by chance on our collection day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Furthermore, we can see the number of likes only for tweets that have been tweeted and retweeted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' In any case, the retweeted number is interesting—the 374 million tweets have been retweeted 401 billion times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' In other words, significant parts of historic Twitter get retweeted on a daily basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Languages Twitter annotates a language variable for every tweet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2 shows those languages that were annotated on at least 1% of our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Together, these 15 languages make up 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='5% of all tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Besides the most common languages on Twitter, we can also find interesting language codes in this list: und stands for undefined and represents tweets for which Twitter was not able to identify a language;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' qme and zxx seem to be used by Twitter for tweets consisting of only media or a Twitter card.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Media There are 112,779,266 media attachments in our data collec- tion (76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='9% photos, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='7% videos, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='4% animated GIFs), of which 37,803,473 have unique media keys (83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='8% photos, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='0% videos, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='2% animated GIFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Geo-tags We found only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='5% of tweets to be geo-tagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' This is not surprising as previous works have shown that the per- centage of geo-tagging in Twitter has been declining (Ajao, Hong, and Liu 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 3 shows the distribution of the geo-tagged tweets across the world, with USA (20%), Brazil (11%), Japan (8%), Saudi Arabia (6%) and India (4%) being the top five countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Estimating prevalence of bot accounts Twitter has a pivotal role in public discourse and entities that are after power and influence often utilize this platform Figure 3: Choropleth map of the geo-tagged tweets across the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' through social bots and other means of automated activi- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Since the early days of Twitter, researchers have been studying bot behavior, and it has become an active research area (Ferrara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Cresci 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' The first estimation of bot prevalence on Twitter indicates that 9-15% of Twit- ter accounts exhibit automated behavior (Varol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2017), while others have observed significantly higher percentages of tweets produced by bot-likely accounts on specific dis- courses (Uyheng and Carley 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Antenore, Camacho Ro- driguez, and Panizzi 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' One major challenge in estimat- ing bot prevalence is the variety of definitions, datasets, and models used for detection (Varol 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' In this study, we employed BotometerLite (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2020), a scalable and light-weight version of the Botome- ter (Sayyadiharikandeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2020), for computing bot scores for unique accounts in our collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 4a, we present the distribution of bot scores and nearly 20% of the 40 million active accounts have scores above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='5 suggesting bot-likely behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' While identification of bots is a complex and possi- bly controversial challenge, plotting the distributions of BotometerLite scores grouped by account age in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 4b sug- gests the proportions of accounts that show bot-like behavior has increased dramatically in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' This result may also suggest that the longevity of simpler bot accounts is limited and they are no longer active on the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 4c, we also present the distribution of bot scores for differ- ent rates of activities in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Accounts that have over 1,000 posts exhibit higher rates of bot-like behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' It is important to mention that accounts studied in this pa- per were identified due to their content creation activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Our collection cannot capture passive accounts that are sim- ply used to boost follower counts without visible activity on tweet streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Fair assessment of bot prevalence is only pos- sible with complete access to Twitter’s internal database;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' since activity streams, network data, and historical tweet archives can capture different sets of accounts (Varol 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Content on Twitter The top 500 hashtags occurred 81,468,508 times in the tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Via manual inspection, we were able to identify the meaning of 95% of these top hashtags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' They can be aggre- gated into ten the categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Table 2 suggests that a large proportion of tweets referred to entertainment, which together comprised about 30% of tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' These included mentions of celebrities (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='5%) and other entertainment-related tweets (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='4%) such as mentions of South Korean boy band members, and other references to music, movies, and TV shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Our data collection time window occurred during Fall/Winter 2022, when the world was discussing the protests in Iran after the death of Mahsa Amini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Therefore, the Iranian protests also comprised a large proportion of the hashtag volume at 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Finally, and perhaps surprisingly, the category sex com- prised over a quarter of all content covered by the top hash- tags, and was almost completely related to escorts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' “Other” topics reflect that on “regular” Twitter days, sports, tech, and art may take up only about 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='3% of Twitter volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 5 is a hashtag visualization that attempts to provide an overview of the entire content on Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' We first removed all tweets from accounts with more than 240 tweets to re- duce the noise from bots using random trending hashtags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' From the remaining tweets, we extracted the 10,000 most often used hashtags in our dataset and created a hashtag sim- ilarity matrix with the number of accounts that have used a pair of two hashtags on the day of data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Every el- ement in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 5 represents a hashtag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' The position is the re- sult of Multidimensional Scaling (MDS) and the color shows the dominant language that was used in the tweets with the particular hashtag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' In this figure, we can see how languages separate the Twitter universe but that there are also topical sub-communities within languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Discussion and Potential Applications Twitter is a social media platform with a worldwide user- base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Open access to its data also makes it attractive to a large community of researchers, journalists, technologists, and policymakers who are interested in examining social and civic behavior online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Early studies of Twitter explored who says what to whom on Twitter (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2011), char- acterizing its primary use as a communication tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Other early work mapped follower communities through ego net- works (Gruzd, Wellman, and Takhteyev 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' However, Twitter has since expanded into its own universe, with a plethora of users, uses, modalities, communities, and real- life implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Twitter is increasingly the source of break- Table 2: The categories of the top 500 hashtags in the dataset Category Hashtags Occurrence Celebrities 159 20,809,742 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='5% Sex 104 20,529,196 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='2% Iranian Protests 15 13,488,295 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='6% Entertainment 45 4,392,227 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='4% Advertisement 32 4,644,540 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='7% Politics 38 3,858,550 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='7% Finance 30 3,549,107 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='4% Games 21 3,348,128 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='1% Other 31 2,672,291 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='3% Unknown 25 4,176,432 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='1% Sum 500 81,468,508 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='0% GeotaggedTweets 500k 400k 300k 200k 100k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='0 BotometerLite 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='0 CDF (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='0 BotometerLite score 0 1 2 3 4 5 6 Density 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 0 2 4 6 8 # of accounts 1e6 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='0 BotometerLite score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='5 Density Nt = Tweet count Nt < 101 101 Nt < 102 102 Nt < 103 103 Nt (c) Figure 4: BotometerLite scores distribution: (a) histogram and cumulative distribution, (b) by account age, (c) by tweet counts in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' ing news, and many studies from the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' and Europe have reported that Twitter is one of the primary sources of news for their citizens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Twitter has been used for political engage- ment and citizen activism worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' During the COVID- 19 pandemic, Twitter even assumed the role of the official mouthpiece and crisis communication tool for many gov- ernments to contact their citizens, and from which citizens could seek help and information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 3 confirms prior reports that geotagging practices are limited in many low- and middle-income countries (Malik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' however, this should not deter scholars from ex- ploring alternative methods of triangulating the location of users (Schwartz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2013), and creating post-stratified es- timates of regional language use (Jaidka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Giorgi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' In prior studies, the difficulties in widespread data collection and analyses have so far implied that most answers are based on smaller samples (usually constrained by geography, for convenience) of a burgeoning Twitter pop- ulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 5 and Table 2 also impressively illustrate that Twitter is about so much more than US politics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' We hope that our dataset is the first step in creating al- ternatives for conducting a representative and truly inclusive analysis of the Twitterverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Temporal snapshots are invalu- able to map the national and international migration patterns that increasingly blur geopolitical boundaries (Zagheni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' The increasing popularity of Twitter has led it into issues of scale, where its moderation can no longer check the large proportion of bots on the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Our findings in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 4 indicate that the infestation of bots may be more pernicious than previously imagined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' We are especially concerned that the escalation of the war on Ukraine by Russia may reflect a spike (in our dataset) in the online activity of bots from Rus- sia operated either by the Russian government or its allied intelligence agencies (Badawy, Ferrara, and Lerman 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' These and other bots serve to amplify trending topics and facilitate the spread of misinformation (though, perhaps, at a rate less than humans do (Vosoughi, Roy, and Aral 2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' They may also misuse hashtags to divert attention away from social or political topics (Earl, Maher, and Pan 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Broni- atowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2018) or strategically target influential users (Shao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Varol and Uluturk 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' We hope that our work will spur more studies on these topics, and we wel- come researchers to explore our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' By observing bursts of discussions around politically charged events and characterizing the temporal spikes in Twitter topics, we can better rationalize how our experience of Twitter as a political hotbed differs from the simplified understanding of the American Twitter landscape reported in Mukerjee, Jaidka, and Lelkes (2022), which suggested that politics is largely a sideshow on Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' It is worth con- sidering that these politically active users may not be rep- resentative of social media users at large (McClain 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Wojcieszak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Twitter is also under scrutiny for how its platform gover- nance may conflict with users’ interests and rights (Van Di- jck, Poell, and De Waal 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Concerns have been raised about alleged biases in the algorithmic amplification (and deamplification) of content, with evidence from France, Germany, Turkey, and the United States, among other coun- tries (Maj´o-V´azquez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Tanash et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Jaidka, Mukerjee, and Lelkes 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Other scholars have also criti- cized Twitter’s use as a censorship weapon by governments and political propagandists worldwide (Varol 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Elmas, Overdorf, and Aberer 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Jakesch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' They, and others, may be interested in examining the trends in the en- forcement of content moderation policies by Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Besides answering questions of data, representativeness, access, and censorship, we anticipate that our dataset is suited to explore the temporal dynamics of online (mis)information in the following directions: Content characteristics: We have provided a high-level exploration of the topics on Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' However, more can Figure 5: MDS of top 10,000 hashtags based on co-usage by same accounts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' colors represent dominant language in tweets using a hashtag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' be done with regard to understanding users’ concerns and priorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' While hashtags act as signposts for the broader Twitter community to find and engage in topics of mu- tual interest (Cunha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2011), tweets without hashtags may offer a different understanding of Twitter discourse, where users may engage in more interpersonal discus- sions of news, politics, and sports than the numbers sug- gest (Rajadesingan, Budak, and Resnick 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Patterns of information dissemination: Informational exchanges occurring on Twitter can overcome spatio- temporal limitations as they essentially reconfigure user connections to create newly emergent communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' However, these communities may vanish as quickly as they are created, as the lifecycle of a tweet determines how long it continues to circulate on Twitter timelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' To the best of our knowledge, no prior research has re- ported on the average “age” of a tweet, and we hope that a 24-hour snapshot will enable us to answer this question empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Content moderation and fake news: Prior research suggests that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='1% of Twitter users accounted for 80% of all fake news sources shared in the lead-up to a US election (Grinberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' However, we ex- pect there to be cross-lingual differences in this distri- bution, especially for low- or under-resourced languages with fewer open tools for fact-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Similarly, we expect that the quality of moderation and hate speech will vary by geography and language, and recommend the use of multilingual large language models to explore these trends (with attention to persisting representative- fa hi ko th tr un ar en it de es ja pt und zh frness caveats (Wu and Dredze 2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Mass mobilization: Twitter is increasingly the hotbed of protest, which has led to some activists donning the role of “movement spilloverers” (Zhou and Yang 2021) or se- rial activists (Bastos and Mercea 2016) who broker infor- mation across different online movements, thereby acting as key coordinators, itinerants, or gatekeepers in the ex- change of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Such users, as well as the constant communities in which they presumably reside (Chowd- hury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2022), may be easier to study through tempo- ral snapshots, as facilitated by this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Echo chambers and filter bubbles: On Twitter, algo- rithms can affect the information diets of users in over 200 countries, with an estimated 396.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='5 million monthly users (Kemp 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Recent surveys of the literature have considered the evidence on how platforms’ designs and affordances influence users behaviors, attitudes, and be- liefs (Gonz´alez-Bail´on and Lelkes 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Studies of the structural and informational networks based on snapshots of Twitter can offer clues to solving these puzzles with- out the constraints of data selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Ethics Statement and Data Availability Ethics statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' We acknowledge that privacy and ethi- cal concerns are associated with collecting and using social media data for research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' However, we took several steps to avoid risks to human subjects since participants no longer opt into being part of our study, in a traditional sense (Zim- mer 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' In our analysis, we only studied and reported population level, and aggregated observations of our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' We share publicly only the tweet IDs with the research com- munity to account for privacy issues and Twitter’s TOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' For this purpose, we use a data sharing and long-term archiving service provided by GESIS - Leibniz Institute for the Social Sciences, a German infrastructure institute for the social sci- ences 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' With regards to data availability, this repository adheres to the FAIR principles (Wilkinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2016) as follows: Findability: In compliance with Twitter’s terms of ser- vice, only tweet IDs are made publicly available at DOI: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='7802/2516.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' A unique Document Ob- ject Identifier (DOI) is associated with the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Its metadata and licenses are also readily available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Accessibility: The dataset can be downloaded using stan- dard APIs and communications protocol (the REST API and OAI-PMH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Interoperability: The data is provided in raw text for- mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Reusability: The CC BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='0 license implies that re- searchers are free to use the data with proper attribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Furthermore, we want to invite the broader research com- munity to approach one or more of the authors and collab- orators (see Acknowledgments) of this paper with research ideas about what can be done with this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' We will be very happy to collaborate with you on your ideas!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' 2https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='gesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content='org/en/data-services/share-data Acknowledgments The data collection effort described in this paper could not have been possible without the great collaboration of a large number of scholars,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' here are some of them (in random order): Chris Schoenherr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Leonard Husmann,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Diyi Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Benedict Witzenberger,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Joan Rodriguez-Amat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Flo- rian Angermeir,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Stefanie Walter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Laura Mahrenbach,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Isaac Bravo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Anahit Sargsyan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Luca Maria Aiello,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Sophie Brandt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Wienke Strathern,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Bilal C¸ akir,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' David Schoch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Yuliia Holu- bosh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Savvas Zannettou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' Kyriaki Kalimeri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=' References Ajao, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFJT4oBgHgl3EQfCSzh/content/2301.11429v1.pdf'} +page_content=';' metadata={'source': 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Let (R, m) be a one dimensional local Cohen-Macaulay ring. An m-primary +ideal I of R is Elias if the types of I and of R/I are equal. Canonical and principal ideals +are Elias, and Elias ideals are closed under inclusion. We give multiple characterizations +of Elias ideals and concrete criteria to identify them. We connect Elias ideals to other +well-studied definitions: Ulrich, m-full, integrally closed, trace ideals, etc. Applications are +given regarding canonical ideals, conductors and the Auslander index. +Introduction +Let (R, m) be a local Cohen-Macaulay ring of dimension one and I be an m-primary ideal +of R. We say that I is Elias if the Cohen-Macaulay types of I and R/I coincide. From +standard facts, principal ideals or canonical ideals are Elias, and we will soon see that this +property begets a rather rich and interesting theory. +Our work is heavily influenced by a nice result in [7], where Elias proves that any ideal +ω that lies inside a high enough power of m and such that R/ω is Gorenstein must be a +canonical ideal. Although not stated explicitly there, the proof showed that any ideal that +lies in a high enough power of m is Elias, in our sense. Another inspiration for the present +work is [2], where De Stefani studies, in our language, powers of m that are Elias in a +Gorenstein local ring, and gives a counter-example to a conjecture by Ding (see Section 4 +for the precise connection). +In this note, we study Elias ideals in depth. They admit many different characterizations, +and enjoy rather useful properties. For instance, they are closed under inclusion, and +principal or canonical ideals are Elias. On the other hand, conductor ideals or regular trace +ideals are not Elias. When R is Gorenstein, they are precisely ideals such that the Auslander +index δ(R/I) is 1. +We are able to obtain many criteria to check whether an ideal is Elias, using very accessible +information such as the minimal number or valuations of generators. +Combining them +immediately gives sharp bounds and information on conductor or canonical ideals, which +can be tricky to obtain otherwise. +There are several obvious ways to extend the present definitions and results to higher +dimension rings or to modules. However, we choose to focus on the ideals in dimension one +case here as they are already interesting enough, and also to keep the paper short. We hope +to address the more general theory in future works. +We now describe briefly the structure and key results of the paper. +• In section 1 we give the formal definition of Elias ideals and prove several key results. +Theorem 1.2 contains several equivalent characterizations of Elias ideals. Corollary +1.3 collects important consequences, for instance that Elias ideals are closed under +ideal containment. Also, criteria for Elias ideals using colon ideals are given. Next, +2020 Mathematics Subject Classification. Primary: 13D02, 13H10. Secondary: 14B99. +1 + +Proposition 1.4 establishes the fundamental change of rings result that are used +frequently in the sequence. +• Section 2 connects Elias ideals to several well-studied class of ideals: Ulrich ideals, +m-full ideals, full ideals, integrally closed ideals, etc. After some basic observations, +(2.3, 2.4, 2.5), we give Theorems 2.7 and Proposition 2.14, which contain concrete +ways to recognize Elias ideals using basic information such as number of generators +or valuations. We also derive that conductor ideals or regular trace ideals are not +Elias (Corollary 2.13). +This indicates one of the useful application: if we know, +for instance, that m2 is Elias, then the conductor or any regular trace ideal must +contains an element of m-adic order 1. +• Given the previous section, it is natural to study the Elias index eli(R), namely +the first power of m that is Elias, and we do so in Section 3. The first main result +here is Theorem 3.2, connecting this index to the generalized L¨oewy length and the +regularity of the associated graded ring. Next, in Theorem 3.3, we characterize rings +with small indexes: eli(R) = 1 if and only if R is regular, and eli(R) = 2 plus R is +Gorenstein is equivalent to e(R) = 2. We give a large class of non-Gorenstein rings +with Elias index 2 (3.4). +• Lastly, in Section 4 we focus on the special case of Gorenstein rings. In such situation, +we observe that Elias ideals are precisely ones whose quotient has Auslander δ- +invariant one. This immediately allows us to apply what we have to recover old +results about the Auslander invariant and Auslander index in 4.1 and 4.3. We give +a counter-example to a Theorem by Ding and also revisit a counter-example to a +conjecture by Ding given in [2] (Examples 4.4 and 4.5). +Acknowledgements: It is a pleasure to thank Juan Elias and Alessandro Di Stefani for +helpful comments and encouragements. The author is partially supported by the Simons +Collaboration Grant FND0077558. +1. Elias ideals: definitions and basic results +Throughout the paper, let (R, m, k) be Cohen-Macaulay local ring of dimension one. For +a module M, set typeR(M) = dimk Extdim M +R +(k, M). Set Q = Q(R) to be the total ring of +fractions of R. Set e = e(R), the Hilbert-Samuel multiplicity of R. For an element x ∈ R, +the m-adic order of x, denoted ord(x) is the smallest a such that x ∈ ma. The order of an +ideal I, denoted ord(I), is the minimum order of its elements. +Definition 1.1. We say that a m-primary ideal I is an Elias ideal if it satisfies type(I) = +type(R/I). +Theorem 1.2. We always have type(I) ≥ type(R/I). The following are equivalent. +(1) type(I) = type(R/I). +(2) For any NZD x ∈ m, xI : m ⊆ (x). +(3) For any NZD x ∈ m, xI : m = x(I : m). +(4) For some NZD x ∈ m, xI : m ⊆ (x). +(5) For some NZD x ∈ m, xI : m = x(I : m). +(6) I :Q m ⊆ R. +(7) K ⊆ m(K :Q I) (assuming R admits a canonical ideal K). +Proof. Let x be a NZD. Then +type(I) = type(I/xI) = dimk +xI : m +xI +≥ dimk +x(I : m) +xI += dimk +I : m +I += type(R/I) +2 + +Thus, type(I) = type(R/I) if and only if xI : m = x(I : m). Now, xI : m ⊆ (x) is equivalent +to xI : m = xJ for some ideal J, as x is a NZD. Rewriting it as xJm ⊆ xI, which is +equivalent to Jm ⊆ I, we get J ⊆ I : m. On the other hand x(I : m) ⊆ xI : m, thus +J = I : m. That establishes the equivalence of first five items. +Note that for any NZD x ∈ m, xI : m = x(I :Q m). Thus, (6) is equivalent to (3). +Let K be a canonical ideal. +Apply HomR(−, K) to the sequence 0 → I → R → +R/I → 0, and indentifying HomR(I, K) with K :Q I, we get 0 → K → K :Q I → +Ext1 +R(R/I, K) = ωR/I → 0. +Since type(I) = µ(K :Q I) and type(R/I) = µ(ωR/I), the +equivalence of (7) and (1) follows. +□ +Corollary 1.3. We have: +(1) If I is isomorphic to R or the canonical module of R (assuming its existence), then +I is Elias. +(2) If I is Elias, then so is J for any ideal J ⊆ I. (being Elias is closed under inclusion) +(3) Let K be a canonical ideal of R and I be an ideal containing K. Then I is Elias if +and only if K ⊆ m(K :R I). +(4) Let K be a canonical ideal of R and I be an ideal such that K ⊆ I. Then K : I is +Elias if and only if K ⊆ mI. +(5) Suppose that I contains a canonical ideal K such that ord(K) = 1. Then I is Elias +if and only if I = K. +Proof. For the first claim, I :Q m ⊂ I :Q I = R. For the second claim, we have J :Q m ⊂ +I :Q m. For (3), first note that K :Q I ⊂ K :Q K = R, so K :Q I = K :R I, and we can use +part (7) of Theorem 1.2. +For part (4), note that K : (K : I) = I hence we can apply part (3). +For part (5), we again apply part (3): if K ⊊ I, then m(K :R I) ⊆ m2, contradicting +ord(K) = 1. +□ +The following change of rings result would be used frequently in what follows. +Proposition 1.4. Let (R, m) → (S, n) be a local, flat rings extension such that dim S = 1 +and S is Noetherian. Then I is an Elias ideal of R if and only if IS is an Elias ideal of S. +Proof. Under the assumption we have typeR(M) typeS/mS(S/mS) = typeS(M ⊗R S) for any +finitely generated R-module M (see for instance [11]), thus the result follows. +□ +2. Elias ideals and other special ideals +Definition 2.1. Let I be an m-primary ideal. +• I is called Ulrich (as an R-module) if µ(I) = e(R). Assuming k is infinite, then I is +Ulrich if and only if xI = mI for some x ∈ m (equivalently, for any x ∈ m such that +ℓ(R/xR) = e(R)). +• I is called m-full if Im : x = I for some x ∈ m. +• I is called full (or basically full) if Im : m = I. +Remark 2.2. When the definition of special ideals such as Ulrich or m-full ones involves +an element x, we say that the property is witnessed by x. Note that being such x is a +Zariski-open condition (for the image of x in the vector space m/m2). For more on these +ideals, see [3, 10, 9, 12]. +3 + +Proposition 2.3. Let I be an m-primary ideal. Let e be the Hilbert-Samuel multiplicity of +R. The following are equivalent. +(1) I is Ulrich. +(2) type(I) = e. +Proof. We can assume k is infinite by making the flat extension R → R[t](m,t). Let x ∈ m be +such that ℓ(R/xR) = e. Then ℓ(I/xI) = e. Note that type(I) = ℓ(soc(I/xI)) ≤ ℓ(I/xI) = +e, and equality happens precisely when m(I/xI) = 0, in other words, I is Ulrich. +□ +Proposition 2.4. Let I be an m-primary ideal. +(1) Suppose k is infinite. If I is Ulrich, then it is m-full. +(2) Suppose k is infinite. If I is integrally closed, then it is m-full. +(3) If I is m-full, then it is full. +Proof. (1): We can find a NZD x such that Ix = Im, so Im : x = Ix : x = I. +(2): see [8, Theorem 2.4]. +(3): We have I ⊆ Im : m ⊆ Im : x, from which the assertion is clear. +□ +Proposition 2.5. If I is m-full, witnessed by a NZD x ∈ m. The following are equivalent: +(1) I is Elias. +(2) I = xJ for some Ulrich ideal J. +Proof. Assume I is Elias, witnessed by a NZD x, so Im : x = I. +We will show that +I ⊆ (x). If not, then I contains an element s whose image in R/(x) is in the socle. Thus +sm ⊂ Im ∩ (x) = x(Im : x) = xI, so s ∈ xI : m ⊂ (x), a contradiction. +Since I ⊆ (x) we must have I = xJ for some J. We have Jx = I = Im : x = Jxm : x = +Jm, so J is Ulrich. +Assume (2). Then I is Ulrich and also full by 2.4, so xI : m = mI : m = I = xJ ⊂ (x), +thus I is Elias. +□ +Corollary 2.6. If e = 2 and k is infinite, then I is Elias if and only if I ⊆ (x) for some +NZD x ∈ m. +Proof. Since e = 2, any ideal is either principal or Ulrich, and 2.4 together with 2.5 give +what we want. +□ +Theorem 2.7. The following hold for an m primary ideal I. +(1) If µ(I) < e and type(R/I) ≥ e − 1, then I is Elias. +(2) Assume µ(mI) ≤ µ(I) = e − 1. Then Im is Elias and Im : m = I. +(3) Furthermore, assume R = S/(f) is a hypersurface, here S is a regular local ring of +dimension 2. Let J be an S ideal minimally generated by e elements, one of them is +f. Then JR is Elias. +Proof. By the inequality type(I) ≥ type(R/I), we must have type(I) is e or e − 1. But if +type(I) = e, then µ(I) = e by 2.3, contradiction. +Next, we have: +type(R/Im) = dimk +Im : m +Im +≥ dimk +I +Im = µ(I) ≥ e − 1 +and Im is not Ulrich by assumption. So Im is Elias and type(Im) = e − 1, which by the +chain above implies that Im : m = I. +4 + +For the last part, let I = JR. Then µR(I) = e − 1 and type(R/I) = type(S/J) = e − 1, +and we can apply the first part. +□ +Example 2.8. Let R = k[[t4, t5, t11]] ∼= k[[a, b, c]]/(a4 − bc, b3 − ac, c2 − a3b2). Then m2 is +Elias: one can check directly or note that µ(m) = µ(m2) = 3 = e(R) − 1 and use 2.7. But +m2 is not contained in (x) for any (x). +Example 2.9. Let R = k[[t6, t7, t15]] ∼= k[[a, b, c]]/(a5 − c2, b3 − ac). +Then the Hilbert +function is {1, 3, 4, 5, 5, 6, . . .}, thus m4 is Elias. In this case, m4 ⊆ (a), so m4 is trivially +Elias. +Let R ⊂ S be a finite birational extension. We recall that the conductor of S in R, +denoted cR(S), is R :Q(R) S. +Proposition 2.10. Let R ⊂ S be a finite birational extension. If IS = I (i.e, I is an +S-module) and I is Elias, then I : m ⊆ cR(S). +Proof. Let Q = Q(R). We have R ⊃ I :Q m = IS :Q mS ⊃ (I : m)S, so I : m ⊆ R :Q S = +cR(S) as desired. +□ +Note that if IS = I, then trace(I) ⊆ cR(S). So naturally, one can ask to extend 2.10 as +follows: +Question 2.11. If I is Elias, do we have I : m ⊆ trace(I)? +The answer is no. In Example 2.8 above, Let R = k[[t4, t5, t11]] ∼= k[[a, b, c]]/(a4 − bc, b3 − +ac, c2 − a3b2). One can check that trace(m2) = (a2, ab, b2, c) while m2 : m = m. +Corollary 2.12. Suppose m2 is Elias (e.g., if R has minimal multiplicity) and is integrally +closed. If m2 ⊆ cR(R) then m ⊆ cR(R). +Proof. Apply 2.10 to I = m2. +□ +Corollary 2.13. Assume that the integral closure R is finite. Then the conductor of R in +R is not Elias. A regular trace ideal is not Elias. +Proof. Let c = cR(R). Then c is a R-module, so if it is Elias we would have c : m ⊆ c, +absurd! Any regular trace ideal must contain c, see for instance [3], so it can not be Elias +either by 1.3. +□ +The following is simple but quite useful for constructing Elias ideals from minimal gener- +ators of Ulrich ideals. See the examples that follow. +Proposition 2.14. Let I ⊂ J be regular ideals with J Ulrich. Let x ∈ m be a minimal +reduction of m. Assume that my ̸⊆ xI for any minimal generator of J. Then I is Elias. +Proof. The assumption implies that xI : m ⊆ mJ = xJ ⊂ (x). +□ +Example 2.15. Let R = k[[a1, . . . , an]]/(aiaj)1≤i 0, so the last assertion follows from 2.5. +□ +Theorem 3.3. We have: +(1) eli(R) = 1 if and only if R is regular. +(2) Assume R is Gorenstein, then eli(R) = 2 if and only if e(R) = 2. +(3) Let (A, n) be a Gorenstein local ring of dimension one. Suppose that R = n :Q(A) n +is local. Then eli(R) ≤ 2. +Proof. (1): Assume m is Elias. +To show that R is regular, we can make the extension +R → R[t](m,t) and assume k is infinite. Choose a NZD x ∈ m − m2, we have m2 : x = m, +that is m is m-full witnessed by x. Then 2.5 shows that m ⊂ (x), thus m is principal. +(2): We can assume again by 1.4 that k is infinite. If e = 2, then m2 ⊂ (x) for a minimal +reduction x of m, thus m2 is Elias. Now, suppose m2 is Elias and e ≥ 3, and we need a +contradiction. We first claim that any Ulrich ideal I of R must lie in m2. Take any minimal +reduction x of m. Then Im = xI ⊆ (x), so I ⊂ (x) : m ⊆ (x) + m2 (otherwise the socle of +R′ = R/xR has order 1, impossible as R′ is Gorenstein of length at least 3). As x is general, +working inside the vector space m/m2, we see that I ⊆ m2. +The set of m-primary Ulrich ideals in R is not empty, as it contains high enough powers +of m. Thus, we can pick an element I in this set maximal with respect to inclusion. By the +last claim, I ⊆ m2, and hence I is also Elias by 1.3. Now 2.4 and 2.5 imply that I = xJ for +some NZD x ∈ m, so J is an Ulrich ideal strictly containing I, and that’s the contradiction +we need. +(3): If R = A, then n is Elias by 1.2, hence A is regular by part (1). Thus R is also +regular, and eli(R) = 1. If R strictly contains A, then cA(R) = A :Q(A) R = n, hence +6 + +n ∼= HomA(R, A) ∼= ωR. So n is a canonical ideal of R. On the other hand, as A is not +regular, µA(R) = 2 (dualize the exact sequence 0 → n → A → A/n → 0 and identify R with +n∗ = HomA(n, A)). Thus ℓA(R/n) = 2, so ℓR(R/n) ≤ 2, which forces m2 ⊂ n, and since n is +Elias, so is m2 by 1.3. +□ +Example 3.4. We give some examples of item (3) in the previous Theorem. +First let +A = R[[t, it]] with i2 = −1. Then R = C[[t]]. +Next, let H = ⟨a1, . . . , an⟩ be any symmetric semigroup and b be the Frobenius number +of H. Let A = k[[H]] be the complete Gorenstein numerical semigroup ring of H. Then +R = k[[⟨a1, . . . , an, b⟩]] has Elias index 2, unless if H = ⟨2, 3⟩, in which case eli(R) = 1. +Examples are R = k[[te, te+1, te2−e−1]] for e ≥ 3. For such ring we have type(R) = 2, +e(R) = e, gll(R) = e − 1, ulr(R) = e − 1, yet eli(R) = 2. These examples show that one can +not hope to get upper bounds for gll(R) or ulr(R) just using eli(R). +4. Elias ideals in Gorenstein rings and Auslander index +In this section we focus on Gorenstein rings. Throughout this section, let (R, m, k) be +a local Gorenstein ring of dimension one and I ⊂ R an m-primary ideal. Recall that for +a finitely generated module M, the Auslander δ invariant of M, δ(M) is defined to be the +smallese number s such that there is a surjection Rs ⊕ N → M. The first s such that +δ(R/ms) = 1 is called the Auslander index of R, denoted index(R). +It turns out that Elias ideals are precisely those who quotient has Auslander invariant +one. We collect here this fact and a few others. They are mostly known or can be deduced +easily from results in previous sections, or both. +Proposition 4.1. Let (R, m, k) be a local Gorenstein ring of dimension one and I ⊂ R an +m-primary ideal. We have: +(1) δ(R/I) = 1 if and only if I is Elias. +(2) Suppose R is Gorenstein. +Then I is Elias if and only if for each NZD x ∈ I, +x ∈ m(x : I). +(3) Suppose R is Gorenstein. For a NZD x ∈ I, x : I is Elias if and only if x ∈ mI. In +particular, if x ∈ m2, then x : m is Elias. +(4) I is Elias if and only if 1 ∈ mI−1, where I−1 = R :Q I. If I is Elias, then I ⊆ +m trace(I). +Proof. Part (1) is a special case of a result by Ding, [6, Proposition 1.2] and our definition +of Elias ideal. Part (2) and (3) are special cases of (3) and (4) of 1.3, as in that case (x) is +isomorphic to the canonical module. +Part (4) is [6, 2.4, 2.5], and also follows easily from results above: the first assertion is +just a rewriting of (2). For the second assertion, it follows from the first that I ⊆ mII−1 = +m trace(I). +□ +There have been considerable interest in the following question: +Question 4.2. Given an ideal I with δ(R/I) = 1, when can one say that I ⊂ (x) for some +NZD x ∈ m? +For instance, a conjecture of Ding asks whether index(R) = gll(R) always. From our +point of view, this is of course just a question about Elias ideals and Elias index. Thus, one +immediately obtains the following. +7 + +Corollary 4.3. Let (R, m, k) be a local Gorenstein ring of dimension one and I ⊂ R an +m-primary ideal. +(1) If I contains a NZD x of order 1, then I is Elias if and only if I = (x). +(2) index(R) = eli(R). +(3) index(R) = gll(R) = ulr(R) + 1 if k is infinite and grm(R) is Cohen-Macaulay (this +happens for instance if R is standard graded or if R is a hypersurface). +Proof. For part (1), we apply (5) of Corollary 1.3. Part (2) is trivial from part (1) of 4.1. +Part (3) is [5, Theorem 2.1], [2, Corollary 2.11], and is also a consequence of 3.2. +□ +Example 4.4. (Counter-examples to a result by Ding) In this example, we construct ex- +amples of homogenous Elias ideals that are not inside principal ideals. +Let S = k[[x1 . . . , xn]], and J be a homogenous ideal such that R = S/J is Gorenstein. +Let f ∈ S be an irreducible element of degree at least 2 but lower than the initial degree of +J, and such that the image of f in R is a NZD. Then I = fR : m is Elias by 4.1 but I is +not inside any principal ideal. For by the irreducibility of f, we must have fR : m = (f), +absurd. +This class of examples contradicts Theorem 3.1 in [6], which claims that for I homogenous +in a graded Gorenstein R, δ(R/I) = 1 (equivalently, I is Elias) if and only if I ⊆ (x) for +some x ∈ m. +For concrete examples, one can take S = Q[[a, b]], J = (a3 − b3), and f = a2 + b2. If one +wants algebraically closed field, one can take S = C[[a, b, c]], J is a complete intersection of +two general cubics, and f = a2 + b2 + c2. +The mistake in [6, Theorem 3.1] is as follows. First, one derives that 1 = � ziyi +xi +with +zi ∈ m and yi +xi ∈ I−1 and hence there is i such that deg(ziyi) = deg(xi), which is correct. +Then Ding claimed that there is u ∈ k such that ziyi = uxi. But this is not true. In the +first example above we have z1 = y1 = a, z2 = y2 = b, x1 = x2 = a2 + b2. +Example 4.5. (De Stefani’s counter-example to a conjecture of Ding, revisited) As men- +tioned above, Ding conjectured that index(R) = gll(R) always when R is Gorenstein. De Ste- +fani gives a clever counter-example in [2]. Let S = k[x, y, z](x,y,z), I = (x2−y5, xy2+yz3−z5). +Then index(R) = 5 but gll(R) = 6. We now show how some parts of the proof in [2], which +is quite involved, can be shortened using our results. +We note that since the Hilbert functions of R are (1, 3, 5, 6, 7, 7, 8, 8...) and e(R) = 8, we +get that m5 is Elias by Theorem 2.7. To conclude we need to show that m5 is not contained +in (y) for any NZD y ∈ m. Note that m6 is Ulrich by Hilbert functions. We first show one +can assume ord(y) = 1. Assume m5 ⊂ (y), m5 = yI, then m5 ∼= I. If ord(y) ≥ 2, then +ym3 ⊂ m5 = yI, so m3 ⊂ I. But as mI ∼= m6 is Ulrich, we get m2I ⊂ (x) for some minimal +reduction of m, thus m5 ⊂ m2I ⊂ (x). For the rest, one can follow [2]. +References +[1] W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, 39, +Cambridge, Cambridge University Press, 1993. +[2] A. De Stefani, A counterexample to conjecture of Ding, J. Algebra, 452, pp. 324–337, 2016. +[3] H. Dao, S. Maitra, P. Sridhar, On reflexive and I-Ulrich modules over curves, arXiv:2101.02641, Trans. +of Amer. Math. Soc., to appear. +[4] H. Dao, T. Kobayashi and R. Takahashi, Burch ideals and Burch rings, Algebra Number Theory, +Algebra Number Theory 14 (2020), no. 8, 2121–2150. +8 + +[5] S. Ding, The associated graded ring and the index of a Gorenstein local ring, Proc. Amer. Math. Soc., +120 (4) (1994),1029–1033. +[6] S. Ding, Auslander’s δ-invariants of Gorenstein local rings, Proc. Amer. Math. Soc., 122 (3) (1994), +649–656. +[7] J. Elias, On the canonical ideals of one-dimensional Cohen-Macaulay local rings, Proc. Edinb. Math. +Soc. (2) 59 (2016), no. 1, 77–90. +[8] S. Goto, Integral closedness of complete-intersection ideals, J. Algebra 108 (1987), no. 1, 151–160. +[9] W. Heinzer, L.J Ratliff and D.E. Rush, Basically full ideals in local rings, Journal of Algebra 250 +(2002), 371–396. +[10] C. Huneke and I. Swanson, Integral closures of ideals, rings and modules, London Math. Society Lecture +Note Series 336, Cambridge University Press, 2006. +[11] H-B. Foxby and A. Thorup, Minimal injective resolutions under flat base change, Proc. Amer. Math. +Soc., 67 (1): 27–31, 1977. +[12] J. Watanabe, m-full ideals, Nagoya Math. J. 106 (1987), 101–111. +Hailong Dao, Department of Mathematics, University of Kansas, 405 Snow Hall, 1460 +Jayhawk Blvd., Lawrence, KS 66045 +Email address: hdao@ku.edu +9 + diff --git a/IdAyT4oBgHgl3EQfr_me/content/tmp_files/load_file.txt b/IdAyT4oBgHgl3EQfr_me/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..851693ad7bb86055f6db3c270c1a477450f7e18d --- /dev/null +++ b/IdAyT4oBgHgl3EQfr_me/content/tmp_files/load_file.txt @@ -0,0 +1,519 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf,len=518 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='00569v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='AC] 2 Jan 2023 ELIAS IDEALS HAILONG DAO Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Let (R, m) be a one dimensional local Cohen-Macaulay ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' An m-primary ideal I of R is Elias if the types of I and of R/I are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Canonical and principal ideals are Elias, and Elias ideals are closed under inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' We give multiple characterizations of Elias ideals and concrete criteria to identify them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' We connect Elias ideals to other well-studied definitions: Ulrich, m-full, integrally closed, trace ideals, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Applications are given regarding canonical ideals, conductors and the Auslander index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Introduction Let (R, m) be a local Cohen-Macaulay ring of dimension one and I be an m-primary ideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' We say that I is Elias if the Cohen-Macaulay types of I and R/I coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' From standard facts, principal ideals or canonical ideals are Elias, and we will soon see that this property begets a rather rich and interesting theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Our work is heavily influenced by a nice result in [7], where Elias proves that any ideal ω that lies inside a high enough power of m and such that R/ω is Gorenstein must be a canonical ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Although not stated explicitly there, the proof showed that any ideal that lies in a high enough power of m is Elias, in our sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Another inspiration for the present work is [2], where De Stefani studies, in our language, powers of m that are Elias in a Gorenstein local ring, and gives a counter-example to a conjecture by Ding (see Section 4 for the precise connection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' In this note, we study Elias ideals in depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' They admit many different characterizations, and enjoy rather useful properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' For instance, they are closed under inclusion, and principal or canonical ideals are Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' On the other hand, conductor ideals or regular trace ideals are not Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' When R is Gorenstein, they are precisely ideals such that the Auslander index δ(R/I) is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' We are able to obtain many criteria to check whether an ideal is Elias, using very accessible information such as the minimal number or valuations of generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Combining them immediately gives sharp bounds and information on conductor or canonical ideals, which can be tricky to obtain otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' There are several obvious ways to extend the present definitions and results to higher dimension rings or to modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' However, we choose to focus on the ideals in dimension one case here as they are already interesting enough, and also to keep the paper short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' We hope to address the more general theory in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' We now describe briefly the structure and key results of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' In section 1 we give the formal definition of Elias ideals and prove several key results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='2 contains several equivalent characterizations of Elias ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='3 collects important consequences, for instance that Elias ideals are closed under ideal containment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Also, criteria for Elias ideals using colon ideals are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Next, 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Primary: 13D02, 13H10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Secondary: 14B99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' 1 Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='4 establishes the fundamental change of rings result that are used frequently in the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Section 2 connects Elias ideals to several well-studied class of ideals: Ulrich ideals, m-full ideals, full ideals, integrally closed ideals, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' After some basic observations, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='5), we give Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='7 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='14, which contain concrete ways to recognize Elias ideals using basic information such as number of generators or valuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' We also derive that conductor ideals or regular trace ideals are not Elias (Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' This indicates one of the useful application: if we know, for instance, that m2 is Elias, then the conductor or any regular trace ideal must contains an element of m-adic order 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Given the previous section, it is natural to study the Elias index eli(R), namely the first power of m that is Elias, and we do so in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' The first main result here is Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='2, connecting this index to the generalized L¨oewy length and the regularity of the associated graded ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Next, in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='3, we characterize rings with small indexes: eli(R) = 1 if and only if R is regular, and eli(R) = 2 plus R is Gorenstein is equivalent to e(R) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' We give a large class of non-Gorenstein rings with Elias index 2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Lastly, in Section 4 we focus on the special case of Gorenstein rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' In such situation, we observe that Elias ideals are precisely ones whose quotient has Auslander δ- invariant one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' This immediately allows us to apply what we have to recover old results about the Auslander invariant and Auslander index in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' We give a counter-example to a Theorem by Ding and also revisit a counter-example to a conjecture by Ding given in [2] (Examples 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Acknowledgements: It is a pleasure to thank Juan Elias and Alessandro Di Stefani for helpful comments and encouragements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' The author is partially supported by the Simons Collaboration Grant FND0077558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Elias ideals: definitions and basic results Throughout the paper, let (R, m, k) be Cohen-Macaulay local ring of dimension one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' For a module M, set typeR(M) = dimk Extdim M R (k, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Set Q = Q(R) to be the total ring of fractions of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Set e = e(R), the Hilbert-Samuel multiplicity of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' For an element x ∈ R, the m-adic order of x, denoted ord(x) is the smallest a such that x ∈ ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' The order of an ideal I, denoted ord(I), is the minimum order of its elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' We say that a m-primary ideal I is an Elias ideal if it satisfies type(I) = type(R/I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' We always have type(I) ≥ type(R/I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' The following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' (1) type(I) = type(R/I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' (2) For any NZD x ∈ m, xI : m ⊆ (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' (3) For any NZD x ∈ m, xI : m = x(I : m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' (4) For some NZD x ∈ m, xI : m ⊆ (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' (5) For some NZD x ∈ m, xI : m = x(I : m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' (6) I :Q m ⊆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' (7) K ⊆ m(K :Q I) (assuming R admits a canonical ideal K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Let x be a NZD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Then type(I) = type(I/xI) = dimk xI : m xI ≥ dimk x(I : m) xI = dimk I : m I = type(R/I) 2 Thus, type(I) = type(R/I) if and only if xI : m = x(I : m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Now, xI : m ⊆ (x) is equivalent to xI : m = xJ for some ideal J, as x is a NZD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Rewriting it as xJm ⊆ xI, which is equivalent to Jm ⊆ I, we get J ⊆ I : m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' On the other hand x(I : m) ⊆ xI : m, thus J = I : m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' That establishes the equivalence of first five items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Note that for any NZD x ∈ m, xI : m = x(I :Q m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Thus, (6) is equivalent to (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Let K be a canonical ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Apply HomR(−, K) to the sequence 0 → I → R → R/I → 0, and indentifying HomR(I, K) with K :Q I, we get 0 → K → K :Q I → Ext1 R(R/I, K) = ωR/I → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Since type(I) = µ(K :Q I) and type(R/I) = µ(ωR/I), the equivalence of (7) and (1) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' □ Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' We have: (1) If I is isomorphic to R or the canonical module of R (assuming its existence), then I is Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' (2) If I is Elias, then so is J for any ideal J ⊆ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' (being Elias is closed under inclusion) (3) Let K be a canonical ideal of R and I be an ideal containing K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Then I is Elias if and only if K ⊆ m(K :R I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' (4) Let K be a canonical ideal of R and I be an ideal such that K ⊆ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Then K : I is Elias if and only if K ⊆ mI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' (5) Suppose that I contains a canonical ideal K such that ord(K) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Then I is Elias if and only if I = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' For the first claim, I :Q m ⊂ I :Q I = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' For the second claim, we have J :Q m ⊂ I :Q m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' For (3), first note that K :Q I ⊂ K :Q K = R, so K :Q I = K :R I, and we can use part (7) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' For part (4), note that K : (K : I) = I hence we can apply part (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' For part (5), we again apply part (3): if K ⊊ I, then m(K :R I) ⊆ m2, contradicting ord(K) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' □ The following change of rings result would be used frequently in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Let (R, m) → (S, n) be a local, flat rings extension such that dim S = 1 and S is Noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Then I is an Elias ideal of R if and only if IS is an Elias ideal of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Under the assumption we have typeR(M) typeS/mS(S/mS) = typeS(M ⊗R S) for any finitely generated R-module M (see for instance [11]), thus the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Elias ideals and other special ideals Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Let I be an m-primary ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' I is called Ulrich (as an R-module) if µ(I) = e(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Assuming k is infinite, then I is Ulrich if and only if xI = mI for some x ∈ m (equivalently, for any x ∈ m such that ℓ(R/xR) = e(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' I is called m-full if Im : x = I for some x ∈ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' I is called full (or basically full) if Im : m = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' When the definition of special ideals such as Ulrich or m-full ones involves an element x, we say that the property is witnessed by x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Note that being such x is a Zariski-open condition (for the image of x in the vector space m/m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' For more on these ideals, see [3, 10, 9, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' 3 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Let I be an m-primary ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Let e be the Hilbert-Samuel multiplicity of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' The following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' (1) I is Ulrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' (2) type(I) = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' We can assume k is infinite by making the flat extension R → R[t](m,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Let x ∈ m be such that ℓ(R/xR) = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Then ℓ(I/xI) = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Note that type(I) = ℓ(soc(I/xI)) ≤ ℓ(I/xI) = e, and equality happens precisely when m(I/xI) = 0, in other words, I is Ulrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Let I be an m-primary ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' (1) Suppose k is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' If I is Ulrich, then it is m-full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' (2) Suppose k is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' If I is integrally closed, then it is m-full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' (3) If I is m-full, then it is full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' (1): We can find a NZD x such that Ix = Im, so Im : x = Ix : x = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' (2): see [8, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' (3): We have I ⊆ Im : m ⊆ Im : x, from which the assertion is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' If I is m-full, witnessed by a NZD x ∈ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' The following are equivalent: (1) I is Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' (2) I = xJ for some Ulrich ideal J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Assume I is Elias, witnessed by a NZD x, so Im : x = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' We will show that I ⊆ (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' If not, then I contains an element s whose image in R/(x) is in the socle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Thus sm ⊂ Im ∩ (x) = x(Im : x) = xI, so s ∈ xI : m ⊂ (x), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Since I ⊆ (x) we must have I = xJ for some J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' We have Jx = I = Im : x = Jxm : x = Jm, so J is Ulrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Assume (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Then I is Ulrich and also full by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='4, so xI : m = mI : m = I = xJ ⊂ (x), thus I is Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' If e = 2 and k is infinite, then I is Elias if and only if I ⊆ (x) for some NZD x ∈ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Since e = 2, any ideal is either principal or Ulrich, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='4 together with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='5 give what we want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' □ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' The following hold for an m primary ideal I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' (1) If µ(I) < e and type(R/I) ≥ e − 1, then I is Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' (2) Assume µ(mI) ≤ µ(I) = e − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Then Im is Elias and Im : m = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' (3) Furthermore, assume R = S/(f) is a hypersurface, here S is a regular local ring of dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Let J be an S ideal minimally generated by e elements, one of them is f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Then JR is Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' By the inequality type(I) ≥ type(R/I), we must have type(I) is e or e − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' But if type(I) = e, then µ(I) = e by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='3, contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Next, we have: type(R/Im) = dimk Im : m Im ≥ dimk I Im = µ(I) ≥ e − 1 and Im is not Ulrich by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' So Im is Elias and type(Im) = e − 1, which by the chain above implies that Im : m = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' 4 For the last part, let I = JR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Then µR(I) = e − 1 and type(R/I) = type(S/J) = e − 1, and we can apply the first part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' □ Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Let R = k[[t4, t5, t11]] ∼= k[[a, b, c]]/(a4 − bc, b3 − ac, c2 − a3b2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Then m2 is Elias: one can check directly or note that µ(m) = µ(m2) = 3 = e(R) − 1 and use 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' But m2 is not contained in (x) for any (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Let R = k[[t6, t7, t15]] ∼= k[[a, b, c]]/(a5 − c2, b3 − ac).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Then the Hilbert function is {1, 3, 4, 5, 5, 6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' }, thus m4 is Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' In this case, m4 ⊆ (a), so m4 is trivially Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Let R ⊂ S be a finite birational extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' We recall that the conductor of S in R, denoted cR(S), is R :Q(R) S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Let R ⊂ S be a finite birational extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' If IS = I (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='e, I is an S-module) and I is Elias, then I : m ⊆ cR(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Let Q = Q(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' We have R ⊃ I :Q m = IS :Q mS ⊃ (I : m)S, so I : m ⊆ R :Q S = cR(S) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' □ Note that if IS = I, then trace(I) ⊆ cR(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' So naturally, one can ask to extend 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='10 as follows: Question 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' If I is Elias, do we have I : m ⊆ trace(I)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' The answer is no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' In Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='8 above, Let R = k[[t4, t5, t11]] ∼= k[[a, b, c]]/(a4 − bc, b3 − ac, c2 − a3b2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' One can check that trace(m2) = (a2, ab, b2, c) while m2 : m = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Suppose m2 is Elias (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=', if R has minimal multiplicity) and is integrally closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' If m2 ⊆ cR(R) then m ⊆ cR(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Apply 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='10 to I = m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Assume that the integral closure R is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Then the conductor of R in R is not Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' A regular trace ideal is not Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Let c = cR(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Then c is a R-module, so if it is Elias we would have c : m ⊆ c, absurd!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Any regular trace ideal must contain c, see for instance [3], so it can not be Elias either by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' □ The following is simple but quite useful for constructing Elias ideals from minimal gener- ators of Ulrich ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' See the examples that follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Let I ⊂ J be regular ideals with J Ulrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Let x ∈ m be a minimal reduction of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Assume that my ̸⊆ xI for any minimal generator of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Then I is Elias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' The assumption implies that xI : m ⊆ mJ = xJ ⊂ (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' □ Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' Let R = k[[a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfr_me/content/2301.00569v1.pdf'} +page_content=' , an]]/(aiaj)1≤i T , s ∈ Sh +� +. +(6) +We iteratively execute the above procedures (i.e., Eqs. 2, 3, +4, 5, and 6) until S ∈ ∅, i.e., there is no correct object can +be detected after the attack. The attack for all objects of x is +thus successful, and we output the final generated adversarial +example. +The SCA algorithm is summarized in Algorithm 2. In an +iteration, if SCA fails to attack any pixels of Sh in the inner +loop, SCA will attack the same Sh in the next iteration. During +this process, SCA keeps accumulating perturbations on these +pixels, with the probability score of each pixel in Sh keeping +reducing, until the probability score of every pixel in Sh is +lower than T . By then, Sh is attacked successfully. +Hyperplane +𝑥! +𝑥" +𝑥# +𝛽! +𝛽" +𝑥! +$ +𝑥" +$ +𝑥! +" − 𝑥! +𝑥$ − 𝑥! +Fig. 4: Illustration of the ApproxBoundary and the LinearSolver of +SCA. The black solid line denotes the real decision boundary of the +object detector. Blue points denote adversarial examples that have +not attacked all objects successfully. Red point denotes adversarial +examples that have already attacked all objects successfully. This +figure illustrates two iterations of the attack, x0 → x1 and x1 → x2. +Take x0 → x1 as an example, SCA first generates a dense adver- +sarial example xB +0 (green point) by CWDF and approximated linear +decision boundary β0 (black dash line). Then it uses LinearSolver +to add a sparse perturbation to support x0 to approximate decision +boundary β0 by satisfying β0 = {x′ : wT (x′ − xB +0 ) = 0} until a +valid sparse adversarial example x1 is obtained. The right image +represents the perturbation after applying CWDF. The left image +represents the perturbation after applying the ApproxBoundary and +the LinearSolver. Comparing these two images, it is clear that the +perturbation becomes sparse. +B. Dense Category-wise Attack (DCA) +It is interesting to investigate our optimization problem +Eq. 1 for p = ∞. In this case, our adversarial perturbation +generation procedure is based on PGD [36] and is called Dense +Category-wise Attack (DCA) since it generates dense perturba- +tions compared with SCA. Note that our DCA framework can +also be based on other adversarial attacks, e.g., FGSM [37]. +We propose three methods based on DCA, i.e., DCA-G, DCA- +L, and DCA-S, depending on the targeted region. They are +described in details next. +1) DCA on Global Region (DCA-G): Given an input image +x and category-wise target pixel sets S, DCA applies two +iterative loops to generate adversarial perturbations. In each +inner loop iteration j, DCA first computes the total loss of all +pixels in target pixel set Sj corresponding to each available +category Cj with � +s∈Sj CE (f(x, s), Cj), where CE(·,·) is +the traditional cross-entropy loss. Then, it computes local +adversarial gradient gj of the loss with respect to the current +image x and normalize it with L∞ norm: +gj = +∇x +� +s∈Sj CE (f(x, s), Cj) +∥∇x +� +s∈Sj CE (f(x, s), Cj)∥∞ +. +After that, DCA adds up all gj to generate a total adversarial +gradient G. In an outer loop iteration p, DCA computes the +global perturbation (GP) by applying sign operation to the + +6 +Algorithm 2 Sparse Category-wise Attack (SCA) +Input: image x, S, max iter outer, max iter inner +Output: adversarial example x∗, perturbation r +Initialize: x1 ← x, p ← 1 +while S ̸∈ ∅ and p ≤ max iter outer do +Compute Sh with Eq. (2) and xp, q ← 0, xp,q ← xp; +while q ≤ max iter inner and Sh /∈ ∅ do +xB +p,q = CWDF(xp,q, Sh, T ) /*See Alg. 1*/ +w = ApproxBoundary(xB +p,q, xp,q, Sh) /*See Eq. 4*/ +xadv +p,q = LinearSolver(xp,q, w, xB +p,q) +/*See Eq. 5*/ +Sh = RemovePixels(xp,q, xadv +p,q , Sh, T ) +/*See Eq. 6*/ +q = q + 1 +end while +xp+1 ← xadv +p,q−1, p = p + 1 +end while +return x∗ = xp, r = xp − x +total adversarial gradient G [36] as follows, +GP = +ϵ +max iter · sign(G), +(7) +where max iter denotes the maximum number of iterations +of the outer loop and the term +ϵ +max iter is perturbation size in +each iteration [37]. At the end of the outer loop, DCA uses +RemovePixels of Eq. 6 to remove from S the target pixels +that have already been attacked successfully on the perturbed +image. Since DCA works on the global (whole) region in the +original image, we call this method, DCA-G, in short. The +pseudo-code of DCA-G is described in Algorithm 3. +2) DCA on Local Region (DCA-L): As we discussed be- +fore, runner-up pixels are usually located near keypoints in the +heatmap. They are the most important pixels for the object. In +addition, perturbations in the background from an image may +not impact the detection results since they are not related to +the objects. To attack these objects, we only need to attack +these significant pixels (i.e., runner-up and keypoints). +To fulfil the goal, we can use attack mask MASKL to restrict +perturbation around detected objects in order to disallow per- +turbation in the background. Attack mask MASKL is generated +from S with the following process, called the GenerateMask +procedure. MASKL is initialized with a zero matrix of the +same size as the input image. We locate each pixel point +s ∈ Si, ∀i ∈ [k], on the input image and set the same location +point of MASKL to be 1. After processing all s, for each pixel +s = 1 in MASKL, we set all the points in the square box +centered at s’s with the size of radius (side length) R∗ to +be 1 too. The resulting MASKL is used as the local attack +mask. The attack performance depends on the value of R∗. We +will quantitatively analyze this dependence in the experimental +studies reported in Section V-C. +A local perturbation (LP) is obtained by applying MASKL +Algorithm 3 Dense Category-wise Attack (DCA) +Input: image x, S, C, ϵ, max iter, T +Output: adversarial example x∗, perturbation r +Initialize: x1 ← x, p ← 1 +while S ̸∈ ∅ and p ≤ max iter do +G ← 0, j ← 1 +while j ≤ k do +if Sj ̸= ∅ then +gj = +∇xp +� +s∈Sj CE (f(xp,s), Cj) +∥∇xp +� +s∈Sj CE (f(xp,s), Cj)∥∞ , G ← G+gj +end if +j ← j + 1 +end while +GP ← +ϵ +max iter · sign(G) +if DCA-G then +xp+1 ← xp + GP +/*Refer to Eq. 7*/ +else if DCA-L then +xp+1 ← xp + GP ∗ MASKL +/*Refer to Eq. 8*/ +else if DCA-S then +xp+1 ← xp + GP ∗ MASKS +/*Refer to Eq. 9*/ +end if +for Si in S do +Si = RemovePixels(xp, xp+1, Si, T ) /*Refer to Eq. 6*/ +end for +p ← p + 1 +end while +return x∗ = xp, r = xp − x +on the global perturbation (GP): +LP = GP ∗ MASKL. +(8) +After generating LP, we update S by removing the points that +have been attacked successfully with RemovePixels (Eq. 6). +The pseudo-code of DCA-L is shown in Algorithm 3. +3) DCA on Semantic Region (DCA-S): Using DCA on +the local region may not get a perfect perturbation around +objects because run-up pixels may not be all around objects +since some of them may not represent semantic information +of objects. In addition, DCA-L uses a regular square mask +around each pixel in S,∀i ∈ [k], which may not precisely select +important perturbation signals from the global perturbation, as +Fig. 5 shows. To address this issue, we propose the DCA-S +method that applies DCA to the semantic region of the image. +Since convolutional layers of the CNN-based anchor-free +model contain abundant information, especially the spatial +information [38], [39], which can indicate which regions of +an input image have a higher response to the output of the +classifier. This characteristic demonstrates that the generated +perturbation will be more effective if we only attack pixels that +are related to the semantic region of objects. A natural strategy + +7 +Original image +Masked image by DCA-L +Masked image by DCA-S +R*= 50 +R*= 120 +Fig. 5: Comparison of masks from DCA-L and DCA-S. A proper +radius R∗ of the DCA-L mask is hard to be determined. A undersized +R∗ (i.e., R∗=50) will miss some important regions (e.g., human and +dog faces) while an oversized R∗ (i.e., R∗=120) will introduce more +useless local perturbations (e.g., perturbations on background), which +may lead to worse results. In contrast, the mask generated by DCA-S +is more precise. +for getting this semantic region is to extract the feature map +based on the convolutional layer outputs [40]–[42]. Shallower +layers usually keep much spatial information, leading to a +sparse and discontinuous high response region in the feature +map. On the other hand, deeper layers contain more global +information due to their larger receptive fields, resulting in a +large and continuous high response region. +In prior work [39], the last convolutional layer is used +to adopt the key region. Discarding spatial information from +shallower layers makes this method hard to accurately target +locations of small objects. Inspired by [40]–[42], we propose +a multi-layer semantic information region selection (MSIRE) +method, to be described in detail next, to extract and integrate +semantic information from both deep and shallow layers. By +integrating the gradient information from these layers, we can +construct a semantic mask that contains the most informative +semantic region of objects, which is then combined with the +global perturbation (GP) to generate the final perturbation. +Now we describe the MSIRE method. Let L = {li}n +i=1 be +the set of layers that we consider to extract key region, where +li is the i-th layer containing feature map activations hi. In +our experiments, n is set to 4. For layer li, we calculate and +sum the gradient of the score for category Cj of all pixels in +target pixel set Sj with respect to its feature map activation +hi, and sum them up across all categories, +Gi = +� +j∈[k] +∂ � +s∈Sj fj(x, s) +∂hi +Then normalize gradient Gi to [0,1] and upsample to the size +of image x: +ˆGi = Φ +� +Gi − min(Gi) +max(Gi) − min(Gi) +� +, +where Φ(·) denotes the operation of upsampling and min(Gi) +and max(Gi) are the minimal and maximal value of elements +in Gi, respectively. The mask corresponding to layer li can be +obtained using a threshold Ts: +maski = δ( ˆGi > Ts), +where δ(·) is an impulse response that turns a pixel greater +than Ts to 1, otherwise to 0. Combining maski, ∀i ∈ [n] as +MASKS = +n +� +i=1 +maski, +where MASKS denotes the final semantic region attack mask. +After obtaining MASKS, we combine it with the global +perturbation (GP) to get a semantic region guided perturbation +(SP) as follows, +SP = GP ∗ MASKS. +(9) +Like its siblings, after generating the perturbation, DCA-S +updates S with RemovePixels (Eq. 6). The pseudo-code of +DCA-S is shown in Algorithm 3. +V. EXPERIMENTS +In this section, we will evaluate the performance of the +proposed adversarial attack (i.e., SCA, DCA-G, DCA-L, and +DCA-S) for both object detection and human pose estimation +based on anchor-free detector CenterNet [20]. +A. Experimental Settings +1) Detectors and Datasets: For object detection, we evalu- +ate our proposed attack on two public datasets: PascalVOC +[32] and MS-COCO [33]. We use the two pre-trained de- +tectors (CenterNet with two different backbones: ResNet18 +(R18) [43] and DLA34 [44]) from [20] in our experiments. +These two detectors are pre-trained on the training set of +PascalVOC (including PascalVOC 2007 and PascalVOC 2012) +and MS-COCO 2017, respectively. Our adversarial examples +are generated on the test set of PascalVOC 2007, which +consists of 4,592 images and 20 categories, and the validation +set of MS-COCO 2017, which contains 5,000 images and +80 object categories. For human pose estimation, we use the +pre-trained detector (CenterNet with DLA34 [44] backbone) +from [20] that is trained on the training set of MS-COCO +Keypoints 2017. Our adversarial examples are generated on its +validation set, which contains 5,000 images and 17 categories +of keypoints for humans. +2) Evaluation Metrics: We evaluate the performance of +both white-box and black-box attacks with the following +metrics. +• The attack performance is evaluated by computing the +decreased percentage of mean average precision (mAP), +referred to as the mAP Score Degradation Ratio (ASR), +which is defined as, +ASR = 1 − mAPattack +mAPclean +, +where mAPattack denotes the mAP of the targeted object +detector on adversarial examples, and mAPclean denotes +the mAP on clean samples. Higher ASR means better +white-box attack performance. +• We also use the L0 and L2 norms of perturbation r, PL0 +and PL2, respectively, to quantify the perceptibility of +the adversarial perturbation. PL0 = ∥r∥0 quantifies the +proportion of perturbed pixels. A lower value means that +fewer image pixels are perturbed. For PL2 = ∥r∥2, a +greater value usually signifies a more perceptible pertur- +bation to humans. +• For black-box attacks, transferability of adversarial ex- +amples generated with other detectors and tested with the + +8 +target detector is used to measure the attack performance. +More specifically, the attack performance of block-box +attacks is measured by the ratio, referred to as the +Attack Transfer Ratio (ATR), of the ASR on the target +model, ASRtarget, to the ASR on the generating model, +ASRorigin: +ATR = ASRtarget +ASRorigin +, +Higher ATR means better transferability. +In our experiments, for object detection, black-box ad- +versarial examples are generated on CenterNet with +one backbone (R18 or DLA34) and tested on Cen- +terNet with a different backbone (DLA34, R18, and +ResNet101 (R101)). We also test these adversarial ex- +amples on other detectors, including anchor-free (Cor- +nerNet [26] with backbone Hourglass [45]) and anchor- +based detectors (Faster-RCNN [19] and SSD [21]). For +human pose estimation, all adversarial examples are gen- +erated on CenterNet with backbone DLA34 and tested on +CenterNet with backbone Hourglass. +Additionally, we follow [46] to simulate a real-world at- +tack transferring scenario, wherein generated adversarial +examples are saved in the JPEG format and then reloaded +to attack the target model. In real-world applications, +images are usually saved in a compression format. JPEG +is a most commonly used lossy compression standard for +images. The transferability test in this way requires ad- +versarial examples to be robust to the JPEG compression +like in most real-world applications. +3) Comparison Methods and Implementation Details: For +object detection, since there is no existing attack dedicated to +anchor-free detectors, we use an existing state-of-the-art attack +designed for attacking anchor-based detectors, i.e., DAG [13] +with VGG16 [47] backbone for attacking Faster-RCNN (FR) +as the attack to compare with. For human pose estimation, we +compare with existing methods proposed in [48] for attacking +human pose estimation systems, called FHPE and PHPE, +which are based on FGSM and PGD, respectively. +Our methods are implemented with Python 3.6 and Pytorch +1.1.0. All the experiments are conducted with an Intel Core +i9-7960 and an Nvidia GeForce GTX-1070Ti GPU. We set +max iter outer, max iter inner, and max iter to 50, 20, and +30, respectively. The default values of R∗, Ts, T , and ϵ are +60, 0.5, 0.1, and 5% of the maximum value of the pixels in an +image, respectively. All input images are resized to 512×512. +B. Experimental Results on Object Detection +1) White-Box Attack Performance: Table I shows the white- +box attack results on both PascalVOC and MS-COCO. +• For PascalVOC, we can see that DCA-G with the DLA34 +backbone achieves the best ASR. SCA with both back- +bones produces much smaller perturbations than other +methods and DCA-G with the R18 is 14 times faster +than DAG. Furthermore, while the ASR performance +of both DCA-L and DCA-S are very close to that of +DAG, DCA-L and DCA-S produce smaller perturbations +Data +Method +Network +mAPclean +mAPattack +ASR +PL2 (×10−3) +PL0 +Time (s) +PascalVOC +DAG +FR +0.70 +0.050 +0.92 +3.20 +0.990 +9.8 +SCA +R18 +0.71 +0.060 +0.91 +0.41 +0.002 +32.5 +SCA +DLA34 +0.79 +0.110 +0.86 +0.44 +0.003 +117.3 +DCA-G +R18 +0.71 +0.070 +0.90 +5.20 +0.990 +0.7 +DCA-G +DLA34 +0.79 +0.050 +0.94 +5.10 +0.990 +1.2 +DCA-L +R18 +0.71 +0.080 +0.89 +2.70 +0.320 +1.6 +DCA-L +DLA34 +0.79 +0.080 +0.90 +2.80 +0.320 +3.1 +DCA-S +R18 +0.71 +0.070 +0.90 +2.40 +0.260 +2.2 +DCA-S +DLA34 +0.79 +0.060 +0.92 +2.20 +0.280 +4.0 +MS-COCO +DAG +FR +0.35 +0.040 +0.89 +5.00 +0.990 +20.4 +SCA +R18 +0.28 +0.027 +0.91 +0.48 +0.004 +50.4 +SCA +DLA34 +0.37 +0.030 +0.92 +0.49 +0.007 +216.0 +DCA-G +R18 +0.28 +0.002 +0.99 +5.80 +0.990 +2.4 +DCA-G +DLA34 +0.37 +0.002 +0.99 +5.90 +0.990 +4.9 +DCA-L +R18 +0.28 +0.006 +0.98 +3.20 +0.380 +3.4 +DCA-L +DLA34 +0.37 +0.008 +0.98 +3.30 +0.390 +6.3 +DCA-S +R18 +0.28 +0.003 +0.99 +2.80 +0.300 +4.7 +DCA-S +DLA34 +0.37 +0.004 +0.99 +3.00 +0.310 +8.5 +TABLE I: White-box performance comparison. “Time” is the av- +erage time to generate an adversarial example. The best results are +shown in bold. +and have lower time complexity than DAG in generating +adversarial examples. +• For MS-COCO, both DCA-G and DCA-S achieve the +highest ASR, 99.0%, which is significantly higher than +that of DAG. Like on PascalVOC, the ASR performance +of SCA with both R18 and DLA34 is in the same +ballpark as that of DAG, SCA produces much smaller +perturbations than other methods in terms of both PL2 +and PL0, and DCA-G is the fastest in generating an +adversarial example. +In general, both SCA and DCA achieve state-of-the-art attack +performance. Specifically, DCA can achieve high ASR and +low time complexity, while SCA can produce much smaller +perturbations without degrading ASR much. We can see that +PL0 of SCA is lower than 1%, implying that SCA can success- +fully attack detectors by perturbing only a small percentage +of pixels of the original image. Comparing DCA-L and DCA- +G, while it may slightly decrease the performance of ASR, +DCA-L using a local region significantly decreases the size +of perturbation. Comparing DCA-S and DCA-L, DCA-S not +only improves the performance of ASR but also reduces the +perturbation size in terms of both PL2 and PL0. This implies +that the DCA on the semantic region is better than it on +the local region. A qualitative comparison between DAG and +our methods is shown in Fig. 6. It is clear that perturbations +generated by SCA and DCA-based methods are hard to be +perceived by humans. +We also show in Fig. 7 the Average Precision (AP) of +each object category on clean inputs and adversarial examples +generated by our methods on PascalVOC with Centernet using +both R18 and DLA34 backbones. The AP drops by a roughly +similar percentage for all the object categories. Fig. 8 shows +the AP and Average Recall (AR) of SCA and DCA-based +methods on MS-COCO with CenterNet. We can notice that +small objects are more vulnerable to adversarial examples +than bigger ones. One possible explanation is that bigger +objects usually have more keypoints than smaller objects on +the heatmap and our algorithms need to attack all of them. +2) Black-Box Attack Performance: In evaluating black-box +attack performance, adversarial examples are generated with +Centernet using the R18 or DLA34 backbone for our proposed +methods or with Faster-RCNN for DAG and tested with other + +9 +Fig. 6: Qualitative comparison between DAG and our methods on the object detection task. Three examples are presented. Column 1: +Detection results of clean inputs. Column 2: DAG’s attack results and perturbations. Column 3: SCA’s attack results and perturbations. +Columns 4 - 6: the attack results and perturbations of DCA-G, DCA-L, and DCA-S, respectively. The perturbations are magnified by a +factor of 30 for better visibility. +AP for aeroplane +AP for bicycle +AP for bird +AP for boat +AP for bottle +AP for bus +AP for car +AP for cat +AP for chair +AP for cow +AP for diningtable +AP for dog +AP for horse +AP for motorbike +AP for person +AP for pottedplant +AP for sheep +AP for sofa +AP for train +AP for tvmonitor +mAP +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 + Clean + SCA + DCA-G + DCA-L + DCA-S +AP for aeroplane +AP for bicycle +AP for bird +AP for boat +AP for bottle +AP for bus +AP for car +AP for cat +AP for chair +AP for cow +AP for diningtable +AP for dog +AP for horse +AP for motorbike +AP for person +AP for pottedplant +AP for sheep +AP for sofa +AP for train +AP for tvmonitor +mAP +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Clean + SCA +DCA-G + DCA-L +DCA-S +Fig. 7: The AP of each object category on clean inputs and adver- +sarial examples generated by SCA and DCA-methods on PascalVOC +with CenterNet using R18 (left) and DLA34 (right) backbones on +PascalVOC. +object detection models. Table II and Table III show the +black-box attack performance on PascalVOC and MS-COCO, +respectively. +• Attack transferability on PascalVOC. From Table II, we +can see that adversarial examples generated by our meth- +ods can successfully transfer to not only CenterNet with +AP[IoU=0.50:0.95] +all +AP [IoU=0.50] +all +AP [IoU=0.75] +all +AP [IoU=0.50:0.95] +small +AP [IoU=0.50:0.95] +medium +AP [IoU=0.50:0.95] +large +AR [IoU=0.50:0.95] +all +AR [IoU=0.50:0.95] +small +AR [IoU=0.50:0.95] +medium +AR [IoU=0.50:0.95] +large +0.0 +0.2 +0.4 +0.6 +0.8 + Clean + SCA + DCA-G + DCA-S + DCA-L +AP[IoU=0.50:0.95] +all +AP [IoU=0.50] +all +AP [IoU=0.75] +all +AP [IoU=0.50:0.95] +small +AP [IoU=0.50:0.95] +medium +AP [IoU=0.50:0.95] +large +AR [IoU=0.50:0.95] +all +AR [IoU=0.50:0.95] +small +AR [IoU=0.50:0.95] +medium +AR [IoU=0.50:0.95] +large +0.0 +0.2 +0.4 +0.6 +0.8 + Clean + SCA + DCA-G + DCA-L + DCA-S +Fig. 8: The AR (top four) and AP (bottom six) performance of +different sizes of objects on clean inputs and adversarial examples +generated SCA and DCA-based methods on MS-COCO with Cen- +terNet using R18 (left) and DLA34 (right) backbones. +different backbones but also completely different types of +object detectors such as Faster-RCNN and SSD. Specif- +ically, we can see that SCA with the DLA34 backbone +achieves the best ATR and SCA with the R18 backbone +achieves similar ATR to the DCA-based methods with +both the R18 and DLA34 backbones. on the other hand, + +10 +20 +40 +60 +80 +100 +120 +0.90 +0.95 +1.00 +ASR +R* +2.5 +3.0 +3.5 +4.0 +Time (s) +(a) +20 +40 +60 +80 +100 +120 +0.2 +0.3 +0.4 +0.5 +0.6 +PL0 +R* +1.0 +2.0 +3.0 +4.0 +5.0 +PL2(´10-3) +(b) +0.3 +0.5 +0.7 +0.9 +0.80 +0.85 +0.90 +ASR +Ts +(c) +0.1 +0.2 +0.3 +0.4 +0.5 +0.50 +0.60 +0.70 +0.80 +0.90 +ASR +T +(d) +0.05 +0.10 +0.15 +0.20 +0.90 +0.95 +1.00 +ASR +0 +(e) +Fig. 9: Sensitivity analysis of hyperparameters. +From +To +R18 +DLA34 +R101 +Faster-RCNN +SSD +mAP +ATR +mAP +ATR +mAP +ATR +mAP +ATR +mAP +ATR +Clean +0.71 +– +0.77 +– +0.79 +– +0.71 +– +0.74 +– +DAG +0.65 +0.09 +0.75 +0.03 +0.72 +0.10 +0.05 +– +0.72 +0.03 +R18-SCA +0.06 +– +0.62 +0.21 +0.61 +0.25 +0.55 +0.25 +0.70 +0.10 +DLA34-SCA +0.42 +0.47 +0.11 +– +0.53 +0.38 +0.44 +0.44 +0.62 +0.19 +R18-DCA-G +0.07 +– +0.62 +0.22 +0.65 +0.20 +0.61 +0.16 +0.72 +0.03 +DLA34-DCA-G +0.50 +0.31 +0.05 +– +0.62 +0.23 +0.53 +0.27 +0.67 +0.10 +R18-DCA-L +0.08 +– +0.61 +0.23 +0.63 +0.23 +0.55 +0.25 +0.68 +0.09 +DLA34-DCA-L +0.48 +0.35 +0.06 +– +0.60 +0.24 +0.51 +0.37 +0.66 +0.12 +R18-DCA-S +0.07 +– +0.65 +0.17 +0.65 +0.17 +0.61 +0.16 +0.73 +0.01 +DLA34-DCA-S +0.52 +0.29 +0.06 +– +0.62 +0.22 +0.57 +0.21 +0.70 +0.06 +TABLE II: Black-box attack results on the PascalVOC dataset. +From: the leftmost column denotes the models which adversarial +examples are generated from. To: the top row denotes the target +models that adversarial examples transfer to (i.e., are tested on). “–” +means the value is unavailable (not defined). +From +To +R18 +DLA34 +R101 +CornerNet +mAP +ATR +mAP +ATR +mAP +ATR +mAP +ATR +Clean +0.29 +– +0.37 +– +0.37 +– +0.43 +– +DAG +0.24 +0.19 +0.33 +0.12 +0.31 +0.18 +0.40 +0.08 +R18-SCA +0.02 +– +0.27 +0.30 +0.24 +0.39 +0.35 +0.20 +DLA34-SCA +0.07 +0.82 +0.03 +– +0.09 +0.82 +0.12 +0.78 +R18-DCA-G +0.00 +– +0.29 +0.21 +0.28 +0.25 +0.38 +0.12 +DLA34-DCA-G +0.10 +0.67 +0.00 +– +0.12 +0.69 +0.13 +0.72 +R18-DCA-L +0.01 +– +0.27 +0.28 +0.27 +0.28 +0.36 +0.17 +DLA34-DCA-L +0.10 +0.67 +0.01 +– +0.12 +0.69 +0.13 +0.71 +R18-DCA-S +0.00 +– +0.31 +0.16 +0.29 +0.22 +0.39 +0.09 +DLA34-DCA-S +0.12 +0.59 +0.00 +– +0.13 +0.64 +0.13 +0.72 +TABLE III: Black-box attack results on the MS-COCO dataset. +From: the leftmost column denotes the models which adversarial +examples are generated from. To: the top row denotes the target +models that adversarial examples transfer to (i.e., are tested on). “–” +means the value is unavailable (not defined). +adversarial examples generated by DAG with Faster- +RCNN have a much poorer transferability in attacking +CenterNet and SSD than our proposed methods, esp. SCA +with the DLA34 backbone. In addition, we can also see +that our proposed methods have better transferability in +attacking CenterNet and Faster-RCNN than in attacking +SSD, which implies that SSD is less reliable on highly +informational points in the CenterNet-generated heatmap +than Faster-RCNN and CenterNet with a different back- +bone. +• Attack Transferability on MS-COCO. From Table III, we +can see that adversarial examples generated with our +proposed methods with the DLA34 backbone have sig- +nificantly higher ATR than other methods including our +proposed methods with the R18 backbone. Our proposed +methods with the DLA34 backbone can attack not only +CenterNet with different backbones but also CornerNet. +Like on PascalVOC, DAG has a poor transferability in +attacking CenterNet and CornerNet on MS-COCO too. +C. Sensitivity Analysis of Hyperparameters +In our algorithms, we have four hyperparameters that need +to be tuned. To study their sensitivity, we generate adversarial +examples by attacking CenterNet with the R18 backbone +on MS-COCO for evaluating R∗ and on PascalVOC for +evaluating Ts, T , and ϵ. Then we report the performance with +respect to each hyperparameter. More details are provided as +follows. +1) Sensitivity Analysis of R∗: In DCA-L, we need to use +R∗ to control the size of extracted local regions. It is clear +that R∗ correlates with the attack performance ASR, PL0, +PL2, and time consumption. We show these relationships in +Fig. 9 (a) and (b). From these figures, we can draw three +conclusions. First, the ASR of DCA-L correlates positively +with R∗ when R∗ is less than 60. However, ASR is stable or +slightly decreased afterward. The reason is that the oversized +mask may introduce more useless perturbation, resulting in +a worse effect. Second, PL0 and PL2 of the perturbation +correlates positively with R∗. This is because higher R∗ means +bigger attack masks. Finally, the average attack time of DCA- +L correlates negatively with R∗ when R∗ is lower than 48 and +becomes stable afterward. +2) Sensitivity Analysis of Ts: In DCA-S, hyperparameter Ts +is used to select semantic regions or pixels that contain more +informative gradients. Fig. 9 (c) shows attack performance +ASR with different Ts. We can see that the best ASR occurs +when Ts is 0.5. ASR decreases when Ts increases from 0.5, +which can be explained by the fact that a too large value of +Ts makes the mask too small, resulting in many useful pixels +related to objects excluded from the mask and thus degraded +attack performance. +3) Sensitivity Analysis of T : The default value of the visual +threshold is 0.3 in the Centernet. We use DCA-G to evaluate +the sensitivity of T . The results are reported in Fig. 9 (d). We +can see that ASR shows an obvious decreasing trend when T +increases. In particular, the decreasing trend becomes sharper +when T is greater than 0.3, which can be explained that more +runner-up points (pixels) are included in the target pixel sets if +T is below 0.3, resulting in improved attacking performance. +We can also see that the performance gap is more than 0.1 +when T changes from 0.3 to 0.1. This implies that runner-up +points have a significant impact on the attack performance. +On the other hand, if T is higher than 0.3, some keypoints +are excluded from the target pixel sets and thus unattacked, +leading to faster degraded ASR. +4) Sensitivity Analysis of ϵ: We analyze the sensitivity of +the amplitude of the perturbation ϵ with DCA-G. The results +are reported in Fig. 9 (e). A large ϵ means a larger perturbation + +11 +Data +Method +Network +mAPclean +mAPattack +ASR +PL2 (×10−3) +PL0 +Time (s) +MS-COCO +Keypoints +FHPE +DLA34 +0.53 +0.31 +0.42 +0.33 +0.990 +0.6 +PHPE +DLA34 +0.53 +0.11 +0.79 +0.20 +0.990 +4.2 +SCA +DLA34 +0.53 +0.03 +0.93 +0.17 +0.002 +130.1 +DCA-G +DLA34 +0.53 +0.00 +1.00 +0.62 +0.990 +11.2 +DCA-L +DLA34 +0.53 +0.04 +0.92 +0.45 +0.250 +12.7 +DCA-S +DLA34 +0.53 +0.03 +0.94 +0.36 +0.180 +15.2 +TABLE IV: White-box attack results on the MS-COCO Keypoints +dataset. “Time” is the average time to generate an adversarial exam- +ple. +To +From +DLA34 +Clean +FHPE +PHPE +SCA +DCA-G +DCA-L +DCA-S +Hourglass +mAP +0.58 +0.50 +0.36 +0.18 +0.26 +0.23 +0.24 +ATR +– +0.33 +0.48 +0.74 +0.55 +0.66 +0.63 +TABLE V: Black-box attack results on the MS-COCO Keypoints +dataset. To: the leftmost column denotes the target models that +adversarial examples transfer to (i.e., are tested on). From: the top +row denotes the models which adversarial examples are generated +from. +size. We expect ASR increases with increasing ϵ. This is +confirmed by the experimental results shown in the figure. +D. Experimental Results on Human Pose Estimation +In the MS-COCO Keypoints dataset, there are 17 categories +of keypoints for humans. We attack target pixels from each +category and report the attacking performance next. +1) White-Box Attack: The white-box attack results on the +MS-COCO Keypoints dataset are summarized in Table IV. We +can see that our DCA-G has the highest ASR, which is 1. This +means DCA-G has attacked all the testing images successfully. +Since FHPE and PHPE are based on the conventional FGSM +and PGD methods and thus very simple in terms of complexity, +we expect them to have less attack time, which is confirmed by +the results shown in Table IV. However, these two comparison +methods have much lower ASR than our proposed methods. +All of our proposed methods outperform these two baselines +on the ASR metric. +On the other hand, it is clear that our SCA has the lowest +PL0 and PL2 values, which is consistent with the observation +on the object detection task. In addition, our DCA-L and DCA- +S also have lower PL0 and PL2 values than DCA-G, which is +due to the role of masks used in both DCA-L and DCA-S. A +qualitative comparison between the comparison methods and +our proposed methods is shown in Fig. 10. +2) Black-Box Attack and Transferability: The black-box +attack results are reported in Table V. We can see that our +SCA achieves the best transferability, which is the same as +on the object detection task. Furthermore, all of our proposed +methods outperform the baseline methods, which implies that +our category-wise attack can improve the transferability of +adversarial examples. +VI. CONCLUSION +In this work, we propose the first adversarial attack on +anchor-free detectors. It is a category-wise attack that attacks +important pixels of all instances of a category simultaneously. +Our attack manifests in two forms, sparse category-wise attack +(SCA) and dense category-wise attack (DCA), when minimiz- +ing the L0 and L∞ norm-based perturbations, respectively. +For SCA, it can generate sparse-imperceptible adversarial +samples. For DCA, we further provide three variants, DCA- +G, DCA-L, and DCA-S, to enable a flexible selection of a +specific attacking region: the global region, the local region, +and the semantic region, respectively, to improve the attack +effectiveness and efficiency. Our experiments on large-scale +benchmark datasets including PascalVOC, MS-COCO, and +MS-COCO Keypoints indicate that the proposed methods +achieve state-of-the-art attack performance and transferability +on both object detection and human pose estimation tasks. +Limitations. There are two limitations in our proposed +methods. The first limitation is that SCA has significantly +higher time complexity since its Algorithm 2 includes CWDF +and Linear Solver procedures. Our experimental results con- +firm it: SCA needs a much longer time to attack an image for +both object detection and human pose estimation. The second +limitation is that the transferability of our proposed methods +on attacking SSD is much lower than that of our methods on +attacking other models. +Future Work. First, we will try to address the aforemen- +tioned limitations of our proposed methods. Second, we plan +to incorporate a scheme to automatically determine the hyper- +parameters. Third, we will try to develop effective defenses +against the proposed attacks, which is important for practical +applications of anchor-free detectors. +APPENDIX A +ALGORITHM OF LINEARSOLVER +The LinearSolver algorithm is shown in Algorithm 4. In +each iteration, we project towards only one single coordinate +of w. If projecting x to a specific direction does not provide a +solution, it will be ignored in the next iteration. More details +can be found in [11]. Note that the projection operator of Q +in Algorithm 4 controls the pixel values between 0 and 255. +Algorithm 4 LinearSolver +Input: image x, normal vector w, boundary point xB, pro- +jection operator Q +Output: perturbated point xadv +Initialize: x0 ← x, i ← 0, H = {} +while wT (xi − xB) ̸= 0 do +r ← 0 +d ← arg maxj∈H |wj| +rd ← |wT (xi−xB)| +|wd| +· sign(wd) +x(i+1) ← Q(xi + r) +H ← H ∪ {d} +i ← i + 1 +end while +return xadv ← xi +REFERENCES +[1] Y. Xiang et al., “Rmbench: Benchmarking deep reinforcement learning +for robotic manipulator control,” arXiv preprint arXiv:2210.11262, 2022. + +12 +Fig. 10: Qualitative comparison between the comparison methods and our proposed methods on the human pose estimation task. Three +examples are presented. 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Jain et al., “On the robustness of human pose estimation,” in CVPR +Workshops, 2019, pp. 29–38. +Yunxu Xie is currently a postgraduate at the School +of Computer Science, Chengdu University of of +Information Technology. His research directions are +adversarial attack, deep learning, and computer vi- +sion. +Dr. Shu Hu is a Postdoc at Carnegie Mellon Uni- +versity. He received his Ph.D. degree in Computer +Science and Engineering from University at Buffalo, +the State University of New York (SUNY) in 2022. +He received his M.A. degree in Mathematics from +University at Albany, SUNY in 2020, and M.Eng. +degree in Software Engineering from University of +Science and Technology of China in 2016. His +research interests include machine learning, digital +media forensics, and computer vision. +Dr. Xin Wang (SM’2020) is a research affiliate +at University at Buffalo, State University of New +York. He received his Ph.D. degree in Computer +Science from the University at Albany, State Uni- +versity of New York in 2015. His research interests +are in machine learning, reinforcement learning, +deep learning, and their applications. He is a senior +member of IEEE. +Quanyu Liao is a graduate student at the School +of Computer Science, Chengdu University of of +Information Technology. His research directions are +adversarial attack, deep learning, and computer vi- +sion. +Bin B. Zhu received the B.S. degree in physics +from the University of Science and Technology of +China, Hefei, China, in 1986, and the M.S. and +Ph. D. degrees in electrical engineering from the +University of Minnesota, Minneapolis, MN, in 1993 +and 1998, respectively. He is currently a Principal +Researcher with Microsoft Research Asia, Beijing, +China. His research interests include DNN security +and privacy, AI applications, Internet and system +security, privacy-preserving processing, content pro- +tection, and signal and multimedia processing. +Dr. Xi Wu is currently the dean of the School of +Computer Science, Chengdu University of Informa- +tion Technology, and the Chinese director of the +International Joint Research Center for Image and +Vision, Chengdu University of Information Technol- +ogy. The main research directions are: image anal- +ysis and computational imaging, high-performance +and parallel distributed computing, smart meteorol- +ogy and numerical weather computing. +Dr. Siwei Lyu is an SUNY Empire Innovation +Professor at the Department of Computer Science +and Engineering, the Director of UB Media Forensic +Lab (UB MDFL), and the founding Co-Director of +Center for Information Integrity (CII) of University +at Buffalo, State University of New York. Dr. Lyu +received his Ph.D. degree in Computer Science from +Dartmouth College in 2005, and his M.S. degree +in Computer Science in 2000 and B.S. degree in +Information Science in 1997, both from Peking Uni- +versity, China. Dr. Lyu’s research interests include +digital media forensics, computer vision, and machine learning. Dr. Lyu is a +Fellow of IEEE. + diff --git a/PdFJT4oBgHgl3EQfIywm/content/tmp_files/load_file.txt b/PdFJT4oBgHgl3EQfIywm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..54e51bdf77a695ac0d49a538b07939ac3be67fe7 --- /dev/null +++ b/PdFJT4oBgHgl3EQfIywm/content/tmp_files/load_file.txt @@ -0,0 +1,1328 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf,len=1327 +page_content='1 Attacking Important Pixels for Anchor-free Detectors Yunxu Xie∗, Shu Hu∗, Xin Wang‡, Senior Member, IEEE, Quanyu Liao, Bin Zhu, Xi Wu‡, Siwei Lyu Fellow, IEEE Abstract—Deep neural networks have been demonstrated to be vulnerable to adversarial attacks: subtle perturbation can completely change the prediction result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Existing adversarial attacks on object detection focus on attacking anchor-based detectors, which may not work well for anchor-free detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In this paper, we propose the first adversarial attack dedicated to anchor-free detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' It is a category-wise attack that attacks important pixels of all instances of a category simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Our attack manifests in two forms, sparse category-wise attack (SCA) and dense category-wise attack (DCA), that minimize the L0 and L∞ norm-based perturbations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' For DCA, we present three variants, DCA-G, DCA-L, and DCA- S, that select a global region, a local region, and a semantic region, respectively, to attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Our experiments on large-scale benchmark datasets including PascalVOC, MS-COCO, and MS- COCO Keypoints indicate that our proposed methods achieve state-of-the-art attack performance and transferability on both object detection and human pose estimation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Index Terms—Adversarial Attack, Object Detection, Human Pose Estimation, Category-wise Attack, Anchor-free Detector I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' INTRODUCTION T HE development of deep neural networks (DNNs) [1]– [5] enables researchers to achieve unprecedented high performance on various computer vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' However, DNN models are vulnerable to adversarial examples: intentionally crafted subtle disturbance on a clean image makes a DNN model predict incorrectly [6], [6]–[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' As a typical type DNNs, convolutional neural networks (CNNs) achieve state- of-the-art (SOTA) performance on classification, objection detection, segmentation, and other computer vision tasks but also suffer from adversarial examples [17], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Studies on generating adversarial examples have attracted increasing attention because they help identify vulnerabilities of trained DNN models before they are launched in services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Object detection is essential in many vision tasks like instance segmentation, pose estimation, and action recog- nition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Existing object detectors can be classified into two groups according to the way they locate objects: anchor- based [19], [21]–[25] and anchor-free detectors [20], [26]– [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' An anchor-based detector first determines many preset anchors on the image and then refines their coordinates and Yunxu Xie, Quanyu Liao, and Xi Wu are with the Chengdu University of Information Technology, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' e-mail:({xieyunxu, xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='wu}@imde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='cn) Shu Hu is with Carnegie Mellon University, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' e-mail:shuhu@cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='edu Xin Wang and Siwei Lyu are with University at Buffalo, SUNY, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' e-mail:({xwang264, siweilyu}@buffalo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='edu) Bin Zhu is with Microsoft Research Asia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' e-mail:(binzhu@microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='com) ∗ Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' ‡ Corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Overall Heatmap Traffic Light Person Car Stop Sign Truck Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 1: First row: The detected results (left) and the proposals (right) of Faster-RCNN [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Second row: The detected results (left) and the overall heatmap (right) of CenterNet [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Third row: Selected target pixels (red) that will be attacked for each category by our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' predicts their categories before outputting final detection re- sults (see the first row in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' An anchor-free detector finds objects without using preset anchors: it detects keypoints of objects and then bounds their spatial extent (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 1 second row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' There are many works on adversarial attacks against anchor-based detectors, such as DAG [13] and UEA [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' However, to the best of our knowledge, there is no published work except our published conference papers [30], [31] on investigating vulnerabilities of anchor-free detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Attacking an anchor-free detector is very different from attacking an anchor-based detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Adversarial attacks on anchor-based detectors work on selected top proposals from a set of anchors of objects, while anchor-free object detectors return only objects’ keypoints via the heatmap mechanism (see the second row in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' These keypoints are used to gener- ate corresponding bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' This detection procedure is completely different from anchor-based detectors, making anchor-based adversarial attacks unable to directly adapt to attack anchor-free detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In addition, many attack methods for anchor-based detectors such as DAG and UEA suffer from the following shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' First, their generated adversarial examples have a poor transferability, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=', adversarial examples generated by DAG on Faster-RCNN can hardly be transferred to attack other object detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Second, some of them attack only one proposal at a time, which is extremely costly in computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Thus, it is desired to investigate new efficient arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='11457v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='CV] 26 Jan 2023 stopsian0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='3 traffic ligltroffic liahto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='5 trafficliahto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='3 cor0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='5 trof truck0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='4 pepers cor coro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='8 鄂WD-5010car 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='987 caro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='9s12 car0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='840 car0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='996 cearuccar0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='990 鄂WO-5010car1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='000 car 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='999 car0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='903 car0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='8870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='997 WO-50108018 WD-50102 and effective attack schemes for anchor-free detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In this work, we propose a novel untargeted adversarial attack, called Category-wise Attack, to attack anchor-free detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Our proposed attack focuses on categories and can attack all objects from the same category simultaneously by attacking a set of important target pixels (see the third row in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The important target pixel set includes detected pixels that are highly informative (higher-level information of objects) as well as undetected pixels that have a high probability to be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Our category-wise attack is formulated as a general frame- work that minimizes Lp norm-based perturbations, where p ∈ {0, ∞}, to flexibly generate sparse and dense pertur- bations, called sparse category-wise attack (SCA) and dense category-wise attack (DCA), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Moreover, in DCA, we explore three attack strategies based on different attack regions: DCA on the global region (DCA-G) that attacks the whole region of an image, DCA on the local region (DCA-L) that attacks only important regions around objects, and DCA on the semantic region (DCA-S) that attacks semantic-rich regions of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Attacking only specific important regions can effectively reduce the number of pixels disturbed while retaining high attack performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We demonstrate the effectiveness of our methods in at- tacking anchor-free detector CenterNet [20] with different backbones in both object detection and human pose estimation tasks using large scale benchmark datasets: PascalVOC [32], MS-COCO [33], and COCO-keypoints [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Our experimental results show that our attack methods outperform existing SOTA methods with high attack performance and low visibility of perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The main contributions of our work can be summarized as follows: 1) We present the first algorithms of untargeted adversarial attacks on anchor-free detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' They attack all objects in the same category simultaneously instead of only one object at a time, which avoids perturbation over-fitting on one object and increases the transferability of generated adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 2) Our category-wise attack is designed to attack important pixels in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' On one hand, it can generate sparse adversarial perturbations to increase imperceptibility of generated adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' On the other hand, it can generate dense adversarial perturbations to improve attacking effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 3) Our method generates more transferable adversarial ex- amples than existing attacks in both object detection and human pose estimation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Our experiments on large- scale benchmark datasets indicate that it achieves the SOTA performance for both white-box and black-box attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' This paper extends our published conference papers [30] (ICME 2021) and [31] (IJCNN 2020) substantially in the following aspects: 1) We provide a general DCA algorithm that includes three attacking approaches based on the tar- geted attacking region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In particular, we propose a new DCA method called DCA-S that restricts perturbations in the se- mantic region, which is extracted according to informative gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' DCA-S can generate more semantically meaningful perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 2) We show the applicability of our SCA and DCA attack methods in a practical human pose estimation task on the MS-COCO Keypoints dataset and demonstrate that they outperform existing attack methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 3) We add more experiments to verify the effectiveness of our proposed attacking methods, especially for DCA-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We also add more studies for analyzing the sensitivity of hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In Section II, we discuss the difference between anchor-free and anchor- based detectors and summarize existing adversarial attacks that are related to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In Section III, we formulate our attack- ing problem in the category-wise attack setting for anchor-free detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In Section IV, we provide a detailed description of our proposed category-wise attack in terms of sparse and dense settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In Section V, we conduct experiments to demonstrate the efficiency and effectiveness of our proposed algorithms for attacking anchor-free detectors in both object detection and human pose estimation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Section VI concludes the paper with discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Anchor-based and Anchor-free Detectors A great progress has been made in object detection in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' With the development of deep convolutional neural net- works, many object detection approaches have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' One of the most popular groups of object detection methods is the RCNN [34] family, such as Faster-RCNN [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The first process of RCNN’s pipeline is to generate a large number of proposals based on anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Then a different classifier is used to classify the proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' At last, a post-processing algorithm, such as non-maximum suppression (NMS) [35], is used to reduce redundancy proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Other typical anchor- based object detectors include YOLOv2 [23] and SSD [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' They need to place a set of rectangles with pre-defined sizes during the training and then put them in some desired positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Anchor-base detectors have high detection accuracy but also have the following shortcomings: anchor boxes should be manually defined before the training and these anchor boxes may not coincide and be consistent with ground-truth boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' To solve these shortcomings, anchor-free object detectors have been proposed, such as CornerNet [26] and Center- Net [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' These object detectors detect objects by detecting their keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Specifically, CornerNet detects an object by detecting the two corners of the object, and CenterNet detects objects by finding their center points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' No anchor is used in both detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' These two detectors not only are faster and simpler to train than anchor-based detectors but also achieve SOTA detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Both CornerNet and CenterNet can use multiple convolutional neural networks as the backbone network to extract semantic features of an input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Keypoints of objects are then located through these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Normally, a keypoint includes the size and category (or class) information of its object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' At last, some post- processing algorithm is used to remove redundancy keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' With its good keypoint estimation, CenterNet works not only for object detection but also for human pose estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 3 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Adversarial Attacks on Object Detection Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' [7] first showed the adversarial example problem of deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' An adversarial example is an original sample perturbed with deliberately crafted pertur- bation, typically imperceptible, that makes the DNN model predict incorrectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Adversarial attacks can be classified into two groups: targeted adversarial attacks and untargeted adver- sarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' A former attack aims to make the DNN model to mispredict to a specific label while a latter attack aims to make the DNN model to mispredict to anything different from the original label (including failure to detect objects for object detection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Our proposed attack is an untargeted adversarial attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Most adversarial attacks focus on minimizing the Lp (p ∈ (0, 1, 2, ∞)) norm of the adversarial perturbation, aiming to make the perturbation imperceptible while maintaining a high attack success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Typical adversarial attacks include Fast Gradient Sign Method (FGSM) [7], Project Gradient Descent (PGD) [9], and DeepFool [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Specifically, FGSM computes the gradient of the loss with respect to the input image to generate the adversarial perturbation as follows: x = x + ϵ · sign(∇xJ(f(x), y)), where f is the classifier, J is the loss function, x is the original input image, y is the ground-truth label of the input image, x is the perturbed image, and ϵ is the amplitude of the perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' PGD applies FGSM iteratively with a smaller amplitude of the perturbation α: xt = ΠBϵ(x) � xt−1 + α · sign(∇xJ(f(xt−1), y) � , where x0 = x and ΠBϵ(x)(·) is the projection function that projects the perturbed image back into the ϵ-ball centered at the original input image x if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' DeepFool uses a generated hyperplane to approximate the decision boundary and computes the lowest Euclidean distance between the input image and the hyperplane iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' It then uses the distance to generate the adversarial perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The above attack methods mainly attack classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Adver- sarial attacks on object detectors have also been proposed, such as DAG [13] and UEA [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' They all focus on attacking anchor-base object detectors such as Faster-RCNN and SSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The main shortcomings of these adversarial attacks are slow and not robust for transferring attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Furthermore, they are designed for attacking anchor-based detectors and don’t work well for attacking anchor-free object detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' To the best of our knowledge, there are no published adversarial attack on anchor-free detectors except our two conference papers [30], [31] that this paper extends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' PROBLEM FORMULATION In this section, we define our optimization problem of attacking anchor-free detectors based on category-wise attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Suppose there exist k object categories, C := {C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=', Ck}, with detected object instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Let [k] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=', k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We use Si to denote the target pixel set of category Ci whose detected object instances will be attacked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' There are k target pixel sets, S := {S1, S2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=', Sk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 2: Detection results before and after attacking only detected pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Red boxes denote originally detected keypoints and bounding boxes before the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Blue boxes denote newly detected keypoints after the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' (a) & (c): a detected object and a detected keypoint at the center of the person before the attack in cases 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' (b) & (d): detection results after attacking only detected pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In case 1, after attacking all detected pixels, a neighboring pixel of the previously detected keypoint is detected as the correct category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In case 2, the centers of the top half and the bottom half of the person appear as newly detected keypoints still detected as a person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In both cases, mAP is barely reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' category-wise attack for anchor-free detectors is formulated as the following constrained optimization problem: minimize r ∥r∥p, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' arg maxjfj(x + r, s) ̸= Ci, s ∈ Si, ∀i ∈ [k], (1) where r is an adversarial perturbation, ∥·∥p is the Lp norm, x is a clean input image, x + r is an adversarial example, f(x + r, s) is the classification prediction score vector for pixel s and fj(x + r, s) is its j-th value, where j ∈ [k], and arg maxjfj(x+r, s) denotes the predicted object category on a target pixel s ∈ Si of adversarial example x + r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' While p in the the Lp norm can take different values, we focus on p ∈ {0, ∞} in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 1, it is natural to use all detected pixels of category Ci as target pixel set Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The detected pixels are selected from the heatmap of category Ci generated by an anchor-free detector such as CenterNet [20] with their probability scores higher than the detector’s preset visual threshold and being detected as right objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Unfortunately, it does not work: after attacking all detected pixels to predict into wrong categories, we expect that the detector should not detect any correct object, but our experiments with CenterNet turn out that it still can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Further investigation enables us to explain the unexpected result as follows: 1) Unattacked neighboring background pixels of the heatmap can become detected pixels of the correct cate- gory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Since their detected box is close to the old detected object, CenterNet can still detect the object even though all the previously detected pixels are attacked into wrong categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' An example is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 2 (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 2) CenterNet regards center pixels of an object as keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' After attacking detected pixels located around the center of an object, newly detected pixels may appear in other positions of the object, making the detector still be able to detect multiple local parts of the correct object with barely reduced mAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' An example is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 2 (c) a (b) (c) (d) Case 1 Case 24 Attack results DCA-G DCA-S DCA-L SCA Category-wise adversarial attack methods person bicycle car motorcycle Target pixel selection SCA DCA-G DCA-L DCA-S Local region Total gradient Sign function Gradients Adversarial gradient computation Detection result Input image CenterNet Heatmap Attack Success?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' True False share share MASK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' generation MASK" generation Update CWDF LinearSolver Global perturbation Global perturbation Global perturbation Sparse perturbation Dense perturbation Local perturbation Semantic perturbation GenerateMask MSIRE Normalized Gradient Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 3: Our general attack framework overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Firstly, we extract the heatmap for each object category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Target pixels are selected based on heatmap and attacking threshold T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Secondly, our proposed category-wise adversarial attack methods including sparse category-wise attack (SCA) and dense category-wise Attacks (DCA-G, DCA-L, and DCA-S) are used to attack target pixels from each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' More details about these methods can be found in the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Finally, target pixel sets will be updated and the perturbation will be delicately modified if the attack is not complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Otherwise, the attack results will output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Pixels that can produce one of the above two changes are referred to as runner-up pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We find that almost all runner- up pixels have a common characteristic: their probability scores are only a little below the visual threshold used in the object detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Based on this characteristic, our category-wise attack methods set an attacking threshold, T , lower than the visual threshold, and select all the pixels from the heatmap whose probability score is above T into Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Therefore, Si includes all detected and runner-up pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In this way, we can improve the transferability of generated adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' METHODOLOGY In this section, we provide a detailed description of our category-wise attack, which is a sparse category-wise attack (SCA) if L0 is used and a dense category-wise attack (DCA) if L∞ is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In DCA, we explore and design three attack strategies called DCA on the global region (DCA-G), DCA on the local region (DCA-L), and DCA on the semantic region (DCA-S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' An overview of the proposed category-wise attack (CA) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In the following description of our methods, we focus on untargeted adversarial attack tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' If the task is a targeted adversarial attack, our methods can be described in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Sparse Category-wise Attack (SCA) The goal of the sparse category-wise attack is to fool the detector while perturbing a minimum number of pixels in the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' It is equivalent to setting p = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Unfortunately, this is an NP-hard problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' To solve this problem, SparseFool [11] relaxes this NP-hard problem by iteratively approximating the classifier as a local linear function in generating sparse adversarial perturbation for im- age classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Motivated by the success of SparseFool on Algorithm 1 Category-Wise DeepFool (CWDF) Input: image x, target pixel set Sh, T Output: dense adversarial example xB Initialize: x1 ← x, p ← 1 while Sh ̸= ∅ do for j ̸= h do vj ← ∇ � s∈Sh fj(xp, s) − ∇ � s∈Sh fh(xp, s) scorej ← � s∈Sh fj(xp, s) end for o ← argminj̸=h |scorej| ∥vj∥2 , xp+1 ← xp + |scoreo| ∥vo∥2 2 vo Sh = RemovePixels(xp, xp+1, Sh, T ) /*Refer to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 6*/ p ← p + 1 end while return xB ← xp image classification, we propose Sparse Category-wise Attack (SCA) to generate sparse perturbations for anchor-free object detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' It is an iterative process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In each iteration, one target pixel set is selected to attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' More specifically, given an input image x and current category-wise target pixel sets S, the pixel set Sh that has the highest probability score from S is selected, where h ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We can formulate Sh as Sh = arg max Si � � s∈Si fi(x, s) �����Si ∈ S, i ∈ [k] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' (2) Then we apply the Category-Wise DeepFool (CWDF) method, whose algorithm is described in Algorithm 1, to generate a dense adversarial example xB for x by computing person bicycle car motorcycleXoseltraffic lighto trgffic lighto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='5 caro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='6jr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='6 aro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' truck0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='3 SOT bers per person0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='5ersr motor motorcycleo persbnd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='7 persono4 persono ipersono4rshhn5 ersihb6 handbaa0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='3 hancbaa0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='41vaseu vase0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='4 vaseo vaseo vose vaseo vaseo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='3 vosel vase0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='4 oseo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='vaseo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='5 vaseo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' voseQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='35 perturbation on Sh as follows, xB = CWDF(x, Sh, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' (3) CWDF is adapted from DeepFool [8] to enable it to attack all pixels from target pixel set Sh simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' After the generation, we remove successfully attacked pixels from Sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Next, SCA uses the ApproxBoundary from [11] to ap- proximate the decision boundary locally with a hyperplane β passing through xB: β △= {x′ : wT (x′ − xB) = 0}, where w is the normal vector of hyperplane β: w= ApproxBoundary(xB, x, Sh) = ∇ � s∈Sh[fargmaxjfj(xB,s)(xB, s) − fargmaxjfj(x,s)(xB, s)] ∥∇ � s∈Sh[fargmaxjfj(xB,s)(xB, s) − fargmaxjfj(x,s)(xB, s)]∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' (4) Sparse adversarial example xadv can then be computed via the LinearSolver process from [11]: xadv = LinearSolver(x, w, xB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' (5) Specifically, in each iteration, we project x towards one single coordinate of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' If the projection in a specific direction has no solutions, then we will ignore that direction in the next iteration because it cannot provide a significant contribution to the final perturbed image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' More details can be found in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' For completeness, we include the algorithm of LinearSolver in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The process of generating perturbation through the ApproxBoundary and the LinearSolver of SCA is illus- trated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' After attacking Sh through the above operations, SCA updates Sh by removing pixels that are no longer detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' This process is called RemovePixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Specifically, taking x, xadv, and Sh as input, RemovePixels first generates a new heatmap for perturbed image xadv with the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Then, it checks whether the probability score of each pixel in Sh is still higher than T on the new heatmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Pixels whose probability score is lower than T are removed from Sh, while the remaining pixels are retained in Sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Target pixel set Sh is thus updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We can formulate the RemovePixels procedure as follows, Sh = RemovePixels(x, xadv, Sh, T ) = � s ��� arg max j fj(xadv, s)=arg max j fj(x, s), farg maxj fj(xadv,s)(xadv, s) > T , s ∈ Sh � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' (6) We iteratively execute the above procedures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=', Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 2, 3, 4, 5, and 6) until S ∈ ∅, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=', there is no correct object can be detected after the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The attack for all objects of x is thus successful, and we output the final generated adversarial example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The SCA algorithm is summarized in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In an iteration, if SCA fails to attack any pixels of Sh in the inner loop, SCA will attack the same Sh in the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' During this process, SCA keeps accumulating perturbations on these pixels, with the probability score of each pixel in Sh keeping reducing, until the probability score of every pixel in Sh is lower than T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' By then, Sh is attacked successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Hyperplane 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 𝑥" 𝑥# 𝛽!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 𝛽" 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' $ 𝑥" $ 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' " − 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 𝑥$ − 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 4: Illustration of the ApproxBoundary and the LinearSolver of SCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The black solid line denotes the real decision boundary of the object detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Blue points denote adversarial examples that have not attacked all objects successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Red point denotes adversarial examples that have already attacked all objects successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' This figure illustrates two iterations of the attack, x0 → x1 and x1 → x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Take x0 → x1 as an example, SCA first generates a dense adver- sarial example xB 0 (green point) by CWDF and approximated linear decision boundary β0 (black dash line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Then it uses LinearSolver to add a sparse perturbation to support x0 to approximate decision boundary β0 by satisfying β0 = {x′ : wT (x′ − xB 0 ) = 0} until a valid sparse adversarial example x1 is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The right image represents the perturbation after applying CWDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The left image represents the perturbation after applying the ApproxBoundary and the LinearSolver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Comparing these two images, it is clear that the perturbation becomes sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Dense Category-wise Attack (DCA) It is interesting to investigate our optimization problem Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 1 for p = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In this case, our adversarial perturbation generation procedure is based on PGD [36] and is called Dense Category-wise Attack (DCA) since it generates dense perturba- tions compared with SCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Note that our DCA framework can also be based on other adversarial attacks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=', FGSM [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We propose three methods based on DCA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=', DCA-G, DCA- L, and DCA-S, depending on the targeted region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' They are described in details next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 1) DCA on Global Region (DCA-G): Given an input image x and category-wise target pixel sets S, DCA applies two iterative loops to generate adversarial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In each inner loop iteration j, DCA first computes the total loss of all pixels in target pixel set Sj corresponding to each available category Cj with � s∈Sj CE (f(x, s), Cj), where CE(·,·) is the traditional cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Then, it computes local adversarial gradient gj of the loss with respect to the current image x and normalize it with L∞ norm: gj = ∇x � s∈Sj CE (f(x, s), Cj) ∥∇x � s∈Sj CE (f(x, s), Cj)∥∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' After that, DCA adds up all gj to generate a total adversarial gradient G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In an outer loop iteration p, DCA computes the global perturbation (GP) by applying sign operation to the 6 Algorithm 2 Sparse Category-wise Attack (SCA) Input: image x, S, max iter outer, max iter inner Output: adversarial example x∗, perturbation r Initialize: x1 ← x, p ← 1 while S ̸∈ ∅ and p ≤ max iter outer do Compute Sh with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' (2) and xp, q ← 0, xp,q ← xp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' while q ≤ max iter inner and Sh /∈ ∅ do xB p,q = CWDF(xp,q, Sh, T ) /*See Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 1*/ w = ApproxBoundary(xB p,q, xp,q, Sh) /*See Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 4*/ xadv p,q = LinearSolver(xp,q, w, xB p,q) /*See Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 5*/ Sh = RemovePixels(xp,q, xadv p,q , Sh, T ) /*See Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 6*/ q = q + 1 end while xp+1 ← xadv p,q−1, p = p + 1 end while return x∗ = xp, r = xp − x total adversarial gradient G [36] as follows, GP = ϵ max iter · sign(G), (7) where max iter denotes the maximum number of iterations of the outer loop and the term ϵ max iter is perturbation size in each iteration [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' At the end of the outer loop, DCA uses RemovePixels of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 6 to remove from S the target pixels that have already been attacked successfully on the perturbed image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Since DCA works on the global (whole) region in the original image, we call this method, DCA-G, in short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The pseudo-code of DCA-G is described in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 2) DCA on Local Region (DCA-L): As we discussed be- fore, runner-up pixels are usually located near keypoints in the heatmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' They are the most important pixels for the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In addition, perturbations in the background from an image may not impact the detection results since they are not related to the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' To attack these objects, we only need to attack these significant pixels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=', runner-up and keypoints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' To fulfil the goal, we can use attack mask MASKL to restrict perturbation around detected objects in order to disallow per- turbation in the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Attack mask MASKL is generated from S with the following process, called the GenerateMask procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' MASKL is initialized with a zero matrix of the same size as the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We locate each pixel point s ∈ Si, ∀i ∈ [k], on the input image and set the same location point of MASKL to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' After processing all s, for each pixel s = 1 in MASKL, we set all the points in the square box centered at s’s with the size of radius (side length) R∗ to be 1 too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The resulting MASKL is used as the local attack mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The attack performance depends on the value of R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We will quantitatively analyze this dependence in the experimental studies reported in Section V-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' A local perturbation (LP) is obtained by applying MASKL Algorithm 3 Dense Category-wise Attack (DCA) Input: image x, S, C, ϵ, max iter, T Output: adversarial example x∗, perturbation r Initialize: x1 ← x, p ← 1 while S ̸∈ ∅ and p ≤ max iter do G ← 0, j ← 1 while j ≤ k do if Sj ̸= ∅ then gj = ∇xp � s∈Sj CE (f(xp,s), Cj) ∥∇xp � s∈Sj CE (f(xp,s), Cj)∥∞ , G ← G+gj end if j ← j + 1 end while GP ← ϵ max iter · sign(G) if DCA-G then xp+1 ← xp + GP /*Refer to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 7*/ else if DCA-L then xp+1 ← xp + GP ∗ MASKL /*Refer to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 8*/ else if DCA-S then xp+1 ← xp + GP ∗ MASKS /*Refer to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 9*/ end if for Si in S do Si = RemovePixels(xp, xp+1, Si, T ) /*Refer to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 6*/ end for p ← p + 1 end while return x∗ = xp, r = xp − x on the global perturbation (GP): LP = GP ∗ MASKL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' (8) After generating LP, we update S by removing the points that have been attacked successfully with RemovePixels (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The pseudo-code of DCA-L is shown in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 3) DCA on Semantic Region (DCA-S): Using DCA on the local region may not get a perfect perturbation around objects because run-up pixels may not be all around objects since some of them may not represent semantic information of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In addition, DCA-L uses a regular square mask around each pixel in S,∀i ∈ [k], which may not precisely select important perturbation signals from the global perturbation, as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 5 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' To address this issue, we propose the DCA-S method that applies DCA to the semantic region of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Since convolutional layers of the CNN-based anchor-free model contain abundant information, especially the spatial information [38], [39], which can indicate which regions of an input image have a higher response to the output of the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' This characteristic demonstrates that the generated perturbation will be more effective if we only attack pixels that are related to the semantic region of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' A natural strategy 7 Original image Masked image by DCA-L Masked image by DCA-S R*= 50 R*= 120 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 5: Comparison of masks from DCA-L and DCA-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' A proper radius R∗ of the DCA-L mask is hard to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' A undersized R∗ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=', R∗=50) will miss some important regions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=', human and dog faces) while an oversized R∗ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=', R∗=120) will introduce more useless local perturbations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=', perturbations on background), which may lead to worse results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In contrast, the mask generated by DCA-S is more precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' for getting this semantic region is to extract the feature map based on the convolutional layer outputs [40]–[42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Shallower layers usually keep much spatial information, leading to a sparse and discontinuous high response region in the feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' On the other hand, deeper layers contain more global information due to their larger receptive fields, resulting in a large and continuous high response region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In prior work [39], the last convolutional layer is used to adopt the key region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Discarding spatial information from shallower layers makes this method hard to accurately target locations of small objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Inspired by [40]–[42], we propose a multi-layer semantic information region selection (MSIRE) method, to be described in detail next, to extract and integrate semantic information from both deep and shallow layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' By integrating the gradient information from these layers, we can construct a semantic mask that contains the most informative semantic region of objects, which is then combined with the global perturbation (GP) to generate the final perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Now we describe the MSIRE method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Let L = {li}n i=1 be the set of layers that we consider to extract key region, where li is the i-th layer containing feature map activations hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In our experiments, n is set to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' For layer li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' we calculate and sum the gradient of the score for category Cj of all pixels in target pixel set Sj with respect to its feature map activation hi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' and sum them up across all categories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Gi = � j∈[k] ∂ � s∈Sj fj(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' s) ∂hi Then normalize gradient Gi to [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='1] and upsample to the size of image x: ˆGi = Φ � Gi − min(Gi) max(Gi) − min(Gi) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' where Φ(·) denotes the operation of upsampling and min(Gi) and max(Gi) are the minimal and maximal value of elements in Gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The mask corresponding to layer li can be obtained using a threshold Ts: maski = δ( ˆGi > Ts), where δ(·) is an impulse response that turns a pixel greater than Ts to 1, otherwise to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Combining maski, ∀i ∈ [n] as MASKS = n � i=1 maski, where MASKS denotes the final semantic region attack mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' After obtaining MASKS, we combine it with the global perturbation (GP) to get a semantic region guided perturbation (SP) as follows, SP = GP ∗ MASKS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' (9) Like its siblings, after generating the perturbation, DCA-S updates S with RemovePixels (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The pseudo-code of DCA-S is shown in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' EXPERIMENTS In this section, we will evaluate the performance of the proposed adversarial attack (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=', SCA, DCA-G, DCA-L, and DCA-S) for both object detection and human pose estimation based on anchor-free detector CenterNet [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Experimental Settings 1) Detectors and Datasets: For object detection, we evalu- ate our proposed attack on two public datasets: PascalVOC [32] and MS-COCO [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We use the two pre-trained de- tectors (CenterNet with two different backbones: ResNet18 (R18) [43] and DLA34 [44]) from [20] in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' These two detectors are pre-trained on the training set of PascalVOC (including PascalVOC 2007 and PascalVOC 2012) and MS-COCO 2017, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Our adversarial examples are generated on the test set of PascalVOC 2007, which consists of 4,592 images and 20 categories, and the validation set of MS-COCO 2017, which contains 5,000 images and 80 object categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' For human pose estimation, we use the pre-trained detector (CenterNet with DLA34 [44] backbone) from [20] that is trained on the training set of MS-COCO Keypoints 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Our adversarial examples are generated on its validation set, which contains 5,000 images and 17 categories of keypoints for humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 2) Evaluation Metrics: We evaluate the performance of both white-box and black-box attacks with the following metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The attack performance is evaluated by computing the decreased percentage of mean average precision (mAP), referred to as the mAP Score Degradation Ratio (ASR), which is defined as, ASR = 1 − mAPattack mAPclean , where mAPattack denotes the mAP of the targeted object detector on adversarial examples, and mAPclean denotes the mAP on clean samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Higher ASR means better white-box attack performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We also use the L0 and L2 norms of perturbation r, PL0 and PL2, respectively, to quantify the perceptibility of the adversarial perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' PL0 = ∥r∥0 quantifies the proportion of perturbed pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' A lower value means that fewer image pixels are perturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' For PL2 = ∥r∥2, a greater value usually signifies a more perceptible pertur- bation to humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' For black-box attacks, transferability of adversarial ex- amples generated with other detectors and tested with the 8 target detector is used to measure the attack performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' More specifically, the attack performance of block-box attacks is measured by the ratio, referred to as the Attack Transfer Ratio (ATR), of the ASR on the target model, ASRtarget, to the ASR on the generating model, ASRorigin: ATR = ASRtarget ASRorigin , Higher ATR means better transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In our experiments, for object detection, black-box ad- versarial examples are generated on CenterNet with one backbone (R18 or DLA34) and tested on Cen- terNet with a different backbone (DLA34, R18, and ResNet101 (R101)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We also test these adversarial ex- amples on other detectors, including anchor-free (Cor- nerNet [26] with backbone Hourglass [45]) and anchor- based detectors (Faster-RCNN [19] and SSD [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' For human pose estimation, all adversarial examples are gen- erated on CenterNet with backbone DLA34 and tested on CenterNet with backbone Hourglass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Additionally, we follow [46] to simulate a real-world at- tack transferring scenario, wherein generated adversarial examples are saved in the JPEG format and then reloaded to attack the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In real-world applications, images are usually saved in a compression format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' JPEG is a most commonly used lossy compression standard for images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The transferability test in this way requires ad- versarial examples to be robust to the JPEG compression like in most real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 3) Comparison Methods and Implementation Details: For object detection, since there is no existing attack dedicated to anchor-free detectors, we use an existing state-of-the-art attack designed for attacking anchor-based detectors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=', DAG [13] with VGG16 [47] backbone for attacking Faster-RCNN (FR) as the attack to compare with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' For human pose estimation, we compare with existing methods proposed in [48] for attacking human pose estimation systems, called FHPE and PHPE, which are based on FGSM and PGD, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Our methods are implemented with Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='6 and Pytorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' All the experiments are conducted with an Intel Core i9-7960 and an Nvidia GeForce GTX-1070Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We set max iter outer, max iter inner, and max iter to 50, 20, and 30, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The default values of R∗, Ts, T , and ϵ are 60, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='1, and 5% of the maximum value of the pixels in an image, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' All input images are resized to 512×512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Experimental Results on Object Detection 1) White-Box Attack Performance: Table I shows the white- box attack results on both PascalVOC and MS-COCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' For PascalVOC, we can see that DCA-G with the DLA34 backbone achieves the best ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' SCA with both back- bones produces much smaller perturbations than other methods and DCA-G with the R18 is 14 times faster than DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Furthermore, while the ASR performance of both DCA-L and DCA-S are very close to that of DAG, DCA-L and DCA-S produce smaller perturbations Data Method Network mAPclean mAPattack ASR PL2 (×10−3) PL0 Time (s) PascalVOC DAG FR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='92 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='990 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='8 SCA R18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='002 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='5 SCA DLA34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='003 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='3 DCA-G R18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='90 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='7 DCA-G DLA34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='94 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='990 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='2 DCA-L R18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='89 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='320 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='6 DCA-L DLA34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='90 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='320 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='1 DCA-S R18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='90 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='260 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='2 DCA-S DLA34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='92 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='280 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='0 MS-COCO DAG FR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='89 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='990 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='4 SCA R18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='004 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='4 SCA DLA34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='007 216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='0 DCA-G R18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='99 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='990 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='4 DCA-G DLA34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='99 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='990 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='9 DCA-L R18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='98 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='380 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='4 DCA-L DLA34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='98 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='390 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='3 DCA-S R18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='99 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='300 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='7 DCA-S DLA34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='99 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='310 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='5 TABLE I: White-box performance comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' “Time” is the av- erage time to generate an adversarial example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The best results are shown in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' and have lower time complexity than DAG in generating adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' For MS-COCO, both DCA-G and DCA-S achieve the highest ASR, 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='0%, which is significantly higher than that of DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Like on PascalVOC, the ASR performance of SCA with both R18 and DLA34 is in the same ballpark as that of DAG, SCA produces much smaller perturbations than other methods in terms of both PL2 and PL0, and DCA-G is the fastest in generating an adversarial example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In general, both SCA and DCA achieve state-of-the-art attack performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Specifically, DCA can achieve high ASR and low time complexity, while SCA can produce much smaller perturbations without degrading ASR much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We can see that PL0 of SCA is lower than 1%, implying that SCA can success- fully attack detectors by perturbing only a small percentage of pixels of the original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Comparing DCA-L and DCA- G, while it may slightly decrease the performance of ASR, DCA-L using a local region significantly decreases the size of perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Comparing DCA-S and DCA-L, DCA-S not only improves the performance of ASR but also reduces the perturbation size in terms of both PL2 and PL0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' This implies that the DCA on the semantic region is better than it on the local region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' A qualitative comparison between DAG and our methods is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' It is clear that perturbations generated by SCA and DCA-based methods are hard to be perceived by humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We also show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 7 the Average Precision (AP) of each object category on clean inputs and adversarial examples generated by our methods on PascalVOC with Centernet using both R18 and DLA34 backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The AP drops by a roughly similar percentage for all the object categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 8 shows the AP and Average Recall (AR) of SCA and DCA-based methods on MS-COCO with CenterNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We can notice that small objects are more vulnerable to adversarial examples than bigger ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' One possible explanation is that bigger objects usually have more keypoints than smaller objects on the heatmap and our algorithms need to attack all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 2) Black-Box Attack Performance: In evaluating black-box attack performance, adversarial examples are generated with Centernet using the R18 or DLA34 backbone for our proposed methods or with Faster-RCNN for DAG and tested with other 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 6: Qualitative comparison between DAG and our methods on the object detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Three examples are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Column 1: Detection results of clean inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Column 2: DAG’s attack results and perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Column 3: SCA’s attack results and perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Columns 4 - 6: the attack results and perturbations of DCA-G, DCA-L, and DCA-S, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The perturbations are magnified by a factor of 30 for better visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' AP for aeroplane AP for bicycle AP for bird AP for boat AP for bottle AP for bus AP for car AP for cat AP for chair AP for cow AP for diningtable AP for dog AP for horse AP for motorbike AP for person AP for pottedplant AP for sheep AP for sofa AP for train AP for tvmonitor mAP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='9 Clean SCA DCA-G DCA-L DCA-S AP for aeroplane AP for bicycle AP for bird AP for boat AP for bottle AP for bus AP for car AP for cat AP for chair AP for cow AP for diningtable AP for dog AP for horse AP for motorbike AP for person AP for pottedplant AP for sheep AP for sofa AP for train AP for tvmonitor mAP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='9 Clean SCA DCA-G DCA-L DCA-S Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 7: The AP of each object category on clean inputs and adver- sarial examples generated by SCA and DCA-methods on PascalVOC with CenterNet using R18 (left) and DLA34 (right) backbones on PascalVOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' object detection models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Table II and Table III show the black-box attack performance on PascalVOC and MS-COCO, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Attack transferability on PascalVOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' From Table II, we can see that adversarial examples generated by our meth- ods can successfully transfer to not only CenterNet with AP[IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='50:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='95] all AP [IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='50] all AP [IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='75] all AP [IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='50:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='95] small AP [IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='50:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='95] medium AP [IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='50:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='95] large AR [IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='50:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='95] all AR [IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='50:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='95] small AR [IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='50:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='95] medium AR [IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='50:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='95] large 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='8 Clean SCA DCA-G DCA-S DCA-L AP[IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='50:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='95] all AP [IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='50] all AP [IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='75] all AP [IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='50:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='95] small AP [IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='50:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='95] medium AP [IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='50:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='95] large AR [IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='50:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='95] all AR [IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='50:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='95] small AR [IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='50:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='95] medium AR [IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='50:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='95] large 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='8 Clean SCA DCA-G DCA-L DCA-S Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 8: The AR (top four) and AP (bottom six) performance of different sizes of objects on clean inputs and adversarial examples generated SCA and DCA-based methods on MS-COCO with Cen- terNet using R18 (left) and DLA34 (right) backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' different backbones but also completely different types of object detectors such as Faster-RCNN and SSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Specif- ically, we can see that SCA with the DLA34 backbone achieves the best ATR and SCA with the R18 backbone achieves similar ATR to the DCA-based methods with both the R18 and DLA34 backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' on the other hand, 10 20 40 60 80 100 120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='00 ASR R* 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='0 Time (s) (a) 20 40 60 80 100 120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='6 PL0 R* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='0 PL2(´10-3) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='90 ASR Ts (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='90 ASR T (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='00 ASR 0 (e) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 9: Sensitivity analysis of hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' From To R18 DLA34 R101 Faster-RCNN SSD mAP ATR mAP ATR mAP ATR mAP ATR mAP ATR Clean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='71 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='77 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='79 – 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='17 DLA34-DCA-L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='01 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='71 R18-DCA-S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='00 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='09 DLA34-DCA-S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='00 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='72 TABLE III: Black-box attack results on the MS-COCO dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' From: the leftmost column denotes the models which adversarial examples are generated from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' To: the top row denotes the target models that adversarial examples transfer to (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=', are tested on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' “–” means the value is unavailable (not defined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' adversarial examples generated by DAG with Faster- RCNN have a much poorer transferability in attacking CenterNet and SSD than our proposed methods, esp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' SCA with the DLA34 backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In addition, we can also see that our proposed methods have better transferability in attacking CenterNet and Faster-RCNN than in attacking SSD, which implies that SSD is less reliable on highly informational points in the CenterNet-generated heatmap than Faster-RCNN and CenterNet with a different back- bone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Attack Transferability on MS-COCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' From Table III, we can see that adversarial examples generated with our proposed methods with the DLA34 backbone have sig- nificantly higher ATR than other methods including our proposed methods with the R18 backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Our proposed methods with the DLA34 backbone can attack not only CenterNet with different backbones but also CornerNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Like on PascalVOC, DAG has a poor transferability in attacking CenterNet and CornerNet on MS-COCO too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Sensitivity Analysis of Hyperparameters In our algorithms, we have four hyperparameters that need to be tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' To study their sensitivity, we generate adversarial examples by attacking CenterNet with the R18 backbone on MS-COCO for evaluating R∗ and on PascalVOC for evaluating Ts, T , and ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Then we report the performance with respect to each hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' More details are provided as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 1) Sensitivity Analysis of R∗: In DCA-L, we need to use R∗ to control the size of extracted local regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' It is clear that R∗ correlates with the attack performance ASR, PL0, PL2, and time consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We show these relationships in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 9 (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' From these figures, we can draw three conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' First, the ASR of DCA-L correlates positively with R∗ when R∗ is less than 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' However, ASR is stable or slightly decreased afterward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The reason is that the oversized mask may introduce more useless perturbation, resulting in a worse effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Second, PL0 and PL2 of the perturbation correlates positively with R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' This is because higher R∗ means bigger attack masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Finally, the average attack time of DCA- L correlates negatively with R∗ when R∗ is lower than 48 and becomes stable afterward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 2) Sensitivity Analysis of Ts: In DCA-S, hyperparameter Ts is used to select semantic regions or pixels that contain more informative gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 9 (c) shows attack performance ASR with different Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We can see that the best ASR occurs when Ts is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' ASR decreases when Ts increases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='5, which can be explained by the fact that a too large value of Ts makes the mask too small, resulting in many useful pixels related to objects excluded from the mask and thus degraded attack performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 3) Sensitivity Analysis of T : The default value of the visual threshold is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='3 in the Centernet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We use DCA-G to evaluate the sensitivity of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The results are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 9 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We can see that ASR shows an obvious decreasing trend when T increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In particular, the decreasing trend becomes sharper when T is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='3, which can be explained that more runner-up points (pixels) are included in the target pixel sets if T is below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='3, resulting in improved attacking performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We can also see that the performance gap is more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='1 when T changes from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='3 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' This implies that runner-up points have a significant impact on the attack performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' On the other hand, if T is higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='3, some keypoints are excluded from the target pixel sets and thus unattacked, leading to faster degraded ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 4) Sensitivity Analysis of ϵ: We analyze the sensitivity of the amplitude of the perturbation ϵ with DCA-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The results are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 9 (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' A large ϵ means a larger perturbation 11 Data Method Network mAPclean mAPattack ASR PL2 (×10−3) PL0 Time (s) MS-COCO Keypoints FHPE DLA34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='6 PHPE DLA34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='990 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='2 SCA DLA34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='002 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='1 DCA-G DLA34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='990 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='2 DCA-L DLA34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='250 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='7 DCA-S DLA34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='180 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='2 TABLE IV: White-box attack results on the MS-COCO Keypoints dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' “Time” is the average time to generate an adversarial exam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' To From DLA34 Clean FHPE PHPE SCA DCA-G DCA-L DCA-S Hourglass mAP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='24 ATR – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='63 TABLE V: Black-box attack results on the MS-COCO Keypoints dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' To: the leftmost column denotes the target models that adversarial examples transfer to (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=', are tested on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' From: the top row denotes the models which adversarial examples are generated from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We expect ASR increases with increasing ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' This is confirmed by the experimental results shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Experimental Results on Human Pose Estimation In the MS-COCO Keypoints dataset, there are 17 categories of keypoints for humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We attack target pixels from each category and report the attacking performance next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 1) White-Box Attack: The white-box attack results on the MS-COCO Keypoints dataset are summarized in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We can see that our DCA-G has the highest ASR, which is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' This means DCA-G has attacked all the testing images successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Since FHPE and PHPE are based on the conventional FGSM and PGD methods and thus very simple in terms of complexity, we expect them to have less attack time, which is confirmed by the results shown in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' However, these two comparison methods have much lower ASR than our proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' All of our proposed methods outperform these two baselines on the ASR metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' On the other hand, it is clear that our SCA has the lowest PL0 and PL2 values, which is consistent with the observation on the object detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In addition, our DCA-L and DCA- S also have lower PL0 and PL2 values than DCA-G, which is due to the role of masks used in both DCA-L and DCA-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' A qualitative comparison between the comparison methods and our proposed methods is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 2) Black-Box Attack and Transferability: The black-box attack results are reported in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' We can see that our SCA achieves the best transferability, which is the same as on the object detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Furthermore, all of our proposed methods outperform the baseline methods, which implies that our category-wise attack can improve the transferability of adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' CONCLUSION In this work, we propose the first adversarial attack on anchor-free detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' It is a category-wise attack that attacks important pixels of all instances of a category simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Our attack manifests in two forms, sparse category-wise attack (SCA) and dense category-wise attack (DCA), when minimiz- ing the L0 and L∞ norm-based perturbations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' For SCA, it can generate sparse-imperceptible adversarial samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' For DCA, we further provide three variants, DCA- G, DCA-L, and DCA-S, to enable a flexible selection of a specific attacking region: the global region, the local region, and the semantic region, respectively, to improve the attack effectiveness and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Our experiments on large-scale benchmark datasets including PascalVOC, MS-COCO, and MS-COCO Keypoints indicate that the proposed methods achieve state-of-the-art attack performance and transferability on both object detection and human pose estimation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' There are two limitations in our proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The first limitation is that SCA has significantly higher time complexity since its Algorithm 2 includes CWDF and Linear Solver procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Our experimental results con- firm it: SCA needs a much longer time to attack an image for both object detection and human pose estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The second limitation is that the transferability of our proposed methods on attacking SSD is much lower than that of our methods on attacking other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Future Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' First, we will try to address the aforemen- tioned limitations of our proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Second, we plan to incorporate a scheme to automatically determine the hyper- parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Third, we will try to develop effective defenses against the proposed attacks, which is important for practical applications of anchor-free detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' APPENDIX A ALGORITHM OF LINEARSOLVER The LinearSolver algorithm is shown in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' In each iteration, we project towards only one single coordinate of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' If projecting x to a specific direction does not provide a solution, it will be ignored in the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' More details can be found in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Note that the projection operator of Q in Algorithm 4 controls the pixel values between 0 and 255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Algorithm 4 LinearSolver Input: image x, normal vector w, boundary point xB, pro- jection operator Q Output: perturbated point xadv Initialize: x0 ← x, i ← 0, H = {} while wT (xi − xB) ̸= 0 do r ← 0 d ← arg maxj∈H |wj| rd ← |wT (xi−xB)| |wd| sign(wd) x(i+1) ← Q(xi + r) H ← H ∪ {d} i ← i + 1 end while return xadv ← xi REFERENCES [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Xiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=', “Rmbench: Benchmarking deep reinforcement learning for robotic manipulator control,” arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='11262, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 12 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 10: Qualitative comparison between the comparison methods and our proposed methods on the human pose estimation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Three examples are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Column 1: Detection results of clean inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 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ICLR, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' [48] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Jain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=', “On the robustness of human pose estimation,” in CVPR Workshops, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' 29–38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Yunxu Xie is currently a postgraduate at the School of Computer Science, Chengdu University of of Information Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' His research directions are adversarial attack, deep learning, and computer vi- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Shu Hu is a Postdoc at Carnegie Mellon Uni- versity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' He received his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' degree in Computer Science and Engineering from University at Buffalo, the State University of New York (SUNY) in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' He received his M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' degree in Mathematics from University at Albany, SUNY in 2020, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' degree in Software Engineering from University of Science and Technology of China in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' His research interests include machine learning, digital media forensics, and computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Xin Wang (SM’2020) is a research affiliate at University at Buffalo, State University of New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' He received his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' degree in Computer Science from the University at Albany, State Uni- versity of New York in 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' His research interests are in machine learning, reinforcement learning, deep learning, and their applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' He is a senior member of IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Quanyu Liao is a graduate student at the School of Computer Science, Chengdu University of of Information Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' His research directions are adversarial attack, deep learning, and computer vi- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Bin B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Zhu received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' degree in physics from the University of Science and Technology of China, Hefei, China, in 1986, and the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' and Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' degrees in electrical engineering from the University of Minnesota, Minneapolis, MN, in 1993 and 1998, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' He is currently a Principal Researcher with Microsoft Research Asia, Beijing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' His research interests include DNN security and privacy, AI applications, Internet and system security, privacy-preserving processing, content pro- tection, and signal and multimedia processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Xi Wu is currently the dean of the School of Computer Science, Chengdu University of Informa- tion Technology, and the Chinese director of the International Joint Research Center for Image and Vision, Chengdu University of Information Technol- ogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' The main research directions are: image anal- ysis and computational imaging, high-performance and parallel distributed computing, smart meteorol- ogy and numerical weather computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Siwei Lyu is an SUNY Empire Innovation Professor at the Department of Computer Science and Engineering, the Director of UB Media Forensic Lab (UB MDFL), and the founding Co-Director of Center for Information Integrity (CII) of University at Buffalo, State University of New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Lyu received his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' degree in Computer Science from Dartmouth College in 2005, and his M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' degree in Computer Science in 2000 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' degree in Information Science in 1997, both from Peking Uni- versity, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Lyu’s research interests include digital media forensics, computer vision, and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} +page_content=' Lyu is a Fellow of IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQfIywm/content/2301.11457v1.pdf'} diff --git a/PtAzT4oBgHgl3EQflf1o/content/tmp_files/2301.01548v1.pdf.txt b/PtAzT4oBgHgl3EQflf1o/content/tmp_files/2301.01548v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b6709b9b9e4c7ba1b47a6d5585835653d2925a0f --- /dev/null +++ b/PtAzT4oBgHgl3EQflf1o/content/tmp_files/2301.01548v1.pdf.txt @@ -0,0 +1,895 @@ +arXiv:2301.01548v1 [astro-ph.GA] 4 Jan 2023 +RAA Vol.0 (20xx) No.0, 000–000 +http://www.raa-journal.org +http://iopscience.iop.org/raa +Research in +Astronomy and +Astrophysics +Three new spiral galaxies with active nuclei producing double radio lobes +X. Y. Gao1,2,3, Z. S. Yuan1,2, J. L. Han1,2,3, Z. L. Wen1,2,3 and S. S. Shan1,3 +1 National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China; xygao@nao.cas.cn; +hjl@nao.cas.cn +2 CAS Key Laboratory of FAST, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, +China; +3 School of Astronomy, University of Chinese Academy of Sciences, Beijing 100049, China +Received 20xx month day; accepted 20xx month day +Abstract Double radio lobes are generally believed to be produced by active nuclei of elliptical galaxies. +However, several double-lobed radio sources have been solidly found to be associated with spiral galaxies. +By cross-matching ∼ 9 × 105 spiral galaxies selected from the SDSS DR8 data with the full 1.4 GHz radio +source catalogs of NVSS and FIRST, we identify three new spiral galaxies: J0326−0623, J1110+0321 and +J1134+3046 that produce double radio lobes, in addition to five double-lobed spirals previously known. By +combining the newly discovered and all the other known cases in literature, we find that most of these spiral +galaxies are located in a galaxy group or a poor cluster, in which the environment is denser than in the field, +and about half of them are the central brightest galaxies in their parent system. We therefore suggest that +the environment is one of the key factors for a spiral to produce double radio lobes. +Key words: galaxies: general — galaxies: – active — galaxies: spiral — radio continuum: galaxies — +galaxies: jets +1 INTRODUCTION +The radio lobes powered by the supermassive black hole +(SMBH) in the center of galaxies are usually hosted by el- +liptical galaxies. They extend to several hundreds of kilo- +parsecs which are much larger than their optical counter- +parts. The point of view that double radio lobes are exclu- +sively hosted by elliptical galaxies has been challenged by +the discovery of the double-lobed radio source J0313−192 +(J0315−1906), which was found to be hosted by a spiral +galaxy (Ledlow et al. 1998, 2001; Keel et al. 2006). +In the last two decades, a few more spiral galaxies +hosting double radio lobes have been revealed. Hota et al. +(2011) discovered the second case Speca (J1409−0302) +with episodic radio emission. Bagchi et al. (2014) iden- +tified J2345−0449 with lobes extending to an extraordi- +nary length of ∼1.6 Mpc. Mulcahy et al. (2016) reported +the serendipitous discovery of the double radio source of +MCG+07–47–10 (J2318+4314) with a low luminosity of +P1.4GHz ∼ 1022 W Hz−1. Very recently, Vietri et al. +(2022) identified a new source J0354−1340 with the dou- +ble radio lobes extending approximately 240 kpc. +Searches for spiral galaxies with double radio lobes +have also been made systematically by several groups. +Mao et al. (2015) cross-matched the optical Galaxy Zoo +“superclean” sample (Lintott et al. 2008) with the Unified +Radio Catalog of Kimball & Ivezi´c (2008), which includes +radio sources from both the Faint Images of the Radio Sky +at Twenty-centimeters (FIRST, Becker et al. 1995) and the +NRAO VLA Sky Survey (NVSS, Condon et al. 1998) data. +They reported a new spiral J1649+2635 with double ra- +dio lobes, with a radio power of about 1024 W Hz−1 at +1.4 GHz. Singh et al. (2015) cross-matched the FIRST +catalog (Becker et al. 1995) with 187 005 spiral galaxies +(Meert et al. 2015) in the Sloan Digital Sky Survey (SDSS, +York et al. 2000) Data Release 7 (DR7) to search for the +coincidence of core source within a radius of 3′′ and also +the double lobes within a radius of 3′. They made addi- +tional search for the extended radio lobes with the NVSS +data (Condon et al. 1998) for the obtained FIRST–SDSS +matched objects and identified four spiral galaxies with +double radio lobes, among which J1159+5820 (Kozieł- +Wierzbowska et al. 2012), J1352+3126 (Donzelli et al. +2007), and J1649+2635 (Mao et al. 2015) have already +been reported previously, while J0836+0532 was found for +the first time. Ortiz Mart´ınez & Andernach (2016) col- +lected 675 874 spiral galaxies from several spiral galaxy +samples (e.g., Huertas-Company et al. 2011; Willett et al. +2013; Kuminski & Shamir 2016), and searched for the as- +sociated FIRST sources (Becker et al. 1995) within a larger +radius of 6′ under constraints of angular distances, position +angles, and arm-length ratio of the double radio sources +with respect to the central optical galaxy. Concentrating +on the extremely symmetric and aligned radio lobes, they + +2 +X. Y. Gao et al. +finally reported the re-discovery of the known case of +J1649+2635 (Mao et al. 2015). Though these efforts have +been dedicated to search for the spiral galaxies hosting +double radio lobes, only a handful of cases have been con- +firmed to date. +It is also unclear why these spirals hold double ra- +dio lobes while the vast majority of other spirals do not. +Often radio lobes may be triggered by the accretion of +host galaxies from the over-dense environments in their +vicinity. For example, the source J0315−1906 is a mem- +ber galaxy of the cluster Abell 428 (Ledlow et al. 1998). +J1409−0302 and J1649+2635 are the brightest galaxies +of their parent systems (Hota et al. 2011; Mao et al. +2015), and J2318+4314 is located close to the galaxy +groups NGC 7618 and UGC 12491 (Mulcahy et al. 2016). +However, Singh et al. (2015) found that J0836+0532 +and J1352+3126 are in galaxy groups with very limited +members and listed them as field galaxies together with +J1159+5820 (see their Table. 7). Therefore a large sample +of such galaxies are needed to investigate the environmen- +tal effect on radio lobes. Wu et al. (2022) recently analyzed +the optical images from the Hubble Space Telescope of a +sample of galaxies with extended double radio lobes seen +from FIRST (Becker et al. 1995), and found that 18 disk +galaxies are of high probability to have the genuine associ- +ation. Some of these disk galaxies have a small inclination +angle and show clear spiral patterns. +We noticed that Kuminski & Shamir (2016) classified +the broad morphological types of ∼ 3 × 106 galaxies in +the SDSS (York et al. 2000) DR8 by analyzing images +of galaxies with computer programs, and their pipeline +picked out ∼ 9 × 105 spiral galaxies. Here, we take this +large sample as the optical basis of spiral galaxies, and +cross-match them with the full radio source catalogs of +NVSS (Condon et al. 1998) and FIRST (Becker et al. +1995). We discover three new spiral galaxies with double +radio lobes. The paper is organized as follows. In Sect. 2, +we introduce the data sets used to identify the double-lobed +spiral galaxies, and also the procedure of identification. We +show the results and discuss the properties of the galaxies +and their environments in Sect. 3. The concluding remarks +are given in Sect. 4. +Throughout the paper, we adopted a flat ΛCDM cos- +mology with H0 = 70 km s−1 Mpc−1, Ωm = 0.3 and ΩΛ += 0.7. +2 DATA AND SEARCH STRATEGY +2.1 Data +By automatic computer program, Kuminski & Shamir +(2016) classified approximately 9×105 spiral galaxies out +of ∼3 × 106 galaxies observed in the SDSS (York et al. +2000) DR8. The classification for spiral galaxies and el- +liptical galaxies shows good agreement with those of the +Galaxy Zoo debiased “superclean” sample (Lintott et al. +2008) and the agreement rate is claimed to reach 98% +when the “classification certainty” p ≥ 0.54. +Two catalogs were released by Kuminski & Shamir +(2016). The “catalog.dat”1 is the catalog of the broad mor- +phology of SDSS galaxies with only the classification cer- +tainty p indicated, and the “spec.dat” including both in- +dications of “p” and morphological remarks (Elliptical, +Spiral, Star) is the morphological catalog of the SDSS ob- +jects with spectra. In this work, we first take all the spiral +galaxies in the two catalogs with p ≥ 0.54. However, we +noticed that some known spirals such as J0836+0532 and +J1649+2635 as shown in Table 1 have a classification cer- +tainty of p = 0.234 and p = 0.371, respectively, less than +0.54. In order not to miss many real spirals, we simply in- +cluded all galaxies marked as “Spiral” in the “spec.dat” +of Kuminski & Shamir (2016), regardless of the values of +p. Therefore, all 366 836 entries in the “spec.dat” marked +as “Spiral” and 1 184 922 entries with p ≥ 0.54 in “cat- +alog.dat” are used to search for associated double radio +lobes. The duplicates in the two catalogs are treated at the +final stage when inspecting the association between optical +and radio images. +The radio counterparts of the optical spiral galaxies are +searched in the NVSS (Condon et al. 1998) and the FIRST +(Becker et al. 1995) source catalogs. The NVSS was car- +ried out with the Very Large Array (VLA) – D configura- +tion at the frequency of 1.4 GHz with an angular resolution +of ∼ 45′′. The survey covers the entire sky north of the dec- +lination of δ = −40◦. Over 1.8 million discrete sources +brighter than ∼2.5 mJy (5σ level) were compiled into a +catalog2. The positional accuracy of NVSS is about 1′′ for +strong sources and 7′′ for faint sources. The FIRST obser- +vations were conducted at the same frequency, but with a +much better angular resolution of ∼ 5′′ by using the VLA – +B configuration array. It has a sensitivity of about 0.13 mJy +beam−1. The sky coverage of FIRST is limited within the +northern and southern Galactic cap regions of about 10 000 +square degrees in total, which is less than one third of that +of the NVSS. The FIRST catalog contains over 9.4 × 105 +entries. For the sources whose flux density is higher than +1 mJy, the radius of the 90% positional confidence error +circle is less than 1′′. The latest FIRST source catalog of +Version 14Dec173 was used in this study. +2.2 Search strategy +Based on the experiences gained from previous work (e.g. +Yuan et al. 2016), we learned that the 5′′ angular resolution +of FIRST could resolve out extended radio structures so +that diffuse radio lobes can be missed, such as the case for +J1409–0302 (Speca, Hota et al. 2011). Therefore we took +the NVSS data as the fundamental basis and the FIRST +data were used as an auxiliary database for radio-lobe iden- +tification. The radius to search for the radio counterpart of +the central optical spiral galaxy is tricky: the larger the ra- +dius is set, the more radio sources around the central galaxy +1 https://cdsarc.cds.unistra.fr/viz-bin/cat/J/ApJS/223/20#/browse +2 https://www.cv.nrao.edu/nvss/NVSSlist.shtml +3 http://sundog.stsci.edu/first/catalogs/readme.html + +Three spiral galaxies with double radio lobes +3 +one would get, and then the more time would be consumed +for distinguishing the true association. Therefore a trade- +off should be made on setting the searching area to bal- +ance the time consumption. By reviewing the known cases +to date (e.g. Ledlow et al. 1998; Hota et al. 2011; Bagchi +et al. 2014; Mao et al. 2015; Singh et al. 2015; Mulcahy +et al. 2016), we searched for the NVSS and FIRST sources +within 800 kpc around the central optical spirals if the red- +shift information is available; otherwise a radius of 3.5′ +around the spiral is set for the NVSS sources and 30′′ for +the FIRST sources if the redshift is unknown. +Spiral galaxies with double radio lobes may have var- +ious appearances in the radio images of NVSS and FIRST. +In the low resolution image of the NVSS, they could have +(1) a central core with distinct double radio lobes, e.g. +J1352+3126 (see Fig. 4 in +Singh et al. 2015); (2) un- +resolved central core and double radio lobes, i.e. show- +ing a structure that is elongated, such as J1649+2635 (see +Fig. 5 in Singh et al. 2015); and (3) distinct double-lobe +structure without a core component intrinsically or extrin- +sically. In the high resolution image of FIRST, the spi- +rals that host double radio lobes could show (1) a cen- +tral core with distinct double radio lobes, e.g. J1649+2635 +(Singh et al. 2015); (2) only a central core, because the ex- +tended lobes are resolved out by the small synthesis beam; +and (3) distinct double-lobe structures without a core, e.g. +J1409−0302 (Hota et al. 2011). The real cases can be any +reasonable combinations of the above possibilities for the +NVSS and FIRST data in their common surveyed area. +However, such loose constraints will yield too many out- +put images, which are very difficult to be checked man- +ually. According to the observational fact that the associ- +ated radio lobes are generally among the closest sources +to the optical center, we therefore only considered the four +closest radio sources to the central galaxy for association. +Unlike Ortiz Mart´ınez & Andernach (2016) to chase for +the extremely symmetric and collimated jets, we allowed +the angle between the two radio lobes (any pair of the four +closest radio sources) to vary in the range of 180±20◦ with +respect to the central galaxy. To avoid missing the cases of +blended core and lobes, we also accepted the cases which +have one or more radio sources close enough (⩽ 22.5′′, +half of the beam size of the NVSS) to the central spirals. +Finally, the quantity of the images that qualified the above +conditions was largely reduced and became suitable for eye +inspections. About 200 000 images in total were finally left +and inspected manually. +The probable candidates were then picked out and +further examined in composite images which combined +both information from radio and optical. The optical im- +ages were taken from the Dark Energy Spectroscopic +Instrument (DESI) Legacy Imaging Survey5 (Dey et al. +2019), which is deeper and has a better quality than the +SDSS. More importantly, the galaxy images observed by +DESI can be well modelled by “The Tractor” with the +point spread function considered (see Dey et al. 2019, for +5 http://legacysurvey.org/ +details). The model-subtracted image, namely, the residual +image, can be conveniently used to determine the existence +of spiral patterns of galaxies. +As in previous discoveries introduced in Sect. 1, the +identified host galaxies with double radio lobes all show +spiral patterns. Even for J0315−1906, which is somehow +edge-on, Ledlow et al. (1998) claimed the detection of a +spiral structure through a deep B-band exposure. We fol- +low the same discipline in this work that a spiral pattern +must be visible for the central optical galaxy. Edge-on +galaxies, which appear as a disk are therefore not consid- +ered. +3 RESULT AND DISCUSSION +By cross-matching the spiral sample taken from Kuminski +& Shamir (2016) with the radio catalogs of the NVSS +(Condon et al. 1998) and the FIRST (Becker et al. +1995), we successfully identify eight double-lobed spiral +galaxies, three of which, J0326−0623, J1110+0321, and +J1134+3046 are revealed for the first time. J1128+2417 +is another case that we independently discovered in this +work. However, it has recently been reported by Wu et al. +(2022) when we are preparing this manuscript for submis- +sion, and we have to list it as a known case. We add a +note for this object in Sect. 3.1. Another four previously +known cases: J1159+5820 (Kozieł-Wierzbowska et al. +2012), J1352+3126 (Donzelli et al. 2007), J1649+2635 +(Mao et al. 2015), and the Speca (J1409−0302)(Hota et al. +2011) have also been re-identified. Their recurrences val- +idate our searching strategy. By combining the radio and +optical data, we show the composite images for these eight +spiral galaxies hosting double radio lobes identified in this +work in Fig. 1. +For the other five known cases, J0836+0532 iden- +tified by Singh et al. (2015) is included in the cata- +log of Kuminski & Shamir (2016), but without morpho- +logical remarks and the classification certainty is p = +0.234, therefore it is missed. J0315−1906 (Ledlow et al. +1998), J0354−1340 (Vietri et al. 2022), and J2318+4314 +(Mulcahy et al. 2016) are not in the SDSS sky, hence we +cannot get them. J2345−0449 (Bagchi et al. 2014) is not +included in Kuminski & Shamir (2016). +3.1 Notes on the newly-identified spirals with double +radio lobes +3.1.1 J0326−0623 +J0326−0623 is a face-on galaxy at a redshift of z = 0.18 +with two major spiral arms, clearly shown in the zoomed +DESI image and the model-subtracted residual image in +Fig. 1. This galaxy is the brightest cluster galaxy (BCG) +of a galaxy cluster in the catalog of Yang et al. (2007), +which contains 13 bright member galaxies of M e +r ≤ −20.5 +mag. The total flux density detected by NVSS at 1.4 GHz +is about 6 mJy. However, the upper lobe is superimposed +by a point-like radio source as detected by FIRST, which is + +4 +X. Y. Gao et al. +Table 1 Parameters for 13 spiral galaxies that host double radio lobes. +Name +Ref.1 +RA +DEC +z +Stot +P1.4 GHz +p +log10M∗ environment Ngal Charc. Ref.2 +(J2000) +(J2000) +(mJy) +(W Hz−1) +(M⊙) +(1) +(2) +(3) +(4) +(5) +(6) +(7) +(8) +(9) +(10) +(11) +(12) +(13) +J0315−1906 +[1] +48.96708 −19.11233 0.067 +100 +1.05 × 1024 +— +10.88 +cluster +11 +member +[1] +J0354−1340 +[2,3] +58.63688 −13.66868 0.076 +15 +2.05 × 1023 +— +11.13 +— +– +– +– +J0836+0532 +[4] +129.23278 +5.54502 0.099 +62 +1.50 × 1024 0.23 +10.91 +group +4 +BGG +[4] +J2318+4314 +[5] +349.63650 +43.24692 0.012 +17 +5.26 × 1021 +— +9.12 +group +– +member +[5] +J2345−0449 +[6] +356.38625 +−4.82372 0.076 +181 +2.47 × 1024 +— +11.20 +group +7 +BGG +[13] +J1128+2417 +[7,0] +172.04848 +24.29636 0.169b +69 +5.34 × 1024 0.66 +10.57 +group +4 +member +[0] +J1159+5820 +[8,4,0] +179.77361 +58.34330 0.054 +338 +2.25 × 1024 0.85 +10.88 +group +1 +BGG +[14] +J1352+3126 +[9,4,0] +208.07450 +31.44625 0.045 4844 +2.21 × 1025 0.98 +10.91 +group +1 +BGG +[15] +J1409−0302 +[10,0] +212.45355 +−3.04237 0.138 +139 +6.91 × 1024 0.67s +11.14 +cluster +10 +BCG +[15] +J1649+2635 [11,4,12,0] 252.35005 +26.58405 0.055 +157 +1.09 × 1024 0.37s +10.88 +cluster +12 +BCG +[14] +J0326−0623 +[0] +51.59929 +−6.38431 0.180 +6⋄ +3.12 × 1023⋄ 0.67 +10.87 +cluster +13 +BCG +[16] +J1110+0321 +[0] +167.60458 +3.36078 0.030 +587 +1.17 × 1024 0.95 +9.14 +group +7 +member [15] +J1134+3046 +[0] +173.58466 +30.77959 0.046 +380 +1.82 × 1024 0.94 +9.51 +group +4 +member [15] +Notes: +Column (1) – (2): source name and the reference to find double lobes; Here Ref.1 is indicated by numbers: [0]= this work; +[1] = Ledlow et al. (1998); [2] = Chen et al. (2020); [3] = Vietri et al. (2022); [4] = Singh et al. (2015); [5] = Mulcahy et al. (2016); [6] += Bagchi et al. (2014); [7] = Wu et al. (2022); [8] = Kozieł-Wierzbowska et al. (2012); [9] = Donzelli et al. (2007); [10] = Hota et al. +(2011); [11] = Mao et al. (2015); [12] = Ortiz Mart´ınez & Andernach (2016); Column (3) – (5): right ascension, declination, and redshift +of spiral galaxies. The redshift information is taken from NED4 expect J1128+2417 labelled with “b”, which is from the photoZ from +SDSS DR8; Column (6): total radio continuum flux density measured by the NVSS at 1.4 GHz, “⋄” indicates that the flux density is +uncertain due to the mixture of an ir-relevant source (see Sect. 3.1). Column (7): radio powers calculated according to Equation 1. Column +(8): classification certainty taken from the “catalog.dat” of Kuminski & Shamir (2016) and the marker “s” indicates the classification +certainty from the “spec.dat” of Kuminski & Shamir (2016); Column (9): stellar mass of the galaxy; Column (10) – (12) : environment +as being a group or a cluster, the number of galaxies in the group/cluster and the character of the spiral in the environment, as being the +BCG/BGG or just a member. Column (13): Reference indicating spirals are in galaxy groups/clusters: [0, 1, 4, and 5] are the same as in +Column (2), together with [13] = Saulder et al. (2016); [14] = Tempel et al. (2018); [15] = Tempel et al. (2012); [16] = Yang et al. (2007). +associated with J032624−062212, a foreground galaxy at +z ∼ 0.16. The morphology of the radio lobes are slightly +bent. It has a size of ∼430 kpc as inferred by the NVSS 5σ +contour. +3.1.2 J1110+0321 +J1110+0321 is a blue galaxy at z = 0.03. The optical +observational and residual images shown in Fig. 1 indi- +cate spiral-arm structures. This galaxy belongs to a galaxy +group (Tempel et al. 2012), which contains seven mem- +bers brighter than −20.5 mag. The NVSS image shows an +elongated morphology with two lobes close to each other, +and FIRST detects bright sources at the peak in each lobes. +The inner-west component detected by the FIRST is prob- +ably partially associated with a background quasar QSO +B1107+0337 at z = 0.965. The overall scale of the radio +lobes measured based on the NVSS 5σ contour is about +100 kpc. +3.1.3 J1128+2417 +J1128+2417 is a blue galaxy at z = 0.169. The optical +residual image for this galaxy presents faint imprint of spi- +ral patterns, while the high-quality deeper image from the +Hubble Space Telescope (Wu et al. 2022) clearly shows +the existence of the spiral structures. With the method in- +troduced in Section 3.3, we find this galaxy is a satellite +galaxy in a galaxy group, which contains four members +with M e +r ≤ −20.5 mag. The NVSS map shows unresolved +radio lobes with an elongated morphology, but FIRST de- +tects two bright jets with some bridge emission. The scale +for the radio emission indicated by the NVSS is around +380 kpc. +3.1.4 J1134+3046 +J1134+3046 is also a blue galaxy at z = 0.046. The +optical residual map of this galaxy presents clear struc- +tures of spiral arms. This galaxy is a member in a galaxy +group (Tempel et al. 2012), which contains four bright +member galaxies with M e +r +≤ −20.5 mag. Similar to +the J1110+0321 and J1128+2417, the NVSS map of +J1134+3046 shows unresolved radio lobes with an elon- +gated morphology, while the FIRST image presents clear +jets. The overall scale of radio emission presented by the +NVSS map is approximately 190 kpc. +3.2 Relation between radio power and stellar mass of +the galaxy +The double radio lobes of the spiral galaxies come from +the central super massive black hole. It is therefore natu- +ral to speculate that the power of these radio lobes could +be related to the mass of the SMBH. The mass of the +SMBH is difficult to assess directly. However, it is related +to the mass of the host galaxies (e.g., Ferrarese & Merritt +2000; Tremaine et al. 2002; Marconi & Hunt 2003). The +total stellar mass of the host galaxy can be well estimated +based on the infrared luminosity of the galaxy which is +less affected by star formation history than an optical lu- +minosity (Bell et al. 2003; Wen et al. 2013). Wen & Han +(2021) found a good scaling relation between the stellar +mass of the galaxy and the 3.4 µm luminosity from the +Wide-field Infrared Survey Explorer (Wright et al. 2010). + +Three spiral galaxies with double radio lobes +5 +Fig. 1 Images for the eight spiral galaxies hosting double radio lobes identified in this work. The panels with large figures +show the optical DESI images (Dey et al. 2019) overlaid with radio contours, green for the NVSS (Condon et al. 1998) +and red for FIRST (Becker et al. 1995). Both the NVSS and FIRST contours satisfy ⟨Sbg⟩ + 5 × 2n/2σ mJy beam−1, +here n = 0, 1, 2, .... The source name and redshift are labeled on top of each plot. The cross indicates the center of the +radio images. The physical scale is shown at the bottom-right corner. The panels with small figures are the zoomed-in +DESI images and the model-subtracted residual images (Dey et al. 2019) for the eight spiral galaxies, with the image size +marked at the top-left corner. + +image size=301imagelsize=20J1134+3046(Z=0.046 ++30:48:00 +J2000 +DEC ++30:46:00 ++30:44:00 +100kpc +11:34:30 +11:34:20 +11:34:10 +RA(J2000)J1128+2417(Z=0.169 ++24:20:00 ++24:18:00 ++24:16:00 +500kpc +11:28:20 +11:28:10 +RA (J2000)Image_ size=40"image size=30J1110+0321(z=0.030 ++03:24:00 ++03:22:00 ++03:20:00 +100 kpc +11:10:30 +11:10:20 +RA (J2000)J0326-0623(Z=0.180 +-06:22:00 +-06:24:00 +-06:26:00 +500kpc +03:26:30 +03:26:20 +RA(J20006 +X. Y. Gao et al. +Fig.1 - continued +Following their procedure, we estimated the stellar mass of +each galaxy listed in Table 1. +On the other hand, the 1.4 GHz flux densities for the +radio lobes of all these galaxies were obtained from the +NVSS catalog, and the radio powers were calculated via +P1.4 GHz = 4πD2 +L × S1.4 GHz × (1 + z)1−β, +(1) +here P1.4 GHz is the radio power in the unit of 1023 +W Hz−1, DL = (1 + z) c +H0 +� z +0 +dz′ +√ +Ωm(1+z′)3+ΩΛ is the lu- + +Image_sze-40°imagesize=30J1649+2635(Z=0.055 ++26:37:00 +DEC(J2000) ++26:35:00 ++26:33:00 +100kpc +16:49:30 +16:49:20 +RA(J2000)J1409-0302 (Z=0.138 +-03:00:00 +DEC(J2000) +-03:03:00 +03:06:00 +500kpc +14:10:00 +14:09:50 +14:09:40 +RA (J2000)imagesize-907image size=100+31:30:00 +J1352+3126(Z=0.045 ++31:27:00 +DEC(J2000) ++31:24:00 +100 kpc +13:52:30 +13:52:20 +13:52:10 +RA (J2000)J1159+5820(z=0.054 ++58:23:00 +DEC(J2000 ++58:20:00 ++58:17:00 +100 kpc +11:59:30 +11:59:10 +11:58:50 +RA (J2000)Three spiral galaxies with double radio lobes +7 +−2 +−1 + 0 + 1 + 2 + 3 + 9 + 10 + 11 +J0326−0623 +J1110+0321 +J1134+3046 +log10(P1.4 GHz) (1023 W/Hz) +log10(M∗/M⊙) +Fig. 2 +Radio power versus stellar mass for ten known +(open) and three newly-identified (solid, name labelled) +spiral galaxies hosting double radio lobes. +minosity distance of a galaxy at a redshift z. S1.4 GHz is +the 1.4 GHz total flux density of the radio lobes in mJy +extracted from the NVSS catalog. (1 + z)(1−β) is the k- +correction term and β is the spectral index of radio galax- +ies. We adopted the statistical mean of β = 0.74 as ob- +tained by Lin & Mohr (2007). All the radio flux densities +and the corresponding radio powers are listed in Table 1. +The radio powers are further compared with the stellar +mass M∗ of the 10 known (open) and the three newly- +identified (solid) double-lobed spiral galaxies in Figure 2. +As shown in Figure 2, the known case of J2318+4314 +stands in the low-stellar mass and low-radio power cor- +ner, while all the other known cases are concentrated in the +upper-right corner with M∗ > 1010 M⊙ and L1.4 GHz > +1023 W Hz−1. Due to the limited data in the low-stellar +mass and low radio power end and the data between the +low- and high-end, it is difficult to draw a conclusion that +the radio power of the lobes is proportional to the stellar +mass of the galaxy, and further the mass of the SMBH. +We also put the three newly-identified cases in Fig. 2. +J1110+0321 and J1134+3046, with a lower stellar mass +but a higher radio power appear in the upper-left corner of +the plot, which further scatters the data-point distribution. +Wu et al. (2022) showed a positive correlation between +L1.4 GHz and M∗ for the nine previously-known cases +(see the first ten entries in Table 1, except J1128+2417) +with/without their 18 new disk galaxies hosting double ra- +dio lobes (see their Fig. 9). They estimated the stellar mass +of the galaxy M∗ by using the SDSS multi-band photome- +try. +3.3 Environment of the spiral-hosted double-lobed +sources +The mechanism for powering the large-scale double ra- +dio lobes by spiral galaxies is not clear. Physically, radio +lobes may be related to dense environment. Hota et al. +(2011) pointed out that J1409−0302 belongs to a galaxy +cluster of MaxBCG J212.45357−03.04237 and it is the +central BCG. Its relic radio lobes may result from the +accretion of galactic filament. Based on the morphology, +Singh et al. (2015) suggested a merger scenario between +a spiral and an elliptical galaxy for both J1159+5820 +and J1352+3126. They also noticed that J0836+0532 and +J1352+3126 are in galaxy groups with very limited group +members, but listed them as field galaxies together with +J1159+5820. Mao et al. (2015) found that J1649+2635 is +in a group rather than a cluster environment and may in- +teract with another group, where the bright galaxy SDSS +J164933.52+265052.0 resides in. +With all such accumulated samples and the new dis- +coveries as listed in Table 1, we can have a good statis- +tics on the environment of these spiral galaxies. Based on +the galaxy group/cluster catalogs (e.g. Yang et al. 2007; +Tempel et al. 2012; Tully 2015; Tempel et al. 2018), we +find that all of them are located in a galaxy group or a clus- +ter, except J0354−1340, for which the information is not +available. +We further used the SDSS data to evaluate the rich- +ness of their parent system. We followed the procedures of +Wen et al. (2012) and Wen & Han (2015) by counting the +member galaxies with M e +r ≤ −20.5 mag if they have a +velocity difference of 2500 km s−1 from the group or clus- +ter when the spectroscopic redshift is available or have a +redshift difference of 0.04(1 + z) if only photo-metric red- +shifts are available. Here, M e +r is evolution-corrected from +Mr with M e +r = Mr + 1.16z. The number of such bright +member galaxies Ngal is listed in Table 1. We noticed that +the member galaxies in the parent system of these spirals +with double radio lobes are much less than those in the +cluster catalog of Wen et al. (2012). Based on the numbers +of member galaxies counted in the above way, we here call +the system a “cluster” if ten or more members are included, +or a “group” if less than 10 members are found. In addition, +we found that more than half of these spirals with double +radio lobes are the BCG or brightest group galaxy (BGG) +in the parent system. +Among the 18 objects associated with double ra- +dio lobes found in Wu et al. (2022), some of the cen- +tral optical galaxies seen face-on or have small inclina- +tion angles show un-ambiguous spiral patterns, qualify- +ing our selection based on morphology. They should be- +long to the same type as the objects discussed in this +work. Except for J1128+241 in their Table 1, which +is the same as our target J1128+2417 as listed in +Table 1, we picked another seven galaxies from Wu et al. +(2022): J0209+075, J0219+015, J0806+062, J0832+184, +J1328+571, J1656+640, and J1721+262 which present +clear spiral arms. We checked their environment as de- +scribed above. Three of them: J0219+015, J0832+184, +and J1721+262 are found in galaxy group and clus- +ter (Tully 2015; Tempel et al. 2012; Yang et al. 2007), +and all the three are the BGGs (J0219+015: Ngal = 6, +J0832+184: Ngal = 3) and BCG (J1721+262, Ngal = 15). + +8 +X. Y. Gao et al. +For J0209+075 (Ngal = 0), J0806+062 (Ngal = 1), +J1328+571 (Ngal = 0), and J1656+640 (Ngal = 5), we +failed to identify them to be located in galaxy groups +or clusters. Ngal = 5 for J1656+640 may be the result +of the projection effect due to the large redshift slice of +0.04(1 + z). +4 CONCLUDING REMARKS +By cross-matching a large sample of machine-selected +spiral galaxies from the SDSS (York et al. 2000) DR8 +(Kuminski & Shamir 2016) with the full radio source +catalogs of the NVSS (Condon et al. 1998) and the +FIRST (Becker et al. 1995), we identify three new spi- +rals, J0326–0623, J1110+0321, and J1134+3046 hosting +double radio lobes, together with five previously known +double-lobed spirals. +With the largest sample of double-lobed spiral galax- +ies by far, we noticed that most spiral galaxies that host +double radio lobes are usually located in the galaxy groups +or galaxy clusters. More than a half of them are the BGGs +or BCGs, implying that the formation of double radio lobes +may be highly related to their surrounding environment. A +more noteworthy fact is that the galaxy groups or clusters +where these spirals reside in have very limited members, +i.e. the environmental density is denser than the field, but +not so dense and hot as in the center of rich clusters where +spirals may be destroyed. +Acknowledgements We thank the anonymous referee +for helpful comments. The authors are supported by +the National Natural Science Foundation of China +(11988101), the National SKA Program of China (Grant +No. 2022SKA0120103), the National Key R&D Program +of China (No. 2021YFA1600401 and 2021YFA1600400), +and the Open Project Program of the Key Laboratory of +FAST, NAOC, Chinese Academy of Sciences. XYG ac- +knowledges the financial support from the CAS-NWO +cooperation programme (Grant No. GJHZ1865). The +National Radio Astronomy Observatory is a facility of the +National Science Foundation operated under cooperative +agreement by Associated Universities, Inc. 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L. 2016, MNRAS, 460, +3669 2 + diff --git a/PtAzT4oBgHgl3EQflf1o/content/tmp_files/load_file.txt b/PtAzT4oBgHgl3EQflf1o/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..03363dbc8332cea92110908ac3157a25bd3f675f --- /dev/null +++ b/PtAzT4oBgHgl3EQflf1o/content/tmp_files/load_file.txt @@ -0,0 +1,789 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf,len=788 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='01548v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='GA] 4 Jan 2023 RAA Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='0 (20xx) No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='0, 000–000 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='raa-journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='org http://iopscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='iop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='org/raa Research in Astronomy and Astrophysics Three new spiral galaxies with active nuclei producing double radio lobes X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Gao1,2,3, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Yuan1,2, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Han1,2,3, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Wen1,2,3 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Shan1,3 1 National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' xygao@nao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='cas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' hjl@nao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='cas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='cn 2 CAS Key Laboratory of FAST, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 3 School of Astronomy, University of Chinese Academy of Sciences, Beijing 100049, China Received 20xx month day;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' accepted 20xx month day Abstract Double radio lobes are generally believed to be produced by active nuclei of elliptical galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' However, several double-lobed radio sources have been solidly found to be associated with spiral galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' By cross-matching ∼ 9 × 105 spiral galaxies selected from the SDSS DR8 data with the full 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='4 GHz radio source catalogs of NVSS and FIRST, we identify three new spiral galaxies: J0326−0623, J1110+0321 and J1134+3046 that produce double radio lobes, in addition to five double-lobed spirals previously known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' By combining the newly discovered and all the other known cases in literature, we find that most of these spiral galaxies are located in a galaxy group or a poor cluster, in which the environment is denser than in the field, and about half of them are the central brightest galaxies in their parent system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' We therefore suggest that the environment is one of the key factors for a spiral to produce double radio lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Key words: galaxies: general — galaxies: – active — galaxies: spiral — radio continuum: galaxies — galaxies: jets 1 INTRODUCTION The radio lobes powered by the supermassive black hole (SMBH) in the center of galaxies are usually hosted by el- liptical galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' They extend to several hundreds of kilo- parsecs which are much larger than their optical counter- parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The point of view that double radio lobes are exclu- sively hosted by elliptical galaxies has been challenged by the discovery of the double-lobed radio source J0313−192 (J0315−1906), which was found to be hosted by a spiral galaxy (Ledlow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1998, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Keel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' In the last two decades, a few more spiral galaxies hosting double radio lobes have been revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Hota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2011) discovered the second case Speca (J1409−0302) with episodic radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Bagchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2014) iden- tified J2345−0449 with lobes extending to an extraordi- nary length of ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='6 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Mulcahy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2016) reported the serendipitous discovery of the double radio source of MCG+07–47–10 (J2318+4314) with a low luminosity of P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='4GHz ∼ 1022 W Hz−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Very recently, Vietri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2022) identified a new source J0354−1340 with the dou- ble radio lobes extending approximately 240 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Searches for spiral galaxies with double radio lobes have also been made systematically by several groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2015) cross-matched the optical Galaxy Zoo “superclean” sample (Lintott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2008) with the Unified Radio Catalog of Kimball & Ivezi´c (2008), which includes radio sources from both the Faint Images of the Radio Sky at Twenty-centimeters (FIRST, Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1995) and the NRAO VLA Sky Survey (NVSS, Condon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1998) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' They reported a new spiral J1649+2635 with double ra- dio lobes, with a radio power of about 1024 W Hz−1 at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='4 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2015) cross-matched the FIRST catalog (Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1995) with 187 005 spiral galaxies (Meert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2015) in the Sloan Digital Sky Survey (SDSS, York et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2000) Data Release 7 (DR7) to search for the coincidence of core source within a radius of 3′′ and also the double lobes within a radius of 3′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' They made addi- tional search for the extended radio lobes with the NVSS data (Condon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1998) for the obtained FIRST–SDSS matched objects and identified four spiral galaxies with double radio lobes, among which J1159+5820 (Kozieł- Wierzbowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2012), J1352+3126 (Donzelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2007), and J1649+2635 (Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2015) have already been reported previously, while J0836+0532 was found for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Ortiz Mart´ınez & Andernach (2016) col- lected 675 874 spiral galaxies from several spiral galaxy samples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=', Huertas-Company et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Willett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Kuminski & Shamir 2016), and searched for the as- sociated FIRST sources (Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1995) within a larger radius of 6′ under constraints of angular distances, position angles, and arm-length ratio of the double radio sources with respect to the central optical galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Concentrating on the extremely symmetric and aligned radio lobes, they 2 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' finally reported the re-discovery of the known case of J1649+2635 (Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Though these efforts have been dedicated to search for the spiral galaxies hosting double radio lobes, only a handful of cases have been con- firmed to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' It is also unclear why these spirals hold double ra- dio lobes while the vast majority of other spirals do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Often radio lobes may be triggered by the accretion of host galaxies from the over-dense environments in their vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' For example, the source J0315−1906 is a mem- ber galaxy of the cluster Abell 428 (Ledlow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' J1409−0302 and J1649+2635 are the brightest galaxies of their parent systems (Hota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2015), and J2318+4314 is located close to the galaxy groups NGC 7618 and UGC 12491 (Mulcahy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' However, Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2015) found that J0836+0532 and J1352+3126 are in galaxy groups with very limited members and listed them as field galaxies together with J1159+5820 (see their Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Therefore a large sample of such galaxies are needed to investigate the environmen- tal effect on radio lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2022) recently analyzed the optical images from the Hubble Space Telescope of a sample of galaxies with extended double radio lobes seen from FIRST (Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1995), and found that 18 disk galaxies are of high probability to have the genuine associ- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Some of these disk galaxies have a small inclination angle and show clear spiral patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' We noticed that Kuminski & Shamir (2016) classified the broad morphological types of ∼ 3 × 106 galaxies in the SDSS (York et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2000) DR8 by analyzing images of galaxies with computer programs, and their pipeline picked out ∼ 9 × 105 spiral galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Here, we take this large sample as the optical basis of spiral galaxies, and cross-match them with the full radio source catalogs of NVSS (Condon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1998) and FIRST (Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' We discover three new spiral galaxies with double radio lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2, we introduce the data sets used to identify the double-lobed spiral galaxies, and also the procedure of identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' We show the results and discuss the properties of the galaxies and their environments in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The concluding remarks are given in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Throughout the paper, we adopted a flat ΛCDM cos- mology with H0 = 70 km s−1 Mpc−1, Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='3 and ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2 DATA AND SEARCH STRATEGY 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='1 Data By automatic computer program, Kuminski & Shamir (2016) classified approximately 9×105 spiral galaxies out of ∼3 × 106 galaxies observed in the SDSS (York et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2000) DR8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The classification for spiral galaxies and el- liptical galaxies shows good agreement with those of the Galaxy Zoo debiased “superclean” sample (Lintott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2008) and the agreement rate is claimed to reach 98% when the “classification certainty” p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Two catalogs were released by Kuminski & Shamir (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The “catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='dat”1 is the catalog of the broad mor- phology of SDSS galaxies with only the classification cer- tainty p indicated, and the “spec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='dat” including both in- dications of “p” and morphological remarks (Elliptical, Spiral, Star) is the morphological catalog of the SDSS ob- jects with spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' In this work, we first take all the spiral galaxies in the two catalogs with p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' However, we noticed that some known spirals such as J0836+0532 and J1649+2635 as shown in Table 1 have a classification cer- tainty of p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='234 and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='371, respectively, less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' In order not to miss many real spirals, we simply in- cluded all galaxies marked as “Spiral” in the “spec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='dat” of Kuminski & Shamir (2016), regardless of the values of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Therefore, all 366 836 entries in the “spec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='dat” marked as “Spiral” and 1 184 922 entries with p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='54 in “cat- alog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='dat” are used to search for associated double radio lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The duplicates in the two catalogs are treated at the final stage when inspecting the association between optical and radio images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The radio counterparts of the optical spiral galaxies are searched in the NVSS (Condon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1998) and the FIRST (Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1995) source catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The NVSS was car- ried out with the Very Large Array (VLA) – D configura- tion at the frequency of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='4 GHz with an angular resolution of ∼ 45′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The survey covers the entire sky north of the dec- lination of δ = −40◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Over 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='8 million discrete sources brighter than ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='5 mJy (5σ level) were compiled into a catalog2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The positional accuracy of NVSS is about 1′′ for strong sources and 7′′ for faint sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The FIRST obser- vations were conducted at the same frequency, but with a much better angular resolution of ∼ 5′′ by using the VLA – B configuration array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' It has a sensitivity of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='13 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The sky coverage of FIRST is limited within the northern and southern Galactic cap regions of about 10 000 square degrees in total, which is less than one third of that of the NVSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The FIRST catalog contains over 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='4 × 105 entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' For the sources whose flux density is higher than 1 mJy, the radius of the 90% positional confidence error circle is less than 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The latest FIRST source catalog of Version 14Dec173 was used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='2 Search strategy Based on the experiences gained from previous work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2016), we learned that the 5′′ angular resolution of FIRST could resolve out extended radio structures so that diffuse radio lobes can be missed, such as the case for J1409–0302 (Speca, Hota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Therefore we took the NVSS data as the fundamental basis and the FIRST data were used as an auxiliary database for radio-lobe iden- tification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The radius to search for the radio counterpart of the central optical spiral galaxy is tricky: the larger the ra- dius is set, the more radio sources around the central galaxy 1 https://cdsarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='fr/viz-bin/cat/J/ApJS/223/20#/browse 2 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='cv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='nrao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='edu/nvss/NVSSlist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='shtml 3 http://sundog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='edu/first/catalogs/readme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='html Three spiral galaxies with double radio lobes 3 one would get, and then the more time would be consumed for distinguishing the true association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Therefore a trade- off should be made on setting the searching area to bal- ance the time consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' By reviewing the known cases to date (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Ledlow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Hota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Bagchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Mulcahy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2016), we searched for the NVSS and FIRST sources within 800 kpc around the central optical spirals if the red- shift information is available;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' otherwise a radius of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='5′ around the spiral is set for the NVSS sources and 30′′ for the FIRST sources if the redshift is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Spiral galaxies with double radio lobes may have var- ious appearances in the radio images of NVSS and FIRST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' In the low resolution image of the NVSS, they could have (1) a central core with distinct double radio lobes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' J1352+3126 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 4 in Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2) un- resolved central core and double radio lobes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' show- ing a structure that is elongated, such as J1649+2635 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 5 in Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' and (3) distinct double-lobe structure without a core component intrinsically or extrin- sically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' In the high resolution image of FIRST, the spi- rals that host double radio lobes could show (1) a cen- tral core with distinct double radio lobes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' J1649+2635 (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2) only a central core, because the ex- tended lobes are resolved out by the small synthesis beam;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' and (3) distinct double-lobe structures without a core, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' J1409−0302 (Hota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The real cases can be any reasonable combinations of the above possibilities for the NVSS and FIRST data in their common surveyed area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' However, such loose constraints will yield too many out- put images, which are very difficult to be checked man- ually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' According to the observational fact that the associ- ated radio lobes are generally among the closest sources to the optical center, we therefore only considered the four closest radio sources to the central galaxy for association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Unlike Ortiz Mart´ınez & Andernach (2016) to chase for the extremely symmetric and collimated jets, we allowed the angle between the two radio lobes (any pair of the four closest radio sources) to vary in the range of 180±20◦ with respect to the central galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' To avoid missing the cases of blended core and lobes, we also accepted the cases which have one or more radio sources close enough (⩽ 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='5′′, half of the beam size of the NVSS) to the central spirals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Finally, the quantity of the images that qualified the above conditions was largely reduced and became suitable for eye inspections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' About 200 000 images in total were finally left and inspected manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The probable candidates were then picked out and further examined in composite images which combined both information from radio and optical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The optical im- ages were taken from the Dark Energy Spectroscopic Instrument (DESI) Legacy Imaging Survey5 (Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2019), which is deeper and has a better quality than the SDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' More importantly, the galaxy images observed by DESI can be well modelled by “The Tractor” with the point spread function considered (see Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2019, for 5 http://legacysurvey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='org/ details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The model-subtracted image, namely, the residual image, can be conveniently used to determine the existence of spiral patterns of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' As in previous discoveries introduced in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1, the identified host galaxies with double radio lobes all show spiral patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Even for J0315−1906, which is somehow edge-on, Ledlow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (1998) claimed the detection of a spiral structure through a deep B-band exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' We fol- low the same discipline in this work that a spiral pattern must be visible for the central optical galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Edge-on galaxies, which appear as a disk are therefore not consid- ered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 3 RESULT AND DISCUSSION By cross-matching the spiral sample taken from Kuminski & Shamir (2016) with the radio catalogs of the NVSS (Condon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1998) and the FIRST (Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1995), we successfully identify eight double-lobed spiral galaxies, three of which, J0326−0623, J1110+0321, and J1134+3046 are revealed for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' J1128+2417 is another case that we independently discovered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' However, it has recently been reported by Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2022) when we are preparing this manuscript for submis- sion, and we have to list it as a known case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' We add a note for this object in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Another four previously known cases: J1159+5820 (Kozieł-Wierzbowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2012), J1352+3126 (Donzelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2007), J1649+2635 (Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2015), and the Speca (J1409−0302)(Hota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2011) have also been re-identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Their recurrences val- idate our searching strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' By combining the radio and optical data, we show the composite images for these eight spiral galaxies hosting double radio lobes identified in this work in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' For the other five known cases, J0836+0532 iden- tified by Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2015) is included in the cata- log of Kuminski & Shamir (2016), but without morpho- logical remarks and the classification certainty is p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='234, therefore it is missed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' J0315−1906 (Ledlow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1998), J0354−1340 (Vietri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2022), and J2318+4314 (Mulcahy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2016) are not in the SDSS sky, hence we cannot get them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' J2345−0449 (Bagchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2014) is not included in Kuminski & Shamir (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='1 Notes on the newly-identified spirals with double radio lobes 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='1 J0326−0623 J0326−0623 is a face-on galaxy at a redshift of z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='18 with two major spiral arms, clearly shown in the zoomed DESI image and the model-subtracted residual image in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' This galaxy is the brightest cluster galaxy (BCG) of a galaxy cluster in the catalog of Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2007), which contains 13 bright member galaxies of M e r ≤ −20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='5 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The total flux density detected by NVSS at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='4 GHz is about 6 mJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' However, the upper lobe is superimposed by a point-like radio source as detected by FIRST, which is 4 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Table 1 Parameters for 13 spiral galaxies that host double radio lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Name Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='1 RA DEC z Stot P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='4 GHz p log10M∗ environment Ngal Charc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='2 (J2000) (J2000) (mJy) (W Hz−1) (M⊙) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) J0315−1906 [1] 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='96708 −19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='11233 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='067 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='05 × 1024 — 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='88 cluster 11 member [1] J0354−1340 [2,3] 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='63688 −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='66868 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='076 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='05 × 1023 — 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='13 — – – – J0836+0532 [4] 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='23278 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='54502 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='099 62 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='50 × 1024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='23 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='91 group 4 BGG [4] J2318+4314 [5] 349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='63650 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='24692 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='012 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='26 × 1021 — 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='12 group – member [5] J2345−0449 [6] 356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='38625 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='82372 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='076 181 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='47 × 1024 — 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='20 group 7 BGG [13] J1128+2417 [7,0] 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='04848 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='29636 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='169b 69 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='34 × 1024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='66 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='57 group 4 member [0] J1159+5820 [8,4,0] 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='77361 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='34330 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='054 338 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='25 × 1024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='85 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='88 group 1 BGG [14] J1352+3126 [9,4,0] 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='07450 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='44625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='045 4844 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='21 × 1025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='98 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='91 group 1 BGG [15] J1409−0302 [10,0] 212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='45355 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='04237 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='138 139 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='91 × 1024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='67s 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='14 cluster 10 BCG [15] J1649+2635 [11,4,12,0] 252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='35005 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='58405 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='055 157 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='09 × 1024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='37s 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='88 cluster 12 BCG [14] J0326−0623 [0] 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='59929 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='38431 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='180 6⋄ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='12 × 1023⋄ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='67 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='87 cluster 13 BCG [16] J1110+0321 [0] 167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='60458 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='36078 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='030 587 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='17 × 1024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='95 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='14 group 7 member [15] J1134+3046 [0] 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='58466 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='77959 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='046 380 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='82 × 1024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='94 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='51 group 4 member [15] Notes: Column (1) – (2): source name and the reference to find double lobes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Here Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='1 is indicated by numbers: [0]= this work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' [1] = Ledlow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (1998);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' [2] = Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' [3] = Vietri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' [4] = Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' [5] = Mulcahy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' [6] = Bagchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' [7] = Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' [8] = Kozieł-Wierzbowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' [9] = Donzelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' [10] = Hota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' [11] = Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' [12] = Ortiz Mart´ınez & Andernach (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Column (3) – (5): right ascension, declination, and redshift of spiral galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The redshift information is taken from NED4 expect J1128+2417 labelled with “b”, which is from the photoZ from SDSS DR8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Column (6): total radio continuum flux density measured by the NVSS at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='4 GHz, “⋄” indicates that the flux density is uncertain due to the mixture of an ir-relevant source (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Column (7): radio powers calculated according to Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Column (8): classification certainty taken from the “catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='dat” of Kuminski & Shamir (2016) and the marker “s” indicates the classification certainty from the “spec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='dat” of Kuminski & Shamir (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Column (9): stellar mass of the galaxy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Column (10) – (12) : environment as being a group or a cluster, the number of galaxies in the group/cluster and the character of the spiral in the environment, as being the BCG/BGG or just a member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Column (13): Reference indicating spirals are in galaxy groups/clusters: [0, 1, 4, and 5] are the same as in Column (2), together with [13] = Saulder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' [14] = Tempel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' [15] = Tempel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' [16] = Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' associated with J032624−062212, a foreground galaxy at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The morphology of the radio lobes are slightly bent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' It has a size of ∼430 kpc as inferred by the NVSS 5σ contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='2 J1110+0321 J1110+0321 is a blue galaxy at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The optical observational and residual images shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1 indi- cate spiral-arm structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' This galaxy belongs to a galaxy group (Tempel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2012), which contains seven mem- bers brighter than −20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='5 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The NVSS image shows an elongated morphology with two lobes close to each other, and FIRST detects bright sources at the peak in each lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The inner-west component detected by the FIRST is prob- ably partially associated with a background quasar QSO B1107+0337 at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The overall scale of the radio lobes measured based on the NVSS 5σ contour is about 100 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='3 J1128+2417 J1128+2417 is a blue galaxy at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The optical residual image for this galaxy presents faint imprint of spi- ral patterns, while the high-quality deeper image from the Hubble Space Telescope (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2022) clearly shows the existence of the spiral structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' With the method in- troduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='3, we find this galaxy is a satellite galaxy in a galaxy group, which contains four members with M e r ≤ −20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='5 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The NVSS map shows unresolved radio lobes with an elongated morphology, but FIRST de- tects two bright jets with some bridge emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The scale for the radio emission indicated by the NVSS is around 380 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='4 J1134+3046 J1134+3046 is also a blue galaxy at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The optical residual map of this galaxy presents clear struc- tures of spiral arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' This galaxy is a member in a galaxy group (Tempel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2012), which contains four bright member galaxies with M e r ≤ −20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='5 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Similar to the J1110+0321 and J1128+2417, the NVSS map of J1134+3046 shows unresolved radio lobes with an elon- gated morphology, while the FIRST image presents clear jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The overall scale of radio emission presented by the NVSS map is approximately 190 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='2 Relation between radio power and stellar mass of the galaxy The double radio lobes of the spiral galaxies come from the central super massive black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' It is therefore natu- ral to speculate that the power of these radio lobes could be related to the mass of the SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The mass of the SMBH is difficult to assess directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' However, it is related to the mass of the host galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=', Ferrarese & Merritt 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Tremaine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Marconi & Hunt 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The total stellar mass of the host galaxy can be well estimated based on the infrared luminosity of the galaxy which is less affected by star formation history than an optical lu- minosity (Bell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Wen & Han (2021) found a good scaling relation between the stellar mass of the galaxy and the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='4 µm luminosity from the Wide-field Infrared Survey Explorer (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Three spiral galaxies with double radio lobes 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1 Images for the eight spiral galaxies hosting double radio lobes identified in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The panels with large figures show the optical DESI images (Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2019) overlaid with radio contours, green for the NVSS (Condon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1998) and red for FIRST (Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Both the NVSS and FIRST contours satisfy ⟨Sbg⟩ + 5 × 2n/2σ mJy beam−1, here n = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='. The source name and redshift are labeled on top of each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The cross indicates the center of the radio images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The physical scale is shown at the bottom-right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The panels with small figures are the zoomed-in DESI images and the model-subtracted residual images (Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2019) for the eight spiral galaxies, with the image size marked at the top-left corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' image size=301imagelsize=20J1134+3046(Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='046 +30:48:00 J2000 DEC +30:46:00 +30:44:00 100kpc 11:34:30 11:34:20 11:34:10 RA(J2000)J1128+2417(Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='169 +24:20:00 +24:18:00 +24:16:00 500kpc 11:28:20 11:28:10 RA (J2000)Image_ size=40"image size=30J1110+0321(z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='030 +03:24:00 +03:22:00 +03:20:00 100 kpc 11:10:30 11:10:20 RA (J2000)J0326-0623(Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='180 06:22:00 06:24:00 06:26:00 500kpc 03:26:30 03:26:20 RA(J20006 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='1 - continued Following their procedure, we estimated the stellar mass of each galaxy listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' On the other hand, the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='4 GHz flux densities for the radio lobes of all these galaxies were obtained from the NVSS catalog, and the radio powers were calculated via P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='4 GHz = 4πD2 L × S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='4 GHz × (1 + z)1−β, (1) here P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='4 GHz is the radio power in the unit of 1023 W Hz−1, DL = (1 + z) c H0 � z 0 dz′ √ Ωm(1+z′)3+ΩΛ is the lu- Image_sze-40°imagesize=30J1649+2635(Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='055 +26:37:00 DEC(J2000) +26:35:00 +26:33:00 100kpc 16:49:30 16:49:20 RA(J2000)J1409-0302 (Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='138 03:00:00 DEC(J2000) 03:03:00 03:06:00 500kpc 14:10:00 14:09:50 14:09:40 RA (J2000)imagesize-907image size=100+31:30:00 J1352+3126(Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='045 +31:27:00 DEC(J2000) +31:24:00 100 kpc 13:52:30 13:52:20 13:52:10 RA (J2000)J1159+5820(z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='054 +58:23:00 DEC(J2000 +58:20:00 +58:17:00 100 kpc 11:59:30 11:59:10 11:58:50 RA (J2000)Three spiral galaxies with double radio lobes 7 −2 −1 0 1 2 3 9 10 11 J0326−0623 J1110+0321 J1134+3046 log10(P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='4 GHz) (1023 W/Hz) log10(M∗/M⊙) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2 Radio power versus stellar mass for ten known (open) and three newly-identified (solid, name labelled) spiral galaxies hosting double radio lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' minosity distance of a galaxy at a redshift z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='4 GHz is the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='4 GHz total flux density of the radio lobes in mJy extracted from the NVSS catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (1 + z)(1−β) is the k- correction term and β is the spectral index of radio galax- ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' We adopted the statistical mean of β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='74 as ob- tained by Lin & Mohr (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' All the radio flux densities and the corresponding radio powers are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The radio powers are further compared with the stellar mass M∗ of the 10 known (open) and the three newly- identified (solid) double-lobed spiral galaxies in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' As shown in Figure 2, the known case of J2318+4314 stands in the low-stellar mass and low-radio power cor- ner, while all the other known cases are concentrated in the upper-right corner with M∗ > 1010 M⊙ and L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='4 GHz > 1023 W Hz−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Due to the limited data in the low-stellar mass and low radio power end and the data between the low- and high-end, it is difficult to draw a conclusion that the radio power of the lobes is proportional to the stellar mass of the galaxy, and further the mass of the SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' We also put the three newly-identified cases in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' J1110+0321 and J1134+3046, with a lower stellar mass but a higher radio power appear in the upper-left corner of the plot, which further scatters the data-point distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2022) showed a positive correlation between L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='4 GHz and M∗ for the nine previously-known cases (see the first ten entries in Table 1, except J1128+2417) with/without their 18 new disk galaxies hosting double ra- dio lobes (see their Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' They estimated the stellar mass of the galaxy M∗ by using the SDSS multi-band photome- try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='3 Environment of the spiral-hosted double-lobed sources The mechanism for powering the large-scale double ra- dio lobes by spiral galaxies is not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Physically, radio lobes may be related to dense environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Hota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2011) pointed out that J1409−0302 belongs to a galaxy cluster of MaxBCG J212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='45357−03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='04237 and it is the central BCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Its relic radio lobes may result from the accretion of galactic filament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Based on the morphology, Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2015) suggested a merger scenario between a spiral and an elliptical galaxy for both J1159+5820 and J1352+3126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' They also noticed that J0836+0532 and J1352+3126 are in galaxy groups with very limited group members, but listed them as field galaxies together with J1159+5820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2015) found that J1649+2635 is in a group rather than a cluster environment and may in- teract with another group, where the bright galaxy SDSS J164933.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='52+265052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='0 resides in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' With all such accumulated samples and the new dis- coveries as listed in Table 1, we can have a good statis- tics on the environment of these spiral galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Based on the galaxy group/cluster catalogs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Tempel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Tully 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Tempel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2018), we find that all of them are located in a galaxy group or a clus- ter, except J0354−1340, for which the information is not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' We further used the SDSS data to evaluate the rich- ness of their parent system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' We followed the procedures of Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2012) and Wen & Han (2015) by counting the member galaxies with M e r ≤ −20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='5 mag if they have a velocity difference of 2500 km s−1 from the group or clus- ter when the spectroscopic redshift is available or have a redshift difference of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='04(1 + z) if only photo-metric red- shifts are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Here, M e r is evolution-corrected from Mr with M e r = Mr + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='16z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The number of such bright member galaxies Ngal is listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' We noticed that the member galaxies in the parent system of these spirals with double radio lobes are much less than those in the cluster catalog of Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Based on the numbers of member galaxies counted in the above way, we here call the system a “cluster” if ten or more members are included, or a “group” if less than 10 members are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' In addition, we found that more than half of these spirals with double radio lobes are the BCG or brightest group galaxy (BGG) in the parent system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Among the 18 objects associated with double ra- dio lobes found in Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2022), some of the cen- tral optical galaxies seen face-on or have small inclina- tion angles show un-ambiguous spiral patterns, qualify- ing our selection based on morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' They should be- long to the same type as the objects discussed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Except for J1128+241 in their Table 1, which is the same as our target J1128+2417 as listed in Table 1, we picked another seven galaxies from Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' (2022): J0209+075, J0219+015, J0806+062, J0832+184, J1328+571, J1656+640, and J1721+262 which present clear spiral arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' We checked their environment as de- scribed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Three of them: J0219+015, J0832+184, and J1721+262 are found in galaxy group and clus- ter (Tully 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Tempel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2007), and all the three are the BGGs (J0219+015: Ngal = 6, J0832+184: Ngal = 3) and BCG (J1721+262, Ngal = 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 8 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' For J0209+075 (Ngal = 0), J0806+062 (Ngal = 1), J1328+571 (Ngal = 0), and J1656+640 (Ngal = 5), we failed to identify them to be located in galaxy groups or clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Ngal = 5 for J1656+640 may be the result of the projection effect due to the large redshift slice of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='04(1 + z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 4 CONCLUDING REMARKS By cross-matching a large sample of machine-selected spiral galaxies from the SDSS (York et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2000) DR8 (Kuminski & Shamir 2016) with the full radio source catalogs of the NVSS (Condon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1998) and the FIRST (Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 1995), we identify three new spi- rals, J0326–0623, J1110+0321, and J1134+3046 hosting double radio lobes, together with five previously known double-lobed spirals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' With the largest sample of double-lobed spiral galax- ies by far, we noticed that most spiral galaxies that host double radio lobes are usually located in the galaxy groups or galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' More than a half of them are the BGGs or BCGs, implying that the formation of double radio lobes may be highly related to their surrounding environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' A more noteworthy fact is that the galaxy groups or clusters where these spirals reside in have very limited members, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' the environmental density is denser than the field, but not so dense and hot as in the center of rich clusters where spirals may be destroyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Acknowledgements We thank the anonymous referee for helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The authors are supported by the National Natural Science Foundation of China (11988101), the National SKA Program of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2022SKA0120103), the National Key R&D Program of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2021YFA1600401 and 2021YFA1600400), and the Open Project Program of the Key Laboratory of FAST, NAOC, Chinese Academy of Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' XYG ac- knowledges the financial support from the CAS-NWO cooperation programme (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' GJHZ1865).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Sloan Foundation, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' Department of Energy Office of Science, and the Participating Institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' SDSS acknowledges support and resources from the Center for High-Performance Computing at the University of Utah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' The SDSS web site is www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='sdss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' References Bagchi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=', Vivek, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=', Vikram, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=', et 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} +page_content=' 2016, MNRAS, 460, 3669 2' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQflf1o/content/2301.01548v1.pdf'} diff --git a/QdFRT4oBgHgl3EQf7Dgl/content/2301.13678v1.pdf b/QdFRT4oBgHgl3EQf7Dgl/content/2301.13678v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..cbba5ebb81bc2bde03408f8a02b7f2e25766ca9c --- /dev/null +++ b/QdFRT4oBgHgl3EQf7Dgl/content/2301.13678v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:842587337039a1398dbb662c5e8c06e585586f7357fbfc0d12b0f08dbac54041 +size 589963 diff --git a/QdFRT4oBgHgl3EQf7Dgl/vector_store/index.pkl b/QdFRT4oBgHgl3EQf7Dgl/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..d48475ae3e139bc232a1d6fefb20d119b9c9f935 --- /dev/null +++ b/QdFRT4oBgHgl3EQf7Dgl/vector_store/index.pkl @@ -0,0 +1,3 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Karlsson1 and F. Aryasetiawan2 +1Department of Engineering Sciences, University of Sk¨ovde, SE-541 28 Sk¨ovde, Sweden +2 Department of Physics, Division of Mathematical Physics, +Lund University, Professorsgatan 1, 223 63, Lund, Sweden +The exchange-correlation hole and potential of the homogeneous electron gas have been investi- +gated within the random-phase approximation, employing the plasmon-pole approximation for the +linear density response function. The angular dependence as well as the time dependence of the +exchange-correlation hole are illustrated for a Wigner-Seitz radius rs = 4 (atomic unit). It is found +that there is a substantial cancellation between exchange and correlation potentials in space and +time, analogous to the cancellation of exchange and correlation self-energies. Analysis of the sum +rule explains why it is more advantageous to use a non-interacting Green function than a renor- +malized one when calculating the response function within the random-phase approximation and +consequently the self-energy within the well-established GW approximation. The present study pro- +vides a starting point for more accurate and comprehensive calculations of the exchange-correlation +hole and potential of the electron gas with the aim of constructing a model based on the local density +approximation as in density functional theory. +I. +INTRODUCTION +The homogeneous electron gas has been a long-lasting +and an invaluable model of valence electrons in solids. +Pseudopotential theory1 explains why the behavior of va- +lence electrons in solids, despite the presence of a strong +ionic potential, nevertheless resembles that of the elec- +tron gas. The effects of exchange and correlations of the +interacting homogeneous electron gas have been studied +thoroughly for the last six decades2–4. An important em- +pirical observation is that the main features of the spec- +tral function arising from exchange and correlations are +quite robust and can be carried over to real materials. +For example, setting aside strongly correlated systems5, +it is generally the case that the spectral function of most +materials consists of a quasiparticle peak and an incoher- +ent satellite feature which can be traced back to collec- +tive charge excitations (plasmons), just as found in the +electron gas6. +The effects of exchange and correlations have been tra- +ditionally studied using the concept of self-energy, a non- +local and energy-dependent quantity that acts on the +Green function as a convolution in space and time7,8. In a +recent development, a different framework for represent- +ing exchange and correlations was proposed in the form of +a time-dependent exchange-correlation (xc) potential9,10. +This formalism is fundamentally different from the self- +energy approach in that the potential acts locally or mul- +tiplicatively on the Green function. Most importantly, +the potential arises naturally as a Coulomb potential of +a charge distribution (exchange-correlation hole) which +fulfills a sum rule and some exact properties. +More- +over, due to the special form of the Coulomb interac- +tion, which depends solely on the separation of two point +charges, it can be shown that the exchange-correlation +potential is in fact the first radial moment of the spher- +ical average of the exchange-correlation hole. This ap- +pealing result is the analog of the result found in the ex- +act expression for the ground-state exchange-correlation +energy, which is very much utilized in density functional +theory and partially explains the success of the local den- +sity approximation11,12. +The exchange-correlation potential formalism also pro- +vides a simple physical picture of the propagation of an +added hole or electron in a many-electron system as in +photoemission and inverse photoemission experiments. +The added hole or electron induces a temporal density +fluctuation of the system initially in its ground state, giv- +ing rise to the exchange-correlation potential, which in +turns acts on the Green function representing the prop- +agation of the added hole or electron. +In this paper, the time-dependent exchange-correlation +hole and its corresponding exchange-correlation potential +of the electron gas are studied using the random-phase +approximation (RPA)3,7,13 to understand the salient fea- +tures of the exchange-correlation hole and potential. The +long-term goal is to use the electron gas results as a basis +for a local density approximation in the spirit of density +functional theory12,14–17. +The theory section commences with a short summary +of the exchange-correlation potential framework, which +is outlined in detail in previous publications. Formulas +for the exchange-correlation hole and potential are then +derived for the homogeneous electron gas. An analysis +of the sum rule and its consequences is presented in Sec. +II F followed by computational results in Sec. III and a +summary at the end. +II. +THEORY +In the exchange-correlation potential formalism the +equation of motion of the equilibrium zero temperature +time-ordered Green function is given by9 +� +i ∂ +∂t − h(r) − Vxc(r, r′; t) +� +G(r, r′; t) = δ(r − r′)δ(t), +(1) +arXiv:2301.05590v1 [cond-mat.str-el] 13 Jan 2023 + +2 +where +h(r) = −1 +2∇2 + Vext(r) + VH(r), +(2) +in which Vext and VH are the external field and the +Hartree potential, respectively. +The Green function is +defined according to7 +iG(r, r′; t) = ⟨T[ ˆψ(rt) ˆψ†(r′0)]⟩, +(3) +where r = (r, σ) labels both space and spin variables, +ˆψ(rt) is the Heisenberg field operator, T is the time- +ordering symbol, and ⟨.⟩ denotes expectation value in the +ground state. +The exchange-correlation potential Vxc is the Coulomb +potential of the exchange-correlation hole ρxc: +Vxc(r, r′; t) = +� +dr′′v(r − r′′)ρxc(r, r′, r′′; t). +(4) +The presence of the instantaneous Coulomb interaction +implies that t′′ = t. The exchange-correlation hole fulfills +an important sum rule +� +d3r′′ρxc(r, r′, r′′; t) = −δσσ′′θ(−t) +(5) +and the following exact condition +ρxc(r, r′, r′′ = r; t) = −ρ(r) +(6) +for any r, r′ and t. +A. +General formula for the exchange-correlation +hole +From the definition of the exchange-correlation hole9, +G(2) = ⟨T[ˆρ(3) ˆψ(1) ˆψ†(2)] += i[ρ(3) + ρxc(1, 2, 3)]G(1, 2), +(7) +where 1 = (r1, t1) etc. with t1 = t3 and the relation2,6 +G(2) = iρ(3)G(1, 2) − δG(1, 2) +δϕ(3) , +(8) +an explicit formula for the exchange-correlation hole is +given by10 +ρxc(1, 2, 3) = i +δ +δϕ(3) ln G(1, 2), +(9) +where ϕ is a probing field, which is set to zero after the +functional derivative is taken. The exchange-correlation +hole can thus be regarded as the linear response of i ln G +with respect to an external field. +From the identity +δG = −G(δG−1)G +(10) +and the equation of motion of the Green function, one +obtains +ρxc(1, 2, 3)G(1, 2) += i +� +d4 G(1, 4) +� +δ(3 − 4) + δVH(4) +δϕ(3) +� +G(4, 2) ++ i +� +d4d5 G(1, 4)δΣ(4, 5) +δϕ(3) G(5, 2), +(11) +where Σ is the self-energy. The first term on the right- +hand side, iG(1, 3)G(3, 2), will be referred to as the ex- +change contribution, the second term involving +δVH +δϕ +as +the linear response contribution, and the last term with +δΣ +δϕ as the vertex correction. The second and third terms +together constitute the correlation contribution. Within +the RPA, the vertex correction is neglected. +The quantity in the curly brackets is the inverse dielec- +tric function: +ϵ−1(4, 3) = δ(4 − 3) + δVH(4) +δϕ(3) . +(12) +It is convenient for later purposes to define +K(4, 3) = δVH(4) +δϕ(3) = +� +d5 v(4 − 5)χ(5, 3), +(13) +where v is the Coulomb interaction and χ is the linear +density response function +χ(5, 3) = δρ(5) +δϕ(3). +(14) +Replacing r1 → r, r2 → r′, and r3 → r′′ and taking +into account the fact that t1 = t3 = t and t2 = 0, one +finds +ρxc(r, r′, r′′; t)G(r, r′; t) = iG(r, r′′; 0−)G(r′′, r′; t) ++ i +� +dr4dt4 G(r, r4; t − t4)K(r4, r′′; t4 − t)G(r4, r′; t4). +(15) +The first term on the right-hand side yields the exchange +hole whereas the second term yields the correlation hole. +It is immediately clear that the exact condition in Eq. (6) +is already fulfilled by the exchange hole implying that +ρc(r, r′, r; t) = 0. +(16) +B. +Interacting homogeneous electron gas +The +Green +function +of +the +paramagnetic +non- +interacting homogeneous electron gas is given by +iG0(r, r′; t) = 1 +Ω +� +k>kF +eik·(r−r′)e−iεktθ(t) +− 1 +Ω +� +k≤kF +eik·(r−r′)e−iεktθ(−t), +(17) + +3 +𝒓 +𝒓′ +𝒓’’ +𝑅 +𝑅′ +𝑅′′ +𝜃 +FIG. 1: Definition of the radial variables R, R′, and R′′. They +are related to the angle θ by R′′2 = R2 − 2RR′ cos θ + R′2 +. +where εk = 1 +2k2, kF is the Fermi wave vector, and Ω is +the space volume. It is understood that σ = σ′. For the +homogeneous electron gas, it is convenient to introduce +the variable R = r′ − r. +In spherical coordinates the +equation of motion becomes +� +i ∂ +∂t − h(R) − Vxc(R; t) +� +�G(R, t) = +1 +4πRδ(R)δ(t), +(18) +where +h(R) = −1 +2 +∂2 +∂R2 , +�G(R, t) = R G(R, t). +(19) +Defining +T(R, t) = +1 +�G(R, t) +h(R) �G(R, t), +(20) +the formal solution is given by +G(R, t) = G(R, 0)e−i +� t +0 dt′[T (R,t′)+Vxc(R,t′)], +(21) +in which it is understood that G(R, 0) = G(R, 0+) for +t > 0 and G(R, 0) = G(R, 0−) for t < 0. In general, from +the equation of motion in Eq. (18) +i[G(R, 0+) − G(R, 0−)] = δ(R) +4πR2 . +(22) +For the homogeneous electron gas, r may be chosen to +be the origin, i.e., r = 0, and one defines the variables +R = r′ − r, R′ = r′′ − r, and R′′ = r′′ − r′ as illustrated +in Fig. 1. +C. +Exchange hole +From Eq. (15) the exchange hole is given by +ρx(r, r′, r′′; t)G(r, r′; t) = iG(r, r′′; 0−)G(r′′, r′; t). +(23) +Unlike the static exchange hole18 in quantum chemistry +and density functional theory, the exchange hole in the +present formalism is time dependent. +Using a non-interacting Green function as given in Eq. +(17) and considering the case t < 0 one finds +ρx(r, r′, r′′; t < 0) +� +k≤kF +eik·(r−r′)e−iεkt += − 1 +Ω +� +k′≤kF +eik′·(r−r′′) × +� +k≤kF +eik·(r′′−r′)e−iεkt. +(24) +For t > 0 +ρx(r, r′, r′′; t > 0) +� +k>kF +eik·(r−r′)e−iεkt += − 1 +Ω +� +k′≤kF +eik′·(r−r′′) × +� +k>kF +eik·(r′′−r′)e−iεkt. +(25) +Expressed in radial variables and the angle θ as ex- +plained in Fig. 1, +ρx(R, R′, θ; t) = iG0(R′, 0−)G0(R′′, t) +G0(R, t) +(26) +where R′′ depends on θ. +G0(R, t) is obtained from Eq. (17) by performing the +k-integral over the solid angle, yielding +iG0(R, t < 0) = − 1 +2π2 +1 +R +� kF +0 +dk k sin (kR)e−ik2t/2, +(27) +iG0(R, t > 0) = +1 +2π2 +1 +R +� ∞ +kF +dk k sin (kR)e−ik2t/2, +(28) +iG0(R, 0−) = − 1 +2π2 +1 +R3 [sin (kFR) − kFR cos (kFR)] . +(29) +G0(R, t < 0) can be expressed in terms of the complex +error function or calculated numerically using a standard +quadrature. The calculation of G0(R, t > 0), however, +needs more care and it is detailed in Appendix A. +D. +Correlation hole +The linear response contribution to ρxc is given by the +second term in Eq. (15). Keeping in mind that t3 = t1 = +t, t2 = 0, one obtains +i +� +d4 G(1, 4)K(4, 3)G(4, 2) += i +� +dr4dt4G(r − r4, t − t4)K(r4 − r′′, t4 − t) +× G(r4 − r′, t4). +(30) + +4 +The details of the calculations using G = G0 are shown +in Appendix B and the results are given by +ρc(R, R′, θ; t < 0) = +A1 + A2 +G0(R, t < 0), +(31) +ρc(R, R′, θ; t > 0) = +B1 + B2 +G0(R, t > 0). +(32) +A1, A2, B1, and B2 are functions of R′ and R′′, and given +by +A1 = γ(R′, R′′, t, 0, t), +(33) +A2 = γ(R′′, R′, t, 0, 0), +(34) +B1 = γ(R′, R′′, 0, t, 0), +(35) +B2 = γ(R′′, R′, 0, t, −t), +(36) +where +γ(R, R′, t, t′, t′′) = 1 +Ω2 +� +k≤kF +e−ik·Re−iεkt +× +� +k′>kF +eik′·R′e−iεk′t′M(|k′ − k|, εk′ − εk, t′′). +(37) +The quantity M is given by +M(q, ω, t) = +� ∞ +0 +dω′ L(q, ω′)−ieiω′t +ω′ + ω , +(38) +where L(q, ω) is the spectral function of K(q, ω) defined +in Eq. (13): +K(k, ω) = +� 0 +−∞ +dω′ +L(k, ω′) +ω − ω′ − iδ + +� ∞ +0 +dω′ +L(k, ω′) +ω − ω′ + iδ . +(39) +K(q, ω) is symmetric in frequency but L(q, ω) is anti- +symmetric and related to K as follows: +L(k, ω) = − 1 +π sign(ω)ImK(k, ω). +(40) +The correlation hole involves coupled integrals over +momenta below and above kF, which are difficult to cal- +culate analytically. +They are six-dimensional integrals +which cannot be easily performed with standard quadra- +tures. To make the computation feasible, a plasmon-pole +approximation for L(k, ω) is employed and described in +the next section. +1. +Plasmon-pole approximation +The plasmon dispersion of the homogeneous electron +gas is given by7 +Ωq = ωp +� +1 + 3 +10 +k2 +Fq2 +ω2p ++ ... +� +, +(41) +where the plasmon frequency ωp in the long-wavelength +limit is given by +ω2 +p = 4πρ0, +(42) +and ρ0 is the electron gas density. The critical momen- +tum, qc, at which the plasmon starts to merge into the +electron-hole excitations is given by the crossing of the +plasmon dispersion with the line εq + kFq yielding +qc = 1 +2a +�√ +1 + 4ac − 1 +� +kF, +(43) +where +a = 1 +2 − +3 +10c, +c = ωp +k2 +F +. +(44) +Great simplification results if a plasmon-pole approx- +imation independent of k is used for L(k, ω) defined in +Eq. (40): +L(k, ω) = ωp +2 [δ(ω − ωp) − δ(ω + ωp)] , +(45) +which corresponds to +K(q → 0, ω) = +ω2 +p +ω2 − ω2p +. +(46) +The approximation is valid for k ≤ qc and for rs = 3, 4 +and 5, which cover most of the average valence densities +in real materials, the critical momenta are qc = 0.86, +0.82, and 0.73 kF, respectively. +Within the plasmon-pole approximation the quantity +M defined in Eq. (38) becomes independent of momenta: +M(q, ω, t) = ωp +2 +−ieiωpt +ωp + ω . +(47) +Then the coupling between k and k′ is partially released. +Using +1 +Ω +� +k +e−ik·R = +1 +2π2R +� +dk k sin (kR) +(48) +yields within the plasmon-pole approximation +γP P (R, R′, t, t′, t′′) = +−2iωp +(2π)4RR′ +� kF +0 +dk k sin (kR)e−iεkt +× +� ∞ +kF +dk′ k′ sin (k′R′) e−iεk′t′eiωpt′′ +ωp + εk′ − εk +. +(49) +Since the plasmon-pole approximation decouples the +angular inter-dependence of k and k′, it is expected to +impart error to the correlation contribution. +To min- +imize error, the upper limit of the integration over k′ +corresponding to unoccupied states is chosen so as to ap- +proximately reproduce the static correlation hole19. This +choice yields a value of ≈ 1.5kF so that integration over +unoccupied states arising from the correlation contribu- +tion is restricted to between kF and 1.5kF. + +5 +E. +Exchange-correlation potential +By making a change of variable R′ = r′′ − r the +exchange-correlation potential in Eq. (4) reduces to the +first radial moment of the spherical average of ρxc: +Vxc(r, r′; t) = +� +dR′R′ ρxc(r, r′, R′; t), +(50) +where ρxc(r, r′, R′; t) for given r, r′, and t depends only +on the radial distance R′ = |r′′ − r|, +ρxc(r, r′, R′; t) = +� +dΩR′ρxc(r, r′, r + R′; t). +(51) +As can be seen from, e.g., Eqs. +(24) and (B5), the +spherical average of ρxc for the electron gas amounts to +performing a solid-angle integration +� +dΩ′′ei(k−k′)·r′′ = 4π sin(∆k R′) +∆k R′ +, +(52) +where ∆k = |k − k′| and R′ = |r′′ − r|, r = 0. +1. +Exchange potential +For t < 0 the exchange hole is given by +¯ρx(R, R′, t < 0)iG0(R, t) += 1 +Ω2 +� +k,k′≤kF +e−ik·r′e−iεkt × 4π sin(∆k R′) +∆k R′ +. +(53) +The exchange potential is the first moment of ¯ρx in R′: +Vx(R, t < 0) = +1 +iG0(R, t) +4π +Ω2 +� +k,k′≤kF +e−ik·Re−iεkt +× +� +dR′ sin(∆k R′) +∆k +. +(54) +Consider the integral over R′ with positive α → 0: +lim +α→0 +� ∞ +0 +dR′ sin(∆k R′)e−αR′ = +1 +∆k . +(55) +One then finds +Vx(R, t < 0) = +1 +iG0(R, t) +4π +Ω2 +� +k,k′≤kF +e−ik·Re−iεkt +1 +(∆k)2 . +(56) +The integral over k′ is given by +f(k) = 1 +Ω +� +k′≤kF +1 +(∆k)2 += +1 +4π2k +� kF +0 +dk′k′ ln +���� +k + k′ +k − k′ +���� , +(57) +which can be performed analytically yielding +f(k) = kF +2π2 F(k/kF), +(58) +where +F(x) = 1 +2 + 1 − x2 +4x +ln +���� +1 + x +1 − x +���� . +(59) +This function is the same as the one appearing in the +static Hartree-Fock theory for the electron gas20. More +explicitly as a function of k, +f(k) = kF +2π2 +�1 +2 + k2 +F − k2 +4kFk +ln +���� +kF + k +kF − k +���� +� +. +(60) +There remains the integral over k which reduces to a +one-dimensional integral over the radial k: +Vx(R, t < 0) = +1 +iG0(R, t) +× 2 +πR +� kF +0 +dk k sin(kR) e−iεktf(k). +(61) +For t > 0 the result is given by +Vx(R, t > 0) = − +1 +iG0(R, t) +× 2 +πR +� ∞ +kF +dk k sin(kR) e−iεktf(k). (62) +2. +Correlation potential +A similar procedure as for the exchange potential can +be applied to the correlation potential using A1, A2, B1, +and B2, given in Eqs. (B18-B21) in Appendix B. +The result is given by +Vc(R, t < 0) = +C1 + C2 +G0(R, t < 0), +(63) +Vc(R, t > 0) = +D1 + D2 +G0(R, t > 0), +(64) +where +C1 = Γ(0, R, t, 0, t), +(65) +C2 = Γ(R, 0, t, 0, 0), +(66) +D1 = Γ(0, R, 0, t, 0), +(67) +D2 = Γ(R, 0, 0, t, −t), +(68) +and +Γ(R, R′, t, t′, t′′) = 4π +Ω2 +� +k≤kF +e−iεkte−ik·R +× +� +k′>kF +e−ik′·R′e−iεk′t′ × M(|k′ − k|, εk′ − εk, t′′) +|k′ − k|2 +. +(69) + +6 +According to Eq. (47), within the plasmon-pole ap- +proximation, +M(|k′ − k|, εk′ − εk, t′′) = ωp +2 +−ieiωpt′′ +ωp + εk′ − εk +, +(70) +which partially decouples the interdependence of k and +k′, allowing for analytical integration over the solid an- +gles of both variables, yielding +C1 = P1(R, t, 0, t), +(71) +C2 = P2(R, t, 0, 0), +(72) +D1 = P1(R, 0, t, 0), +(73) +D2 = P2(R, 0, t, −t), +(74) +where +P1(R, t, t′, t′′) = −iωpeiωpt′′ +4π3R +� kF +0 +dk +� ∞ +kF +dk′ k sin (k′R) +× e−iεkte−iεk′t′ +ωp + εk′ − εk +ln +���� +k + k′ +k − k′ +����, +(75) +P2(R, t, t′, t′′) = −iωpeiωpt′′ +4π3R +� kF +0 +dk +� ∞ +kF +dk′ k′ sin (kR) +× e−iεkte−iεk′t′ +ωp + εk′ − εk +ln +���� +k + k′ +k − k′ +����. +(76) +Due to the use of the plasmon-pole approximation, the +upper limit of the integral over k′ is restricted to 1.5kF +to reproduce approximately the static correlation hole, +as described earlier in Sec. II D 1. +F. +Sum rule +In this section, the sum rule and its consequences are +discussed and a simple vertex approximation respecting +the sum rule is proposed. The results and conclusions +reached in this section are quite general and supported +by the electron gas results. +1. +Exchange hole +One has for a non-interacting G and t < 0 +i +� +dr′′G0(r, r′′; 0−)G0(r′′, r′; t < 0) = −G0(r, r′; t < 0), +(77) +which can be shown as follows: +i +� +dr′′G0(r, r′′; 0−)G0(r′′, r′; t < 0) += −i +� +dr′′ � +k≤kF +ϕk(r)ϕ∗ +k(r′′) +� +k′≤kF +ϕk′(r′′)ϕ∗ +k′(r′)e−iεk′t += −i +� +k≤kF +ϕk(r)ϕ∗ +k(r′)e−iεkt += −G0(r, r′; t < 0). +(78) +It is also quite clear that +i +� +dr′′G0(r, r′′; 0−)G0(r′′, r′; t > 0) = 0. +(79) +It then follows from Eq. +(23) that the exchange hole +fulfills the sum rule when G0 is used: +� +dr′′ρx(r, r′, r′′; t < 0) = −1. +(80) +Explicitly for the electron gas, it follows from Eq. (24) +that the sum rule for t < 0 is fulfilled by the exchange +hole since +� +d3r′′ei(k−k′)·r′′ = (2π)3δ(k − k′). +(81) +The sum rule for t > 0 is zero since k ̸= k′ as can be seen +from Eq. (25). +In general +− i +� +dr′′G(r, r′′; 0−)G(r′′, r′; t < 0) ̸= G(r, r′; t < 0) +(82) +and also +i +� +dr′′G(r, r′′; 0−)G(r′′, r′; t > 0) ̸= 0. +(83) +unless G = G0. This implies that if only the exchange +part is considered, neglecting the correlation and vertex +terms, then in general the sum rule is not fulfilled when +a renormalized G is used. +2. +Correlation hole +It is also evident that the correlation part of the +exchange-correlation hole gives no contribution to the +sum rule. The reason for this can be seen by considering +the change in the charge density under a perturbation: +δρ(1) = +� +d2 χ(1, 2)δϕ(2), +(84) +where χ is the linear density response function as defined +in Eq. (14). A constant perturbation, δϕ = 1, does not +alter the density so that +� +d2 χ(1, 2) = 0. +(85) +This property is fulfilled by the RPA response function +calculated using G0 as shown below. +It is interesting to observe that in the case of the elec- +tron gas, it can be seen explicitly from Eq. (B5) in Ap- +pendix B that the integral of A1 over r′′ is zero since +k ̸= k′ and the same conclusion holds for A2, B1, and +B2. Hence the sum-rule is fulfilled, irrespective of the +approximation used for K(q, ω). + +7 +Since the response function can be expanded in powers +of the polarization P, +χ = P + PvP + ... +(86) +it follows that if the polarization function fulfills +� +d2P(1, 2) = 0 +(87) +then Eq. (85) is satisfied. +The polarization in the RPA is given by +P(r, r′; t) = −iG(r, r′; t)G(r′, r; −t). +(88) +If a non-interacting G is used, then Eq. (87) and conse- +quently Eq. (85) are fulfilled. This can be understood as +follows. It can be shown that +� +dr′′G0(r, r′′; t)G0(r′′, r′; −t) = 0 +(89) +by using the definition of G0: +� +dr′′G0(r, r′′; t)G0(r′′, r′; −t) += θ(t) +� +dr′′⟨ ˆψ(rt) ˆψ†(r′′)⟩⟨ ˆψ†(r′) ˆψ(r′′, −t)⟩ ++ θ(−t) +� +dr′′⟨ ˆψ†(r′′) ˆψ(rt)⟩⟨ ˆψ(r′′, −t) ˆψ†(r′)⟩. +(90) +Since the expectation value is taken with respect to a +single Slater determinant, the integral over r′′ amounts +to +� +dr′′ϕ∗ +k(r′′)ϕk′(r′′) = 0, +(91) +where {ϕk} is a set of orbitals defining the single Slater +determinant, and if k corresponds to an occupied state +then k′ corresponds to an unoccupied state and vice +versa. +However, if a renormalized G is used to calculate P in +the RPA as in Eq. (88), the conditions in Eq. (87) and +consequently Eq. (85) are no longer fulfilled in general. +Thus, with a renormalized G, both the exchange and +correlation terms would violate the sum rule, and it seems +unlikely that the sum of these two terms would fulfill the +sum rule. This might explain why it is not favorable to +use a renormalized G in the RPA and consequently in +the GW approximation2,6. It also implies that inclusion +of the vertex δΣ/δϕ would restore the sum rule. Hence +to preserve the sum rule, the use of a renormalized G +should be accompanied by inclusion of the vertex. +3. +A simple vertex correction +The preceding consideration suggests a scheme for an +approximate vertex correction that would preserve the +sum rule. Consider the following screened-exchange self- +energy, +Σ(1, 2) = iG0(1, 2)W0(1, 2), +(92) +where W0(1, 2) is an instantaneous interaction, +W0(1, 2) = W0(r1, r2)δ(t1 − t2), +(93) +with W0(r1, r2) being some screened interaction, e.g., the +Thomas-Fermi screened interaction or the static screened +interaction calculated within RPA. By using G0 through- +out, the vertex correction is given by +δΣ(1, 2) +δϕ(3) += iδG0(1, 2) +δϕ(3) +W0(1, 2) += −i +� +d4d5 G0(1, 4)δG−1 +0 (4, 5) +δϕ(3) +G0(5, 2)W0(1, 2), (94) +where the identity δG = −G(δG−1)G has been utilised +and δW0 +δϕ = 0 since it is assumed that W0 is fixed. In the +presence of the probing field ϕ, G0 fulfills the equation +of motion +� +i ∂ +∂t1 +− h(1) − ϕ(1) +� +G0(1, 2) = δ(1 − 2), +(95) +so that +δG−1 +0 (4, 5) +δϕ(3) += −δ(4 − 5) +� +δ(3 − 4) + δVH(4) +δϕ(3) +� +. +(96) +This leads to +δΣ(1, 2) +δϕ(3) += iG0(1, 3)G0(3, 2)W0(1, 2) ++ i +� +d4 G0(1, 4)δVH(4) +δϕ(3) G0(4, 2)W0(1, 2), (97) +and hence according to Eqs. (85) and (89), +� +d3δΣ(1, 2) +δϕ(3) += 0, +(98) +noting that t1 = t2 due to the instantaneous interaction. +It follows that the vertex in Eq. (97) preserves the sum +rule. +G. +A model Green function for the interacting +electron gas +To assist in analyzing the results of the calculations +of Vxc of the electron gas, a model Green function has +been constructed. A physically motivated model for the +Green function of the interacting electron gas is given by +the following: +G(R, t < 0) = i +Ω +� +k≤kF +� +Zk + (1 − Zk)eiωkt� +× e−iEkteik·R, +(99) + +8 +G(R, t > 0) = − i +Ω +� +k>kF +� +Zk + (1 − Zk)e−iωkt� +× e−iEkteik·R, +(100) +where Ek is the quasiparticle energy, Zk is the quasi- +particle renormalization factor, and ωk is the plasmon +energy. To allow for analytic derivation of the exchange- +correlation potential, Zk and ωk are assumed to be in- +dependent of k and Ek is taken to be a renormalized +free-electron gas dispersion: +Zk = Z, +ωk = ωp, +Ek = αεk = α +2 k2. +(101) +For an electron gas of density ρ0 the plasmon energy is +given in Eq. (42). +The model Green function can be improved by includ- +ing the possibility of having spectral weight above or be- +low the Fermi level for k ≤ kF or k > kF, respectively: +G(R, t < 0) = i +Ω +� +k≤kF +(C1 + C2 + C3)eik·R, +(102) +G(R, t > 0) = − i +Ω +� +k>kF +(D1 + D2 + D3)eik·R, +(103) +where +C1 = Ze−iEkt, +(104) +C2 = (1 − β− +k )(1 − Z)e−i(Ek−ωp)t, +(105) +C3 = β− +k (1 − Z)e−i(EF+ωp)t, +(106) +D1 = Zpe−iEkt, +(107) +D2 = (1 − β+ +k )(1 − Zp)e−i(Ek+ωp)t, +(108) +D3 = β+ +k (1 − Zp)e−i(EF−ωp)t, +(109) +in which +β− +k = +k +2kF +θ(kF − k), +(110) +β+ +k = 1 +2 +kmax − k +kmax − kF +θ(kmax − k). +(111) +For k > kmax, the dispersion is taken to be that of the +free-electron gas for otherwise the integration over k to +infinity would not converge. +It means physically that +high-energy electrons are free since they do not experi- +ence exchange and correlations. To conserve electronic +charge, the weight above the Fermi level coming from +states smaller than kF is equated to the weight below the +Fermi level coming from states larger than kF, yielding +an upper limit, neglecting the k-dependence of β±, +kmax ≈ +� +1 + 1 − Z +1 − Zp +�1/3 +kF +(112) +above which the spectrum does not have weight below +the Fermi level. +For t ̸= 0, the exchange-correlation potential can be +obtained from the equation of motion: +Vxc(R, t) = +1 +G(R, t) [i∂t − h] G(R, t). +(113) +Since +h exp (ik · R) = k2 +2 exp (ik · R) +(114) +one finds for t < 0 +[i∂t − h] G(R, t) += i 1 +Ω +� +k≤kF +(A1 + A2 + A3)eik·R, +(115) +where +A1 = Z(Ek − εk)e−iEkt, +(116) +A2 = (1 − β− +k )(1 − Z)(Ek − εk − ωp)e−i(Ek−ωp)t, (117) +A3 = β− +k (1 − Z)(EF − εk + ωp)e−i(EF+ωp)t. +(118) +For t > 0 +[i∂t − h] G(R, t) += −i 1 +Ω +� +k>kF +(B1 + B2 + B3)eik·R, +(119) +where +B1 = Zp(Ek − εk)e−iEkt, +(120) +B2 = (1 − β+ +k )(1 − Zp)(Ek − εk + ωp)e−i(Ek+ωp)t, +(121) +B3 = β+ +k (1 − Zp)(EF − εk − ωp)e−i(EF−ωp)t. +(122) +III. +RESULTS +The results shown in this section are all expressed in +atomic units (a.u.) and correspond to rs = 4. Two pairs +of representative times, t = ±34.75 corresponding to 1/¯ε, +where ¯ε is the center of the occupied band, and t = ±4.62 +corresponding to the inverse of the plasmon energy, 1/ωp, +have been chosen for illustrations. Additional times are +also considered as appropriate. +A. +Angular dependence of the +exchange-correlation hole +In Fig. 2 the real part of the exchange holes for the +case R = 10, t = −34.75, and for three different angles +θ = 0, π/4, π/2 as defined in Fig. 1 are shown. These +density fluctuations are due to exchange-only interaction + +9 +and correspond to the case when a hole created at R = 0 +at t = −34.75 is to be annihilated later at R = 10 at +t = 0. It can be seen that there are more fluctuations +for θ = 0 representing the direction along the line con- +necting the location where the hole is created at R = 0 +and the location where the hole is annihilated at R = 10. +The fluctuations are stronger in the direction towards +R = 10 from the origin than in the opposite direction, +which reflect the preponderance of the hole presence and +its effects in the former direction. Indeed, the density +fluctuations along the direction perpendicular to the line +joining R = 0 and R = 10 (θ = π/2) are found to be the +least whereas for the direction θ = π/4 they are some- +where in between those of θ = 0 and π/2. In accordance +with discussion in the paragraph following Eq. (15), the +exchange hole already fulfills the exact condition in Eq. +(6), i.e., ρx(0) = −ρ0, where ρ0 the homogeneous electron +gas density. +A similar behavior can be observed for the correla- +tion holes shown in Fig. 2. These density fluctuations +arise from the classical Coulomb interaction in response +to the creation of a hole and its accompanying exchange +hole. The fluctuations are particularly strong along the +direction towards R = 10 from the origin. According to +Eq. (16), the correlation hole at the origin should vanish, +which is clearly not the case here. There are two possible +explanations for this discrepancy, one being the use of +the plasmon-pole approximation and another being the +neglect of the vertex term. One may speculate that the +vertex term is needed even when the exact response func- +tion is used, but to answer this issue in the definitive, cal- +culations employing the full RPA response function are +required. It is, however, to be noted that discrepancy +may not be as severe as it might appear at first sight be- +cause what enters into the exchange-correlation poten- +tial is the first moment of the spherical average of the +exchange-correlation hole so that contributions at small +R are significantly cut down. Numerical calculations of +the sum rule show that in accordance with theory the +exchange hole alone fulfills the sum rule for t < 0 and +the correlation hole integrates to zero whereas for t > 0 +both the exchange and the correlation holes integrate to +zero. +B. +Time dependence of the exchange-correlation +hole +To illustrate the time dependence of the exchange- +correlation hole, the exchange holes for two representa- +tive times t = −4.62 and t = −34.75 are shown in Fig. +3 for the case R = 10 and θ = 0. A similar result is +found for the correlation hole. +For a short time scale +in which the hole is annihilated at a location well sepa- +rated from the location where it was created, the fluctu- +ations are much stronger than those of a long period. As +a guide, it is useful to consider the limit of R = 0 and +t = 0−, which yields the static exchange-correlation hole. +-20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +-2 +-1.5 +-1 +-0.5 +0 +0.5 +1 +10-3 +-20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +-2 +-1.5 +-1 +-0.5 +0 +0.5 +1 +10-3 +FIG. 2: The real part of the exchange hole (top) and the +correlation hole (bottom) for θ = 0 (solid black), π/4 (dashed +blue), and π/2 (dotted red) for the case R = 10 and t = +−34.75. +There is a correlation between the spatial separation R +and the time period, which determines the behavior of +the exchange-correlation hole. Within a short period of +time but with large R, the system may not have suffi- +cient time to relax so that the density fluctuations arising +from a sudden creation of a hole at t = −4.62 have not +stabilized. These large density fluctuations for small θ, +however, contribute little to the exchange-correlation po- +tential since only the spherical average of the exchange- +correlation hole is needed and when taking the spherical +average each angular contribution is multiplied by sin θ. +On the other hand, for a relatively short spatial sep- +aration R = 2, the time-dependence has a much less in- +fluence on the exchange hole, as can be seen in Fig. 4. + +10 +-20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +-4 +-2 +0 +2 +4 +6 +8 +10 +10-3 +FIG. 3: +The real part of the exchange hole for t = −4.62 +and −34.75 and the case R = 10, θ = 0. +The difference between the exchange holes for t = −4.62 +and −34.75 is much less pronounced than for the case +of R = 10. The case of small R and small t approaches +the static exchange-correlation hole limit of R = 0 and +t = 0−. +For t > 0 corresponding to an electron addition, the +exchange hole exhibits a more oscillatory behavior than +for t < 0. This can be understood from the exact sum +rule which dictates that the exchange hole must integrate +to zero for t > 0. For larger t, however, the oscillations +become damped as the system stabilizes itself. +C. +Spherical average of the exchange-correlation +hole +The relevant quantity determining the exchange- +correlation potential is the first radial moment of the +spherical average of the exchange-correlation holes. In +the upper panel of Fig. 5 the spherical average of the +real part of the exchange hole multiplied by R′ is shown +for several values of R and t. When taking a spherical +average the factor sin θ implies that the angular region +around θ ≈ 0 contributes significantly less than the re- +gion around θ ≈ π/2. This is confirmed by comparing +Fig. 5 and Fig. 4 for R = 2 in which it can be seen that +the behavior of the spherical average essentially follows +that of the exchange hole for θ = π/2. +For R = 2 and t = −4.62 the exchange hole is virtually +indistinguishable from the static Hartree-Fock exchange +hole. As expected, the exchange hole for small R and t +should approach the static exchange hole. However, even +for large values of R and t the exchange hole still follows +closely the static one, as can be seen in the top panel of +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +-2 +-1.5 +-1 +-0.5 +0 +0.5 +1 +10-3 +FIG. 4: +The real part of the exchange hole for t = ±4.62 +(blue) and ±34.75 (black) for R = 2, θ = π/2. The solid and +dashed lines correspond to t < 0 and t > 0, respectively. +Fig. 5. As R is increased, some weight is transferred from +the small to the large regions of R′ and the exchange hole +appears to become less dependent on t for large R. +The spherical average of the correlation hole for several +values of R and t is shown in the lower panel of Fig. 5. +The general behavior of the correlation hole is similar to +that of the exchange hole, except for R = 2 in which the +correlation hole fluctuates strongly around the origin as +a function of time than the corresponding exchange hole. +As also expected from the sum rule, which integrates +to zero for the correlation hole, more oscillations can be +observed than in the corresponding exchange holes. For +t > 0 the correlation hole tends to be positive around the +origin and vanishes as R increases. +The exchange and the correlation holes exhibit strong +fluctuations, deviating greatly from the static one at +t = −4.62, which is found to correlate with the radial +distance. To investigate further, the exchange and the +correlation holes in the vicinity of R = 10 are calcu- +lated and shown in Fig. 6. It is quite evident that the +large fluctuations at R = 9 − 10 correlate with the time +t = −4.62. These large fluctuations originate from the +small values of |G0(R, t)| at these values of R and t, and +are carried over to the exchange-correlation potentials as +discussed in the next section. + +11 +0 +5 +10 +15 +-0.04 +-0.02 +0 +0.02 +0 +5 +10 +15 +-0.04 +-0.02 +0 +0.02 +0 +5 +10 +15 +-0.04 +-0.02 +0 +0.02 +0 +5 +10 +15 +-0.04 +-0.02 +0 +0.02 +0 +5 +10 +15 +-0.04 +-0.02 +0 +0.02 +0 +5 +10 +15 +-0.04 +-0.02 +0 +0.02 +0 +5 +10 +15 +-0.04 +-0.02 +0 +0.02 +0 +5 +10 +15 +-0.04 +-0.02 +0 +0.02 +0 +5 +10 +15 +-0.1 +-0.05 +0 +0.05 +0 +5 +10 +15 +-0.1 +-0.05 +0 +0.05 +0 +5 +10 +15 +-0.1 +-0.05 +0 +0.05 +0 +5 +10 +15 +-0.1 +-0.05 +0 +0.05 +FIG. 5: +The real part of the spherical average of the +exchange hole (upper panel), the correlation hole (middle +panel), and the exchange-correlation hole (lower panel) mul- +tiplied by R′ for t = ±4.62 (blue), ±34.75 (black), ±69.5 +(red), and R = 2, 10, 30, 50. The solid and dashed curves cor- +respond to t < 0 and t > 0, respectively. The green curve +is the static exchange hole from the Hartree-Fock approxima- +tion. For t = −4.62 (solid blue) and R = 2 the exchange hole +is virtually indistinguishable from the static one. +0 +5 +10 +15 +-0.05 +0 +0.05 +0 +5 +10 +15 +-0.05 +0 +0.05 +0 +5 +10 +15 +-0.05 +0 +0.05 +0 +5 +10 +15 +-0.05 +0 +0.05 +0 +5 +10 +15 +-0.05 +0 +0.05 +0 +5 +10 +15 +-0.05 +0 +0.05 +0 +5 +10 +15 +-0.05 +0 +0.05 +0 +5 +10 +15 +-0.05 +0 +0.05 +0 +5 +10 +15 +-0.1 +-0.05 +0 +0.05 +0 +5 +10 +15 +-0.1 +-0.05 +0 +0.05 +0 +5 +10 +15 +-0.1 +-0.05 +0 +0.05 +0 +5 +10 +15 +-0.1 +-0.05 +0 +0.05 +FIG. 6: +The real part of the spherical average of the exchange +hole (upper panel), the correlation hole (middle panel), and +the exchange-correlation hole (lower panel) multiplied by R′ +for t = ±4.62 (blue), ±34.75 (black), ±69.5 (red), and R = +8, 9, 10, 11. The solid and dashed curves correspond to t < 0 +and t > 0, respectively. The green curve is the static exchange +hole from the Hartree-Fock approximation. + +12 +D. +Exchange-correlation potentials +From the spherical average of the exchange-correlation +hole the exchange-correlation potential can be deter- +mined. +The real and imaginary parts of the exchange +potential for t = ±4.62 and ±34.75 are shown in Figs. 7 +and 8, respectively. For t < 0 there is a striking differ- +ence between the exchange potentials corresponding to +the two times. The exchange potential corresponding to +t = −4.62 exhibits a pronounced structure around R = 9 +and 16, whereas the one corresponding to t = −34.75 is +almost constant with a relatively weak feature. As men- +tioned earlier, this strong structure at t = −4.62 arises +from the small value of |G0(R, t)| at R = 9 and 16 as +shown in Fig. 9. In fact, the exchange potential has a +strong structure at those positions for a range of t, as +can be seen in Fig. 10. The structure is very sharp for +very small t and as the magnitude of t increases, it be- +comes weaker. A plausible explanation is that when a +hole is introduced into the system, large density fluctu- +ations arise and within a short period of time the sys- +tem does not have enough time to relax, generating as +a consequence a strong spatial variation in the exchange +potential. However, strong spatial variations also appear +at a large-time scale as shown in Fig. 11. The common +denominator for the presence of strong structures in the +exchange-correlation potential is the diminishing value of +|G0(R, t)| at the corresponding position R and time t, as +can be seen in Fig. 12. +For t > 0 the exchange potentials are generally weaker +and less attractive than for t < 0 as can be seen in Fig. 7. +The difference between the two cases may be traced back +to the fundamental physical difference between removing +an electron (hole creation, t < 0) and adding an elec- +tron (t > 0), in that the removed electron is part of the +system in the ground state whereas the added electron +is not. This is also reflected in the sum rule which inte- +grates to −1 for a hole addition but zero for an electron +addition. Thus, the exchange potential associated with +a hole creation must be stronger than that associated +with an electron addition. +Aside from special systems +that possess a particle-hole symmetry, the creation of a +hole can be expected to create a stronger disturbance to +the system in the ground state than the addition of an +electron. +The exchange potential does not take into account the +classical Coulomb response of the electrons which results +in screening of the density fluctuations arising from the +exchange interaction. In Fig. 7 (middle) the real parts +of the correlation potentials associated with the linear +density response of the system are shown for the two pairs +of representative times t = ±4.62 and ±34.75. As in the +case of exchange, the correlation potential corresponding +to t = −4.62 has a pronounced structure in comparison +with the one corresponding to t = −34.75. +The real and imaginary parts of the exchange- +correlation potentials are displayed in Figs. 7 and 8, re- +spectively. Peaks in the imaginary part of the exchange- +correlation potential, which is the analog of the imagi- +nary part of the self-energy, signals the presence of damp- +ing. +It is evident that there is a strong cancellation +of exchange and correlation. +The Kramers-Kronig-like +structures in the exchange potential between R = 8 and +10 as well as between 15 and 17 in Fig. 7 induce po- +larization in which electrons and holes are accumulated +in opposite regions where the minimum and maximum +of Vx are located. +This exchange polarization is neu- +tralized by Coulomb screening, which is affected by the +presence of inverted structures in Vc at the corresponding +regions. This results in a strong cancellation between ex- +change and correlation potentials and the resulting total +exchange-correlation potential has a much less structure +than both the exchange and the correlation potentials. +To illustrate further the large cancellation between ex- +change and correlation, the real part of the exchange, cor- +relation, and exchange-correlation potentials for t = −1 +are displayed in Fig. +13. +It is tempting to speculate +that the remaining structure in the exchange-correlation +potential might be smoothed out when full RPA calcula- +tions are carried out without relying on the plasmon-pole +approximation. A similar cancellation is also found for +t = −34.75 (Fig. 7) but to a much lesser extent since +both the exchange and the correlation potentials have +little structures to begin with. +The strong cancellation between exchange and corre- +lation is well known in the self-energy formalism. The +cancellation manifests itself in the one-particle disper- +sion in which the too large occupied bandwidth within +the Hartree-Fock approximation is significantly reduced +by correlation effects, resulting in a dispersion close to +the non-interacting one. The Vxc formalism, however, of- +fers a different perspective in which the cancellation can +be explicitly seen in the exchange-correlation potential in +space and time. +R +9 +5 +9 +16 +5 +t 0 to -5 -60 -84 -102 -114 +TABLE I: +The approximate values of R ≤ 20 and the cor- +responding times t ≤ 0 around which strong structures are +found in the exchange and the correlation potentials and +strong deviations in the exchange and the correlation holes +from their static counterparts are observed. + +13 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +-0.4 +-0.35 +-0.3 +-0.25 +-0.2 +-0.15 +-0.1 +-0.05 +0 +0.05 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +-0.2 +-0.15 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +-0.4 +-0.35 +-0.3 +-0.25 +-0.2 +-0.15 +-0.1 +-0.05 +0 +0.05 +FIG. 7: +The real part of the exchange potential (top), the +correlation potential (middle), and the exchange-correlation +potential for t = ±4.62 (blue) and ±34.75 (black). The solid +and dashed curves correspond to t < 0 and t > 0, respectively. +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +-0.35 +-0.3 +-0.25 +-0.2 +-0.15 +-0.1 +-0.05 +0 +0.05 +0.1 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +-0.15 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +-0.25 +-0.2 +-0.15 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +0.2 +FIG. 8: +The imaginary part of the exchange potential +(top), the correlation potential (middle), and the exchange- +correlation potential (bottom) for t = ±4.62 (blue) and +±34.75 (black). The solid and dashed curves correspond to +t < 0 and t > 0, respectively. + +14 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +-1 +-0.5 +0 +0.5 +1 +1.5 +2 +10-3 +FIG. 9: +The real (solid) and the imaginary (dashed) parts of +the non-interacting Green function for t = −4.62 (blue) and +−34.75 (black). For t = −4.62, the value of |G0| is close to +zero at positions R = 9 and 16, whereas for t = −34.75, it +remains relatively substantial. +E. +Comparison with the model Green function +To check the electron gas results and to gain more in- +sight into the salient features of the exchange-correlation +potentials, a model Green function has been constructed +as described in Sec. II G. The exchange-correlation po- +tentials derived from the model for t = ±4.62 and t = +±34.75 are shown in Fig. 14, which should be compared +with Fig. 7. Apart from the absolute position, which +cannot be determined without calculating the quasipar- +ticle dispersion from the resulting electron gas exchange- +correlation potential, the agreement for t = ±34.75 is +quite striking. There is a discrepancy in the separation +between Vxc(t > 0) and Vxc(t < 0), which may indicate +an overestimation of correlation within the plasmon-pole +approximation. Increasing the range of integration from +2kF to 4kF in the integration over the unoccupied states +results in a down shift of the correlation potential while +the shape remains stable. +There is less agreement for +t = ±4.62, both concerning the shape as well as the sep- +aration between Vxc(t > 0) and Vxc(t < 0). Again, this +may be due to the plasmon-pole approximation, or, spec- +ulatively speaking, to the model itself, which may not be +accurate for small t. The small time behavior is deter- +mined by the quality of the model at high energy. The +model assumes a sharp plasmon excitation, which may +be valid for small momentum but becomes less accurate +at large momenta beyond the critical momentum. +To make a separate comparison of both Vx and Vc, the +exchange potential deduced from the Hartree-Fock Green +function is calculated and used to determine the correla- +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +-1 +-0.5 +0 +0.5 +FIG. 10: +The real part of the exchange potential for t = −1 +(blue), −4 (black) and −8 (red). +The structure at R = 9 +and 16 becomes sharper as t → 0−, which, however, is largely +cancelled by the correlation potential, as illustrated in Fig. +13. +tion potential of the model by subtracting the exchange +potential from the total exchange-correlation potential. +The results are shown in Fig. +15 for t = ±4.62 and +t = ±34.75. Here, there is no ambiguity regarding the +absolute position of the exchange potentials. The close +agreement between the exchange potential deduced from +the Hartree-Fock Green function and the one calculated +from Eqs. (61) and (62) for the electron gas attests that +the physical interpretation of the exchange-correlation +hole in Eq. (15), in which the first term on the right- +hand side is associated with the exchange hole, is well +motivated. +IV. +CONCLUSION AND SUMMARY +The exchange-correlation hole and potential of the ho- +mogeneous electron gas have been studied within the +RPA for rs = 4 and several representative time periods. +The angular dependence of the exchange-correlation hole +for t < 0 shows a stronger oscillation along the line join- +ing the location where the hole is created and the location +where the hole is annihilated (θ = 0) whereas along the +perpendicular direction (θ = π/2) it shows the least oscil- +lation. This behavior can be attributed to the degree of +preponderance of the hole along the respective directions. +The behavior of the exchange and the correlation holes +reveals a correlation between the separation R and the +time period t. +For certain values of R large fluctua- +tions are found within a range of time. This behaviour is +mimicked by the exchange and the correlation potentials + +15 +0 +5 +10 +15 +20 +-1 +-0.5 +0 +0.5 +0 +5 +10 +15 +20 +-1 +-0.5 +0 +0.5 +0 +5 +10 +15 +20 +-1 +-0.5 +0 +0.5 +FIG. 11: +The real part of the exchange potential (top), the +correlation potential (middle), and the exchange-correlation +potential (bottom) for t = −60 (blue) and −84 (red). +and is found to originate from the diminishing value of +|G0(R, t)| at the respective position and time. +The spherical average of the exchange-correlation hole +for t < 0 is generally larger than for t > 0, which is +a consequence of the sum rule being −1 for the former +and 0 for the latter. This property is inherited by the +exchange-correlation potential, since it is determined by +the first radial moment of the spherical average of the +exchange-correlation hole. It is found that the exchange +potential in space and time is substantially cancelled by +the correlation potential, which maybe seen as the analog +of the well-known cancellation of the Fock exchange and +the correlation self-energy of the electron gas in momen- +tum and frequency. The strong cancellation results in an +exchange-correlation potential with much less structure +than in the corresponding exchange and correlation po- +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +10-4 +FIG. 12: +The real (solid) and the imaginary (dashed) parts +of the non-interacting Green function for t = −60 (blue) and +−84 (red). +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +FIG. 13: +The real part of the exchange potential (blue), +the correlation potential (black), and the exchange-correlation +potential (red) for t = −1. The large cancellation between +exchange and correlation can be clearly seen. +tentials. This encouraging result lends support for the +feasibility of applying the local density approximation. +Analysis of the sum rule offers a physical explanation +why using a non-interacting Green function is more ad- +vantageous than using a renormalized one when calcu- +lating the response function in RPA and consequently +in calculating the self-energy within the GW approxima- +tion. A simple vertex correction that preserves the sum + +16 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +-0.4 +-0.35 +-0.3 +-0.25 +-0.2 +-0.15 +-0.1 +-0.05 +0 +0.05 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +-0.4 +-0.35 +-0.3 +-0.25 +-0.2 +-0.15 +-0.1 +-0.05 +0 +0.05 +FIG. 14: +The real part of the exchange-correlation potential +for t = ±4.62 (top) and t = ±34.75 (bottom) from the model +Green function. The solid and dashed curves correspond to +t < 0 and t > 0, respectively. The parameters used are indi- +cated in the figure. Two different values of renormalization +factors Z are used, one for t < 0 (Z = 0.67) and one for +t > 0 (Zp = 0.8). The choice of the parameters is based on +the GW results. +rule is then proposed. +The present work provides a starting point for more ac- +curate calculations of the exchange-correlation hole and +potential of the electron gas. The plasmon-pole approx- +imation is not expected to yield accurate results since +it neglects the limits in the solid angles. Nevertheless, +comparison with results obtained from a model Green +function suggests that it captures the main features of +the exchange-correlation potential. It would be highly +desirable to perform full RPA calculations without em- +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +FIG. 15: +The real part of the exchange potential Vx (blue) de- +rived from the Hartree-Fock Green function and the exchange- +correlation potential Vxc (black) from the model Green func- +tion for t = −4.62 (top) and t = −34.75 (bottom). +The +correlation potential (red) is defined as Vc = Vxc − Vx. +ploying the plasmon-pole approximation with the aim of +constructing a local-density type approximation for the +exchange-correlation potential, which can be applied to +calculate the Green function of real materials, circum- +venting computationally expensive traditional self-energy +calculations based on Feynman diagrams and path inte- +gral techniques. +Acknowledgments +Financial support from the Knut and Alice Wallen- +berg (KAW) Foundation (Grant number 2017.0061) and + +17 +the Swedish Research Council (Vetenskapsr˚adet, VR, +Grant number 2021 04498) is gratefully acknowledged. +We thank Rex Godby for valuable discussions. +Appendix A: Calculation of G0(R, t > 0) +The integral over k in G0(R, t > 0) can be decomposed +as follows: +� ∞ +kF +dk = +� ∞ +0 +dk − +� kF +0 +dk. +(A1) +iG0(R, t > 0) = +1 +2π2R {I(0, ∞) − I(0, kF)} , +(A2) +where +I(a, b) = +� b +a +dk k sin(kR)e−ik2t/2. +(A3) +Consider the following integral +� ∞ +−∞ +dk keikRe−ik2t/2 = eiR2/2t +� ∞ +−∞ +dk ke−i(k−R/t)2t/2. +(A4) +By making a change of variable, +k − R +t = +� +2 +t x, +(A5) +the integral I(0, ∞) can be performed analytically yield- +ing +I(0, ∞) = +� π +2it +R +it eiR2/2t. +(A6) +For t → 0+, by introducing a converging factor e−|α|q +one finds, +lim +α→0 +� ∞ +0 +dk k sin(kR)e−|α|q = 0 +(A7) +implying that indeed G0(R, 0+) = G0(R, 0−) for R ̸= 0. +Appendix B: Correlation hole +From Eq. +(15), the correlation hole is given by the +following equation: +ρc(r, r′, r′′; t)G(r, r′; t) = +i +� +dr4dt4 G(r, r4; t − t4)K(r4, r′′; t4 − t)G(r4, r′; t4). +(B1) +A non-interacting G is used, +G0(r − r4, t − t4) = i +Ω +� +k≤kF +eik·(r−r4)e−iεk(t−t4)θ(t4 − t) +− i +Ω +� +k>kF +eik·(r−r4)e−iεk(t−t4)θ(t − t4), +(B2) +G0(r4 − r′, t4) = i +Ω +� +k′≤kF +eik′·(r4−r′)e−iεk′t4θ(−t4) +− i +Ω +� +k′>kF +eik′·(r4−r′)e−iεk′t4θ(t4), +(B3) +and +K(r4 − r′′, t4 − t) += 1 +Ω +� +q +� dω +2π e−iq·(r4−r′′)e−iω(t4−t)K(q, ω). +(B4) +Consider the right-hand side of Eq. (B1) and the case +t < 0. From the product of the two Green functions one +obtains four terms but only two survive. The first non- +zero term is (note the additional factor of i from iGKG) +A1 = − i3 +Ω3 +� +k≤kF +� +k′>kF +eik·re−ik′·r′ +× +� +q +eiq·r′′ � dω +2π ei(ω−εk)tK(q, ω) +× +� +dr4 e−i(k−k′+q)·r4 +� ∞ +0 +dt4 e−i(ω−εk+εk′−iη)t4 += 1 +Ω2 +� +k≤kF +� +k′>kF +eik·re−ik′·r′ei(k′−k)·r′′ +× +� dω +2π K(|k′ − k|, ω) +ei(ω−εk)t +ω − εk + εk′ − iη , +(B5) +which can be rewritten in terms of the radial variables: +A1 = 1 +Ω2 +� +k≤kF +e−ik·R′ � +k′>kF +eik′·R′′ +× +� dω +2π K(|k′ − k|, ω) +ei(ω−εk)t +ω − εk + εk′ − iη . +(B6) +The second non-zero term is similar to the first and given +by +A2 = − 1 +Ω2 +� +k>kF +e−ik·R′ � +k′≤kF +eik′·R′′ +× +� dω +2π K(|k′ − k|, ω) +e−iεk′t +ω − εk + εk′ + iη . +(B7) +For the case t > 0 similar consideration yields +B1 = 1 +Ω2 +� +k≤kF +e−ik·R′ � +k′>kF +eik′·R′′ +× +� dω +2π K(|k′ − k|, ω) +e−iεk′t +ω − εk + εk′ − iη +(B8) + +18 +B2 = − 1 +Ω2 +� +k>kF +e−ik·R′ � +k′≤kF +eik′·R′′ +× +� dω +2π K(|k′ − k|, ω) +ei(ω−εk)t +ω − εk + εk′ + iη . +(B9) +To calculate the integral over ω one utilizes the spectral +representation of K: +K(k, ω) = +� 0 +−∞ +dω′ +L(k, ω′) +ω − ω′ − iδ + +� ∞ +0 +dω′ +L(k, ω′) +ω − ω′ + iδ , +(B10) +where +L(k, ω) = − 1 +π sign(ω)ImK(k, ω). +(B11) +The spectral function L(k, ω) is an odd function in ω. +For the case t < 0 the contour integral for A1 in the +complex ω plane is closed in the lower-half plane yielding +� dω +2π +1 +ω − ω′ + iδ × +ei(ω−εk)t +ω − εk + εk′ − iη += +−iei(ω′−εk−iδ)t +ω′ − εk + εk′ − iη . +(B12) +One obtains, using L(k, −ω) = −L(k, ω), +A1 = 1 +Ω2 +� +k≤kF +e−ik·R′ � +k′>kF +eik′·R′′e−iεkt +× +� ∞ +0 +dω′ L(|k′ − k|, ω′) +−ieiω′t +ω′ + εk′ − εk +(B13) +and +A2 = 1 +Ω2 +� +k>kF +e−ik·R′ � +k′≤kF +eik′·R′′e−iεk′t +× +� ∞ +0 +dω′ L(|k′ − k|, ω′) +−i +ω′ + εk − εk′ . +(B14) +For the case t > 0 one obtains +B1 = 1 +Ω2 +� +k≤kF +e−ik·R′ � +k′>kF +eik′·R′′e−iεk′t +× +� ∞ +0 +dω′ L(|k′ − k|, ω′) +−i +ω′ + εk′ − εk +(B15) +B2 = 1 +Ω2 +� +k>kF +e−ik·R′ � +k′≤kF +eik′·R′′e−iεkt +× +� ∞ +0 +dω′ L(|k′ − k|, ω′) +−ie−iω′t +ω′ + εk − εk′ . +(B16) +For each t the integral over ω′ can be parametrized as +follows: +M(q, ω, t) = +� ∞ +0 +dω′ L(q, ω′)−ieiω′t +ω′ + ω , +(B17) +so that +A1 = 1 +Ω2 +� +k≤kF +e−ik·R′e−iεkt � +k′>kF +eik′·R′′ +× M(|k′ − k|, εk′ − εk, t); +(B18) +A2 = 1 +Ω2 +� +k>kF +e−ik·R′ � +k′≤kF +eik′·R′′e−iεk′t +× M(|k′ − k|, εk − εk′, 0); +(B19) +B1 = 1 +Ω2 +� +k≤kF +e−ik·R′ � +k′>kF +eik′·R′′e−iεk′t +× M(|k′ − k|, εk′ − εk, 0); +(B20) +B2 = 1 +Ω2 +� +k>kF +e−ik·R′ � +k′≤kF +eik′·R′′e−iεkt +× M(|k′ − k|, εk − εk′, −t). +(B21) +By defining +γ(R, R′, t, t′, t′′) = 1 +Ω2 +� +k≤kF +e−ik·Re−iεkt +× +� +k′>kF +eik′·R′e−iεk′t′M(|k′ − k|, εk′ − εk, t′′), +(B22) +A1, A2, B1, and B2 can be written as +A1 = γ(R′, R′′, t, 0, t), +(B23) +A2 = γ(R′′, R′, t, 0, 0), +(B24) +B1 = γ(R′, R′′, 0, t, 0), +(B25) +B2 = γ(R′′, R′, 0, t, −t). +(B26) +The correlation hole is given by +ρc(R, R′, θ; t < 0) = +A1 + A2 +G0(R, t < 0), +ρc(R, R′, θ; t > 0) = +B1 + B2 +G0(R, t > 0). +(B27) +1 R. M. Martin, Electronic Structure: +Basic Theory and +Practical Methods (Cambridge University Press, Cam- +bridge, 2004). + +19 +2 L. Hedin, Phys. Rev. 139, A796 (1965). +3 G. D. Mahan, Many-Particle Physics (Springer New York, +NY, 2000). +4 R. M. Martin, L. Reining, and D. M. Ceperley, Interacting +Electrons: Theory and Computational Approaches (Cam- +bridge University Press, Cambridge, 2016). +5 F. Aryasetiawan and F. Nilsson, Downfolding Methods +in Many-Electron Theory (AIP Publishing, Melville, New +York, 2022). +6 F. Aryasetiawan and O. Gunnarsson, Rep. Prog. Phys. 61, +237 (1998). +7 A. L Fetter and J. D Walecka, Quantum Theory of Many- +Particle Systems, (Dover, Mineola, New York, 2003). +8 J. W. Negele and H. Orland, Quantum Many-Particle Sys- +tems, (Westview Press, Boulder, Colorado, 1998). +9 F. Aryasetiawan, Phys. Rev. B 105, 075106 (2022). +10 F. Aryasetiawan and T. Sj¨ostrand, Phys. Rev. B 106, +045123 (2022). +11 O. Gunnarsson and B. I. Lundqvist, Phys. Rev. B 13, +(4274 1976). +12 R. O. Jones and O. Gunnarsson, Rev. Mod. Phys. 61, 689 +(1989). +13 D. Pines and D. Bohm, Phys. Rev. 85, 338 (1952). +14 P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). +15 W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). +16 A. D. Becke, J. Chem. Phys. 140, 18A301 (2014). +17 R. O. Jones, Rev. Mod. Phys. 87, 897 (2015). +18 J. C. Slater, Phys. Rev. 81, 385 (1951). +19 Y. Wang and J. P. Perdew, Phys. Rev. B 44, 13298 (1991). +20 N. W. Ashcroft and N. D. Mermin, Solid State Physics, +(Saunders College Publishing, NY, 1976) + diff --git a/TNE5T4oBgHgl3EQfaw8F/content/tmp_files/load_file.txt b/TNE5T4oBgHgl3EQfaw8F/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a71127e8e510bd6fc17f710ac21cb546270a324c --- /dev/null +++ b/TNE5T4oBgHgl3EQfaw8F/content/tmp_files/load_file.txt @@ -0,0 +1,864 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf,len=863 +page_content='Time-dependent exchange-correlation hole and potential of the electron gas K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Karlsson1 and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Aryasetiawan2 1Department of Engineering Sciences, University of Sk¨ovde, SE-541 28 Sk¨ovde, Sweden 2 Department of Physics, Division of Mathematical Physics, Lund University, Professorsgatan 1, 223 63, Lund, Sweden The exchange-correlation hole and potential of the homogeneous electron gas have been investi- gated within the random-phase approximation, employing the plasmon-pole approximation for the linear density response function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The angular dependence as well as the time dependence of the exchange-correlation hole are illustrated for a Wigner-Seitz radius rs = 4 (atomic unit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' It is found that there is a substantial cancellation between exchange and correlation potentials in space and time, analogous to the cancellation of exchange and correlation self-energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Analysis of the sum rule explains why it is more advantageous to use a non-interacting Green function than a renor- malized one when calculating the response function within the random-phase approximation and consequently the self-energy within the well-established GW approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The present study pro- vides a starting point for more accurate and comprehensive calculations of the exchange-correlation hole and potential of the electron gas with the aim of constructing a model based on the local density approximation as in density functional theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' INTRODUCTION The homogeneous electron gas has been a long-lasting and an invaluable model of valence electrons in solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Pseudopotential theory1 explains why the behavior of va- lence electrons in solids, despite the presence of a strong ionic potential, nevertheless resembles that of the elec- tron gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The effects of exchange and correlations of the interacting homogeneous electron gas have been studied thoroughly for the last six decades2–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' An important em- pirical observation is that the main features of the spec- tral function arising from exchange and correlations are quite robust and can be carried over to real materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' For example, setting aside strongly correlated systems5, it is generally the case that the spectral function of most materials consists of a quasiparticle peak and an incoher- ent satellite feature which can be traced back to collec- tive charge excitations (plasmons), just as found in the electron gas6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The effects of exchange and correlations have been tra- ditionally studied using the concept of self-energy, a non- local and energy-dependent quantity that acts on the Green function as a convolution in space and time7,8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' In a recent development, a different framework for represent- ing exchange and correlations was proposed in the form of a time-dependent exchange-correlation (xc) potential9,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' This formalism is fundamentally different from the self- energy approach in that the potential acts locally or mul- tiplicatively on the Green function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Most importantly, the potential arises naturally as a Coulomb potential of a charge distribution (exchange-correlation hole) which fulfills a sum rule and some exact properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' More- over, due to the special form of the Coulomb interac- tion, which depends solely on the separation of two point charges, it can be shown that the exchange-correlation potential is in fact the first radial moment of the spher- ical average of the exchange-correlation hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' This ap- pealing result is the analog of the result found in the ex- act expression for the ground-state exchange-correlation energy, which is very much utilized in density functional theory and partially explains the success of the local den- sity approximation11,12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The exchange-correlation potential formalism also pro- vides a simple physical picture of the propagation of an added hole or electron in a many-electron system as in photoemission and inverse photoemission experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The added hole or electron induces a temporal density fluctuation of the system initially in its ground state, giv- ing rise to the exchange-correlation potential, which in turns acts on the Green function representing the prop- agation of the added hole or electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' In this paper, the time-dependent exchange-correlation hole and its corresponding exchange-correlation potential of the electron gas are studied using the random-phase approximation (RPA)3,7,13 to understand the salient fea- tures of the exchange-correlation hole and potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The long-term goal is to use the electron gas results as a basis for a local density approximation in the spirit of density functional theory12,14–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The theory section commences with a short summary of the exchange-correlation potential framework, which is outlined in detail in previous publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Formulas for the exchange-correlation hole and potential are then derived for the homogeneous electron gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' An analysis of the sum rule and its consequences is presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' II F followed by computational results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' III and a summary at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' THEORY In the exchange-correlation potential formalism the equation of motion of the equilibrium zero temperature time-ordered Green function is given by9 � i ∂ ∂t − h(r) − Vxc(r, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t) � G(r, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t) = δ(r − r′)δ(t), (1) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05590v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='str-el] 13 Jan 2023 2 where h(r) = −1 2∇2 + Vext(r) + VH(r), (2) in which Vext and VH are the external field and the Hartree potential, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The Green function is defined according to7 iG(r, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t) = ⟨T[ ˆψ(rt) ˆψ†(r′0)]⟩, (3) where r = (r, σ) labels both space and spin variables, ˆψ(rt) is the Heisenberg field operator, T is the time- ordering symbol, and ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='⟩ denotes expectation value in the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The exchange-correlation potential Vxc is the Coulomb potential of the exchange-correlation hole ρxc: Vxc(r, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t) = � dr′′v(r − r′′)ρxc(r, r′, r′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (4) The presence of the instantaneous Coulomb interaction implies that t′′ = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The exchange-correlation hole fulfills an important sum rule � d3r′′ρxc(r, r′, r′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t) = −δσσ′′θ(−t) (5) and the following exact condition ρxc(r, r′, r′′ = r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t) = −ρ(r) (6) for any r, r′ and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' General formula for the exchange-correlation hole From the definition of the exchange-correlation hole9, G(2) = ⟨T[ˆρ(3) ˆψ(1) ˆψ†(2)] = i[ρ(3) + ρxc(1, 2, 3)]G(1, 2), (7) where 1 = (r1, t1) etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' with t1 = t3 and the relation2,6 G(2) = iρ(3)G(1, 2) − δG(1, 2) δϕ(3) , (8) an explicit formula for the exchange-correlation hole is given by10 ρxc(1, 2, 3) = i δ δϕ(3) ln G(1, 2), (9) where ϕ is a probing field, which is set to zero after the functional derivative is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The exchange-correlation hole can thus be regarded as the linear response of i ln G with respect to an external field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' From the identity δG = −G(δG−1)G (10) and the equation of motion of the Green function, one obtains ρxc(1, 2, 3)G(1, 2) = i � d4 G(1, 4) � δ(3 − 4) + δVH(4) δϕ(3) � G(4, 2) + i � d4d5 G(1, 4)δΣ(4, 5) δϕ(3) G(5, 2), (11) where Σ is the self-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The first term on the right- hand side, iG(1, 3)G(3, 2), will be referred to as the ex- change contribution, the second term involving δVH δϕ as the linear response contribution, and the last term with δΣ δϕ as the vertex correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The second and third terms together constitute the correlation contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Within the RPA, the vertex correction is neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The quantity in the curly brackets is the inverse dielec- tric function: ϵ−1(4, 3) = δ(4 − 3) + δVH(4) δϕ(3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (12) It is convenient for later purposes to define K(4, 3) = δVH(4) δϕ(3) = � d5 v(4 − 5)χ(5, 3), (13) where v is the Coulomb interaction and χ is the linear density response function χ(5, 3) = δρ(5) δϕ(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (14) Replacing r1 → r, r2 → r′, and r3 → r′′ and taking into account the fact that t1 = t3 = t and t2 = 0, one finds ρxc(r, r′, r′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t)G(r, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t) = iG(r, r′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 0−)G(r′′, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t) + i � dr4dt4 G(r, r4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t − t4)K(r4, r′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t4 − t)G(r4, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (15) The first term on the right-hand side yields the exchange hole whereas the second term yields the correlation hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' It is immediately clear that the exact condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (6) is already fulfilled by the exchange hole implying that ρc(r, r′, r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (16) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Interacting homogeneous electron gas The Green function of the paramagnetic non- interacting homogeneous electron gas is given by iG0(r, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t) = 1 Ω � k>kF eik·(r−r′)e−iεktθ(t) − 1 Ω � k≤kF eik·(r−r′)e−iεktθ(−t), (17) 3 𝒓 𝒓′ 𝒓’’ 𝑅 𝑅′ 𝑅′′ 𝜃 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 1: Definition of the radial variables R, R′, and R′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' They are related to the angle θ by R′′2 = R2 − 2RR′ cos θ + R′2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' where εk = 1 2k2, kF is the Fermi wave vector, and Ω is the space volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' It is understood that σ = σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' For the homogeneous electron gas, it is convenient to introduce the variable R = r′ − r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' In spherical coordinates the equation of motion becomes � i ∂ ∂t − h(R) − Vxc(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t) � �G(R, t) = 1 4πRδ(R)δ(t), (18) where h(R) = −1 2 ∂2 ∂R2 , �G(R, t) = R G(R, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (19) Defining T(R, t) = 1 �G(R, t) h(R) �G(R, t), (20) the formal solution is given by G(R, t) = G(R, 0)e−i � t 0 dt′[T (R,t′)+Vxc(R,t′)], (21) in which it is understood that G(R, 0) = G(R, 0+) for t > 0 and G(R, 0) = G(R, 0−) for t < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' In general, from the equation of motion in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (18) i[G(R, 0+) − G(R, 0−)] = δ(R) 4πR2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (22) For the homogeneous electron gas, r may be chosen to be the origin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=', r = 0, and one defines the variables R = r′ − r, R′ = r′′ − r, and R′′ = r′′ − r′ as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Exchange hole From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (15) the exchange hole is given by ρx(r, r′, r′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t)G(r, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t) = iG(r, r′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 0−)G(r′′, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (23) Unlike the static exchange hole18 in quantum chemistry and density functional theory, the exchange hole in the present formalism is time dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Using a non-interacting Green function as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (17) and considering the case t < 0 one finds ρx(r, r′, r′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t < 0) � k≤kF eik·(r−r′)e−iεkt = − 1 Ω � k′≤kF eik′·(r−r′′) × � k≤kF eik·(r′′−r′)e−iεkt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (24) For t > 0 ρx(r, r′, r′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t > 0) � k>kF eik·(r−r′)e−iεkt = − 1 Ω � k′≤kF eik′·(r−r′′) × � k>kF eik·(r′′−r′)e−iεkt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (25) Expressed in radial variables and the angle θ as ex- plained in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 1, ρx(R, R′, θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t) = iG0(R′, 0−)G0(R′′, t) G0(R, t) (26) where R′′ depends on θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' G0(R, t) is obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (17) by performing the k-integral over the solid angle, yielding iG0(R, t < 0) = − 1 2π2 1 R � kF 0 dk k sin (kR)e−ik2t/2, (27) iG0(R, t > 0) = 1 2π2 1 R � ∞ kF dk k sin (kR)e−ik2t/2, (28) iG0(R, 0−) = − 1 2π2 1 R3 [sin (kFR) − kFR cos (kFR)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (29) G0(R, t < 0) can be expressed in terms of the complex error function or calculated numerically using a standard quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The calculation of G0(R, t > 0), however, needs more care and it is detailed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Correlation hole The linear response contribution to ρxc is given by the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Keeping in mind that t3 = t1 = t, t2 = 0, one obtains i � d4 G(1, 4)K(4, 3)G(4, 2) = i � dr4dt4G(r − r4, t − t4)K(r4 − r′′, t4 − t) × G(r4 − r′, t4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (30) 4 The details of the calculations using G = G0 are shown in Appendix B and the results are given by ρc(R, R′, θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t < 0) = A1 + A2 G0(R, t < 0), (31) ρc(R, R′, θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t > 0) = B1 + B2 G0(R, t > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (32) A1, A2, B1, and B2 are functions of R′ and R′′, and given by A1 = γ(R′, R′′, t, 0, t), (33) A2 = γ(R′′, R′, t, 0, 0), (34) B1 = γ(R′, R′′, 0, t, 0), (35) B2 = γ(R′′, R′, 0, t, −t), (36) where γ(R, R′, t, t′, t′′) = 1 Ω2 � k≤kF e−ik·Re−iεkt × � k′>kF eik′·R′e−iεk′t′M(|k′ − k|, εk′ − εk, t′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (37) The quantity M is given by M(q, ω, t) = � ∞ 0 dω′ L(q, ω′)−ieiω′t ω′ + ω , (38) where L(q, ω) is the spectral function of K(q, ω) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (13): K(k, ω) = � 0 −∞ dω′ L(k, ω′) ω − ω′ − iδ + � ∞ 0 dω′ L(k, ω′) ω − ω′ + iδ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (39) K(q, ω) is symmetric in frequency but L(q, ω) is anti- symmetric and related to K as follows: L(k, ω) = − 1 π sign(ω)ImK(k, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (40) The correlation hole involves coupled integrals over momenta below and above kF, which are difficult to cal- culate analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' They are six-dimensional integrals which cannot be easily performed with standard quadra- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' To make the computation feasible, a plasmon-pole approximation for L(k, ω) is employed and described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Plasmon-pole approximation The plasmon dispersion of the homogeneous electron gas is given by7 Ωq = ωp � 1 + 3 10 k2 Fq2 ω2p + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' � , (41) where the plasmon frequency ωp in the long-wavelength limit is given by ω2 p = 4πρ0, (42) and ρ0 is the electron gas density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The critical momen- tum, qc, at which the plasmon starts to merge into the electron-hole excitations is given by the crossing of the plasmon dispersion with the line εq + kFq yielding qc = 1 2a �√ 1 + 4ac − 1 � kF, (43) where a = 1 2 − 3 10c, c = ωp k2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (44) Great simplification results if a plasmon-pole approx- imation independent of k is used for L(k, ω) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (40): L(k, ω) = ωp 2 [δ(ω − ωp) − δ(ω + ωp)] , (45) which corresponds to K(q → 0, ω) = ω2 p ω2 − ω2p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (46) The approximation is valid for k ≤ qc and for rs = 3, 4 and 5, which cover most of the average valence densities in real materials, the critical momenta are qc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='86, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='82, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='73 kF, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Within the plasmon-pole approximation the quantity M defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (38) becomes independent of momenta: M(q, ω, t) = ωp 2 −ieiωpt ωp + ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (47) Then the coupling between k and k′ is partially released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Using 1 Ω � k e−ik·R = 1 2π2R � dk k sin (kR) (48) yields within the plasmon-pole approximation γP P (R, R′, t, t′, t′′) = −2iωp (2π)4RR′ � kF 0 dk k sin (kR)e−iεkt × � ∞ kF dk′ k′ sin (k′R′) e−iεk′t′eiωpt′′ ωp + εk′ − εk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (49) Since the plasmon-pole approximation decouples the angular inter-dependence of k and k′, it is expected to impart error to the correlation contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' To min- imize error, the upper limit of the integration over k′ corresponding to unoccupied states is chosen so as to ap- proximately reproduce the static correlation hole19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' This choice yields a value of ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5kF so that integration over unoccupied states arising from the correlation contribu- tion is restricted to between kF and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5kF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 5 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Exchange-correlation potential By making a change of variable R′ = r′′ − r the exchange-correlation potential in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (4) reduces to the first radial moment of the spherical average of ρxc: Vxc(r, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t) = � dR′R′ ρxc(r, r′, R′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t), (50) where ρxc(r, r′, R′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t) for given r, r′, and t depends only on the radial distance R′ = |r′′ − r|, ρxc(r, r′, R′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t) = � dΩR′ρxc(r, r′, r + R′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (51) As can be seen from, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=', Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (24) and (B5), the spherical average of ρxc for the electron gas amounts to performing a solid-angle integration � dΩ′′ei(k−k′)·r′′ = 4π sin(∆k R′) ∆k R′ , (52) where ∆k = |k − k′| and R′ = |r′′ − r|, r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Exchange potential For t < 0 the exchange hole is given by ¯ρx(R, R′, t < 0)iG0(R, t) = 1 Ω2 � k,k′≤kF e−ik·r′e−iεkt × 4π sin(∆k R′) ∆k R′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (53) The exchange potential is the first moment of ¯ρx in R′: Vx(R, t < 0) = 1 iG0(R, t) 4π Ω2 � k,k′≤kF e−ik·Re−iεkt × � dR′ sin(∆k R′) ∆k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (54) Consider the integral over R′ with positive α → 0: lim α→0 � ∞ 0 dR′ sin(∆k R′)e−αR′ = 1 ∆k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (55) One then finds Vx(R, t < 0) = 1 iG0(R, t) 4π Ω2 � k,k′≤kF e−ik·Re−iεkt 1 (∆k)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (56) The integral over k′ is given by f(k) = 1 Ω � k′≤kF 1 (∆k)2 = 1 4π2k � kF 0 dk′k′ ln ���� k + k′ k − k′ ���� , (57) which can be performed analytically yielding f(k) = kF 2π2 F(k/kF), (58) where F(x) = 1 2 + 1 − x2 4x ln ���� 1 + x 1 − x ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (59) This function is the same as the one appearing in the static Hartree-Fock theory for the electron gas20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' More explicitly as a function of k, f(k) = kF 2π2 �1 2 + k2 F − k2 4kFk ln ���� kF + k kF − k ���� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (60) There remains the integral over k which reduces to a one-dimensional integral over the radial k: Vx(R, t < 0) = 1 iG0(R, t) × 2 πR � kF 0 dk k sin(kR) e−iεktf(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (61) For t > 0 the result is given by Vx(R, t > 0) = − 1 iG0(R, t) × 2 πR � ∞ kF dk k sin(kR) e−iεktf(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (62) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Correlation potential A similar procedure as for the exchange potential can be applied to the correlation potential using A1, A2, B1, and B2, given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (B18-B21) in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The result is given by Vc(R, t < 0) = C1 + C2 G0(R, t < 0), (63) Vc(R, t > 0) = D1 + D2 G0(R, t > 0), (64) where C1 = Γ(0, R, t, 0, t), (65) C2 = Γ(R, 0, t, 0, 0), (66) D1 = Γ(0, R, 0, t, 0), (67) D2 = Γ(R, 0, 0, t, −t), (68) and Γ(R, R′, t, t′, t′′) = 4π Ω2 � k≤kF e−iεkte−ik·R × � k′>kF e−ik′·R′e−iεk′t′ × M(|k′ − k|, εk′ − εk, t′′) |k′ − k|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (69) 6 According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (47),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' within the plasmon-pole ap- proximation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' M(|k′ − k|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' εk′ − εk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t′′) = ωp 2 −ieiωpt′′ ωp + εk′ − εk ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (70) which partially decouples the interdependence of k and k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' allowing for analytical integration over the solid an- gles of both variables,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' yielding C1 = P1(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (71) C2 = P2(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (72) D1 = P1(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (73) D2 = P2(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' −t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (74) where P1(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t′′) = −iωpeiωpt′′ 4π3R � kF 0 dk � ∞ kF dk′ k sin (k′R) × e−iεkte−iεk′t′ ωp + εk′ − εk ln ���� k + k′ k − k′ ����,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (75) P2(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t′′) = −iωpeiωpt′′ 4π3R � kF 0 dk � ∞ kF dk′ k′ sin (kR) × e−iεkte−iεk′t′ ωp + εk′ − εk ln ���� k + k′ k − k′ ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (76) Due to the use of the plasmon-pole approximation, the upper limit of the integral over k′ is restricted to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5kF to reproduce approximately the static correlation hole, as described earlier in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' II D 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Sum rule In this section, the sum rule and its consequences are discussed and a simple vertex approximation respecting the sum rule is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The results and conclusions reached in this section are quite general and supported by the electron gas results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Exchange hole One has for a non-interacting G and t < 0 i � dr′′G0(r, r′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 0−)G0(r′′, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t < 0) = −G0(r, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t < 0), (77) which can be shown as follows: i � dr′′G0(r, r′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 0−)G0(r′′, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t < 0) = −i � dr′′ � k≤kF ϕk(r)ϕ∗ k(r′′) � k′≤kF ϕk′(r′′)ϕ∗ k′(r′)e−iεk′t = −i � k≤kF ϕk(r)ϕ∗ k(r′)e−iεkt = −G0(r, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (78) It is also quite clear that i � dr′′G0(r, r′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 0−)G0(r′′, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t > 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (79) It then follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (23) that the exchange hole fulfills the sum rule when G0 is used: � dr′′ρx(r, r′, r′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t < 0) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (80) Explicitly for the electron gas, it follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (24) that the sum rule for t < 0 is fulfilled by the exchange hole since � d3r′′ei(k−k′)·r′′ = (2π)3δ(k − k′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (81) The sum rule for t > 0 is zero since k ̸= k′ as can be seen from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' In general − i � dr′′G(r, r′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 0−)G(r′′, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t < 0) ̸= G(r, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t < 0) (82) and also i � dr′′G(r, r′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 0−)G(r′′, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t > 0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (83) unless G = G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' This implies that if only the exchange part is considered, neglecting the correlation and vertex terms, then in general the sum rule is not fulfilled when a renormalized G is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Correlation hole It is also evident that the correlation part of the exchange-correlation hole gives no contribution to the sum rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The reason for this can be seen by considering the change in the charge density under a perturbation: δρ(1) = � d2 χ(1, 2)δϕ(2), (84) where χ is the linear density response function as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' A constant perturbation, δϕ = 1, does not alter the density so that � d2 χ(1, 2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (85) This property is fulfilled by the RPA response function calculated using G0 as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' It is interesting to observe that in the case of the elec- tron gas, it can be seen explicitly from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (B5) in Ap- pendix B that the integral of A1 over r′′ is zero since k ̸= k′ and the same conclusion holds for A2, B1, and B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Hence the sum-rule is fulfilled, irrespective of the approximation used for K(q, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 7 Since the response function can be expanded in powers of the polarization P, χ = P + PvP + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (86) it follows that if the polarization function fulfills � d2P(1, 2) = 0 (87) then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (85) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The polarization in the RPA is given by P(r, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t) = −iG(r, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t)G(r′, r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' −t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (88) If a non-interacting G is used, then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (87) and conse- quently Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (85) are fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' This can be understood as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' It can be shown that � dr′′G0(r, r′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t)G0(r′′, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' −t) = 0 (89) by using the definition of G0: � dr′′G0(r, r′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t)G0(r′′, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' −t) = θ(t) � dr′′⟨ ˆψ(rt) ˆψ†(r′′)⟩⟨ ˆψ†(r′) ˆψ(r′′, −t)⟩ + θ(−t) � dr′′⟨ ˆψ†(r′′) ˆψ(rt)⟩⟨ ˆψ(r′′, −t) ˆψ†(r′)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (90) Since the expectation value is taken with respect to a single Slater determinant, the integral over r′′ amounts to � dr′′ϕ∗ k(r′′)ϕk′(r′′) = 0, (91) where {ϕk} is a set of orbitals defining the single Slater determinant, and if k corresponds to an occupied state then k′ corresponds to an unoccupied state and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' However, if a renormalized G is used to calculate P in the RPA as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (88), the conditions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (87) and consequently Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (85) are no longer fulfilled in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Thus, with a renormalized G, both the exchange and correlation terms would violate the sum rule, and it seems unlikely that the sum of these two terms would fulfill the sum rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' This might explain why it is not favorable to use a renormalized G in the RPA and consequently in the GW approximation2,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' It also implies that inclusion of the vertex δΣ/δϕ would restore the sum rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Hence to preserve the sum rule, the use of a renormalized G should be accompanied by inclusion of the vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' A simple vertex correction The preceding consideration suggests a scheme for an approximate vertex correction that would preserve the sum rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Consider the following screened-exchange self- energy, Σ(1, 2) = iG0(1, 2)W0(1, 2), (92) where W0(1, 2) is an instantaneous interaction, W0(1, 2) = W0(r1, r2)δ(t1 − t2), (93) with W0(r1, r2) being some screened interaction, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=', the Thomas-Fermi screened interaction or the static screened interaction calculated within RPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' By using G0 through- out, the vertex correction is given by δΣ(1, 2) δϕ(3) = iδG0(1, 2) δϕ(3) W0(1, 2) = −i � d4d5 G0(1, 4)δG−1 0 (4, 5) δϕ(3) G0(5, 2)W0(1, 2), (94) where the identity δG = −G(δG−1)G has been utilised and δW0 δϕ = 0 since it is assumed that W0 is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' In the presence of the probing field ϕ, G0 fulfills the equation of motion � i ∂ ∂t1 − h(1) − ϕ(1) � G0(1, 2) = δ(1 − 2), (95) so that δG−1 0 (4, 5) δϕ(3) = −δ(4 − 5) � δ(3 − 4) + δVH(4) δϕ(3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (96) This leads to δΣ(1, 2) δϕ(3) = iG0(1, 3)G0(3, 2)W0(1, 2) + i � d4 G0(1, 4)δVH(4) δϕ(3) G0(4, 2)W0(1, 2), (97) and hence according to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (85) and (89), � d3δΣ(1, 2) δϕ(3) = 0, (98) noting that t1 = t2 due to the instantaneous interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' It follows that the vertex in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (97) preserves the sum rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' A model Green function for the interacting electron gas To assist in analyzing the results of the calculations of Vxc of the electron gas, a model Green function has been constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' A physically motivated model for the Green function of the interacting electron gas is given by the following: G(R, t < 0) = i Ω � k≤kF � Zk + (1 − Zk)eiωkt� × e−iEkteik·R, (99) 8 G(R, t > 0) = − i Ω � k>kF � Zk + (1 − Zk)e−iωkt� × e−iEkteik·R, (100) where Ek is the quasiparticle energy, Zk is the quasi- particle renormalization factor, and ωk is the plasmon energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' To allow for analytic derivation of the exchange- correlation potential, Zk and ωk are assumed to be in- dependent of k and Ek is taken to be a renormalized free-electron gas dispersion: Zk = Z, ωk = ωp, Ek = αεk = α 2 k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (101) For an electron gas of density ρ0 the plasmon energy is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The model Green function can be improved by includ- ing the possibility of having spectral weight above or be- low the Fermi level for k ≤ kF or k > kF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' respectively: G(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t < 0) = i Ω � k≤kF (C1 + C2 + C3)eik·R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (102) G(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t > 0) = − i Ω � k>kF (D1 + D2 + D3)eik·R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (103) where C1 = Ze−iEkt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (104) C2 = (1 − β− k )(1 − Z)e−i(Ek−ωp)t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (105) C3 = β− k (1 − Z)e−i(EF+ωp)t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (106) D1 = Zpe−iEkt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (107) D2 = (1 − β+ k )(1 − Zp)e−i(Ek+ωp)t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (108) D3 = β+ k (1 − Zp)e−i(EF−ωp)t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (109) in which β− k = k 2kF θ(kF − k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (110) β+ k = 1 2 kmax − k kmax − kF θ(kmax − k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (111) For k > kmax, the dispersion is taken to be that of the free-electron gas for otherwise the integration over k to infinity would not converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' It means physically that high-energy electrons are free since they do not experi- ence exchange and correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' To conserve electronic charge, the weight above the Fermi level coming from states smaller than kF is equated to the weight below the Fermi level coming from states larger than kF, yielding an upper limit, neglecting the k-dependence of β±, kmax ≈ � 1 + 1 − Z 1 − Zp �1/3 kF (112) above which the spectrum does not have weight below the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' For t ̸= 0, the exchange-correlation potential can be obtained from the equation of motion: Vxc(R, t) = 1 G(R, t) [i∂t − h] G(R, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (113) Since h exp (ik · R) = k2 2 exp (ik · R) (114) one finds for t < 0 [i∂t − h] G(R, t) = i 1 Ω � k≤kF (A1 + A2 + A3)eik·R, (115) where A1 = Z(Ek − εk)e−iEkt, (116) A2 = (1 − β− k )(1 − Z)(Ek − εk − ωp)e−i(Ek−ωp)t, (117) A3 = β− k (1 − Z)(EF − εk + ωp)e−i(EF+ωp)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (118) For t > 0 [i∂t − h] G(R, t) = −i 1 Ω � k>kF (B1 + B2 + B3)eik·R, (119) where B1 = Zp(Ek − εk)e−iEkt, (120) B2 = (1 − β+ k )(1 − Zp)(Ek − εk + ωp)e−i(Ek+ωp)t, (121) B3 = β+ k (1 − Zp)(EF − εk − ωp)e−i(EF−ωp)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (122) III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' RESULTS The results shown in this section are all expressed in atomic units (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=') and correspond to rs = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Two pairs of representative times, t = ±34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='75 corresponding to 1/¯ε, where ¯ε is the center of the occupied band, and t = ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62 corresponding to the inverse of the plasmon energy, 1/ωp, have been chosen for illustrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Additional times are also considered as appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Angular dependence of the exchange-correlation hole In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 2 the real part of the exchange holes for the case R = 10, t = −34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='75, and for three different angles θ = 0, π/4, π/2 as defined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 1 are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' These density fluctuations are due to exchange-only interaction 9 and correspond to the case when a hole created at R = 0 at t = −34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='75 is to be annihilated later at R = 10 at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' It can be seen that there are more fluctuations for θ = 0 representing the direction along the line con- necting the location where the hole is created at R = 0 and the location where the hole is annihilated at R = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The fluctuations are stronger in the direction towards R = 10 from the origin than in the opposite direction, which reflect the preponderance of the hole presence and its effects in the former direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Indeed, the density fluctuations along the direction perpendicular to the line joining R = 0 and R = 10 (θ = π/2) are found to be the least whereas for the direction θ = π/4 they are some- where in between those of θ = 0 and π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' In accordance with discussion in the paragraph following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (15), the exchange hole already fulfills the exact condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (6), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=', ρx(0) = −ρ0, where ρ0 the homogeneous electron gas density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' A similar behavior can be observed for the correla- tion holes shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' These density fluctuations arise from the classical Coulomb interaction in response to the creation of a hole and its accompanying exchange hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The fluctuations are particularly strong along the direction towards R = 10 from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (16), the correlation hole at the origin should vanish, which is clearly not the case here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' There are two possible explanations for this discrepancy, one being the use of the plasmon-pole approximation and another being the neglect of the vertex term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' One may speculate that the vertex term is needed even when the exact response func- tion is used, but to answer this issue in the definitive, cal- culations employing the full RPA response function are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' It is, however, to be noted that discrepancy may not be as severe as it might appear at first sight be- cause what enters into the exchange-correlation poten- tial is the first moment of the spherical average of the exchange-correlation hole so that contributions at small R are significantly cut down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Numerical calculations of the sum rule show that in accordance with theory the exchange hole alone fulfills the sum rule for t < 0 and the correlation hole integrates to zero whereas for t > 0 both the exchange and the correlation holes integrate to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Time dependence of the exchange-correlation hole To illustrate the time dependence of the exchange- correlation hole, the exchange holes for two representa- tive times t = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62 and t = −34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='75 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 3 for the case R = 10 and θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' A similar result is found for the correlation hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' For a short time scale in which the hole is annihilated at a location well sepa- rated from the location where it was created, the fluctu- ations are much stronger than those of a long period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' As a guide, it is useful to consider the limit of R = 0 and t = 0−, which yields the static exchange-correlation hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 20 15 10 5 0 5 10 15 20 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5 1 10-3 20 15 10 5 0 5 10 15 20 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5 1 10-3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 2: The real part of the exchange hole (top) and the correlation hole (bottom) for θ = 0 (solid black), π/4 (dashed blue), and π/2 (dotted red) for the case R = 10 and t = −34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' There is a correlation between the spatial separation R and the time period, which determines the behavior of the exchange-correlation hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Within a short period of time but with large R, the system may not have suffi- cient time to relax so that the density fluctuations arising from a sudden creation of a hole at t = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62 have not stabilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' These large density fluctuations for small θ, however, contribute little to the exchange-correlation po- tential since only the spherical average of the exchange- correlation hole is needed and when taking the spherical average each angular contribution is multiplied by sin θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' On the other hand, for a relatively short spatial sep- aration R = 2, the time-dependence has a much less in- fluence on the exchange hole, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 10 20 15 10 5 0 5 10 15 20 4 2 0 2 4 6 8 10 10-3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 3: The real part of the exchange hole for t = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62 and −34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='75 and the case R = 10, θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The difference between the exchange holes for t = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62 and −34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='75 is much less pronounced than for the case of R = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The case of small R and small t approaches the static exchange-correlation hole limit of R = 0 and t = 0−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' For t > 0 corresponding to an electron addition, the exchange hole exhibits a more oscillatory behavior than for t < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' This can be understood from the exact sum rule which dictates that the exchange hole must integrate to zero for t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' For larger t, however, the oscillations become damped as the system stabilizes itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Spherical average of the exchange-correlation hole The relevant quantity determining the exchange- correlation potential is the first radial moment of the spherical average of the exchange-correlation holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' In the upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 5 the spherical average of the real part of the exchange hole multiplied by R′ is shown for several values of R and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' When taking a spherical average the factor sin θ implies that the angular region around θ ≈ 0 contributes significantly less than the re- gion around θ ≈ π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' This is confirmed by comparing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 4 for R = 2 in which it can be seen that the behavior of the spherical average essentially follows that of the exchange hole for θ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' For R = 2 and t = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62 the exchange hole is virtually indistinguishable from the static Hartree-Fock exchange hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' As expected, the exchange hole for small R and t should approach the static exchange hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' However, even for large values of R and t the exchange hole still follows closely the static one, as can be seen in the top panel of 0 2 4 6 8 10 12 14 16 18 20 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5 1 10-3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 4: The real part of the exchange hole for t = ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62 (blue) and ±34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='75 (black) for R = 2, θ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The solid and dashed lines correspond to t < 0 and t > 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' As R is increased, some weight is transferred from the small to the large regions of R′ and the exchange hole appears to become less dependent on t for large R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The spherical average of the correlation hole for several values of R and t is shown in the lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The general behavior of the correlation hole is similar to that of the exchange hole, except for R = 2 in which the correlation hole fluctuates strongly around the origin as a function of time than the corresponding exchange hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' As also expected from the sum rule, which integrates to zero for the correlation hole, more oscillations can be observed than in the corresponding exchange holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' For t > 0 the correlation hole tends to be positive around the origin and vanishes as R increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The exchange and the correlation holes exhibit strong fluctuations, deviating greatly from the static one at t = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62, which is found to correlate with the radial distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' To investigate further, the exchange and the correlation holes in the vicinity of R = 10 are calcu- lated and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' It is quite evident that the large fluctuations at R = 9 − 10 correlate with the time t = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' These large fluctuations originate from the small values of |G0(R, t)| at these values of R and t, and are carried over to the exchange-correlation potentials as discussed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 11 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='02 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='02 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='02 0 5 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+page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='02 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 5: The real part of the spherical average of the exchange hole (upper panel), the correlation hole (middle panel), and the exchange-correlation hole (lower panel) mul- tiplied by R′ for t = ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62 (blue), ±34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='75 (black), ±69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5 (red), and R = 2, 10, 30, 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The solid and dashed curves cor- respond to t < 0 and t > 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The green curve is the static exchange hole from the Hartree-Fock approxima- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' For t = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62 (solid blue) and R = 2 the exchange hole is virtually indistinguishable from the static one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 6: The real part of the spherical average of the exchange hole (upper panel), the correlation hole (middle panel), and the exchange-correlation hole (lower panel) multiplied by R′ for t = ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62 (blue), ±34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='75 (black), ±69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5 (red), and R = 8, 9, 10, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The solid and dashed curves correspond to t < 0 and t > 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The green curve is the static exchange hole from the Hartree-Fock approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 12 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Exchange-correlation potentials From the spherical average of the exchange-correlation hole the exchange-correlation potential can be deter- mined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The real and imaginary parts of the exchange potential for t = ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62 and ±34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='75 are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 7 and 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' For t < 0 there is a striking differ- ence between the exchange potentials corresponding to the two times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The exchange potential corresponding to t = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62 exhibits a pronounced structure around R = 9 and 16, whereas the one corresponding to t = −34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='75 is almost constant with a relatively weak feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' As men- tioned earlier, this strong structure at t = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62 arises from the small value of |G0(R, t)| at R = 9 and 16 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' In fact, the exchange potential has a strong structure at those positions for a range of t, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The structure is very sharp for very small t and as the magnitude of t increases, it be- comes weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' A plausible explanation is that when a hole is introduced into the system, large density fluctu- ations arise and within a short period of time the sys- tem does not have enough time to relax, generating as a consequence a strong spatial variation in the exchange potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' However, strong spatial variations also appear at a large-time scale as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The common denominator for the presence of strong structures in the exchange-correlation potential is the diminishing value of |G0(R, t)| at the corresponding position R and time t, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' For t > 0 the exchange potentials are generally weaker and less attractive than for t < 0 as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The difference between the two cases may be traced back to the fundamental physical difference between removing an electron (hole creation, t < 0) and adding an elec- tron (t > 0), in that the removed electron is part of the system in the ground state whereas the added electron is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' This is also reflected in the sum rule which inte- grates to −1 for a hole addition but zero for an electron addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Thus, the exchange potential associated with a hole creation must be stronger than that associated with an electron addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Aside from special systems that possess a particle-hole symmetry, the creation of a hole can be expected to create a stronger disturbance to the system in the ground state than the addition of an electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The exchange potential does not take into account the classical Coulomb response of the electrons which results in screening of the density fluctuations arising from the exchange interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 7 (middle) the real parts of the correlation potentials associated with the linear density response of the system are shown for the two pairs of representative times t = ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62 and ±34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' As in the case of exchange, the correlation potential corresponding to t = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62 has a pronounced structure in comparison with the one corresponding to t = −34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The real and imaginary parts of the exchange- correlation potentials are displayed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 7 and 8, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Peaks in the imaginary part of the exchange- correlation potential, which is the analog of the imagi- nary part of the self-energy, signals the presence of damp- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' It is evident that there is a strong cancellation of exchange and correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The Kramers-Kronig-like structures in the exchange potential between R = 8 and 10 as well as between 15 and 17 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 7 induce po- larization in which electrons and holes are accumulated in opposite regions where the minimum and maximum of Vx are located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' This exchange polarization is neu- tralized by Coulomb screening, which is affected by the presence of inverted structures in Vc at the corresponding regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' This results in a strong cancellation between ex- change and correlation potentials and the resulting total exchange-correlation potential has a much less structure than both the exchange and the correlation potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' To illustrate further the large cancellation between ex- change and correlation, the real part of the exchange, cor- relation, and exchange-correlation potentials for t = −1 are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' It is tempting to speculate that the remaining structure in the exchange-correlation potential might be smoothed out when full RPA calcula- tions are carried out without relying on the plasmon-pole approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' A similar cancellation is also found for t = −34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='75 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 7) but to a much lesser extent since both the exchange and the correlation potentials have little structures to begin with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The strong cancellation between exchange and corre- lation is well known in the self-energy formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The cancellation manifests itself in the one-particle disper- sion in which the too large occupied bandwidth within the Hartree-Fock approximation is significantly reduced by correlation effects, resulting in a dispersion close to the non-interacting one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The Vxc formalism, however, of- fers a different perspective in which the cancellation can be explicitly seen in the exchange-correlation potential in space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' R 9 5 9 16 5 t 0 to -5 -60 -84 -102 -114 TABLE I: The approximate values of R ≤ 20 and the cor- responding times t ≤ 0 around which strong structures are found in the exchange and the correlation potentials and strong deviations in the exchange and the correlation holes from their static counterparts are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 13 0 2 4 6 8 10 12 14 16 18 20 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='25 0 2 4 6 8 10 12 14 16 18 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 7: The real part of the exchange potential (top), the correlation potential (middle), and the exchange-correlation potential for t = ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62 (blue) and ±34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='75 (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The solid and dashed curves correspond to t < 0 and t > 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 0 2 4 6 8 10 12 14 16 18 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='35 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 8: The imaginary part of the exchange potential (top), the correlation potential (middle), and the exchange- correlation potential (bottom) for t = ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62 (blue) and ±34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='75 (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The solid and dashed curves correspond to t < 0 and t > 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 14 0 2 4 6 8 10 12 14 16 18 20 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5 2 10-3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 9: The real (solid) and the imaginary (dashed) parts of the non-interacting Green function for t = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62 (blue) and −34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='75 (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' For t = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62, the value of |G0| is close to zero at positions R = 9 and 16, whereas for t = −34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='75, it remains relatively substantial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Comparison with the model Green function To check the electron gas results and to gain more in- sight into the salient features of the exchange-correlation potentials, a model Green function has been constructed as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' II G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The exchange-correlation po- tentials derived from the model for t = ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62 and t = ±34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='75 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 14, which should be compared with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Apart from the absolute position, which cannot be determined without calculating the quasipar- ticle dispersion from the resulting electron gas exchange- correlation potential, the agreement for t = ±34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='75 is quite striking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' There is a discrepancy in the separation between Vxc(t > 0) and Vxc(t < 0), which may indicate an overestimation of correlation within the plasmon-pole approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Increasing the range of integration from 2kF to 4kF in the integration over the unoccupied states results in a down shift of the correlation potential while the shape remains stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' There is less agreement for t = ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62, both concerning the shape as well as the sep- aration between Vxc(t > 0) and Vxc(t < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Again, this may be due to the plasmon-pole approximation, or, spec- ulatively speaking, to the model itself, which may not be accurate for small t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The small time behavior is deter- mined by the quality of the model at high energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The model assumes a sharp plasmon excitation, which may be valid for small momentum but becomes less accurate at large momenta beyond the critical momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' To make a separate comparison of both Vx and Vc, the exchange potential deduced from the Hartree-Fock Green function is calculated and used to determine the correla- 0 2 4 6 8 10 12 14 16 18 20 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 10: The real part of the exchange potential for t = −1 (blue), −4 (black) and −8 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The structure at R = 9 and 16 becomes sharper as t → 0−, which, however, is largely cancelled by the correlation potential, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' tion potential of the model by subtracting the exchange potential from the total exchange-correlation potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 15 for t = ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62 and t = ±34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Here, there is no ambiguity regarding the absolute position of the exchange potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The close agreement between the exchange potential deduced from the Hartree-Fock Green function and the one calculated from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (61) and (62) for the electron gas attests that the physical interpretation of the exchange-correlation hole in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (15), in which the first term on the right- hand side is associated with the exchange hole, is well motivated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' CONCLUSION AND SUMMARY The exchange-correlation hole and potential of the ho- mogeneous electron gas have been studied within the RPA for rs = 4 and several representative time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The angular dependence of the exchange-correlation hole for t < 0 shows a stronger oscillation along the line join- ing the location where the hole is created and the location where the hole is annihilated (θ = 0) whereas along the perpendicular direction (θ = π/2) it shows the least oscil- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' This behavior can be attributed to the degree of preponderance of the hole along the respective directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The behavior of the exchange and the correlation holes reveals a correlation between the separation R and the time period t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' For certain values of R large fluctua- tions are found within a range of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' This behaviour is mimicked by the exchange and the correlation potentials 15 0 5 10 15 20 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5 0 5 10 15 20 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5 0 5 10 15 20 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 11: The real part of the exchange potential (top), the correlation potential (middle), and the exchange-correlation potential (bottom) for t = −60 (blue) and −84 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' and is found to originate from the diminishing value of |G0(R, t)| at the respective position and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The spherical average of the exchange-correlation hole for t < 0 is generally larger than for t > 0, which is a consequence of the sum rule being −1 for the former and 0 for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' This property is inherited by the exchange-correlation potential, since it is determined by the first radial moment of the spherical average of the exchange-correlation hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' It is found that the exchange potential in space and time is substantially cancelled by the correlation potential, which maybe seen as the analog of the well-known cancellation of the Fock exchange and the correlation self-energy of the electron gas in momen- tum and frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The strong cancellation results in an exchange-correlation potential with much less structure than in the corresponding exchange and correlation po- 0 2 4 6 8 10 12 14 16 18 20 4 3 2 1 0 1 2 3 10-4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 12: The real (solid) and the imaginary (dashed) parts of the non-interacting Green function for t = −60 (blue) and −84 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 0 2 4 6 8 10 12 14 16 18 20 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 13: The real part of the exchange potential (blue), the correlation potential (black), and the exchange-correlation potential (red) for t = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The large cancellation between exchange and correlation can be clearly seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' tentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' This encouraging result lends support for the feasibility of applying the local density approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Analysis of the sum rule offers a physical explanation why using a non-interacting Green function is more ad- vantageous than using a renormalized one when calcu- lating the response function in RPA and consequently in calculating the self-energy within the GW approxima- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' A simple vertex correction that preserves the sum 16 0 2 4 6 8 10 12 14 16 18 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 2 4 6 8 10 12 14 16 18 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='05 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 14: The real part of the exchange-correlation potential for t = ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62 (top) and t = ±34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='75 (bottom) from the model Green function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The solid and dashed curves correspond to t < 0 and t > 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The parameters used are indi- cated in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Two different values of renormalization factors Z are used, one for t < 0 (Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='67) and one for t > 0 (Zp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The choice of the parameters is based on the GW results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' rule is then proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The present work provides a starting point for more ac- curate calculations of the exchange-correlation hole and potential of the electron gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The plasmon-pole approx- imation is not expected to yield accurate results since it neglects the limits in the solid angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Nevertheless, comparison with results obtained from a model Green function suggests that it captures the main features of the exchange-correlation potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' It would be highly desirable to perform full RPA calculations without em- 0 2 4 6 8 10 12 14 16 18 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='4 0 2 4 6 8 10 12 14 16 18 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' 15: The real part of the exchange potential Vx (blue) de- rived from the Hartree-Fock Green function and the exchange- correlation potential Vxc (black) from the model Green func- tion for t = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='62 (top) and t = −34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='75 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The correlation potential (red) is defined as Vc = Vxc − Vx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' ploying the plasmon-pole approximation with the aim of constructing a local-density type approximation for the exchange-correlation potential, which can be applied to calculate the Green function of real materials, circum- venting computationally expensive traditional self-energy calculations based on Feynman diagrams and path inte- gral techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Acknowledgments Financial support from the Knut and Alice Wallen- berg (KAW) Foundation (Grant number 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content='0061) and 17 the Swedish Research Council (Vetenskapsr˚adet, VR, Grant number 2021 04498) is gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' We thank Rex Godby for valuable discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Appendix A: Calculation of G0(R, t > 0) The integral over k in G0(R, t > 0) can be decomposed as follows: � ∞ kF dk = � ∞ 0 dk − � kF 0 dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (A1) iG0(R, t > 0) = 1 2π2R {I(0, ∞) − I(0, kF)} , (A2) where I(a, b) = � b a dk k sin(kR)e−ik2t/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (A3) Consider the following integral � ∞ −∞ dk keikRe−ik2t/2 = eiR2/2t � ∞ −∞ dk ke−i(k−R/t)2t/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (A4) By making a change of variable, k − R t = � 2 t x, (A5) the integral I(0, ∞) can be performed analytically yield- ing I(0, ∞) = � π 2it R it eiR2/2t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (A6) For t → 0+, by introducing a converging factor e−|α|q one finds, lim α→0 � ∞ 0 dk k sin(kR)e−|α|q = 0 (A7) implying that indeed G0(R, 0+) = G0(R, 0−) for R ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' Appendix B: Correlation hole From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (15), the correlation hole is given by the following equation: ρc(r, r′, r′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t)G(r, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t) = i � dr4dt4 G(r, r4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t − t4)K(r4, r′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t4 − t)G(r4, r′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (B1) A non-interacting G is used, G0(r − r4, t − t4) = i Ω � k≤kF eik·(r−r4)e−iεk(t−t4)θ(t4 − t) − i Ω � k>kF eik·(r−r4)e−iεk(t−t4)θ(t − t4), (B2) G0(r4 − r′, t4) = i Ω � k′≤kF eik′·(r4−r′)e−iεk′t4θ(−t4) − i Ω � k′>kF eik′·(r4−r′)e−iεk′t4θ(t4), (B3) and K(r4 − r′′, t4 − t) = 1 Ω � q � dω 2π e−iq·(r4−r′′)e−iω(t4−t)K(q, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (B4) Consider the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (B1) and the case t < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' From the product of the two Green functions one obtains four terms but only two survive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' The first non- zero term is (note the additional factor of i from iGKG) A1 = − i3 Ω3 � k≤kF � k′>kF eik·re−ik′·r′ × � q eiq·r′′ � dω 2π ei(ω−εk)tK(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' ω) × � dr4 e−i(k−k′+q)·r4 � ∞ 0 dt4 e−i(ω−εk+εk′−iη)t4 = 1 Ω2 � k≤kF � k′>kF eik·re−ik′·r′ei(k′−k)·r′′ × � dω 2π K(|k′ − k|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' ω) ei(ω−εk)t ω − εk + εk′ − iη ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (B5) which can be rewritten in terms of the radial variables: A1 = 1 Ω2 � k≤kF e−ik·R′ � k′>kF eik′·R′′ × � dω 2π K(|k′ − k|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' ω) ei(ω−εk)t ω − εk + εk′ − iη .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (B6) The second non-zero term is similar to the first and given by A2 = − 1 Ω2 � k>kF e−ik·R′ � k′≤kF eik′·R′′ × � dω 2π K(|k′ − k|, ω) e−iεk′t ω − εk + εk′ + iη .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (B7) For the case t > 0 similar consideration yields B1 = 1 Ω2 � k≤kF e−ik·R′ � k′>kF eik′·R′′ × � dω 2π K(|k′ − k|, ω) e−iεk′t ω − εk + εk′ − iη (B8) 18 B2 = − 1 Ω2 � k>kF e−ik·R′ � k′≤kF eik′·R′′ × � dω 2π K(|k′ − k|, ω) ei(ω−εk)t ω − εk + εk′ + iη .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (B9) To calculate the integral over ω one utilizes the spectral representation of K: K(k, ω) = � 0 −∞ dω′ L(k, ω′) ω − ω′ − iδ + � ∞ 0 dω′ L(k, ω′) ω − ω′ + iδ , (B10) where L(k, ω) = − 1 π sign(ω)ImK(k, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (B11) The spectral function L(k, ω) is an odd function in ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' For the case t < 0 the contour integral for A1 in the complex ω plane is closed in the lower-half plane yielding � dω 2π 1 ω − ω′ + iδ × ei(ω−εk)t ω − εk + εk′ − iη = −iei(ω′−εk−iδ)t ω′ − εk + εk′ − iη .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (B12) One obtains, using L(k, −ω) = −L(k, ω), A1 = 1 Ω2 � k≤kF e−ik·R′ � k′>kF eik′·R′′e−iεkt × � ∞ 0 dω′ L(|k′ − k|, ω′) −ieiω′t ω′ + εk′ − εk (B13) and A2 = 1 Ω2 � k>kF e−ik·R′ � k′≤kF eik′·R′′e−iεk′t × � ∞ 0 dω′ L(|k′ − k|, ω′) −i ω′ + εk − εk′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (B14) For the case t > 0 one obtains B1 = 1 Ω2 � k≤kF e−ik·R′ � k′>kF eik′·R′′e−iεk′t × � ∞ 0 dω′ L(|k′ − k|, ω′) −i ω′ + εk′ − εk (B15) B2 = 1 Ω2 � k>kF e−ik·R′ � k′≤kF eik′·R′′e−iεkt × � ∞ 0 dω′ L(|k′ − k|, ω′) −ie−iω′t ω′ + εk − εk′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (B16) For each t the integral over ω′ can be parametrized as follows: M(q, ω, t) = � ∞ 0 dω′ L(q, ω′)−ieiω′t ω′ + ω , (B17) so that A1 = 1 Ω2 � k≤kF e−ik·R′e−iεkt � k′>kF eik′·R′′ × M(|k′ − k|, εk′ − εk, t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (B18) A2 = 1 Ω2 � k>kF e−ik·R′ � k′≤kF eik′·R′′e−iεk′t × M(|k′ − k|, εk − εk′, 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (B19) B1 = 1 Ω2 � k≤kF e−ik·R′ � k′>kF eik′·R′′e−iεk′t × M(|k′ − k|, εk′ − εk, 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (B20) B2 = 1 Ω2 � k>kF e−ik·R′ � k′≤kF eik′·R′′e−iεkt × M(|k′ − k|, εk − εk′, −t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (B21) By defining γ(R, R′, t, t′, t′′) = 1 Ω2 � k≤kF e−ik·Re−iεkt × � k′>kF eik′·R′e−iεk′t′M(|k′ − k|, εk′ − εk, t′′), (B22) A1, A2, B1, and B2 can be written as A1 = γ(R′, R′′, t, 0, t), (B23) A2 = γ(R′′, R′, t, 0, 0), (B24) B1 = γ(R′, R′′, 0, t, 0), (B25) B2 = γ(R′′, R′, 0, t, −t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' (B26) The correlation hole is given by ρc(R, R′, θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t < 0) = A1 + A2 G0(R, t < 0), ρc(R, R′, θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} +page_content=' t > 0) = B1 + B2 G0(R, t > 0).' metadata={'source': 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(Saunders College Publishing, NY, 1976)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfaw8F/content/2301.05590v1.pdf'} diff --git a/UNE3T4oBgHgl3EQfEQnM/content/2301.04295v1.pdf b/UNE3T4oBgHgl3EQfEQnM/content/2301.04295v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..fb60b905a6046a1014e053382475f679a81a9360 --- /dev/null +++ b/UNE3T4oBgHgl3EQfEQnM/content/2301.04295v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:070893cb9c5ab546f3e641371eca726f487665f0145c55c301a2bc6a41bd5570 +size 1366491 diff --git a/UNE3T4oBgHgl3EQfEQnM/vector_store/index.faiss b/UNE3T4oBgHgl3EQfEQnM/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..558ea5046e1ed78c76efd4c85c7e5b1e54e5f8df --- /dev/null +++ b/UNE3T4oBgHgl3EQfEQnM/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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+Abstract +Many policy optimization approaches in reinforce- +ment learning incorporate a Kullback-Leilbler +(KL) divergence to the previous policy, to pre- +vent the policy from changing too quickly. This +idea was initially proposed in a seminal paper on +Conservative Policy Iteration, with approxima- +tions given by algorithms like TRPO and Mun- +chausen Value Iteration (MVI). We continue this +line of work by investigating a generalized KL +divergence—called the Tsallis KL divergence— +which use the q-logarithm in the definition. The +approach is a strict generalization, as q = 1 cor- +responds to the standard KL divergence; q > 1 +provides a range of new options. We character- +ize the types of policies learned under the Tsallis +KL, and motivate when q > 1 could be beneficial. +To obtain a practical algorithm that incorporates +Tsallis KL regularization, we extend MVI, which +is one of the simplest approaches to incorporate +KL regularization. We show that this generalized +MVI(q) obtains significant improvements over the +standard MVI(q = 1) across 35 Atari games. +1. Introduction +There is ample theoretical evidence that it is useful to incor- +porate KL regularization into policy optimization in rein- +forcement learning. The most basic approach is to regularize +towards a uniform policy, resulting in entropy regulariza- +tion. More effective, however, is to regularize towards the +previous policy. By choosing KL regularization between +consecutively updated policies, the optimal policy becomes +a softmax over a uniform average of the full history of ac- +tion value estimates (Vieillard et al., 2020a). This averaging +smooths out noise, allowing for better theoretical results +(Azar et al., 2012; Kozuno et al., 2019; Vieillard et al., +2020a; Kozuno et al., 2022; Abbasi-Yadkori et al., 2019). +Despite these theoretical benefits, there are some issues with +1Department of Computing Science, University of Alberta. +2Osaka University. +3Nara Institute of Science and Technology. +Correspondence to: Lingwei Zhu . +using KL regularization in practice. It is well-known that +the uniform average is susceptible to outliers; this issue is +inherent to KL divergence (Futami et al., 2018). In practice, +heuristics such as assigning vanishing regularization coef- +ficients to some estimates have been implemented widely +to increase robustness and accelerate learning (Grau-Moya +et al., 2019; Haarnoja et al., 2018; Kitamura et al., 2021). +However, theoretical guarantees no longer hold for those +heuristics (Vieillard et al., 2020a; Kozuno et al., 2022). A +natural question is what alternatives we can consider to this +KL divergence regularization, that allows us to overcome +some of these disadvantages while maintaining the bene- +fits associate with restricting aggressive policy changes and +smoothing errors. +In this work, we explore one possible direction by generaliz- +ing to Tsallis KL divergences. Tsallis KL divergences were +introduced for physics (Tsallis, 1988; 2009), using a simple +idea: replacing the use of the logarithm with the deformed +q-logarithm. The implications for policy optimization, how- +ever, are that we get quite a different form for the resulting +policy. Tsallis entropy with q = 2 has actually already been +considered for policy optimization (Chow et al., 2018; Lee +et al., 2018), by replacing Shannon entropy with Tsallis +entropy to maintain stochasticity in the policy. The resulting +policies are called sparsemax policies, because they concen- +trate the probability on higher-valued actions and truncate +the probability to zero for lower-valued actions. Intuitively, +this should have the benefit of maintaining stochasticity, +but only amongst the most promising actions, unlike the +Boltzmann policy which maintains nonzero probability on +all actions. Unfortunately, using only Tsallis entropy did +not provide significant benefits, and in fact often performed +worse than existing methods. We find, however, that using a +Tsallis KL divergence to the previous policy does provide +notable gains. +We first show how to incorporate Tsallis KL regularization +into the standard value iteration updates, and prove that +we maintain convergence under this generalization from +KL regularization to Tsallis KL regularization. We then +characterize the types of policies learned under Tsallis KL, +highlighting that there is now a more complex relationship +to past action-values than a simple uniform average. We +then show how to extend Munchausen Value Iteration (MVI) +(Vieillard et al., 2020b), to use Tsallis KL regularization, +arXiv:2301.11476v1 [cs.LG] 27 Jan 2023 + +Generalized Munchausen Reinforcement Learning using Tsallis KL Divergence +which we call MVI(q). We use this naming convention +to highlight that this is a strict generalization of MVI: by +setting q = 1, we exactly recover MVI. We first investigate +the impact of choosing different q in a small simulation +environment. We then compare MVI(q = 2) with MVI +(namely the standard choice where q = 1), and find that we +obtain significant performance improvements in Atari. +2. Problem Setting +We focus on discrete-time discounted Markov Decision +Processes (MDPs) expressed by the tuple (S, A, d, P, r, γ), +where S and A denote state space and finite action space, +respectively. +d(·) denotes the initial state distribution. +P(·|s, a) denotes transition probability over the state space +given state-action pair (s, a), and r(s, a) defines the re- +ward associated with that transition. When time step t is +of concern, we write rt := r(st, at). γ ∈ (0, 1) is the dis- +count factor. A policy π(·|s) is a mapping from the state +space to distributions over actions. We define the state value +function following policy π and starting from initial state +s0 ∼ d(·) as Vπ(s) = E [�∞ +t=0 rt|s0 = s]. Likewise, we +define the state-action value function given initial action +a0 as Qπ(s, a) = E [�∞ +t=0 rt|s0 = s, a0 = a], where the +expectation is with respect to the policy π and transition +probability P. +A standard approach to find the optimal value function +Q∗ is value iteration. To define the formulas for value +iteration, it will be convenient to write these functions +as vectors, Vπ ∈ R|S| and Qπ ∈ R|S|×|A|. +For nota- +tional convenience, we define the inner product for any +two functions F1, F2 ∈ R|S|×|A| as ⟨F1, F2⟩ ∈ R|S|, +where we only take the inner product over actions. The +Bellman operator acting upon any function Q ∈ R|S|×|A| +can be defined as: TπQ := r + γPπQ, where PπQ ∈ +R|S|×|A| := Es′∼P (·|s,a),a′∼π(·|s′)[Q(s′, a′)] = P⟨π, Q⟩. +When π is greedy with respect to Q, we have the Bell- +man optimality operator defined by T∗Q := r + γP∗Q, +P∗Q=Es′∼P (·|s,a)[maxa′ Q(s′, a′)]. The above definitions +should be understood as component-wise. Repeatedly ap- +plying the Bellman operator TπQ converges to the unique +fixed point Qπ, and T∗Q it converges to the optimal action +value function Q∗ := Qπ∗: +� +πk+1 = arg maxπ⟨π, Qk⟩ , +Qk+1 = (T∗Qk)m , +(1) +where the integer m ≥ 1 denotes repeated applications. +Choosing m = 1, ∞ corresponds value iteration and policy +iteration, respectively. 1 < m < ∞implies the use of ap- +proximate modified policy iteration (Scherrer et al., 2015). +This basic recursion can be modified with the addition of a +regularizer Ω(π): +� +πk+1 = arg maxπ(⟨π, Qk⟩ − τΩ(π)) , +Qk+1 = +� +Tπk+1,ΩQk +�m +(2) +where Tπk+1,ΩQk = r + γP(⟨πk+1, Qk⟩ − τΩ(πk+1)) is +the regularized Bellman operator (Geist et al., 2019). This +modified recursion is guaranteed to converge if Ω is strongly +convex in π. For example, the negative Shannon entropy +Ω(π) = −H (π) = ⟨π, ln π⟩ has the resulting optimal pol- +icy πk+1 ∝ exp +� +τ −1Qk +� +, where ∝ indicates proportional +to up to a constant not depending on actions. +Another popular choice is KL divergence Ω(π) += +DKL(π || µ) = ⟨π, ln π−ln µ⟩, which is more general since +we recover Shannon entropy when we choose µ to be a +uniform distribution, i.e. +1 +|A|. In this work, when we say +KL regularization, we mean the standard choice of setting +µ = πk, the estimate from the previous update. There- +fore, DKL serves as a penalty to penalize aggressive policy +changes. The optimal policy in this case takes the form +πk+1 ∝ πk exp +� +τ −1Qk +� +. By induction, we can show this +KL-regularized optimal policy πk+1 is a softmax over a +uniform average over the history of action value estimates +(Vieillard et al., 2020a): +πk+1 ∝ πk exp +� +τ −1Qk +� +∝· · ·∝exp +� +�τ −1 +k +� +j=1 +Qj +� +� . (3) +Using KL regularization has been shown to be theoretically +superior to entropy regularization, in terms of error tolerance +(Azar et al., 2012; Vieillard et al., 2020a; Kozuno et al., +2022; Chan et al., 2022). +The definitions of H(·) and DKL(·||·) rely on the standard +logarithm and its inverse (the exponential) and both in- +duce softmax policies as an exponential over action-values +(Hiriart-Urruty & Lemar´echal, 2004; Nachum & Dai, 2020). +Convergence properties of the resulting regularized algo- +rithms have been well studied (Kozuno et al., 2019; Geist +et al., 2019; Vieillard et al., 2020a). In this paper, we in- +vestigate Tsallis entropy and Tsallis KL divergence as the +regularizer, which generalize Shannon entropy and KL di- +vergence respectively. +3. Generalizing to Tsallis Regularization +We can easily incorporate other regularizers in to the value +iteration recursion and maintain convergence, as long as +those regularizers are strongly convex in π. This property is +satisfied by Tsallis entropy and the Tsallis KL divergence, +for certain q. In this section, we define the Tsallis KL +divergence and characterize the types of policies that arise +from using this regularizer. + +Generalized Munchausen Reinforcement Learning using Tsallis KL Divergence +3.1. Tsallis Regularization +Tsallis regularization results from generalizing from the +standard logarithm to the q-logarithm. The q-logarithm and +its (unique) inverse function the q-exponential are defined +as follows. For q ∈ R+\{1} +lnqπ := πq−1 − 1 +q − 1 +, +expq Q = [1 + (q − 1)Q] +1 +q−1 ++ +(4) +where [·]+ = max{·, 0}. When q = 1, we define lnq, expq +be ln , exp respectively, because the above approaches these +standard values as q → 1. Tsallis entropy is defined as +Sq(π) := ⟨−π, lnqπ⟩. The Tsallis KL divergence is defined +as Dq +KL(π || µ) := +� +π, lnq π +µ +� +(Prehl et al., 2012). Tsallis +KL is more mass-covering than KL; i.e. its value is pro- +portional to the q-th power of the ratio π +µ, when q > 0 is +big, large values of π +µ are strongly penalized (Wang et al., +2018). In the limit of q → 1, Tsallis entropy recovers Shan- +non entropy and the Tsallis KL divergence recovers the KL +divergence. 1 +We have a similar generalization in the form of the policy +defined by the q-exponential. When Ω(π) = −S2(π), there +is a simple closed-form for the optimal policy called the +sparsemax (Martins & Astudillo, 2016; Lee et al., 2020): +πk+1(a|s) = exp2 +�Qk(s, a) +2τ +− ψ +�Qk(s, ·) +2τ +�� +ψ +�Qk(s, ·) +2τ +� += +� +a∈S(s) +Qk(s,a) +2τ +− 1 +|S(s)| +. +(5) +S(s) is the set of highest-valued actions, satisfying the rela- +tion 1+i +Qk(s,a(i)) +2τ +>�i +j=1 +Qk(s,a(j)) +2τ +, where a(j) indicates +the action with jth largest action value. This sparsemax +sets the probability to zero on the lowest-valued actions, +πk+1(a(i)|s) = 0, i = z + 1, . . . , |A|, where Qk(s, a(z)) > +2τ +� +ψ +� +Qk(s,·) +2τ +� +− 1 +� +> Qk(s, a(z+1)). This truncation is +shown in the bottom of Figure 1. It is worth noting that for +general Tsallis entropy Sq(π), q ̸= 1, 2, ∞, the normaliza- +tion ψ cannot be analytically solved (Lee et al., 2020) and +hence the policy does not have a closed-form expression. +We use first-order expansion to approximate the policy for +those indices. Note that q = 1 corresponds to Shannon +entropy and q = ∞ to no regularization. +3.2. Convergence Results +Now let us turn to formalizing when value iteration with +Tsallis regularization converges. Similar to logarithm, q- +1The q-logarithm has been primarily used in physics. It is +worth noting that q used in RL literature is different from the +statistical physics (denote by q∗). But the RL case can be recovered +by leveraging the duality q = 2 − q∗ (Naudts, 2002; Suyari & +Tsukada, 2005; Lee et al., 2020). +Gaussian Policies +Boltzmann Policies +Figure 1: (Top) Illustration of the components π1 ln π1 +π2 +and π1 lnq π1 +π2 between Gaussian policies and Boltzmann +policies when q = 2. Tsallis KL divergence is more mass- +covering than KL. (Bottom) Illustration of sparsemax acting +upon π1 by truncating actions that have values lower than ψ +defined in Eq. (5). +logarithm has the following properties2. Convexity: lnqπ +is convex for q ≤ 0, concave for q > 0. When q = 2, both +lnq , expq become linear. Monotonicity: lnq π is monotoni- +cally increasing with respect to π. The following similarity +between Shannon entropy (reps. KL) and Tsallis entropy +(resp. Tsallis KL) is highlighted: Bounded entropy: we have +0 ≤ H (π) ≤ ln |A|; and ∀q, 0 ≤ Sq(π) ≤ − lnq +1 +|A|. Gen- +eralized KL property: ∀q, Dq +KL(π || µ) ≥ 0. Dq +KL(π || µ) = 0 +if and only if π = µ almost everywhere, and Dq +KL(π || µ) → +∞ whenever π(a|s) > 0 and µ(a|s) = 0. +However, despite their similarity, a crucial difference is that +lnq is non-extensive, which means it is not additive (Tsallis, +1988). In fact, lnq is only pseudo-additive: +lnq πµ = lnq π + lnq µ + (q − 1) lnq π lnq µ. +(6) +Pseudo-additivity complicates obtaining convergence re- +sults for Eq. (2) with q-logarithm regularizers, since the +techniques used for Shannon entropy and KL divergence are +generally not applicable to their lnq counterparts. Moreover, +deriving the optimal policy may be nontrivial. Convergence +results have only been established for Tsallis entropy regu- +larization with q = 2 (Lee et al., 2018; Chow et al., 2018). +Our first simple result is to show that Eq. (2) with Ω(π) = +Dq +KL(π || µ), for any µ, converges at least for q = 2. In fact, +it converges for any q where Dq +KL(π || µ) is strongly convex, +but we are only able to show this for q = 2. +Theorem 1. The regularized recursion Eq. (2) with Ω(π) = +Dq +KL(π || ·) when q = 2 converges to the unique regularized +optimal policy. +Proof. See Appendix B for the full proof. It simply involves +proving that this regularizer is strongly convex. +2We prove these properties in Appendix A + +TGaussian Policy +T1 +T2 +T 1 +T1 +T2 +T 1 +T2Boltzmann Policy +T1 +T2 +Ti ln +T2 +T1 +八 +2Generalized Munchausen Reinforcement Learning using Tsallis KL Divergence +3.3. TKL Regularized Policies Do More Than +Averaging +Our second result shows the optimal regularized policy un- +der Tsallis KL regularization does more than uniform aver- +aging: it can be seen as performing weighted average whose +degree of weighting is controlled by q. To be specific, we +consider Eq. (2) with m = 1 and Ω(π) = Dq +KL(π || πk). +� +� +� +πk+1 = arg maxπ +� +π, Qk − lnq π +πk +� +, +Qk+1 = r + γP +� +πk+1, Qk − lnq +πk+1 +πk +� +, +(7) +where we dropped the regularization coefficient τ for con- +venience. +Proposition 2. The greedy policy πk+1 in Equation (7) +satisfies +πq−1 +k+1 ∝ +� +expqQ1 · · · expqQk +�q−1 +(8) += expq +� +� +k +� +j=1 +Qj +� +� +q−1 ++ +k +� +j=2 +(q − 1)j +k +� +i1=1<··· 1 empirically. +3.4. Considering q > 1 Is Beneficial +In this section, we demonstrate that selecting q > 1 with +Tsallis KL regularization can be beneficial. We test entropic +indices from [1.0, ∞) ∪ {∞} on Acrobot-v1 from Ope- +nAI gym (Brockman et al., 2016). Hyperparameters of each +q were independently fine-tuned. The result is shown in Fig- +ure 2, with reward +100 for visualization. For q ̸= 1, 2, ∞, +we use the first-order Taylor expansion policy expressions +detailed in Appendix D. Recall that when q = ∞ corre- +sponds to no regularization. For each q we run 5×105 steps +and average over 50 seeds. +It can be seen from Figure 2 that the best performance was +achieved by q = 2. The performance versus the choice of q +has some surprising outcomes. The first is that performance +drops drastically for q = 3, 4. Since the effect of Tsallis + +Acrobot-v1 +30 +20 +Average Return +10 +0 +-10 +-20 +-30 +1.0 +2.0 +3.0 +4.0 +5.0 +10.0 +20.0 +inf +entropic index qGeneralized Munchausen Reinforcement Learning using Tsallis KL Divergence +KL divergence is proportional to q, for large values such as +q ≥ 10 the regularization effect may well be approximating +no regularization q = ∞. In this regard, intermediate values +between 2 and 10 may enforce too strong regularization or +mass-covering property, hence constrain the policy from +changing, leading to poor performance. However, we are +not yet able to fully explain this oscillatory behavior, but +it is clear here that the choice of q can have a significant +impact on performance. +4. A Practical Algorithm for Tsallis KL +Regularization +In this section we provide a practical algorithm for imple- +menting Tsallis regularization. We first explain why this +is not straightforward to simply implement KL-regularized +value iteration, even for q = 2, and how Munchausen Value +Iteration ( MVI) overcomes this issue with a clever implicit +regularization trick. We then explain why we cannot directly +use MVI, and then generalize MVI using a new interpreta- +tion of this trick as advantage learning. +4.1. Implicit Regularization With MVI +Even for the standard KL, it is difficult to implement KL- +regularized value iteration with function approximation. +The difficulty arises from the fact that we cannot exactly +obtain πk+1 ∝ πk exp (Qk). This policy might not be repre- +sentable by our function approximator. For q = 1, one could +store all past Qk and use their average outputs; with neural +networks, however, this is computationally infeasible. +An alternative direction has been to construct a different +value function iteration scheme, which is equivalent to the +original KL regularized value iteration (Azar et al., 2012; +Kozuno et al., 2019). A recent method of this family is +Munchausen VI ( MVI) (Vieillard et al., 2020b). MVI +implicitly enforces KL regularization using the recursion +� +πk+1 = arg maxπ ⟨π, Qk − τ ln π⟩ +Qk+1 =r+ατ ln πk+1+γP ⟨πk+1, Qk−τ ln πk+1⟩ +(11) +We see that Eq. (11) is Eq. (2) with Ω(π) = −H (π) (blue) +plus an additional red Munchausen term, with coefficient +α. Vieillard et al. (2020b) showed that implicit KL regular- +ization was performed under the hood, even though we still +have tractable πk+1 ∝ exp +� +τ −1Qk +� +: +Qk+1 = r + ατ ln πk+1 + γP ⟨πk+1, Qk − τ ln πk+1⟩ +⇔ Qk+1 − ατ ln πk+1 = r + γP +� +⟨πk+1, Qk − ατ ln πk⟩ +− ⟨πk+1, ατ(ln πk+1 − ln πk) − (1 − α)τ ln πk+1⟩ +� +⇔ Q′ +k+1 = r + γP +� +⟨πk+1, Q′ +k⟩ − ατDKL(πk+1||πk) ++ (1 − α)τH (πk+1) +� +(12) +where Q′ +k+1 :=Qk+1 −ατ ln πk+1 is the generalized action +value function. +To extend this to q > 1, a natural choice seems to be to +replace ln with lnq in this recursion. Unfortunately, the +q-logarithm is only pseudo-additive, rather than additive: +ln(ab) = ln a + ln b but lnq(ab) ̸= lnq a + lnq b. Conse- +quently, we cannot use the same derivation above, because +Dq +KL(π || µ) ̸= ⟨π, lnqπ − lnqµ⟩. We show in Figure 3 that +this naive approach, that assumes additivity, indeed fails. +4.2. MVI(q) For General q +We first discuss the alternative derivation for implicit KL +regularization for q > 1, and then discuss two practical ap- +proaches to approximation the recursion. Instead of adding +ατ lnqπk+1, let us consider subtracting ατ lnq +1 +πk+1 . For +q = 1, this is actually equivalent. For q > 1, we no longer +have ln(a−1) = − ln a, but we will see this gives us some- +thing much closer to implicit KL regularization. First, let us +introduce Q′ +k+1 = Qk+1 + ατ lnq +1 +πk+1 and write +Qk+1 + ατ lnq +1 +πk+1 += +r + γP +� +πk+1, Qk + ατ lnq +1 +πk +− ατ lnq +1 +πk +− τ lnqπk+1 +� +⇔Q′ +k+1 = r + γP ⟨πk+1, Q′ +k⟩ +− γP +� +πk+1, ατ lnq +1 +πk ++ τ lnqπk+1 +� +(13) +For q = 1, the term on the last line could be rearranged to +produce the KL and entropy regularization. But, for q > 1, +because of pseudo-additivity (formula in Eq.(6)), we can +only use the relationship +lnq +1 +πk += lnq +πk+1 +πk +− lnqπk+1 − (q − 1) lnqπk+1 lnq +1 +πk +. +We can simplify the final term in Eq.(13) as follows. +ατ +� +πk+1, lnq +πk+1 +πk +− lnqπk+1 − (q − 1) lnqπk+1 lnq +1 +πk +� ++ τ ⟨πk+1, lnqπk+1⟩ += ατDq +KL(πk+1||πk) − τ (1 − α) Sq(πk+1) +− ατ(q − 1) +� +πk+1, lnq +1 +πk +lnqπk+1 +� +Like with q = 1, we obtain an implicit regularization by +adding ατ lnq +1 +πk+1 , though now with an additional term +weighted by q − 1. +This final term poses a problem. It involves both k and +k + 1. If we try to construct a recursion that incorporates +this term, the term that has to be added to Qk+1 relies on +the future policy, which is not yet available. One reasonable + +Generalized Munchausen Reinforcement Learning using Tsallis KL Divergence +Figure 3: MVI(q) (proposed method) and MVI with π being +the sparsemax policy induced by S2(π). +Algorithm 1 MVI(q) for q = 2 +Input: number of iterations K, entropy coefficient τ, +Munchausen coefficient α +Initialize Q0, π0 arbitrarily +Define Cond(i, Qk) = 1+i +Qk(s,a(i)) +2τ +>�i +j=1 +Qk(s,a(j)) +2τ +for k = 1, 2, . . . , K do +# Policy Improvement +for (s, a) ∈ (S, A) do +Sort Qk(s, a(1)) > · · · > Qk(s, a(|A|)) +S(s) = max{i ∈ {1, 2, . . . , |A|} +��Cond(i, Qk)} +Compute ψ( Qk(s,·) +2τ +) = +� +a∈S(s) +Qk(s,a) +2τ +−1 +|S(s)| +πk+1(a|s) = exp2 +� +Qk(s,a) +2τ +− ψ +� +Qk(s,·) +2τ +�� +end for +# Policy Evaluation +for (s, a, s′) ∈ (S, A) do +Qk+1(s, a)=r(s, a)+ατ (Qk(s, a)−M2,τQk(s)) ++γ � +b∈A πk+1(b|s′) (Qk+1(s′, b)−τ ln2 πk+1(b|s′)) +end for +end for +approximation, therefore, is to simply use ατ lnq +1 +πk+1 and +ignore this additional term. +There is another route to consider, to generalize MVI to +q > 1. For the implementation of MVI, the red term is +actually computed using ατ ln πk+1 = α(Qk − MτQk), +where MτQk +:= +1 +Zk +� +exp +� +τ −1Qk +� +, Qk +� +, Zk += +� +1, exp +� +τ −1Qk +�� +is the Boltzmann softmax operator.3 +They chose this route to improve numerical stability. We +can generalize this Mτ operator, to q > 1, and use +Qk − Mq,τQk, where +Mq,τQk = +� +expq +�Qk +qτ − ψ +�Qk +qτ +�� +, Qk +� +. +(14) +3Using MτQ is equivalent to the log-sum-exp operator up to +a constant shift (Azar et al., 2012). +Let us look more carefully at this operator. Assume that +τ absorbs q, and Q(s, a(z)) > τ +� +ψ +� +Q(s,·) +τ +� +− +1 +q−1 +� +> +Q(s, a(z+1)), then expq +� Qk(s,a(i)) +τ +− ψ +� +Qk(s,·) +τ +�� += 0 +for i = z + 1, . . . , |A|. Therefore, Mq,τQk is an expecta- +tion of Q(s, a(1)), · · · , Q(s, a(z)) and Qk − Mq,τQk. The +Tsallis policy has support only on these actions, and so this +corresponds to the expected value under the Tsallis policy. +This is the same property as Qk − MτQk for q = 1, which +also provides the advantage. An interesting thing to note +is that this generalization also indicates an issue with using +lnqπk+1, which equals Qk +τ − ψ +� +Qk +τ +� +. This term indicates +the values compared to the normalization ψ +� +Qk +τ +� +, which is +not the same as the expected values, and so does not equal +the advantage. +Unlike q = 1, however, adding the term Qk − Mq,τQk +for q > 1 no longer has a clear connection to KL reg- +ularization. It is not the case that −ατ lnq +1 +πk+1 equals +Qk − Mq,τQk (see Appendix E). Since ατ lnq +1 +πk+1 is it- +self an approximation and one term is omitted, it is actually +possible that Qk − Mq,τQk more faithfully approximates +KL regularization plus entropy regularization. However, +this remains a big open question. We empirically tested +both approximations—adding ατ lnq +1 +πk+1 versus adding +Qk − Mq,τQk—and found that adding the advantage term +was generally more effective, and numerically stable. This +result mirrors the original MVI, so the final algorithm we +pursue for this extension uses Qk − Mq,τQk. +We summarize this MVI(q) algorithm in Algorithm 1 for +q = 2, and provide the more general version in Algorithm +2 in Appendix D. When q = 1, we get ψ +� +Qk +τ +� += MτQk, +recovering MVI. For q = ∞, we get that M∞,τQk is +maxa Qk(s, a)—no regularization—and we recover advan- +tage learning (Baird & Moore, 1999). Similar to the original +MVI algorithm, MVI(q) enjoys tractable policy expression +with πk+1 ∝ expq +� +τ −1Qk +� +. +5. Experiments +In this section we investigate the utility of MVI(q) in the +Atari 2600 benchmark (Bellemare et al., 2013). We have +seen in Figure 2 that considering q > 1 on simple environ- +ments could be helpful. We test whether this result holds in +more challenging environments. Specifically, we compare +to standard MVI (q = 1), which was already shown to have +competitive performance on Atari (Vieillard et al., 2020b). +We restrict our attention to q = 2, which was generally +effective in other settings and also allows us to contrast to +previous work (Lee et al., 2020) that only used entropy reg- +ularization with KL regularization. For MVI(q = 2), we +take the exact same learning setup—hyperparameters and + +CartPole +500 +MVI(q) +M-VI S2(π) +Average Return +250 +2 +3 +5 +4 +Iteration +1e5Generalized Munchausen Reinforcement Learning using Tsallis KL Divergence +Figure 4: Learning curves of MVI(q) and MVI on the selected Atari games. On some environments MVI(q) significantly +improve upon MVI. Quantitative improvements over MVI and Tsallis-VI are shown in Figures 5 and 6. +MVI(𝑞) improvement over MVI +Figure 5: The percent improvement of MVI(q) with q = 2 +over standard MVI (where q = 1) on the selected Atari +games. The improvement is computed by subtracting the +MVI scores from MVI(q) and normalized by the MVI. +architecture—as MVI(q = 1) and simply modify the term +added to the VI update, as in Algorithm 1. Hyperparameters +and full results are provided in Appendix G. +We also compare MVI(q) several advantage learning algo- +rithms on another benchmark task, called MinAtar (Young +& Tian, 2019). Advantage learning has been under active +investigation due to its appealing theory (Farahmand, 2011) +and superior empirical performance (Ferret et al., 2021). +Performance and experimental settings are detailed in Ap- +pendix F, due to space considerations. As a summary, we +find that MVI(q) with q = 1 and q = 2 are competitive with +these methods, but that the difference between q = 1 and +q = 2 is not nearly as stark in MinAtar as in Atari. +MVI(𝑞) improvement over Tsallis-VI +Figure 6: MVI(q) improvement over Tsallis-VI on Atari +environments, normalized with Tsallis-VI scores. The maxi- +mum improvement for RoadRunner was capped at 1000% +to better visualize performance in the other games. +5.1. Comparing MVI(q) with q = 1 to q = 2 +We provide the overall performance of MVI versus +MVI(q = 2) in Figure 5. Using q = 2 provides a large +improvement in about 5 games, about double the perfor- +mance in the next 5 games, comparable performance in +the next 7 games and then slightly worse performance in 3 +games (PrivateEye, Chooper and Seaquest). Both +PrivateEye and Seaquest are considered harder ex- +ploration games, which might explain this discrepancy. The +Tsallis policy with q = 2 reduces the support on actions, +truncating some probabilities to zero. In general, with a +higher q, the resulting policy is greedier, with q = ∞ corre- +sponding to exactly the greedy policy. It is possible that for +these harder exploration games, the higher stochasticity in +the softmax policy from MVI whre q = 1 promoted more +exploration. A natural next step is to consider incorporating +more directed exploration approaches, into MVI(q = 2), to + +MsPacman +5000 +Average Return +4000 +3000 +2000 +1000 +2 +3 +5 +4 +1e7 +IterationSpacelnvaders +15000 +Average Return +10000 +5000 +O +2 +3 +5 +4 +O +1e7 +IterationAmidar +2500 +MVI(q) +M-VI +2000 +Average Return +W +1500 +1000 +500 +0 +2 +3 +5 +4 +1e7 +IterationAsterix +250000 +200000 +Return +150000 +Average F +100000 +50000 +0 +2 +3 +5 +0 +1 +4 +Iteration +1e7Assault +10000 +8000 +Average Return +6000 +4000 +2000 +0 +2 +3 +4 +5 +1e7 +IterationBeamRider +15000 +Return +10000 +Average F +5000 +2 +3 +5 +4 +1e7 +IterationEnduro +4000 +Average Return +3000 +2000 +1000 +2 +3 +5 +4 +1e7 +IterationHero +20000 +Average Return +15000 +10000 +5000 +2 +3 +5 +4 +1e7 +IterationEVIImprovementoverMDQN +600 +500 +Improvement (%) +400 +300 +200 +100 +0 +-100 +Amidar +Asterix +Assault +Enduro +Space. +Hero +BeamRider +Berzerk +KungFu. +UpNDown +Riverraid +MsPacman +Breakout +Zaxxon +Robotank +Pong +Pitfall +PrivateEye +Chopper. +SeaquestEVI Improvement overTsallisDQN +1000 +800 +Improvement (%) +600 +400 +200 +0 +RoadRun. +Amidar +Assault +Zaxxon +Riverraid +Seaquest +Hero +Robotank +PrivateEye +Alien +Berzerk +Centipede +Kangaroo +Frostbite +KungFu. +MsPacman +Tutankham +Boxing +Breakout +Asteroids +Enduro +pong +Asterix +StarGunner +Chopper. +Jamesbond +Gopher +Space. +BeamRider +BankHeist +Pitfall +AtlantisGeneralized Munchausen Reinforcement Learning using Tsallis KL Divergence +benefit from the fact that lower-value actions are removed +(avoiding taking poor actions) while exploring in a more +directed way when needed. +We examine the learning curves for the games where MVI(q) +had the most significant improvement, in Figure 4. Partic- +ularly notable is how much more quickly MVI(q) learned +with q = 2, in addition to plateauing at a higher point. In +Hero, MVI(q) learned a stably across the runs, whereas +standard MVI with q = 1 clearly has some failures. +These results are quite surprising. The algorithms are oth- +erwise very similar, with the seemingly small change of +using Munchausen term Qk − Mq=2,τQk instead of Qk − +Mq=1,τQk and using the q-logarithm and q-exponential for +the entropy regularization and policy parameterization. Pre- +vious work using q = 2 to get the sparsemax with entropy +regularization generally harmed performance (Lee et al., +2018; 2020). It seems that to get the benefits of the general- +ization to q > 1, the addition of the KL regularization might +be key. We validate this in the next section. +5.2. The Importance of Including KL Regularization +In the policy evaluation step of Algorithm 1, if we set +α = 0 then we recover Tsallis-VI which uses regularization +Ω(π) = −Sq(π) in Eq. (2). In other words, we recover +the algorithm that incorporates entropy regularization using +the q-logarithm and the resulting sparsemax policy. Unlike +MVI, Tsallis-VI has not been comprehensively evaluated on +Atari games, so we include results for the larger benchmark +set comprising 35 Atari games. We plot the percentage +improvement of MVI(q) over Tsallis-VI in Figure 6. +The improvement from including the Munchausen term +(α > 0) is stark. For more than half of the games, MVI(q) +resulted in more than 100% improvement. For the remain- +ing games it was comparable. For 10 games, it provided +more than 400% improvement. Looking more specifically +at which games there was notable improvement, it seems +that exploration may again have played a role. MVI(q) +performs much better on Seaquest and PrivateEye. +Both MVI(q) and Tsallis-VI have policy parameterizations +that truncate action support, setting probabilities to zero for +some actions. The KL regularization term, however, likely +slows this down. It is possible the Tsallis-VI is concentrat- +ing too quickly, resulting in insufficient exploration. +6. Related Work +There is a growing body of literature studying generaliza- +tions of KL divergence in reinforcement learning (Nachum +et al., 2019; Zhang et al., 2020). Futami et al. (2018) dis- +cussed the inherent drawback of KL divergence in gener- +ative modeling and proposed to use β- and γ-divergence +to allow for weighted average of sample contribution. A +more commonly explored generalization is the f-divergence +(Sason & Verd´u, 2016), which is also being used in other +machine learning areas including for generative modeling +(Nowozin et al., 2016; Wan et al., 2020; Yu et al., 2020) +and imitation learning (Ghasemipour et al., 2019; Ke et al., +2019). In reinforcement learning, Wang et al. (2018) dis- +cussed using tail adaptive f-divergence to enforce the mass- +covering property. Belousov & Peters (2019) discussed a +special family of f-divergence known as the α-divergence. +Though the f-divergence offers great generality, it remains +unknown how to find the environment-specific function f +since one cannot sweep functions naturally as we did for the +scalar entropic indices in Figure 2. The same holds true for +another extension known as Bregman divergence (Neu et al., +2017; Huang et al., 2022), which also requires specifying the +generator function. Exploring the generator-environment +relationship is an interesting future direction. +7. Conclusion and Discussion +We investigated the use of the more general q-logarithm +for entropy regularization and KL regularization, instead +of the standard logarithm (q = 1), which gave rise to Tsal- +lis entropy and Tsallis KL regularization. We extended +several results previously shown for q = 1, namely we +proved (a) the form of the Tsallis policy in terms of the past +action-values, (b) convergence of value iteration for q = 2 +and (c) a relationship between adding a q-logarithm to the +action-value update, to provide implicit KL regularization +and entropy regularization, mimicking Munchausen Value +Iteration (MVI). We used these results to propose a general- +ization to MVI, which we call MVI(q), because for q = 1 +we exactly recover MVI. We showed empirically that the +generalization to q > 1 can be beneficial, providing notable +improvements in the Atari 2600 benchmark. +This is the first work to use Tsallis KL regularization for +policy optimization, and naturally there are many open ques- +tions. One key question is to better understand the approx- +imating in MVI(q). When q = 1, the added Munchausen +term is equivalent to adding explicit KL regularization. This +perfect equivalence is lost for q > 1. It is important to +better understand what the Munchausen term is really doing +for q > 1, and when the approximation is very close to +implicit KL regularization and when it is not. Another key +question is to understand the role of the entropic index q +on performance. It was not that case that performance was +very similar for all q, and in fact at times there seemed to +be an oscillating pattern. 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URL https://openreview.net/forum? +id=HkxlcnVFwB. + +Generalized Munchausen Reinforcement Learning using Tsallis KL Divergence +A. Basic facts of Tsallis KL divergence +We present some basic facts about Tsallis KL divergence introduced in Section 3. +Let us recall the definition of q∗-logarithm and Tsallis entropy originally defined in (Tsallis, 1988; 2009): +lnq∗ x = x1−q∗ − 1 +1 − q∗ +, +Sq∗(x) = − +� +xq∗, lnq∗ x +� +. +The definition lnqx used in this paper exploited the q∗ = 2 − q duality (Naudts, 2002; Suyari & Tsukada, 2005): +lnq x = xq−1 − 1 +q − 1 +, +Sq(x) = − ⟨x, lnq x⟩ , +which is common in the RL context (Lee et al., 2020). By the duality we can show (Suyari & Tsukada, 2005, Eq.(12)): +Sq∗(x) := − +� +xq∗, x1−q∗ − 1 +1 − q∗ +� += +� +1, xq∗� +− 1 +1 − q∗ += ⟨1, xq⟩ − 1 +1 − q += − +� +x, xq−1 − 1 +q − 1 +� +=: Sq(x), +i.e. the duality between logarithms lnq∗ x and lnq x allow us to define Tsallis entropy by an alternative notation q that +eventually reaches to the same functional form. +We now come to examine Tsallis KL divergence (or Tsallis relative entropy) defined in the statistical physics literature: +Dq +KL(π || µ) = +� +π, − lnq∗ µ +π +� +(Furuichi et al., 2004). In the main paper we used the definition Dq +KL(π || µ) = +� +π, lnq π +µ +� +(Prehl et al., 2012). We show they are equivalent by the same logic +� +π, − lnq∗ µ +π +� += +� +π, − +� µ +π +�1−q∗ +− 1 +1 − q∗ +� += +� +π, +� +π +µ +�q∗−1 +− 1 +q∗ − 1 +� += +� +π, lnq +π +µ +� +, +(15) +The equivalence allows us to work with lnq∗ to prove the nonnegativity of Tsallis KL divergence easily: +• Nonnegativity Dq +KL(π || µ) ≥ 0: since the function − lnq∗ π is convex, by Jensen’s inequality +� +π, − lnq∗ µ +π +� +≥ − lnq∗ +� +π, µ +π +� += 0, +• Conditions of Dq +KL(π || µ) = 0: directly from the above, in Jensen’s inequality the equality holds only when µ +π = 1 +almost everywhere, i.e. Dq +KL(π || µ) = 0 implies µ = π almost everywhere. +• Conditions of Dq +KL(π || µ) = ∞: Let us work with lnq, following (Cover & Thomas, 2006), let us define +0 lnq +0 +0 = 0, +0 lnq +0 +µ = 0, +π lnq +π +0 = ∞. +We conclude that Dq +KL(π || µ) = ∞ whenever π > 0 and µ = 0. +• Bounded entropy ∀q, 0 ≤ Sq(π) ≤ − lnq +1 +|A|: let µ = +1 +|A|, by the nonnegativity of Tsallis KL divergence: +Dq +KL(π || µ) = ⟨π, lnq (|A| · π)⟩ = +� +π, (|A| · π)q−1 − 1 +q − 1 +� += |A|q−1 +� +⟨1, πq⟩ − 1 +q − 1 +− +1 +|A|q−1 − 1 +q − 1 +� +≥ 0. +Notice that ⟨1,πq⟩−1 +q−1 += ⟨π, lnqπ⟩ = −Sq(π) and +1 +|A|q−1 −1 +q−1 += lnq 1 +|A|, we conclude that +Sq(π) ≤ − lnq +1 +|A|. + +Generalized Munchausen Reinforcement Learning using Tsallis KL Divergence +B. Proof of convergence of Ω(π) = Dq +KL(π || ·) when q = 2 +Let us define ||·||p as the lp-norm. The convergence proof for Ω(π) = Dq +KL(π || ·) when q = 2 comes from that Ω(π) is +strongly convex in π: +Ω(π) = Dq=2 +KL (π||·) = +� +π, ln2 +π +· +� += +� +π, +� π +· +�2−1 − 1 +2 − 1 +� += +��� +���π +· +��� +��� +2 +2 − 1. +(16) +Similarly, the negative Tsallis sparse entropy −S2(π) is also strongly convex. Then the propositions of (Geist et al., 2019) +can be applied, which we restate in the following: +Lemma 3 ((Geist et al., 2019)). Define regularized value functions as: +Qπ,Ω = r + γPVπ,Ω, +Vπ,Ω = ⟨π, Qπ,Ω⟩ − Ω(π). +If Ω(π) is strongly convex, let Ω∗(Q) = maxπ ⟨π, Q⟩ − Ω(π) denote the Legendre-Fenchel transform of Ω(π), then +• ∇Ω∗ is Lipschitz and is the unique maximizer of arg maxπ ⟨π, Q⟩ − Ω(π). +• Tπ,Ω is a γ-contraction in the supremum norm, i.e. ||Tπ,ΩV1 − Tπ,ΩV2||∞ ≤ γ ||V1 − V2||∞. Further, it has a unique +fixed point Vπ,Ω. +• The policy π∗,Ω = arg maxπ ⟨π, Q∗,Ω⟩ − Ω(π) is the unique optimal regularized policy. +Note that in the main paper we dropped the subscript Ω for both the regularized optimal policy and action value function to +lighten notations. It is now clear that Eq. (7) indeed converges for entropic indices that make Dq +KL(π || ·) strongly convex. +But we mostly consider the case q = 2. +C. Derivation of the Tsallis KL Policy +We extend the proof and use the same notations from (Lee et al., 2020, Appendix D) to derive the Tsallis KL reg- +ularized policy. +Define state visitation as ρπ(s) = Eπ,P [�∞ +t=0 1(st = s)] and state-action visitaion ρπ(s, a) = +Eπ,P [�∞ +t=0 1(st = s, at = a)]. The core of the proof resides in establishing the one-to-one correspondence between +the policy and the induced state-action visitation ρπ. For example, Tsallis entropy is written as +Sq(π) = Sq(ρπ) = − +� +s,a +ρπ(s, a) lnq +ρπ(s, a) +� +a ρπ(s, a). +This unique correspondence allows us to replace the optimization variable from π to ρπ. Indeed, one can always restore the +policy by π(a|s) := +ρπ(s,a) +� +a′ ρπ(s,a′). +Let us then write Tsallis KL divergence as Dq +KL(π || µ) = Dq +KL(ρ || ν) = � +s,a ρ(s, a) lnq +ρ(s,a) � +a′ ν(s,a′) +ν(s,a) � +a′ ρ(s,a′) by replacing +the policies π, µ with their state-action visitation ρ, ν. One can then convert the Tsallis MDP problem into the following +problem: +max +ρ +� +s,a +ρ(s, a) +� +s′ +r(s, a)P(s′|s, a) − Dq +KL(ρ || ν) +subject to ∀s, a, +ρ(s, a) > 0, +� +a +ρ(s, a) = d(s) + +� +s′,a′ +P(s|s′, a′)ρ(s′, a′), +(17) +where d(s) is the initial state distribution. Eq. (17) is known as the Bellman Flow Constraints (Lee et al., 2020, Prop. 5) +and is concave in ρ since the first term is linear and the second term is concave in ρ. Then the primal and dual solutions +satisfy KKT conditions sufficiently and necessarily. Following (Lee et al., 2020, Appendix D.2), we define the Lagrangian + +Generalized Munchausen Reinforcement Learning using Tsallis KL Divergence +objective as +L := +� +s,a +ρ(s, a) +� +s′ +r(s, a)P(s′|s, a) − Dq +KL(ρ || ν) + +� +s,a +λ(s, a)ρ(s, a) ++ +� +s +ζ(s) +� +�d(s) + +� +s′,a′ +P(s|s′, a′)ρ(s′, a′) − +� +a +ρ(s, a) +� +� +where λ(s, a) and ζ(s) are dual variables for nonnegativity and Bellman flow constraints. The KKT conditions are: +∀s, a, +ρ∗(s, a) ≥ 0, +d(s) + +� +s′,a′ +P(s|s′, a′)ρ∗(s′, a′) − +� +a +ρ∗(s, a) = 0, +λ∗(s, a) ≤ 0, +λ∗(s, a)ρ∗(s, a) = 0, +0 = +� +s′ +r(s, a)P(s′|s, a) + γ +� +s′ +ζ∗(s′)P(s′|s, a) − ζ∗(s) + λ∗(s, a) − ∂Dq +KL(ρ∗ || ν) +∂ρ(s, a) +, +where − ∂Dq +KL(ρ∗ || ν) +∂ρ(s, a) += − lnq +ρ∗(s, a) � +a′ ν(s, a′) +ν(s, a) � +a′ ρ∗(s, a′) − +�ρ∗(s, a) � +a′ ν(s, a′) +ν(s, a) � +a′ ρ∗(s, a′) +�q−1 ++ +� +a +� +ρ∗(s, a) +� +a′ ρ∗(s, a′) +�q �� +a′ ν(s, a) +ν(s, a) +�q−1 +. +The dual variable ζ∗(s) can be shown to equal to the optimal state value function V ∗(s) following (Lee et al., 2020), and +λ∗(s, a) = 0 whenever ρ∗(s, a) > 0. +By noticing that xq−1 += (q − 1) lnqx + 1, we can show that − ∂Dq +KL(ρ∗||ν) +∂ρ(s,a) += −q lnq +ρ∗(s,a) � +a′ ν(s,a′) +ν(s,a) � +a′ ρ∗(s,a′) − 1 + +� +a +� +ρ∗(s,a) +� +a′ ρ∗(s,a′) +�q � � +a′ ν(s,a) +ν(s,a) +�q−1 +. Substituting ζ∗(s) = V ∗(s), π∗(a|s) = +ρ∗(s,a) +� +a′ ρ∗(s,a), µ∗(a|s) = +ν∗(s,a) +� +a′ ν∗(s,a) into +the above KKT condition and leverage the equality Q∗(s, a) = r(s, a) + Es′∼P [γζ∗(s′)] we have: +Q∗(s, a) − V ∗(s) − q lnq +π(a|s) +µ(a|s) − 1 + +� +a′ +π(a|s) +�π(a|s) +µ(a|s) +�q−1 += 0 +⇔ π∗(a|s) = µ(a|s) expq +� +� +�Q∗(s, a) +q +− +V ∗(s) + 1 − � +a′ π(a|s) +� +π(a|s) +µ(a|s) +�q−1 +q +� +� +� . +By comparing it to the maximum Tsallis entropy policy (Lee et al., 2020, Eq.(49)) we see the only difference lies in +the baseline term µ(a|s)−(q−1), which is expected since we are exploiting Tsallis KL regularization. Let us define the +normalization function as +ψ +�Q∗(s, ·) +q +� += +V ∗(s) + 1 − � +a π(a|s) +� +π(a|s) +µ(a|s) +�q−1 +q +, +then we can write the policy as +π∗(a|s) = µ(a|s) expq +�Q∗(s, a) +q +− ψ +�Q∗(s, ·) +q +�� +. +In a way similar to KL regularized policies, at k + 1-th update, take π∗ = πk+1, µ = πk and Q∗ = Qk, we write +πk+1 ∝ πk expqQk since the normalization function does not depend on actions. We ignored the scaling constant q and +regularization coefficient. Hence one can now expand Tsallis KL policies as: +πk+1 ∝ πk expq(Qk) ∝ πk−1 expq(Qk−1) expq(Qk) ∝ · · · ∝ expqQ1 · · · expqQk, +which proved the first part of Eq. (9). + +Generalized Munchausen Reinforcement Learning using Tsallis KL Divergence +C.0.1. TSALLIS KL POLICIES DO MORE THAN AVERAGE +We now show the second part of Eq. (9), which stated that the Tsallis KL policies do more than average. This follows from +the following lemma: +Lemma 4 (Eq. (25) of (Yamano, 2002)). +� +expq∗ x1 . . . expq∗ xn +�1−q∗ += expq∗ +� +� +k +� +j=1 +xj +� +� +1−q∗ ++ +k +� +j=2 +(1 − q∗)j +k +� +i1=1<··· · · · > Qk(s, a(|A|)) +Find S(s) = max +� +i ∈ {|A|} +�� p + i +Qk(s,a(i)) +qτ +≥ �i +j=1 +Qk(s,a(j)) +qτ ++ j +� +p − +p +q−1 +�� +Compute ˜ψ +� +Qk(s,·) +qτ +� += +� +a∈S(s) +Qk(s,a) +qτ +−p +|S(s)| ++ +� +p − +p +q−1 +� +πk+1(a|s) = expq +� +Qk(s,a) +qτ +− ˜ψ +� +Qk(s,·) +qτ +�� +end for +# Policy Evaluation +for (s, a, s′) ∈ (S, A) do +Qk+1(s, a) = r(s, a) + ατ (Qk(s, a) − Mq,τQk(s)) + γ � +b∈A πk+1(b|s′) (Qk(s′, b) − τ lnq πk+1(b|s′)) +end for +end for +When q = 2, we recover the sparsemax policy. When q ̸= 1, 2, the closed-form expression of π, ψ might not exist. Following +(Chen et al., 2018), we leverage the first order Taylor expansion f(z) + f ′(z)(x − z) on the policy Eq. (21), where we let +z = 1, x = +� +Qπ(s,a) +qτ +− ψ +� +Qπ(s,·) +qτ +�� ++ +q−1 +p , f(x) = x +1 +q−1 , f ′(x) = +1 +q−1x +2−q +q−1 . So that +˜π∗(a|s) ≈ f(z) + f ′(z)(x − z) += 1 + +1 +q − 1 +��Qπ(s, a) +qτ +− ˜ψ +�Qπ(s, ·) +qτ +�� q − 1 +p +− 1 +� +. +(22) +By the constraint � +a π(a|s) = 1 we can approximately obtain the normalization as: +˜ψ +�Qπ(s, ·) +qτ +� +≈ +� +a∈S(s) +Qπ(s,a) +qτ +− p +|S(s)| ++ +� +p − +p +q − 1 +� +. +(23) +The associated set S(s) of allowable actions must satisfy: +p + iQπ(s, a(i)) +qτ +≥ +i +� +j=1 +Qπ(s, a(j)) +qτ ++ j +� +p − +p +q − 1 +� +. +(24) +In this paper, we consider the case p = 1 +2, which leads to the following approximate policy and normalization: +˜ψ +�Qπ(s, ·) +qτ +� += +� +a∈S(s) +Qπ(s,a) +qτ +− 1 +2 +|S(s)| ++ +�1 +2 − +1 +2(q − 1) +� +, +(25) +where the set S(s) then allows actions satisfying 1 +2 + i Qπ(s,a) +qτ +> �i +j=1 +Qπ(s,a(j)) +qτ ++ j( 1 +2 − +1 +2(q−1)). +E. Implicit KL via Regularized Advantage Learning +This section motivates the implementation of MVI(q). Our core observation is the analogy that implicit KL regularization +can be enforced by performing Shannon entropy-regularized advantage learning. To show this, we establish the connection +between MVI and Conservative Value Iteration (CVI) which formulates under what conditions explicit KL can be enforced +via regularized advantage learning. The derivation is different from the connection shown in (Vieillard et al., 2020b, +Appendix A.4) and focuses on justfying our purpose of implementing Eq. (7) with MVI(q). + +Generalized Munchausen Reinforcement Learning using Tsallis KL Divergence +E.1. Implicit KL. +CVI (Kozuno et al., 2019) investigates the scheme maximizing the following objective: +E +� ∞ +� +t=0 +γt (rt + τH (π(·|st)) − σDKL(π(·|st) || ¯π(·|st))) +� +, +(26) +recall that H (π) := − ⟨π, ln π⟩ is the Shannon entropy and DKL(π||¯π) := ⟨π, ln π − ln ¯π⟩ is the KL divergence. By the +Fenchel conjugacy, we can show that the optimal policy takes the form +π∗ ∝ ¯π +σ +σ+τ exp +� +1 +σ + τ (r + γPV ∗ +¯π ) +� +, +(27) +where V ∗ +¯π is the regularized value function. In practice, at the k + 1-th iteration, ¯π is set to the previous policy πk. Hence +Eq. (27) can be written as πk+1 ∝ πk +σ +σ+τ exp +� +1 +σ+τ (r + γPVπk) +� +. While a two-loop policy iteration algorithm can be +employed to obtain πk+1, we can avoid computing the recursively defined πk by defining the action preference function: +Ψk+1 := r + γPVπk + σ ln πk = Qk+1 + σ ln πk. +(28) +Now the optimal policy for k + 1-th iteration can be compactly expressed as +πk+1 = +exp +� +1 +σ+τ Ψk+1 +� +� +1, exp +� +1 +σ+τ Ψk+1 +��. +(29) +Let us define α := +σ +σ+τ ∈ [0, 1] and ζ := +1 +σ+τ ∈ (0, ∞]. CVI converges to the maximum Shannon entropy optimal policy +by iterating upon Ψ: +Ψk+1 = r + γP ⟨πk, Ψk⟩ + α (Ψk − MζΨk) , +(30) +where MζΨk is the value function computed by (Kozuno et al., 2019) +MζΨk = +� +exp (ζΨk+1) +⟨1, exp (ζΨk+1)⟩, Ψk +� +. +(31) +Eq. (30) implies that CVI performs a recursion which evaluates the preference function first: +� +Ψk+1 = r + γP ⟨πk, Ψk⟩ + α (Ψk − MζΨk) +, +πk+1 = exp (ζΨk+1) / ⟨1, exp (ζΨk+1)⟩ +. +(32) +Since the order of policy evaluation/improvement does not matter, we can always reverse them: +� +πk+1 = exp (ζΨk) / ⟨1, exp (ζΨk)⟩ , +Ψk+1 = r + γP ⟨πk+1, Ψk⟩ + α (Ψk − MζΨk) , +(33) +where the definition of the action preference function is modified as: +Ψk+1 := Qk+1 + σ ln πk+1. +(34) +in Eq. (33), we obtain: +Ψk+1 = r + γP ⟨πk+1, Ψk⟩ + α (Ψk − MζΨk) . +(35) +It is worth noting that Eq. (35) is itself a new value iteration scheme, regardless of the definition of Ψ. CVI states that, if the +policy satisfies the Boltzmann assumption, then simply iterating upon Eq. (35) results in KL regularization. +The above analysis implies that any value iteration methods could enjoy the KL regularization properties by replacing +Ψ, MζΨ with Q, MζQ without needing to care about the definition of Ψ, MζΨ since we can arbitrarily initialize Q0 = + +Generalized Munchausen Reinforcement Learning using Tsallis KL Divergence +Ψ0 = 0. The simplest method to ensure the policy meets the Boltzmann assumption is by performing soft Q-learning on top +of Eq. (35). By rewriting Eq. (35) as a soft Q-learning scheme and replacing every appearance of Ψ, MζΨ with Q, MζQ. +Let us start from Eq. (35): +Ψk+1 = r + γP ⟨πk+1, Ψk⟩ + α (Ψk − MζΨk) +(i) +⇔ Qk+1 = r + α (Qk − MζQk) + γP ⟨πk+1, Qk⟩ +(ii) +⇒ Qk+1 = r + α (Qk − ⟨πk+1, Qk⟩) + γP +� +πk+1, Qk − ζ−1 ln πk+1 +� +(iii) +⇔ Qk+1 = r + α (Qk − MζQk) + γP +� +πk+1, Qk − ζ−1 ln πk+1 +� +(36) +where (i) was by a change of variable. Note that this scheme does not specify the conditions of πk+1, which can be +arbitrary. To ensure the policy satisfies the Boltzmann condition, we leveraged (ii) which is performing soft Q-learning. +The regularization ζ−1 ln πk+1 guarantees the policy πk+1 is a softmax with temperature ζ. Finally, (iii) comes from that +MζQk and ⟨πk+1, Qk⟩ are interchangeable under the soft Q-learning context. Let us compare this final scheme to MVI and +observe that α (Qk − MτQk) = ατ ln πk we see this is exactly the MVI update rule Eq. (11). +E.2. Implicit Tsallis KL. +Following a similar logic to implicit KL, let us assume that Qk − Mq,τQk plays the role of −τ lnq +1 +πk+1 . If q = 1, we see +−τ lnq +1 +πk+1 recovers the Munchausen term τ ln πk+1. By the pseudo-additivity (note the subscript is k): +−τ lnq +1 +πk += −τ +� +lnq +πk+1 +πk +− lnqπk+1 − (q − 1) lnqπk+1 lnq +1 +πk +� +(37) +Let us recall the MVI(q) recursion: +� +πk+1 = arg maxπ ⟨π, Qk − τ lnqπ⟩ +Qk+1 = r + α (Qk − Mq,τQk) + γP ⟨πk+1, Qk − τ lnqπk+1⟩ +we can write the policy evaluation step as: +Qk+1 = r + α (Qk − Mq,τQk) + γP ⟨πk+1, Qk − τ lnqπk+1⟩ +(i) +⇔ Qk+1 + ατ lnq +1 +πk+1 += r + γP +� +πk+1, Qk + ατ lnq +1 +πk +− ατ lnq +1 +πk +− τ lnqπk+1 +� +(ii) +⇔ Q′ +k+1 = r + γP +� +⟨πk+1, Q′ +k⟩ − ατ +� +lnq +πk+1 +πk +− lnqπk+1 − (q − 1) lnqπk+1 lnq +1 +πk +� +− τ lnqπk+1 +� += r + γP +� +⟨πk+1, Q′ +k⟩ − ατDq +KL(πk+1||πk) − +� +πk+1, τ(1 − α) lnqπk+1 − ατ(q − 1) lnqπk+1lnq +1 +πk +�� += r + γP +� +⟨πk+1, Q′ +k⟩ − ατDq +KL(πk+1||πk) + τ (1 − α) Sq(πk+1) + ατ +� +πk+1, (q − 1) lnqπk+1 lnq +1 +πk +�� +, +(38) +where in (i) we leveraged the assumption Qk − Mq,τQk = −τ lnq +1 +πk+1 . In (ii) we expanded the term −τ lnq 1 +πk using +Eq. (37) and defined the generalized action value function Q′ +k+1 = Qk+1 + ατ lnq +1 +πk+1 . It is clear that when q = 1 we +recover MVI. However, when q ̸= 1, the last expectation term could be difficult to evaluate. +We now give intuition of why Qk − Mq,τQk could be simplified approximation to the complicated relationship between +−τ lnq +1 +πk+1 and the action gap for q = 2. From (Lee et al., 2018), when Ω(π) = S2(π), the optimal regularized value +function satisfies: +Vk = τ +�� +πk+1, Qk +τ ++ ψ +�Qk +τ +�� ++ 1 +� +⇔ ψ +�Qk +τ +� += Vk +τ − 1 − +� +πk+1, Qk +τ +� +. +(39) + +Generalized Munchausen Reinforcement Learning using Tsallis KL Divergence +Table 1: Parameters used for MinAtar games. +Network Parameter +Value +Algorithmic Parameter +Value +T (total steps) +1 × 107 +γ (discount rate) +0.99 +C (interaction period) +1 +τMVI(q) ( MVI(q) entropy coef.) +0.03 +|B| (buffer size) +1 × 105 +αMVI(q) ( MVI(q) advantage coef.) +0.9 +Bt (batch size) +32 +τ MVI ( MVI entropy coef.) +0.03 +I (update period) +1000 +α MVI ( MVI advantage coef.) +0.9 +Q-network architecture +Conv1 +3,316 - FC128 +αPAL (PAL advantage coef.) +0.9 +activation units +ReLU +ϵ (epsilon greedy threshold) +1.0 → 0.05|10% +optimizer +RMSProp +αSAL (SAL advantage coef.) +Beta(2, 9) + 7 +9 +optimizer learning rate +2.5 × 10−4 +αAL (Smooth AL advantage coef.) +0.2 +(RMSProp) squared momentum +0.95 +wAL (Smooth AL smoothing coef.) +0.3 +(RMSProp) minimum momentum +0.01 +Table 2: Parameters used for Gym environments. +Network Parameter +Value +Algorithm Parameter +Value +T (total steps) +5 × 105 +γ (discount rate) +0.99 +C (interaction period) +4 +ϵ (epsilon greedy threshold) +0.01 +|B| (buffer size) +5 × 104 +τ (Tsallis entropy coefficient) +{0.3, 0.03, 0.003} +Bt (batch size) +128 +α (advantage coefficient) +{0.1, 0.5, 0.9} +I (update period) +100 (Car.) / 2500 (Others) +Q-network architecture +FC512 - FC512 +activation units +ReLU +optimizer +Adam +optimizer learning rate +10−3 +Let us now expand on our Munchausen term − lnq +1 +πk+1 to see how it is connected to Qk − Vk in an involved way. +− ln2 +1 +πk+1 += − ln2 +� +exp2 +�Qk +τ +− ψ +�Qk +τ +���−1 += − +� +exp0 +� +− +�Qk +τ +− ψ +�Qk +τ +��� +− 1 +� += − +�� +1 + Qk +τ +− ψ +�Qk +τ +��−1 +− 1 +� += − +�� +1 + Qk +τ +− Vk +τ + 1 + +� +πk+1, Qk +τ +��−1 +− 1 +� +, += +Qk − Vk +Qk − Vk + ⟨πk+1, Qk⟩ + 2τ + +⟨πk+1, Qk⟩ + τ +Qk − Vk + ⟨πk+1, Qk⟩ + 2τ , +(40) +where the second equation we used the identity expq(f(x))−1 = exp2−q(−f(x)) (Yamano, 2002). It is clear that − ln2 +1 +πk+1 +is a complicated function of Qk − Vk that appears both in the numerator and denominator. When q > 2, it is expected that +the relationship becomes even more complicated. +F. MVI(q) As Advantage Learning +F.1. Comparison and Performance +Similar to MVI, MVI(q) can be seen as a member of advantage learning algorithms. It is therefore interesting to compare +against existing advantage learning algorithms on challenging tasks to illustrate the pros and cons of MVI(q). We choose the +MinAtar environments (Young & Tian, 2019) and perform a grid search for the environment-specific q ∈ [1.0, 5.0] at the 0.5 +resolution. q = 1 is MVI. The results are show in Figure 7. +We also compare MVI(q) against various advantage learning methods: +• Smoothing AL (Gan et al., 2022) propose to perform the interpolation (1 − w)Q + wT∗Q + α(Q − V ) with an +additional weight coefficient w to prevent overly large and harmful action gap. + +Generalized Munchausen Reinforcement Learning using Tsallis KL Divergence +Table 3: Parameters used for Atari games. +Network Parameter +Value +Algorithmic Parameter +Value +T (total steps) +5 × 107 +γ (discount rate) +0.99 +C (interaction period) +4 +τMVI(q) ( MVI(q) entropy coefficient) +10 +|B| (buffer size) +1 × 106 +αMVI(q) ( MVI(q) advantage coefficient) +0.9 +Bt (batch size) +32 +τTsallis (Tsallis-VI entropy coef.) +10 +I (update period) +8000 +α MVI ( MVI advantage coefficient) +0.9 +activation units +ReLU +τ MVI ( MVI entropy coefficient) +0.03 +optimizer +Adam +ϵ (epsilon greedy threshold) +1.0 → 0.01|10% +optimizer learning rate +10−4 +Q-network architecture +Conv4 +8,832 - Conv2 +4,464 - Conv1 +3,364 - FC512 - FC +Figure 7: MVI(q) performance of different q on the MinAtar games. q = 1 corresponding to MVI is performant as expected. +However on Asterix and SpaceInvaders optimal q > 1 emerge. q = 2 is in general reasonably good. +Algorithm/Environment +Asterix +Breakout +Freeway +Seaquest +SpaceInvaders +MVI(q) (proposed) +12.27 ± 0.58 +12.16 ± 1.47 +52.93 ± 0.32 +21.42 ± 4.95 +55.61 ± 3.43 +Smoothing AL (Gan et al., +2022) +10.32 ± 3.28 +14.53 ± 1.78 +50.56 ± 0.83 +16.94 ± 4.45 +54.5 ± 5.99 +MVI (Vieillard et al., 2020b) +9.35 ± 0.29 +12.86 ± 1.54 +54.81 ± 0.55 21.61 ± 1.11 +48.59 ± 3.53 +CVI (Kozuno et al., 2019) +11.21 ± 1.38 +12.38 ± 0.38 +52.64 ± 0.34 +17.74 ± 4.78 +44.67 ± 3.94 +PAL (Bellemare et al., 2016) +10.43 ± 0.62 +13.10 ± 1.86 +53.58 ± 0.34 +21.03 ± 3.14 +43.55 ± 3.36 +SAL (Lu et al., 2019) +1.65 ± 0.92 +7.54 ± 5.32 +17.08 ± 9.32 +6.61 ± 4.32 +30.20 ± 18.95 +Table 4: Comparison between MVI(q) and other advantage learning algorithms on the MinAtar environments. +• Munchausen VI ( MVI) (Vieillard et al., 2020b) implicitly perform KL regularization by Eq. (11). In practice ατ ln π is +computed by α(Q − V ). +• Conservative Value Iteration (CVI) (Kozuno et al., 2019) study under what conditions advantage learning enforces KL +regularization. + +Asterix +12 +10 +reward +8 +9 +4 +2 +0 +5 +S +5 +2 +.C +5 +entropic index qBreakout +12 +10 +reward +8 +9 +4 +2 +0 +5 +5 +5 +C +' +5 +entropic index qFreeway +50 +40 +reward +30 +20 +10 +0 +5 +C +' +5 +entropic index qSeaquest +20 +15 +reward +10 +5 +0 +5 +5 +5 +5 +.C +' +5' +entropic index qSpacelnvaders +50 +40 +reward +30 +20 +10 +0 +5 +5 +5 +.C +5 +entropic index qGeneralized Munchausen Reinforcement Learning using Tsallis KL Divergence +• Persistent Advantage Learning (PAL) (Bellemare et al., 2016) correct bias in the Bellman optimality operator by a +gating maximum operator. +• Stochastic Advantage Learning (SAL) (Lu et al., 2019) study a more general family of advantage learning by allowing +the advantage coefficient to be stochastic. +All algorithms are run for 1 × 107 steps with five seeds. We summarize the scores of all algorithms at the end of learning in +Table 4. It can be seen that MVI(q) and MVI are amongst the top performers. This result is surprising, since it is commonly +perceived that Tsallis entropy based methods could underperform on Atari and MinAtar environments where exploration +is of paramount importance (Lee et al., 2020). The competitive performance of MVI(q) could potentially be attributed to +its unique policy expression Eq. (10). This result is surprising, since it is commonly perceived that Tsallis entropy based +methods typically underperform on challenging tasks such as Atari and MinAtar environments due to insufficient exploration +resulted from truncating actions (Lee et al., 2020). We conjecture the comparable performance of MVI(q) to MVI could be +attributed to its unique policy weighting scheme Eq. (10). +F.2. Setup +For MinAtar games we employed the configuration recommended by (Young & Tian, 2019). The hyperparameters are listed +in Table 1. Since MinAtar games optimize different aspects of Atari games, no frame skipping is necessary and hence the +for every frame we update the network. The epsilon greedy threshold ϵ : 1.0 → 0.05|10% denotes that ϵ is initialized as 1.0 +and gradually decays to 0.05 through the first 10% of learning. +The parameter τ is the Shannon entropy coefficient used for MVI and plays a similar role as α. The advantage learning +coefficient α is also used for PAL. For SAL, we use the empirically most performant choice αSAL of (Lu et al., 2019) which +is drawn from Beta(2, 9) distribution and adds a constant 7 +9. Then E [αSAL] = 1 and Var [αSAL] = +7 +405. For SmoothAL, we +choose αAL = 0.2, wAL = 0.3 that has been demonstrated to have superior performance by (Gan et al., 2022). The network +architecture consists of only one convolutional layer where Convd +a,bc denotes a convolutional layer with c filters of size +a × b and stride d. CVI used the same Boltzmann temperature ζ = 0.03 and advantage coefficient α as in Table 1. +G. Implementation Details of Gym and Atari Environments +We list the hyperparameters for Gym environments in Table 2. The epsilon threshold is fixed at 0.01 from the beginning of +learning. FC n refers to the fully connected layer with n activation units. +For the Atari games we implemented MVI(q), Tsallis-VI and MVI based on the Quantile Regression DQN (Dabney et al., +2018). We leverage the optimized Stable-Baselines3 architecture (Raffin et al., 2021) for best performance. The details +can be seen from Table 3. The Q-network uses 3 convolutional layers. The epsilon greedy threshold is initialized at 1.0 +and gradually decays to 0.01 at the end of first 10% of learning. For conservative learning, we choose the Tsallis entropy +coefficient as α = 10. +We show in Figure 8 the full learning curves of MVI(q) and Tsallis-VI on the selected Atari games. Figures 9 and 10 show +the full learning curves of MVI(q) and MVI on the selected Atari games. + +Generalized Munchausen Reinforcement Learning using Tsallis KL Divergence +Figure 8: Learning curves of MVI(q) and MVI on the selected Atari games. + +MVI(q) +M-VI +Assault +Asterix +Amidar +BeamRider +2500 +10000 +17500 +250000 +8000 +2000 +14000 +200000 +6000 +1500 +150000 +10500 +MMY +4000 +1000 +100000 +7000 +2000 +50000 +500 +3500 +0 +0 +Berzerk +Breakout +ChopperCommand +Enduro +2000 +8000 +500 +1600 +4000 +6400 +400 +3200 +1200 +4800 +300 +2400 +800 +200 +3200 +1600 +400 +100 +1600 +800 +0 +0 +0 +0: +Hero +KungFuMaster +MsPacman +Pitfall +50000 +6000 +Return +200 +25000 +40000 +4800 +19000 +30000 +3600 +-360 +13000 +Average +20000 +7000 +2400 +-640 +1000 W +10000 +1200 +-920 +-5000 +0 +-1200 +Pong +PrivateEye +Riverraid +Robotank +2500 +20 +70 +0000 +1900 +12 +6000 +56 +4 +1300 +42 +2000 +-4 +700 +8000 +28 +100 hwmohwly +-12 +14 +4000 +-20 +-500 +n +UpNDown +Zaxxon +Seaquest +Spacelnvaders +12500 +17500 +25000 +50000 +10000 +20000 +14000 +40000 +7500 +10500 +15000 +30000 +7000 +10000 +5000 +20000 +10000 +3500 +5000 +2500 +0 +0 +70 +1e7 +1 +4 +1e? +0 +1 +4 +0 +1 +4 +0 +1 +2 +4 +Iteration +Iteration +iteration +IterationGeneralized Munchausen Reinforcement Learning using Tsallis KL Divergence +Figure 9: Learning curves of MVI(q) and Tsallis-VI on the selected Atari games. + +MVI(q) +Tsallis-VI +Alien +Amidar +Asteroids +Assault +2500 +2250 +10000 +8000 +2000 +1850 +8000 +6400 +1500 +1450 +6000 +4800 +1000 +1050 +4000 +3200 +500 +650 +1600 +2000 +0 +250 +0 +Asterix +Atlantis +BankHeist +BeamRider +4.0 1e6 +17500 +250000 +1000 +3.2 +800 +14000 +200000 +2.4 +600 +150000 +10500 +1.6 +400 +100000 +7000 +0.8 +200 +50000 +0.0 +3500 +0 +0 +0 +Boxing +Berzerk +Breakout +Centipede +Return +2000 +100 +10000 +500 +1600 +80 +7600 +400 +60 +1200 +5200 +age +300 +40 +800 +200 +2800 +Avera +20 +400 +100 +400 +0 +0 +0 +-2000 +ChopperCommand +Enduro +Frostbite +Gopher +1400 +50000 +6000 +4000 +1160 +4800 +40000 +3200 +920 +3600 +30000 +2400 +680 +1600 +2400 +20000 +1200 +440 +800 +10000 +0 +0 +200 +0 +Hero +Jamesbond +Kangaroo +Krull +8000 +10000 +20000 +14000 +6400 +8400 +16000 +11200 +6800 +12000 +4800 +8400 +5200 +8000 +3200 +5600 +3600 +4000 +1600 +2800 +2000 +0 +0~ +0 +1 +4 +1 +0 +Iteration +0 +4 +Iteration +levs +Iteration +Iteration +1e7Generalized Munchausen Reinforcement Learning using Tsallis KL Divergence +Figure 10: (cont’d) MVI(q) and Tsallis-VI on the selected Atari games. + +MVI(q) +Tsallis-VI +KungFuMaster +MsPacman +Pitfall +PrivateEye +6000 +40000 +200 +1000 +4800 +32000 +-80 +700 +3600 +-360 +24000 +400 +2400 +16000 +-640 +100 +8000 +1200 +-200 +-920 +-500 +0 +0 +1200 +Pong +Riverraid +RoadRunner +Robotank +60000 +70 +20 +20000 +48000 +12 +56 +16000 +36000 +4 +42 +12000 +24000 +-4 +28 +8000 +12000 +-12 +14 +4000 +-20 +0 +n +0 +Seaquest +Spacelnvaders +StarGunner +Tutankham +8000 +17500 +140000 +250 +6400 +14000 +112000 +190 +4800 +10500 +84000 +130 +3200 +7000 +56000 +70 +1600 +3500 +28000 +10 +0 +0 +0 +-50 +4 +UpNDown +Zaxxon +4 +Iteration +14000 +Iteration +40000 +10800 +32000 +7600 +24000 +4400 +16000 +1200 +8000 +0 +-2000 +0 +1 +4 +0 +1 +4 +5 +Iteration +Iteration +1e7? \ No newline at end of file diff --git a/UNFJT4oBgHgl3EQfNCyL/content/tmp_files/load_file.txt b/UNFJT4oBgHgl3EQfNCyL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ae4882bb148c301107b33be5a91a2a145c186375 --- /dev/null +++ b/UNFJT4oBgHgl3EQfNCyL/content/tmp_files/load_file.txt @@ -0,0 +1,1702 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf,len=1701 +page_content='Generalized Munchausen Reinforcement Learning using Tsallis KL Divergence Lingwei Zhu 1 Zheng Chen 2 Takamitsu Matsubara 3 Martha White 1 Abstract Many policy optimization approaches in reinforce- ment learning incorporate a Kullback-Leilbler (KL) divergence to the previous policy, to pre- vent the policy from changing too quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=' This idea was initially proposed in a seminal paper on Conservative Policy Iteration, with approxima- tions given by algorithms like TRPO and Mun- chausen Value Iteration (MVI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=' We continue this line of work by investigating a generalized KL divergence—called the Tsallis KL divergence— which use the q-logarithm in the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=' The approach is a strict generalization, as q = 1 cor- responds to the standard KL divergence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=' q > 1 provides a range of new options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=' We character- ize the types of policies learned under the Tsallis KL, and motivate when q > 1 could be beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=' To obtain a practical algorithm that incorporates Tsallis KL regularization, we extend MVI, which is one of the simplest approaches to incorporate KL regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=' We show that this generalized MVI(q) obtains significant improvements over the standard MVI(q = 1) across 35 Atari games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=' Introduction There is ample theoretical evidence that it is useful to incor- porate KL regularization into policy optimization in rein- forcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=' The most basic approach is to regularize towards a uniform policy, resulting in entropy regulariza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=' More effective, however, is to regularize towards the previous policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=' By choosing KL regularization between consecutively updated policies, the optimal policy becomes a softmax over a uniform average of the full history of ac- tion value estimates (Vieillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=', 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=' This averaging smooths out noise, allowing for better theoretical results (Azar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=' Kozuno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=' Vieillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=' Kozuno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=' Abbasi-Yadkori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=' Despite these theoretical benefits, there are some issues with 1Department of Computing Science, University of Alberta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=' 2Osaka University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=' 3Nara Institute of Science and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFJT4oBgHgl3EQfNCyL/content/2301.11476v1.pdf'} +page_content=' Correspondence to: Lingwei Zhu =15.9) +in NOvA [5], and water (=14.3) in T2K [6]. +Future projects will be based on an +argon target (A=40) in DUNE and a water target in Hyper-Kamiokande, as well as in the +THEIA proposal [7]. In principle, the use of light isoscalar nuclei like carbon or oxygen can +1 The OPERA experiment was designed to detect ντ appearance and did not have a near detector. +2 + +benefit LBL measurements, although the corresponding detector technologies are usually +characterized by somewhat coarser resolutions. In all cases the near detector measurements +are the key factor in determining the ultimate physics sensitivity. In this paper we discuss +a method to obtain an effective oxygen target based on a low-density detector allowing a +precise characterization of nuclear effects and of the (anti)neutrino flux at the near detector +sites [8, 9]. +The paper is organized as follows. Sec. II briefly summarizes the detector technology +designed to offer an accurate control of the neutrino targets. In Sec. III we discuss the +“solid” oxygen concept, while in Sec IV we describe different ways to obtain a corresponding +water target. Section V outlines the main features of those targets together with some of +the physics measurements that they can enable. +II. +CONTROL OF TARGETS +A detector technology designed to offer a control of the configuration, chemical composi- +tion, and mass of the neutrino targets similar to electron scattering experiments is a Straw +Tube Tracker (STT), in which the targets are physically separated from the actual tracking +system. A large number of thin planes – each typically 1-2% of radiation length X0 – of var- +ious passive materials with comparable thickness are alternated and dispersed throughout +active layers – made of four straw planes – of negligible mass in order to guarantee the same +acceptance to final state particles produced in (anti)neutrino interactions. The STT allows +to minimize the thickness of individual active layers and to approximate the ideal case of +a pure target detector – the targets constitute about 97% of the mass – while keeping the +total thickness of the stack comparable to one radiation length. Each target plane can be +removed or replaced with different materials during data taking, providing a flexible target +configuration. +The low average density ρ ≤ 0.17 g/cm3 and the overall dimensions comparable to one X0 +allow an accurate reconstruction of the four-momenta of the visible final state particles, as +well as of the event kinematics in a plane transverse to the beam direction. The lightness of +the tracking straws and the chemical purity of the targets, together with the physical spacing +among the individual target planes, make the vertex resolution less critical in associating the +interactions to the correct target material. For events with a single reconstructed charged +track the corresponding uncertainty is given by the ratio between the thickness of the straw +walls (< 20µm) and the one of a single target layer, typically below 0.5%. For events with at +least two reconstructed charged tracks this uncertainty is reduced to less than 0.1%, thanks +to a vertex resolution (≪ 1 mm [10]) much smaller than the target thickness. +The detector must be placed inside a magnetic field for the momentum measurement and +surrounded by an electromagnetic calorimeter for the detection of neutral particles. The use +of a distributed target mass within a relatively large volume (∼ 40 m3) and a high track +sampling of 0.15-0.30% X0 reduce the impact of multiple scattering on the measurements. +The detector is optimized for the “solid” hydrogen technique, in which ν(¯ν) interactions +on free protons are obtained by subtracting measurements on dedicated graphite (C) tar- +gets from those on polypropylene (CH2) targets [8, 9]. This technique is conceived to be +model-independent, as the data from the graphite targets automatically include all types of +processes, as well as detector effects, relevant for the selection of interactions on H. For CC +interactions the dilution factor with respect to a pure H2 target can be reduced by a factor +5-7 with a kinematic analysis based on energy-momentum conservation [11]. The thickness +3 + +of the two default target materials, as well as the average density of the detector, depend +on the value of the magnetic field available, in order to limit the multiple scattering con- +tribution to the momentum and angular resolutions. For B=0.6 T we can use a thickness +up to about 7 mm for the CH2 targets and 4 mm for the C targets 2. Detector simula- +tions with GEANT4 [12] indicate that a single hit resolution of 200 µm is sufficient for the +various physics measurements. The average momentum resolution expected for muons is +δp/p ∼ 3.5% and the average angular resolution better than 2 mrad with the default CH2 +and C targets. The momentum scale can be calibrated to about 0.2% using reconstructed +K0 → π+π− decays [13, 14]. +III. +OXYGEN TARGET +Since a pure oxygen target in liquid or gaseous form is not feasible due to safety and prac- +tical considerations, we are restricted to the oxygen available within chemical compounds. +The precise control of the targets offered by the STT (Sec. II) allows the implementation +of a “solid” oxygen target from a subtraction between thin polyoxymethylene (CH2O) and +polypropylene (CH2) targets. The former is an engineering thermoplastic (acetal, delrin) +used for precision parts and characterized by high strength, hardness and rigidity, with +X0 = 27.28 cm and ρ = 1.41 g/cm3. Several CH2O planes can be easily integrated into +the detector by replacing some of the default CH2 targets. The distribution of the generic +kinematic variables ⃗x in ν(¯ν)-oxygen interactions can then be obtained as: +NO(⃗x) ≡ NCH2O(⃗x) − MCH2/CH2O +MCH2 +NCH2(⃗x) +(1) +where NCH2O and NCH2 are the numbers of events selected from the polyoxymethylene +and polypropylene targets, respectively. The interactions from this latter are normalized +by the ratio between the total fiducial masses of CH2 within the polypropylene and the +acetal targets, MCH2/CH2O/MCH2. Both targets must have comparable thickness in terms of +radiation and nuclear interaction lengths and must be alternated throughout the detector +volume to guarantee the same acceptance for final state particles. To this end, a solid acetal +slab 4.5 mm thick can be used, corresponding to about 0.016 X0. The oxygen content by +mass within acetal is dominant at 53.3%. We note that polypropylene is the main target +material required for the “solid” hydrogen concept in STT. We therefore expect the statistical +uncertainty on the measured CH2 background to be much smaller compared to the one of +the acetal target. +IV. +WATER TARGETS +In addition to direct measurements on an oxygen target, it can be useful to have a +complementary water target within the same detector. +To this end, we can exploit the +simultaneous presence of polyoxymethylene, polypropylene, and graphite targets in STT. +The distribution of the generic kinematic variables ⃗x in ν(¯ν)-water interactions can then be +2 The C targets can be built from isotropic graphite, which is characterized by good mechanical properties, +a density of about 1.8 g/cm3, and a high purity. +4 + +Target material +Composition +Density +Thickness +Rad. length +Nucl. int. length +Polypropylene +CH2 +0.91 g/cm3 +7.0 mm +0.015 X0 +0.008 λI +Graphite +C +1.80 g/cm3 +4.0 mm +0.016 X0 +0.008 λI +Polyoxymethylene +CH2O +1.41 g/cm3 +4.5 mm +0.016 X0 +0.008 λI +TABLE I. Possible parameters of the individual targets to be alternated within STT (for B=0.6 +T) in the “solid” oxygen and hydrogen techniques. The thickness can be fine-tuned depending on +the specific detector configuration and application. See text for details. +simply obtained from a subtraction between CH2O and C targets: +NH2O(⃗x) ≡ NCH2O(⃗x) − MC/CH2O +MC +NC(⃗x) +(2) +where NCH2O and NC are the numbers of events selected from the polyoxymethylene and +graphite targets, respectively. The interactions from this latter are normalized by the ratio +between the total fiducial masses of C within the graphite and CH2O targets, MC/CH2O/MC. +The advantages of this minimal approach are that we do not need to introduce additional +targets, we can design all targets to have the same acceptance, and we avoid extraneous +materials achieving a high chemical purity. The water content by mass within acetal is +60%. Similarly to the case of the oxygen target discussed above, the available mass of the +graphite target is expected to be significantly larger than the C content within acetal, as it +is an essential component of the “solid” hydrogen technique. We note that the simultaneous +presence of the three materials within STT would allow a complete characterization of the +water target together with its separate constituent elements, O and H. +We can also explicitly integrate thin water targets within STT, replacing some of the +main polypropylene ones. Such passive water targets must be contained within sealed plastic +shells. In order to minimize the total thickness of individual targets in terms of radiation +length, as well as the amount of spurious materials to be subtracted from the shell, we can +use 12 mm water layers encapsulated inside acetal shells 1.5 mm thick. The total effective +thickness of such targets would be equivalent to about 0.044 X0. +The corresponding C +content to be subtracted following Eq.(2) to obtain a pure water target is only about 10.4%. +An interesting application of such water targets in STT is the measurement of ν and ¯ν +interactions off the bound neutron in the deuteron (D), which can be obtained from a +subtraction between heavy water (D2O) and ordinary water (H2O) targets [8]. To this end, +both targets must be enclosed into identical acetal shells, which must be filled in such a way +as to contain the same total mass of oxygen. +V. +MEASURING NUCLEAR EFFECTS +Nuclear effects and the (anti)neutrino flux are the leading sources of systematic un- +certainties in high-energy neutrino scattering measurements [9, 15], as well as in modern +long-baseline oscillation experiments [16, 17]. +Both issues arise because in conventional +(anti)neutrino beams the energy of the incoming neutrino is unknown on an event-by-event +basis. The need to infer the neutrino energy from the detected final state particles consti- +tutes an intrinsic limitation of high-energy neutrino experiments using nuclear targets, as +5 + +the nuclear smearing introduces substantial systematic uncertainties in the process (Sec. I). +The availability of both H and nuclear targets within the same detector can help to mit- +igate such problems in STT [8, 9]. The relative νµ and ¯νµ fluxes as a function of energy +can be determined in-situ with an accuracy around 1% using exclusive νµp → µ−pπ+ and +¯νµp → µ−n processes on H at small energy transfer [14]. The combined use of ν-H and +¯ν-H CC interactions can provide a control sample free from nuclear effects to calibrate the +neutrino energy scale in CC interactions from the nuclear targets [8]. +The STT offers a tool to measure nuclear modifications of cross-sections and to constrain +the systematic uncertainties associated to the nuclear smearing for the various integrated +nuclear targets. Each individual target is designed to be transparent to final state parti- +cles (Tab. I) allowing, together with the low average density of the detector, an accurate +reconstruction and characterization of the various event topologies in ν(¯ν) interactions. Sim- +ulations of the detector response with GEANT4 [12] result in a rather uniform acceptance +over the full 4π angle, with values of 95-99% for µ±, π±, K±, e±. A key requirement is to +guarantee the same acceptance across all nuclear targets, which is achieved by the combined +effect of their thinness (Tab. I) and of their alternation throughout the detector volume. +Detailed detector simulations indicate that in this way the acceptance difference between +targets can be kept within 10−3 for all particles. The subtraction procedure required to +obtain interactions on H, O, and H2O can then be considered model-independent. Further- +more, the detector acceptance effectively cancels out in comparisons among the selected +interactions on the H, C, and O targets. +The high intensity of modern (anti)neutrino beams complements well the relatively small +mass of the various targets in STT. For illustration, a fiducial mass of one tonne of water at +the future Long-Baseline Neutrino Facility (LBNF) [1, 18] will collect about 1.4×106 νµ CC +events/year with the default low-energy spectrum (a factor of two higher with the planned +PIP-II upgrade) and about 6.6×106 νµ CC events/year with the high-energy beam spectrum +and the upgraded beam 3. With such high event rates a limited number of acetal and/or +water targets in STT would suffice to obtain sensible physics measurements. Assuming as +a reference a STT configuration with a “solid” hydrogen mass equivalent to about 10 m3 of +liquid H2 4, about 20 modules equipped with the acetal targets described above would provide +an O target mass similar to the graphite one. An overall water target mass close to one tonne +is therefore relatively easy to achieve. We note that the statistical uncertainties expected +from such a water target at LBNF would be roughly comparable with the systematics +from the 0.2% energy scale uncertainty in STT, and smaller than the ones from the in-situ +determination of the flux using exclusive processes on H [14]. +Comparing measurements of the bound nucleon structure functions F O +2,3 from the “solid” +oxygen with the ones of the free nucleons in H with similar acceptance can provide insights +on the nuclear modifications of the nucleon properties [8, 19–21]. The oxygen target can +also provide complementary measurements with respect to the C and Ca targets to test the +isospin (charge) symmetry [8]. The isotopic content expected for a standard O target is +99.76% of 16O, 0.2% of 18O, and 0.04% of 17O, resulting on average in the smallest isovector +component among stable elements β = (2Z − A)/A = 6 × 10−5. A comparison between ν +and ¯ν interactions on oxygen through the ratios RO +2 = F ¯ν +2 /F ν +2 − 1 and RO +3 = xF ¯ν +3 /xF ν +3 − 1 +for the structure functions F2 and xF3 can provide useful information about the isospin +symmetry in nucleons and nuclei. +3 On-axis rates expected at the near detector site. +4 A fiducial mass of “solid” hydrogen around 700 kg can be obtained from the combination of about 5 tons +of polypropylene and about 600 kg of graphite. +6 + +[1] B. Abi et al. (DUNE), (2020), arXiv:2002.03005 [hep-ex]. +[2] K. Abe et al. (Hyper-Kamiokande), (2018), arXiv:1805.04163 [physics.ins-det]. +[3] R. Acquafredda et al., JINST 4, P04018 (2009). +[4] D. +G. +Michael +et +al. +(MINOS), +Nucl. Instrum. Meth. A 596, 190 (2008), +arXiv:0805.3170 [physics.ins-det]. +[5] D. S. Ayres et al. (NOvA), (2007), 10.2172/935497. +[6] K. +Abe +et +al. +(T2K), +Nucl. Instrum. Meth. A 659, 106 (2011), +arXiv:1106.1238 [physics.ins-det]. +[7] M. Askins et al. (Theia), Eur. Phys. J. C 80, 416 (2020), arXiv:1911.03501 [physics.ins-det]. +[8] R. Petti, Phys. Lett. B 834, 137469 (2022), arXiv:2205.10396 [hep-ph]. +[9] R. +Petti, +in +Proceedings, +27th +International +Workshop +on +Deep +Inelastic +Scatter- +ing +and +Related +Subjects +(DIS +2019): +Torino, +Italy, +April +8-12, +2019 +(2019) +PoS (DIS2019) 235, arXiv:1910.05995 [hep-ex]. +[10] M. Anfreville et al., Nucl. Instrum. Meth. A 481, 339 (2002), arXiv:hep-ex/0104012. +[11] H. Duyang, B. Guo, S. R. Mishra, and R. Petti, (2018), arXiv:1809.08752 [hep-ph]. +[12] S. Agostinelli et al. (GEANT4), Nucl. Instrum. Meth. A506, 250 (2003). +[13] Q. Wu et al. (NOMAD), Phys. Lett. B660, 19 (2008), arXiv:0711.1183 [hep-ex]. +[14] H. +Duyang, +B. +Guo, +S. +R. +Mishra, +and +R. +Petti, +Phys. Lett. B795, 424 (2019), +arXiv:1902.09480 [hep-ph]. +[15] L. +Alvarez-Ruso +et +al. +(NuSTEC), +Prog. Part. Nucl. Phys. 100, 1 (2018), +arXiv:1706.03621 [hep-ph]. +[16] P. Coloma, +P. Huber, C.-M. Jen, +and C. Mariani, Phys. Rev. D 89, 073015 (2014), +arXiv:1311.4506 [hep-ph]. +[17] U. +Mosel, +O. +Lalakulich, +and +K. +Gallmeister, +Phys. Rev. Lett. 112, 151802 (2014), +arXiv:1311.7288 [nucl-th]. +[18] J. Rout, S. Roy, M. Masud, M. Bishai, +and P. Mehta, Phys. Rev. D 102, 116018 (2020), +arXiv:2009.05061 [hep-ph]. +[19] S. A. Kulagin and R. Petti, Nucl. Phys. A765, 126 (2006), arXiv:hep-ph/0412425 [hep-ph]. +[20] S. A. Kulagin and R. Petti, Phys. Rev. D76, 094023 (2007), arXiv:hep-ph/0703033 [hep-ph]. +[21] S. A. Kulagin and R. Petti, Phys. Rev. C90, 045204 (2014), arXiv:1405.2529 [hep-ph]. +7 + diff --git a/XNE3T4oBgHgl3EQf1Qsz/content/tmp_files/load_file.txt b/XNE3T4oBgHgl3EQf1Qsz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4d7860aec43edaa17f525074c88a126c3fc55361 --- /dev/null +++ b/XNE3T4oBgHgl3EQf1Qsz/content/tmp_files/load_file.txt @@ -0,0 +1,304 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf,len=303 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='04744v1 [hep-ex] 11 Jan 2023 An Oxygen Target for (Anti)neutrinos R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Petti∗ Department of Physics and Astronomy, University of South Carolina, Columbia, South Carolina 29208, USA Abstract We discuss a method to obtain an effective oxygen target within a low-density detector allowing an accurate characterization of the various event topologies in ν(¯ν)-oxygen interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Results can be of interest for long-baseline neutrino oscillation experiments utilizing water targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' In particular, the combination of both oxygen and hydrogen targets within the same detector can provide in-situ measurements of nuclear effects and of the (anti)neutrino flux, which are the leading sources of systematic uncertainties in long-baseline oscillation analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' These measurements can also provide useful information about the nuclear modifications of bound nucleons, as well as about the isospin symmetry in nucleons and nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' ∗ Roberto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='Petti@cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='ch 1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' INTRODUCTION Measurements of high-energy neutrino interactions are challenging both at the source and at the detector sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The high intensity of modern (anti)neutrino beams obviates the endemic lack of statistics of older neutrino experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' However, the fact that the energy of the projectile (anti)neutrino is unknown on an event-by-event basis still represents an intrinsic limitation – even when its overall energy spectrum is known with high precision – making the detector itself the critical element in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Besides factors like detector resolutions and energy scale uncertainties, the use of nuclei as (anti)neutrino targets appears ineluctably problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The initial momentum of the target nucleon within the nucleus is unknown and hadrons produced in the primary interaction can undergo an additional unknown modification as they can be absorbed or re-interact within the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Neutrino detectors have to infer the (anti)neutrino energy from the reconstructed final state particles emerging from the nucleus, which are affected by a substantial nuclear smearing and related systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The issues above are exacerbated in long-baseline (LBL) neutrino oscillation experiments, in which the need of a multi-kton mass imposes heavy nuclear targets combined with rela- tively coarse detector resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Their observation of CP violation in the leptonic sector relies on the detection of tiny differences between neutrino and antineutrino Charged Cur- rent (CC) interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Nuclear effects can introduce asymmetries between neutrinos and antineutrinos potentially mimicking the effect of CP violation, since they are in general isospin and flavor dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The physics sensitivity achievable by modern LBL oscillation experiments is thus largely determined by their control of the various systematic uncer- tainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' In particular, the physics promise of next-generation projects like DUNE [1] and Hyper-Kamiokande [2] is accompanied by an impressive percent-level precision required in systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Near detectors are the critical elements taking on the challenge of controlling systematic uncertainties in LBL experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' To this end, they must fulfill two separate tasks charac- terized by conflicting requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' On one side, they need the highest possible resolution – together with a precise calibration of the energy scales – in order to characterize in great details the various event topologies in ν(¯ν) interactions on the same nuclear target used in the far detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The main goal is to provide in-situ measurements of nuclear effects and of the (anti)neutrino flux, which are typically the leading sources of systematic uncertainties in the LBL oscillation analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' On the other side, they must provide a calibration of the event reconstruction in the far detectors, requiring an identical detector technology and nec- essarily a coarser resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' This second task is complicated by the impossibility of having identical detectors at the near and far sites, due to differences in rates, event containment, (anti)neutrino energy spectra, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' In practice the two tasks can be factorized using two separate detector technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Reconstruction effects can also be controlled with dedicated test-beam exposures of the key detector elements, supplemented by appropriate calibration samples in the far detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' In the following we will focus on the first task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Different nuclear targets have been used by LBL experiments, including lead (A=207) in OPERA 1 [3], iron (A=56) in MINOS [4], carbon-based liquid scintillator (=15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='9) in NOvA [5], and water (=14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='3) in T2K [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Future projects will be based on an argon target (A=40) in DUNE and a water target in Hyper-Kamiokande, as well as in the THEIA proposal [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' In principle, the use of light isoscalar nuclei like carbon or oxygen can 1 The OPERA experiment was designed to detect ντ appearance and did not have a near detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' 2 benefit LBL measurements, although the corresponding detector technologies are usually characterized by somewhat coarser resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' In all cases the near detector measurements are the key factor in determining the ultimate physics sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' In this paper we discuss a method to obtain an effective oxygen target based on a low-density detector allowing a precise characterization of nuclear effects and of the (anti)neutrino flux at the near detector sites [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' II briefly summarizes the detector technology designed to offer an accurate control of the neutrino targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' III we discuss the “solid” oxygen concept, while in Sec IV we describe different ways to obtain a corresponding water target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Section V outlines the main features of those targets together with some of the physics measurements that they can enable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' CONTROL OF TARGETS A detector technology designed to offer a control of the configuration, chemical composi- tion, and mass of the neutrino targets similar to electron scattering experiments is a Straw Tube Tracker (STT), in which the targets are physically separated from the actual tracking system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' A large number of thin planes – each typically 1-2% of radiation length X0 – of var- ious passive materials with comparable thickness are alternated and dispersed throughout active layers – made of four straw planes – of negligible mass in order to guarantee the same acceptance to final state particles produced in (anti)neutrino interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The STT allows to minimize the thickness of individual active layers and to approximate the ideal case of a pure target detector – the targets constitute about 97% of the mass – while keeping the total thickness of the stack comparable to one radiation length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Each target plane can be removed or replaced with different materials during data taking, providing a flexible target configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The low average density ρ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='17 g/cm3 and the overall dimensions comparable to one X0 allow an accurate reconstruction of the four-momenta of the visible final state particles, as well as of the event kinematics in a plane transverse to the beam direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The lightness of the tracking straws and the chemical purity of the targets, together with the physical spacing among the individual target planes, make the vertex resolution less critical in associating the interactions to the correct target material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' For events with a single reconstructed charged track the corresponding uncertainty is given by the ratio between the thickness of the straw walls (< 20µm) and the one of a single target layer, typically below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' For events with at least two reconstructed charged tracks this uncertainty is reduced to less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='1%, thanks to a vertex resolution (≪ 1 mm [10]) much smaller than the target thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The detector must be placed inside a magnetic field for the momentum measurement and surrounded by an electromagnetic calorimeter for the detection of neutral particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The use of a distributed target mass within a relatively large volume (∼ 40 m3) and a high track sampling of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='15-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='30% X0 reduce the impact of multiple scattering on the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The detector is optimized for the “solid” hydrogen technique, in which ν(¯ν) interactions on free protons are obtained by subtracting measurements on dedicated graphite (C) tar- gets from those on polypropylene (CH2) targets [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' This technique is conceived to be model-independent, as the data from the graphite targets automatically include all types of processes, as well as detector effects, relevant for the selection of interactions on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' For CC interactions the dilution factor with respect to a pure H2 target can be reduced by a factor 5-7 with a kinematic analysis based on energy-momentum conservation [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The thickness 3 of the two default target materials, as well as the average density of the detector, depend on the value of the magnetic field available, in order to limit the multiple scattering con- tribution to the momentum and angular resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' For B=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='6 T we can use a thickness up to about 7 mm for the CH2 targets and 4 mm for the C targets 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Detector simula- tions with GEANT4 [12] indicate that a single hit resolution of 200 µm is sufficient for the various physics measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The average momentum resolution expected for muons is δp/p ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='5% and the average angular resolution better than 2 mrad with the default CH2 and C targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The momentum scale can be calibrated to about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='2% using reconstructed K0 → π+π− decays [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' OXYGEN TARGET Since a pure oxygen target in liquid or gaseous form is not feasible due to safety and prac- tical considerations, we are restricted to the oxygen available within chemical compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The precise control of the targets offered by the STT (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' II) allows the implementation of a “solid” oxygen target from a subtraction between thin polyoxymethylene (CH2O) and polypropylene (CH2) targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The former is an engineering thermoplastic (acetal, delrin) used for precision parts and characterized by high strength, hardness and rigidity, with X0 = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='28 cm and ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='41 g/cm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Several CH2O planes can be easily integrated into the detector by replacing some of the default CH2 targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The distribution of the generic kinematic variables ⃗x in ν(¯ν)-oxygen interactions can then be obtained as: NO(⃗x) ≡ NCH2O(⃗x) − MCH2/CH2O MCH2 NCH2(⃗x) (1) where NCH2O and NCH2 are the numbers of events selected from the polyoxymethylene and polypropylene targets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The interactions from this latter are normalized by the ratio between the total fiducial masses of CH2 within the polypropylene and the acetal targets, MCH2/CH2O/MCH2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Both targets must have comparable thickness in terms of radiation and nuclear interaction lengths and must be alternated throughout the detector volume to guarantee the same acceptance for final state particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' To this end, a solid acetal slab 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='5 mm thick can be used, corresponding to about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='016 X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The oxygen content by mass within acetal is dominant at 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' We note that polypropylene is the main target material required for the “solid” hydrogen concept in STT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' We therefore expect the statistical uncertainty on the measured CH2 background to be much smaller compared to the one of the acetal target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' WATER TARGETS In addition to direct measurements on an oxygen target, it can be useful to have a complementary water target within the same detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' To this end, we can exploit the simultaneous presence of polyoxymethylene, polypropylene, and graphite targets in STT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The distribution of the generic kinematic variables ⃗x in ν(¯ν)-water interactions can then be 2 The C targets can be built from isotropic graphite, which is characterized by good mechanical properties, a density of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='8 g/cm3, and a high purity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' 4 Target material Composition Density Thickness Rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' length Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' length Polypropylene CH2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='91 g/cm3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='0 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='015 X0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='008 λI Graphite C 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='80 g/cm3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='0 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='016 X0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='008 λI Polyoxymethylene CH2O 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='41 g/cm3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='5 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='016 X0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='008 λI TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Possible parameters of the individual targets to be alternated within STT (for B=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='6 T) in the “solid” oxygen and hydrogen techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The thickness can be fine-tuned depending on the specific detector configuration and application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' simply obtained from a subtraction between CH2O and C targets: NH2O(⃗x) ≡ NCH2O(⃗x) − MC/CH2O MC NC(⃗x) (2) where NCH2O and NC are the numbers of events selected from the polyoxymethylene and graphite targets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The interactions from this latter are normalized by the ratio between the total fiducial masses of C within the graphite and CH2O targets, MC/CH2O/MC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The advantages of this minimal approach are that we do not need to introduce additional targets, we can design all targets to have the same acceptance, and we avoid extraneous materials achieving a high chemical purity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The water content by mass within acetal is 60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Similarly to the case of the oxygen target discussed above, the available mass of the graphite target is expected to be significantly larger than the C content within acetal, as it is an essential component of the “solid” hydrogen technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' We note that the simultaneous presence of the three materials within STT would allow a complete characterization of the water target together with its separate constituent elements, O and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' We can also explicitly integrate thin water targets within STT, replacing some of the main polypropylene ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Such passive water targets must be contained within sealed plastic shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' In order to minimize the total thickness of individual targets in terms of radiation length, as well as the amount of spurious materials to be subtracted from the shell, we can use 12 mm water layers encapsulated inside acetal shells 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='5 mm thick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The total effective thickness of such targets would be equivalent to about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='044 X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The corresponding C content to be subtracted following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' (2) to obtain a pure water target is only about 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' An interesting application of such water targets in STT is the measurement of ν and ¯ν interactions off the bound neutron in the deuteron (D), which can be obtained from a subtraction between heavy water (D2O) and ordinary water (H2O) targets [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' To this end, both targets must be enclosed into identical acetal shells, which must be filled in such a way as to contain the same total mass of oxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' MEASURING NUCLEAR EFFECTS Nuclear effects and the (anti)neutrino flux are the leading sources of systematic un- certainties in high-energy neutrino scattering measurements [9, 15], as well as in modern long-baseline oscillation experiments [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Both issues arise because in conventional (anti)neutrino beams the energy of the incoming neutrino is unknown on an event-by-event basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The need to infer the neutrino energy from the detected final state particles consti- tutes an intrinsic limitation of high-energy neutrino experiments using nuclear targets, as 5 the nuclear smearing introduces substantial systematic uncertainties in the process (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The availability of both H and nuclear targets within the same detector can help to mit- igate such problems in STT [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The relative νµ and ¯νµ fluxes as a function of energy can be determined in-situ with an accuracy around 1% using exclusive νµp → µ−pπ+ and ¯νµp → µ−n processes on H at small energy transfer [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The combined use of ν-H and ¯ν-H CC interactions can provide a control sample free from nuclear effects to calibrate the neutrino energy scale in CC interactions from the nuclear targets [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The STT offers a tool to measure nuclear modifications of cross-sections and to constrain the systematic uncertainties associated to the nuclear smearing for the various integrated nuclear targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Each individual target is designed to be transparent to final state parti- cles (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' I) allowing, together with the low average density of the detector, an accurate reconstruction and characterization of the various event topologies in ν(¯ν) interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Sim- ulations of the detector response with GEANT4 [12] result in a rather uniform acceptance over the full 4π angle, with values of 95-99% for µ±, π±, K±, e±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' A key requirement is to guarantee the same acceptance across all nuclear targets, which is achieved by the combined effect of their thinness (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' I) and of their alternation throughout the detector volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Detailed detector simulations indicate that in this way the acceptance difference between targets can be kept within 10−3 for all particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The subtraction procedure required to obtain interactions on H, O, and H2O can then be considered model-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Further- more, the detector acceptance effectively cancels out in comparisons among the selected interactions on the H, C, and O targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The high intensity of modern (anti)neutrino beams complements well the relatively small mass of the various targets in STT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' For illustration, a fiducial mass of one tonne of water at the future Long-Baseline Neutrino Facility (LBNF) [1, 18] will collect about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='4×106 νµ CC events/year with the default low-energy spectrum (a factor of two higher with the planned PIP-II upgrade) and about 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='6×106 νµ CC events/year with the high-energy beam spectrum and the upgraded beam 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' With such high event rates a limited number of acetal and/or water targets in STT would suffice to obtain sensible physics measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Assuming as a reference a STT configuration with a “solid” hydrogen mass equivalent to about 10 m3 of liquid H2 4, about 20 modules equipped with the acetal targets described above would provide an O target mass similar to the graphite one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' An overall water target mass close to one tonne is therefore relatively easy to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' We note that the statistical uncertainties expected from such a water target at LBNF would be roughly comparable with the systematics from the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='2% energy scale uncertainty in STT, and smaller than the ones from the in-situ determination of the flux using exclusive processes on H [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' Comparing measurements of the bound nucleon structure functions F O 2,3 from the “solid” oxygen with the ones of the free nucleons in H with similar acceptance can provide insights on the nuclear modifications of the nucleon properties [8, 19–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The oxygen target can also provide complementary measurements with respect to the C and Ca targets to test the isospin (charge) symmetry [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' The isotopic content expected for a standard O target is 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='76% of 16O, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='2% of 18O, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content='04% of 17O, resulting on average in the smallest isovector component among stable elements β = (2Z − A)/A = 6 × 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' A comparison between ν and ¯ν interactions on oxygen through the ratios RO 2 = F ¯ν 2 /F ν 2 − 1 and RO 3 = xF ¯ν 3 /xF ν 3 − 1 for the structure functions F2 and xF3 can provide useful information about the isospin symmetry in nucleons and nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE3T4oBgHgl3EQf1Qsz/content/2301.04744v1.pdf'} +page_content=' 3 On-axis rates expected at the near detector site.' metadata={'source': 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b/XdAzT4oBgHgl3EQfmf14/content/tmp_files/2301.01564v1.pdf.txt @@ -0,0 +1,13500 @@ +— DRAFT — +Regularity Theory for +Elliptic PDE +Xavier Fern´andez-Real +Xavier Ros-Oton +EPFL SB MATH, Institute of Mathematics, Station 8, CH- +1015 Lausanne, Switzerland +E-mail address: xavier.fernandez-real@epfl.ch +Universit¨at Z¨urich, Institut f¨ur Mathematik, Winterthur- +erstrasse 190, 8057 Z¨urich, Switzerland, & +ICREA, Pg. Llu´ıs Companys 23, 08010 Barcelona, Spain, & +Universitat de Barcelona, Departament de Matem`atiques i +Inform`atica, Gran Via de les Corts Catalanes 585, 08007 Barcelona, +Spain, & +Centre de Recerca Matem`atica, Edifici C, Campus Bellaterra, +08193 Bellaterra, Spain +E-mail address: xros@icrea.cat +arXiv:2301.01564v1 [math.AP] 4 Jan 2023 + +— DRAFT — +2020 Mathematics Subject Classification. 35J15, 35B65, 35J05, 35J20, +35J60, 35R35. +Key words and phrases. Elliptic PDE, Schauder estimates, Hilbert XIXth +problem, nonlinear elliptic equations, obstacle problem. + +— DRAFT — +Contents +Preface +vii +Chapter 1. +Overview and Preliminaries +1 +§1.1. +Preliminaries: Sobolev and H¨older spaces +3 +§1.2. +A review on the Laplace equation +9 +§1.3. +Probabilistic interpretation of harmonic functions +21 +Chapter 2. +Linear elliptic PDE +25 +§2.1. +Harnack’s inequality +26 +§2.2. +Schauder estimates for the Laplacian +33 +§2.3. +Schauder estimates for operators in non-divergence form +46 +§2.4. +Schauder estimates for operators in divergence form +59 +§2.5. +The case of continuous coefficients +64 +§2.6. +Boundary regularity +68 +Chapter 3. +Nonlinear variational PDE & Hilbert’s XIXth problem +71 +§3.1. +Overview +72 +§3.2. +Existence and basic estimates +75 +§3.3. +De Giorgi’s proof +81 +§3.4. +Solution to Hilbert’s XIXth problem +94 +§3.5. +Further results and open problems +95 +Chapter 4. +Fully nonlinear elliptic PDE +97 +§4.1. +What is ellipticity? +98 +§4.2. +Equations in two variables +102 +v + +— DRAFT — +vi +Contents +§4.3. +Existence of solutions +105 +§4.4. +Regularity of solutions: an overview +115 +§4.5. +Further results and open problems +121 +Chapter 5. +The obstacle problem +125 +§5.1. +Some motivations and applications +128 +§5.2. +Basic properties of solutions I +130 +§5.3. +Basic properties of solutions II +141 +§5.4. +Regularity of free boundaries: an overview +147 +§5.5. +Classification of blow-ups +150 +§5.6. +Regularity of the free boundary +161 +§5.7. +Singular points +171 +§5.8. +On the size of the singular set +175 +Appendix A. +Some properties of H¨older spaces +179 +Appendix B. +Proof of the boundary Harnack inequality +191 +Appendix C. +Probabilistic interpretation of fully nonlinear equations 201 +Appendix D. +Motivations and applications for the obstacle problem +211 +Notation +219 +Bibliography +223 +Index +229 + +— DRAFT — +Preface +One of the most basic and important questions in PDE is that of regularity. +It is also a unifying problem in the field, since it affects all kinds of PDEs. +A classical example is Hilbert’s XIXth problem (1900), which roughly speak- +ing asked to determine whether all solutions to uniformly elliptic variational +PDEs are smooth. The question was answered positively by De Giorgi and +Nash in 1956 and 1957, and it is now one of the most famous and important +theorems in the whole field of PDE. +The question of regularity has been a central line of research in elliptic +PDE since the mid-20th century, with extremely important contributions by +Nirenberg, Caffarelli, Krylov, Evans, Figalli, and many others. Their works +have enormously influenced many areas of Mathematics linked one way or +another with PDE, including: Harmonic Analysis, Calculus of Variations, +Differential Geometry, Geometric Measure Theory, Continuum and Fluid +Mechanics, Probability Theory, Mathematical Physics, and Computational +and Applied Mathematics. +This text emerged from two PhD courses on elliptic PDE given by the +second author at the University of Z¨urich in 2017 and 2019. It aims to pro- +vide a self-contained introduction to the regularity theory for elliptic PDE, +focusing on the main ideas rather than proving all results in their greatest +generality. The book can be seen as a bridge between an elementary PDE +course and more advanced textbooks such as [GT77] or [CC95]. Moreover, +we believe that the present selection of results and techniques complements +nicely other books on elliptic PDE such as [Eva98], [HL97], and [Kry96], +as well as the recent book [ACM18]. For example, we give a different proof +of the Schauder estimates (due to L. Simon) which is not contained in other +textbooks; we prove some basic results for fully nonlinear equations that +vii + +— DRAFT — +viii +Preface +are not covered in [CC95]; and we also include a detailed study of the ob- +stacle problem, often left to more specialized textbooks such as [Fri88] or +[PSU12]. Furthermore, at the end of Chapters 3, 4, and 5 we provide a +review of some recent results and open problems. +We would like to thank Alessio Figalli, Thomas Kappeler, Alexis Michelat, +Joaquim Serra, and Wei Wang, for several comments and suggestions on this +book. +Finally, we acknowledge the support received from the following funding +agencies: X.F. was supported by the European Research Council under the +Grant Agreement No. 721675 “Regularity and Stability in Partial Differen- +tial Equations (RSPDE)”, by the Swiss National Science Foundation (SNF +grants 200021 182565 and PZ00P2 208930), and by the Swiss State Secre- +tariat for Education, Research and lnnovation (SERI) under contract num- +ber M822.00034; X.R. was supported by the European Research Council un- +der the Grant Agreement No. 801867 “Regularity and singularities in elliptic +PDE (EllipticPDE)”, by the Swiss National Science Foundation (SNF grant +200021 178795), by AEI project PID2021-125021NA-I00 (Spain), by the +grant RED2018-102650-T funded by MCIN/AEI/10.13039/501100011033, +and by the Spanish State Research Agency through the Mar´ıa de Maeztu +Program for Centers and Units of Excellence in R&D (CEX2020-001084-M). +Z¨urich, 2020 + +— DRAFT — +Chapter 1 +Overview and +Preliminaries +A beautiful result in Complex Analysis states that because the real part +u(x, y) of any holomorphic function satisfies +uxx + uyy = 0, +it must be real analytic. Moreover, the oscillation of u in any given domain +controls all the derivatives in any (compactly contained) subdomain. +In higher dimensions, the same phenomenon occurs for solutions to +(1.1) +∆u = 0 +in +Ω ⊂ Rn. +These are harmonic functions, and (1.1) is the simplest elliptic partial dif- +ferential equation (PDE). Any solution to this equation is smooth (real an- +alytic), and satisfies +∥u∥Ck(Q) ≤ Ck,Q∥u∥L∞(Ω) +for all +k = 1, 2, 3, ... +for any compact subdomain Q ⊂⊂ Ω. That is, all derivatives are controlled +by the supremum of u. +Here, and throughout the book, Ω is any bounded domain of Rn. +• Regularity for Laplace’s equation: +∆u = 0 +in +Ω ⊂ Rn +=⇒ +u is C∞ inside Ω. +This kind of regularization property is common in elliptic PDEs and is +the topic of the present book. +1 + +— DRAFT — +2 +1. Overview and Preliminaries +One can give three different kinds of explanations for this phenomenon: +(a) Integral representation of solutions: Poisson kernels, fundamental +solutions, etc. +(b) Energy considerations: Harmonic functions are local minimizers of +the Dirichlet energy +E(u) := +� +Ω +|∇u|2 dx +(i.e., if we change u to w in ˜Ω ⊂ Ω, then E(w) ≥ E(u)). +(c) Comparison principle: A harmonic function cannot have any inte- +rior maximum point (maximum principle). +These three approaches are extremely useful in different contexts, as well +as in the development of the regularity theory for nonlinear elliptic PDEs. +The structure of the book is as follows: +⋆ First, in Chapter 2 we will study linear elliptic PDEs +n +� +i,j=1 +aij(x)∂iju = f(x) +in +Ω ⊂ Rn +and +n +� +i,j=1 +∂i +� +aij(x)∂ju +� += f(x) +in +Ω ⊂ Rn, +where the coefficients aij(x) and the right-hand side f(x) satisfy appropriate +regularity assumptions. In the simplest case, (aij)i,j ≡ Id, we have +∆u = f(x) +in +Ω ⊂ Rn. +The type of result we want to prove is: “u is two derivatives more regular +than f”. +⋆ Then, in Chapter 3 we will turn our attention to nonlinear variational +PDEs: +minimizers of +E(u) := +� +Ω +L(∇u)dx, +L smooth and uniformly convex. +The regularity for such kind of nonlinear PDEs was Hilbert’s XIXth problem +(1900). +⋆ In Chapter 4 we will study nonlinear elliptic PDEs in their most +general form +F(D2u, ∇u, u, x) = 0 +in +Ω ⊂ Rn, + +— DRAFT — +1.1. Preliminaries: Sobolev and H¨older spaces +3 +or simply +F(D2u) = 0 +in +Ω ⊂ Rn. +These are called fully nonlinear elliptic equations, and in general they do +not have a variational formulation in terms of an energy functional. +⋆ In Chapter 5 we will study the obstacle problem, a constrained min- +imization problem: +minimize +� +Ω +|∇u|2dx, +among functions u ≥ ϕ in Ω, +where ϕ is a given smooth “obstacle”. This is the simplest and most impor- +tant elliptic free boundary problem. Moreover, it can be seen as a nonlinear +PDE of the type min{−∆u, u − ϕ} = 0 in Ω. +As we will see, in each of these contexts we will use mainly: (b) energy +considerations, or (c) maximum principle. +At the end of the book, we have also included four appendices to com- +plement the theory from the main chapters. +1.1. Preliminaries: Sobolev and H¨older spaces +We next give a quick review on Lp, Sobolev, and H¨older spaces, stating the +results that will be used later in the book. +Lp spaces. Given Ω ⊂ Rn and 1 ≤ p < ∞, the space Lp(Ω) is the set +Lp(Ω) := +� +u measurable in Ω : +� +Ω +|u|pdx < ∞ +� +. +It is a Banach space, with the norm ∥u∥Lp(Ω) := ( +� +Ω |u|p)1/p. +When p = ∞, the space L∞(Ω) is the set of bounded functions (up to +sets of measure zero), with the norm ∥u∥L∞(Ω) := esssupΩ|u|. +A well-known result in this setting is the Lebesgue differentiation theo- +rem (see, for example, [EG92]). +Theorem 1.1. If u ∈ L1(Ω), then for almost every x ∈ Ω we have +lim +r→0 +� +Br(x) +��u(x) − u(y) +��dy = 0. +When this holds at a point x ∈ Ω, we say that x is a Lebesgue point of u. +Here, and throughout the book, +� +A denotes the average +1 +|A| +� +A, where +A ⊂ Rn is any set of finite and positive measure. +A useful consequence of this result is the following. + +— DRAFT — +4 +1. Overview and Preliminaries +Corollary 1.2. Assume u ∈ L1(Ω), and +� +Ω +uv dx = 0 +for all v ∈ C∞ +c (Ω). +Then, u = 0 a.e. in Ω. +Integration by parts. A fundamental identity in the study of PDEs is the +following. +Theorem 1.3 (Integration by parts). Assume Ω ⊂ Rn is any bounded C1 +domain1. Then, for any u, v ∈ C1(Ω) we have +(1.2) +� +Ω +∂iu v dx = − +� +Ω +u ∂iv dx + +� +∂Ω +uv νi dS, +where ν is the unit (outward) normal vector to ∂Ω, and i = 1, 2, ..., n. +Notice that, as an immediate consequence, we find the divergence theo- +rem, as well as Green’s first identity +� +Ω +∇u · ∇v dx = − +� +Ω +u ∆v dx + +� +∂Ω +u ∂v +∂ν dS. +The regularity requirements of Theorem 1.3 can be relaxed. For instance, +the domain Ω need only be Lipschitz, while only u, v ∈ H1(Ω) is necessary +in (1.2) — where H1 is a Sobolev space, defined below. +Sobolev spaces. Given any domain Ω ⊂ Rn and 1 ≤ p ≤ ∞, the Sobolev +spaces W 1,p(Ω) consist of all functions whose (weak) derivatives are in Lp(Ω), +namely +W 1,p(Ω) := {u ∈ Lp(Ω) : ∂iu ∈ Lp(Ω) for i = 1, ..., n} . +We refer to the excellent books [Eva98, Bre11] for the definition of weak +derivatives and a detailed exposition on Sobolev spaces. +A few useful properties of Sobolev spaces are the following (see [Eva98]): +(S1) The spaces W 1,p(Ω) are complete. +(S2) The inclusion W 1,p(Ω) ⊂ Lp(Ω) is compact. +(S3) The space H1(Ω) := W 1,2(Ω) is a Hilbert space with the scalar product +(u, v)H1(Ω) = +� +Ω +uv + +� +Ω +∇u · ∇v. +(S4) Any bounded sequence {uk} in the Hilbert space H1(Ω) contains a +weakly convergent subsequence {ukj}, that is, there exists u ∈ H1(Ω) +such that +(1.3) +(ukj, v)H1(Ω) → (u, v)H1(Ω) +for all v ∈ H1(Ω). +1We refer to the Notation section (page 221) for the definition of C1 domains. + +— DRAFT — +1.1. Preliminaries: Sobolev and H¨older spaces +5 +In addition, such u will satisfy +(1.4) +∥u∥H1(Ω) ≤ lim inf +j→∞ ∥ukj∥H1(Ω), +and since H1(Ω) is compactly embedded in L2(Ω) one has +(1.5) +∥u∥L2(Ω) = lim +j→∞ ∥ukj∥L2(Ω). +(S5) Let Ω be any bounded Lipschitz domain, and 1 ≤ p ≤ ∞. Then, there +is a continuous (and compact for p > 1) trace operator from W 1,p(Ω) +to Lp(∂Ω). For C0 functions, such trace operator is simply u �→ u|∂Ω. +Because of this, for any function u ∈ H1(Ω) we will still denote by +u|∂Ω its trace on ∂Ω. +(S6) For 1 ≤ p < ∞, C∞(Ω) functions are dense in W 1,p(Ω). Moreover, if +Ω is bounded and Lipschitz, C∞(Ω) functions are dense in W 1,p(Ω). +(S7) For 1 ≤ p < ∞, we define the space W 1,p +0 (Ω) as the closure of C∞ +c (Ω) +in W 1,p(Ω). +Similarly, we denote H1 +0(Ω) := W 1,2 +0 (Ω). +When Ω is +bounded and Lipschitz, it is the space of functions u ∈ W 1,p(Ω) such +that u|∂Ω = 0. +(S8) If u ∈ W 1,p(Ω), 1 ≤ p ≤ ∞, then for any subdomain K ⊂⊂ Ω we have +���� +u(x + h) − u(x) +|h| +���� +Lp(K) +≤ C ∥∇u∥Lp(Ω) +for all h ∈ Bδ, with δ > 0 small enough. +Conversely, if u ∈ Lp(Ω), 1 < p ≤ ∞, and +���� +u(x + h) − u(x) +|h| +���� +Lp(K) +≤ C +for every h ∈ Bδ, then u ∈ W 1,p(K) and ∥∇u∥Lp(Ω) ≤ C. (However, +this property fails when p = 1.) +(S9) Given any function u, define u+ = max{u, 0} and u− = max{−u, 0}, +so that u = u+ − u−. Then, for any u ∈ W 1,p(Ω) we have u+, u− ∈ +W 1,p(Ω), and ∇u = ∇u+ − ∇u− a.e. in Ω. +In particular, the gradient of Sobolev functions vanishes almost +everywhere on level sets, ∇u(x) = 0 for a.e. x ∈ {u = 0}. +An important inequality in this context is the following. +Theorem 1.4 (Sobolev inequality). If p < n, then +�� +Rn |u|p∗dx +�1/p∗ +≤ C +�� +Rn |∇u|pdx +�1/p +, +1 +p∗ += 1 +p − 1 +n, +for some constant C depending only on n and p. In particular, we have a +continuous inclusion W 1,p(Rn) ⊂ Lp∗(Rn). + +— DRAFT — +6 +1. Overview and Preliminaries +Notice that, as p ↑ n we have p∗ → ∞. In the limiting case p = n, +however, it is not true that W 1,n functions are bounded. This can be seen +by taking, for example, u(x) = log log +� +1 + 1 +|x| +� +∈ W 1,n(B1). Still, in case +p > n, the following occurs. +Theorem 1.5 (Morrey inequality). If p > n, then +sup +x̸=y +��u(x) − u(y) +�� +|x − y|α +≤ C +�� +Rn |∇u|pdx +�1/p +, +α = 1 − n +p , +for some constant C depending only on n and p. +In particular, when p > n any function in W 1,p is continuous (after +possibly being redefined on a set of measure 0). +Finally, we will also use the following inequalities in bounded domains. +Theorem 1.6 (Poincar´e inequality). Let Ω ⊂ Rn be any bounded Lipschitz +domain, and let p ∈ [1, ∞). Then, for any u ∈ W 1,p(Ω) we have +� +Ω +|u − uΩ|pdx ≤ CΩ,p +� +Ω +|∇u|pdx, +where uΩ := +� +Ω u, and +� +Ω +|u|pdx ≤ C′ +Ω,p +�� +Ω +|∇u|pdx + +� +∂Ω +��u|∂Ω +��pdσ +� +. +The constants CΩ,p and C′ +Ω,p depend only on n, p, and Ω. +H¨older spaces. Given α ∈ (0, 1), the H¨older space C0,α(Ω) is the set of +continuous functions u ∈ C(Ω) such that the H¨older semi-norm is finite, +[u]C0,α(Ω) := sup +x,y∈Ω +x̸=y +��u(x) − u(y) +�� +|x − y|α +< ∞. +The H¨older norm is +∥u∥C0,α(Ω) := ∥u∥L∞(Ω) + [u]C0,α(Ω). +When α = 1, this is the usual space of Lipschitz continuous functions. +More generally, given k ∈ N and α ∈ (0, 1), the space Ck,α(Ω) is the set +of functions u ∈ Ck(Ω) such that the following norm is finite +∥u∥Ck,α(Ω) = ∥u∥Ck(Ω) + [Dku]C0,α(Ω), +where +∥u∥Ck(Ω) := +k +� +j=1 +∥Dju∥L∞(Ω). + +— DRAFT — +1.1. Preliminaries: Sobolev and H¨older spaces +7 +Notice that this yields the inclusions +C0 ⊃ C0,α ⊃ Lip ⊃ C1 ⊃ C1,α ⊃ ... ⊃ C∞. +We will often write ∥u∥Ck,α(Ω) instead of ∥u∥Ck,α(Ω). +Finally, it is sometimes convenient to use the following notation. When +β > 0 is not an integer, we define Cβ(Ω) := Ck,α(Ω), where β = k + α, +k ∈ N, α ∈ (0, 1). +There are many properties or alternative definitions of H¨older spaces +that will be used throughout the book. They are valid for all α ∈ (0, 1), and +are proved in Appendix A. +(H1) Assume +oscBr(x)u ≤ C◦rα +for all Br(x) ⊂ B1, +where oscAu := supA u − infA u. +Then, u ∈ C0,α(B1) and [u]C0,α(B1) ≤ CC◦, with C depending +only on n, α. +(H2) Let ux,r := +� +Br(x) u. Assume +∥u − ux,r∥L∞(Br(x)) ≤ C◦rα +for all Br(x) ⊂ B1. +Then, u ∈ C0,α(B1) and [u]C0,α(B1) ≤ CC◦, with C depending +only on n, α. +(H3) Let ux,r := +� +Br(x) u. Assume +� � +Br(x) +|u − ux,r|2 +�1/2 +≤ C◦rα +for all Br(x) ⊂ B1. +Then, u ∈ C0,α(B1) and [u]C0,α(B1) ≤ CC◦, with C depending only +on n, α. +(H4) Assume that for every x there is a constant Cx such that +∥u − Cx∥L∞(Br(x)) ≤ C◦rα +for all Br(x) ⊂ B1. +Then, u ∈ C0,α(B1) and [u]C0,α(B1) ≤ CC◦, with C depending only +on n, α. +Assume that for every x there is a linear function ℓx(y) = ax + +bx · (y − x) such that +∥u − ℓx∥L∞(Br(x)) ≤ C◦r1+α +for all Br(x) ⊂ B1. +Then, u ∈ C1,α(B1) and [Du]C0,α(B1) ≤ CC◦, with C depending only +on n, α. + +— DRAFT — +8 +1. Overview and Preliminaries +Assume that for every x there is a quadratic polynomial Px(y) +such that +∥u − Px∥L∞(Br(x)) ≤ C◦r2+α +for all Br(x) ⊂ B1. +Then, u ∈ C2,α(B1) and [D2u]C0,α(B1) ≤ CC◦, with C depending only +on n, α. +(H5) Let ρ◦ ∈ (0, 1). +Assume that, for every x ∈ B1/2, there exists a +sequence of quadratic polynomials, (Pk)k∈N, such that +(1.6) +∥u − Pk∥L∞(Bρk◦ (x)) ≤ C◦ρk(2+α) +◦ +for all k ∈ N. +Then, u ∈ C2,α(B1/2) and [D2u]C0,α(B1/2) ≤ CC◦, with C depending +only on n, α, and ρ◦. +(H6) Assume that α ∈ (0, 1), ∥u∥L∞(B1) ≤ C◦, and +sup +x∈B1 +x±h∈B1 +��u(x + h) + u(x − h) − 2u(x) +�� +|h|α +≤ C◦. +Then, u ∈ C0,α(B1) and ∥u∥C0,α(B1) ≤ CC◦, with C depending only +on n, α. +Assume that α ∈ (0, 1), ∥u∥L∞(B1) ≤ C◦, and +sup +x∈B1 +x±h∈B1 +��u(x + h) + u(x − h) − 2u(x) +�� +|h|1+α +≤ C◦. +Then, u ∈ C1,α(B1) and ∥u∥C1,α(B1) ≤ CC◦, with C depending only +on n, α. +However, such property fails when α = 0. +(H7) Assume that α ∈ (0, 1], ∥u∥L∞(B1) ≤ C◦, and that for every h ∈ B1 +we have +(1.7) +���� +u(x + h) − u(x) +|h|α +���� +Cβ(B1−|h|) +≤ C◦, +with C◦ independent of h. +Assume in addition that α + β is not +an integer. Then, u ∈ Cα+β(B1) and ∥u∥Cα+β(B1) ≤ CC◦, with C +depending only on n, α, β. +However, such property fails when α + β is an integer. +(H8) Assume that ui → u0 uniformly in Ω ⊂ Rn, and that ∥ui∥Ck,α(Ω) ≤ C◦, +with α ∈ (0, 1] and for some C◦ independent of i. Then, we have that +u0 ∈ Ck,α(Ω), and +∥u0∥Ck,α(Ω) ≤ C◦. + +— DRAFT — +1.2. A review on the Laplace equation +9 +Finally, an important result in this context is the following particular +case of the Arzel`a–Ascoli theorem. +Theorem 1.7 (Arzel`a–Ascoli). Let Ω ⊂ Rn, α ∈ (0, 1), and let {fi}i∈N be +any sequence of functions fi satisfying +∥fi∥C0,α(Ω) ≤ C◦. +Then, there exists a subsequence fij which converges uniformly to a func- +tion f ∈ C0,α(Ω). +More generally, this result — combined with (H8) — implies that if +∥ui∥Ck,α(Ω) ≤ C◦, +with α ∈ (0, 1), then a subsequence uij will converge in the Ck(Ω) norm to +a function u ∈ Ck,α(Ω). +Interpolation inequalities in H¨older spaces. A useful tool that will be +used throughout the book is the following. For each 0 ≤ γ < α < β ≤ 1 and +every ε > 0, we have +(1.8) +∥u∥C0,α(Ω) ≤ Cε∥u∥C0,γ(Ω) + ε∥u∥C0,β(Ω), +where C is a constant depending only on n and ε. (When γ = 0, C0,γ should +be replaced by L∞.) This follows from the interpolation inequality +∥u∥C0,α(Ω) ≤ ∥u∥t +C0,γ(Ω)∥u∥1−t +C0,β(Ω) +t = β − α +β − γ . +More generally, (1.8) holds for higher-order H¨older norms too. In par- +ticular, we will use that for any ε > 0 and α ∈ (0, 1) +∥∇u∥L∞(Ω) ≤ Cε∥u∥L∞(Ω) + ε[∇u]C0,α(Ω), +and +(1.9) +∥u∥C2(Ω) = ∥u∥C1,1(Ω) ≤ Cε∥u∥L∞(Ω) + ε[D2u]C0,α(Ω). +We refer to [GT77, Lemma 6.35] for a proof of such inequalities. +1.2. A review on the Laplace equation +Elliptic equations are those that share some common properties with the +Laplace equation. (We will be more rigorous about this in the subsequent +chapters.) Thus, we start with a quick review about the Laplace equation +and harmonic functions. +The Dirichlet problem for this equation is the following: +(1.10) +� ∆u += +0 +in Ω +u += +g +on ∂Ω, + +— DRAFT — +10 +1. Overview and Preliminaries +where the boundary condition g is given. The domain Ω ⊂ Rn is bounded +and smooth (or at least Lipschitz). The Dirichlet problem is solvable, and +it has a unique solution. +A useful way to think of the Laplacian ∆ is to notice that, up to a +multiplicative constant, it is the only linear operator of second order which +is translation invariant and rotation invariant. Indeed, it can be seen as an +operator which measures (infinitesimally) the difference between u at x and +the average of u around x, in the following sense: for any C2 function w we +have +∆w(x) = lim +r→0 +cn +r2 +� � +Br(x) +w(y)dy − w(x) +� += lim +r→0 +cn +r2 +� +Br(x) +� +w(y) − w(x) +� +dy, +(1.11) +for some positive constant cn. This can be shown, for example, by using the +Taylor expansion of w(y) around x. Moreover, a similar formula holds with +integrals in ∂Br(x) instead of Br(x). See, for example, [DV21]. +Actually, one can show by using the divergence theorem that +(1.12) +n +r +d +dr +� +∂Br(x) +w dσ = +� +Br(x) +∆w, +from which (1.11) also follows. +Existence of solutions: energy methods. The most classical way to +construct solutions of (1.10) is by “energy methods”. Namely, we consider +the convex functional +E(u) := 1 +2 +� +Ω +|∇u|2dx +among functions satisfying +u|∂Ω = g, +and then look for the function u that minimizes the functional — see Theo- +rem 1.10 below for more details about the existence of a minimizer. Notice +that such minimizer u will clearly satisfy the boundary condition u = g on +∂Ω, so we only have to check that it will satisfy in addition ∆u = 0 in Ω. +If u is the minimizer, then E(u) ≤ E(u+εv) for every v ∈ C∞ +c (Ω). Since, +for every fixed v, such function in ε has a minimum at ε = 0, we have +d +dε +���� +ε=0 +E(u + εv) = 0. + +— DRAFT — +1.2. A review on the Laplace equation +11 +Thus, +0 += +d +dε +���� +ε=0 +E(u + εv) = d +dε +���� +ε=0 +1 +2 +� +Ω +|∇u + εv|2dx += +d +dε +���� +ε=0 +1 +2 +� +Ω +� +|∇u|2 + 2ε∇u · ∇v + ε2|∇v|2� +dx += +� +Ω +∇u · ∇v dx. +Hence, if u is the minimizer of the functional, then +(1.13) +� +Ω +∇u · ∇v dx = 0 +for all +v ∈ C∞ +c (Ω). +If u is regular enough (say, u ∈ C2), then we can integrate by parts (Theo- +rem 1.3) to find that +� +Ω +∆u v dx = 0 +for all +v ∈ C∞ +c (Ω). +Thus, using Corollary 1.2 we deduce that ∆u = 0 in Ω, as wanted. +Remark 1.8. As mentioned above, one should prove regularity of u before +integrating by parts — a priori the minimizer u will only satisfy u ∈ H1(Ω). +We will prove this in Corollary 1.12 below. +If no extra regularity of u is available, then the above argument shows +that any minimizer u of E is a weak solution, in the following sense. +Definition 1.9. We say that u is a weak solution of the Dirichlet problem +(1.10) whenever u ∈ H1(Ω), u|∂Ω = g, and +� +Ω +∇u · ∇v dx = 0 +for all +v ∈ H1 +0(Ω). +Here, u|∂Ω is the trace of u on ∂Ω; recall (S5) above. +More generally, given f ∈ L2(Ω), we say that u satisfies −∆u = f in Ω +in the weak sense whenever u ∈ H1(Ω) and +� +Ω +∇u · ∇v dx = +� +Ω +fv +for all +v ∈ H1 +0(Ω). +Finally, we say that u is weakly superharmonic (resp. weakly subharmonic) +in Ω, or satisfies ∆u ≤ 0 in Ω in the weak sense (resp. ∆u ≥ 0 in the weak +sense) if +� +Ω +∇u·∇v dx ≥ 0 +� +resp. +� +Ω +∇u · ∇v dx ≤ 0 +� +for all +v ∈ H1 +0(Ω), v ≥ 0. + +— DRAFT — +12 +1. Overview and Preliminaries +Notice that, if H1(Ω) ∋ uk ⇀ u ∈ H1(Ω) weakly in H1, and L2(Ω) ∋ +fk ⇀ f ∈ L2(Ω) weakly in L2 are such that ∆uk = fk in Ω in the weak +sense, then ∆u = f in the weak sense as well (by taking the limits in the pre- +vious definitions). Similarly, the weak limit of weakly (sub-)superharmonic +functions is (sub-)superharmonic. +We next show the following: +Theorem 1.10 (Existence and uniqueness of weak solutions). Assume that +Ω ⊂ Rn is any bounded Lipschitz domain, and that +(1.14) +� +w ∈ H1(Ω) : w|∂Ω = g +� +̸= ∅. +Then, there exists a unique weak solution to the Dirichlet problem (1.10). +Proof. Existence. Let +θ◦ := inf +�1 +2 +� +Ω +|∇w|2dx : w ∈ H1(Ω), w|∂Ω = g +� +, +that is, the infimum value of E(w) among all admissible functions w. +Let us take a sequence of functions {uk} such that +• uk ∈ H1(Ω) +• uk|∂Ω = g +• E(uk) → θ◦ as k → ∞. +By the Poincar´e inequality (Theorem 1.6 with p = 2), the sequence {uk} +is uniformly bounded in H1(Ω), and therefore a subsequence {ukj} will +converge to a certain function u strongly in L2(Ω) and weakly in H1(Ω) +(recall (1.3)-(1.5) in (S4)). Moreover, by compactness of the trace operator, +we will have ukj|∂Ω → u|∂Ω in L2(∂Ω), so that u|∂Ω = g. Furthermore, such +function u will satisfy E(u) ≤ lim infj→∞ E(ukj) (by (1.4) and (1.5)), and +therefore it will be a minimizer of the energy functional. +Thus, we have constructed a minimizer u of the energy functional E(u) +satisfying the boundary condition u|∂Ω = g. By the argument above, for +any minimizer u we have that (1.13) holds. Since C∞ +c (Ω) is dense in H1 +0(Ω), +it follows that (1.13) holds for all v ∈ H1 +0(Ω), and thus it is a weak solution +of (1.10). + +— DRAFT — +1.2. A review on the Laplace equation +13 +Uniqueness. If u is any weak solution to (1.10), then for every v ∈ +H1 +0(Ω) we have +E(u + v) += +1 +2 +� +Ω +|∇u + ∇v|2dx += +1 +2 +� +Ω +|∇u|2dx + +� +Ω +∇u · ∇v dx + 1 +2 +� +Ω +|∇v|2dx += +E(u) + 0 + 1 +2 +� +Ω +|∇v|2dx ≥ E(u), +with strict inequality if v ̸≡ 0. Thus, if u solves (1.10), then it is unique. +□ +In other words, we have shown that u is a weak solution of (1.10) if +and only if it minimizes the functional E(u) and, moreover, the minimizer +of such energy functional exists and it is unique. +Remark 1.11. An interesting question is to determine the set of possible +boundary data g : ∂Ω → R such that (1.14) holds. Of course, when Ω is +any bounded Lipschitz domain, and g is Lipschitz, then it is easy to show +that g has a Lipschitz extension inside Ω, and in particular (1.14) holds. +However, if g is very irregular then it might happen that it is not the trace +of any H1(Ω) function, so that (1.14) fails in this case. It turns out that the +right condition on g is the following: Given any bounded Lipschitz domain +Ω, (1.14) holds if and only if +� +∂Ω +� +∂Ω +|g(x) − g(y)|2 +|x − y|n+1 +dx dy < ∞. +We refer to [Eva98] for more details. +Poisson kernel and fundamental solution. The unique weak solution +to the Dirichlet problem in a ball is explicit: +� ∆u += +0 +in B1 +u += +g +on ∂B1, +=⇒ +u(x) = cn +� +∂B1 +(1 − |x|2)g(σ) +|x − σ|n +dσ, +where cn is a positive dimensional constant. By an easy rescaling argument, +a similar formula holds in any ball Br(x◦) ⊂ Rn. +Thus, we deduce that for any harmonic function ∆u = 0 in Ω, with +Br ⊂ Ω, we have +(1.15) +u(x) = cn +r +� +∂Br +(r2 − |x|2)u(y) +|x − y|n +dy. +By taking x = 0, this yields the mean value property u(0) = +� +∂Br u. More- +over, an immediate consequence of the Poisson kernel representation is the +following. + +— DRAFT — +14 +1. Overview and Preliminaries +Corollary 1.12. Let Ω ⊂ Rn be any open set, and u ∈ H1(Ω) be any +function satisfying ∆u = 0 in Ω in the weak sense. Then, u is C∞ inside Ω. +Moreover, if u is bounded and ∆u = 0 in B1 in the weak sense, then we +have the estimates +(1.16) +∥u∥Ck(B1/2) ≤ Ck∥u∥L∞(B1), +for all k ∈ N, and for some constant Ck depending only on k and n. +Proof. For any ball Br(x◦) ⊂ Ω, we will have (1.15). +Thanks to such +representation, it is immediate to see then that u ∈ C∞(Br/2(x◦)) and the +estimates (1.16) hold. Since this can be done for any ball Br(x◦) ⊂ Ω, we +deduce that u is C∞ inside Ω. +□ +On the other hand, we recall that the fundamental solution for the Lapla- +cian is given by +(1.17) +Φ(x) := +� +� +� +� +� +κn +|x|n−2 +if n ≥ 3 +κ2 log 1 +|x| +if n = 2, +for some explicit positive dimensional constant κn. Such function satisfies +∆Φ = 0 in Rn \ {0}, but it is singular at x = 0. In fact, it satisfies +−∆Φ = δ0 +in +Rn, +where δ0 is the Dirac delta function. In particular, we have that w := Φ ∗ f +solves −∆w = f in Rn, for any given f with appropriate decay at infinity. +Maximum principle. The maximum principle states the following: +If +∆u ≥ 0 in Ω, and u ∈ C(Ω), then +max +Ω +u = max +∂Ω u. +In particular, we also deduce the comparison principle: if ∆u ≥ ∆v in Ω, +and u ≤ v on ∂Ω, then u ≤ v in the whole domain Ω. +Recall that a function is said to be subharmonic if −∆u ≤ 0, and super- +harmonic if −∆u ≥ 0. +As shown next, the maximum principle actually holds for any weak +solution u. +Proposition 1.13. Let Ω ⊂ Rn be any bounded open set. +Assume that +u ∈ H1(Ω) satisfies, in the weak sense, +� −∆u +≥ +0 +in Ω +u +≥ +0 +on ∂Ω. +Then, u ≥ 0 in Ω. + +— DRAFT — +1.2. A review on the Laplace equation +15 +Proof. Notice that −∆u ≥ 0 in Ω if and only if +(1.18) +� +Ω +∇u · ∇v dx ≥ 0 +for all +v ≥ 0, v ∈ H1 +0(Ω). +Let us consider u− := max{−u, 0} and u+ := max{u, 0}, so that u = u+ − +u−. By (S9) we have that u± ∈ H1(Ω) whenever u ∈ H1(Ω), and thus +we can choose v = u− ≥ 0 in (1.18). Namely, using that u+u− = 0 and +∇u = ∇u+ − ∇u−, we get +0 ≤ +� +Ω +∇u · ∇u− dx = − +� +Ω +|∇u−|2 dx. +Since u−|∂Ω ≡ 0 this implies u− ≡ 0 in Ω, that is, u ≥ 0 in Ω. +□ +A useful consequence of the maximum principle is the following. +Lemma 1.14. Let u be any weak solution of +� ∆u += +f +in Ω +u += +g +on ∂Ω. +Then, +∥u∥L∞(Ω) ≤ C +� +∥f∥L∞(Ω) + ∥g∥L∞(∂Ω) +� +, +for a constant C depending only on the diameter of Ω. +Proof. Let us consider the function +˜u(x) := u(x)/ +� +∥f∥L∞(Ω) + ∥g∥L∞(∂Ω) +� +. +We want to prove that |˜u| ≤ C in Ω, for some constant C depending only +on the diameter of Ω. +Notice that such function ˜u solves +� +∆˜u += +˜f +in Ω +u += +˜g +on ∂Ω, +with |˜g| ≤ 1 and | ˜f| ≤ 1. +Let us choose R large enough so that BR ⊃ Ω; after a translation, we +can take R = 1 +2diam(Ω). In BR, let us consider the function +w(x) = R2 − x2 +1 +2 ++ 1. +Such function w satisfies� ∆w += +−1 +in Ω +w +≥ +1 +on ∂Ω, +Therefore, by the comparison principle, we deduce that +˜u ≤ w +in +Ω. + +— DRAFT — +16 +1. Overview and Preliminaries +Since w ≤ C (with C depending only on R), we deduce that ˜u ≤ C in Ω. +Finally, repeating the same argument with −˜u instead of ˜u, we find that +|˜u| ≤ C in Ω, and thus we are done. +□ +Finally, another important result which follows from the maximum prin- +ciple is the following. Here, we say that Ω satisfies the interior ball condition +whenever there exists ρ◦ > 0 such that every point on ∂Ω can be touched +from inside with a ball of radius ρ◦ contained in Ω. That is, for any x◦ ∈ ∂Ω +there exists Bρ◦(y◦) ⊂ Ω with x◦ ∈ ∂Bρ◦(y◦). +It is not difficult to see that any C2 domain satisfies such condition, and +also any domain which is the complement of a convex set. +Lemma 1.15 (Hopf Lemma). Let Ω ⊂ Rn be any domain satisfying the +interior ball condition. Let u ∈ C(Ω) be any positive harmonic function in +Ω ∩ B2, with u ≥ 0 on ∂Ω ∩ B2. +Then, u ≥ c◦d in Ω ∩ B1 for some c◦ > 0, where d(x) := dist(x, Ωc). +Proof. Since u is positive and continuous in Ω∩B2, we have that u ≥ c1 > 0 +in {d ≥ ρ◦/2} ∩ B3/2 for some c1 > 0. +Let us consider the solution of ∆w = 0 in Bρ◦ \ Bρ◦/2, with w = 0 on +∂Bρ◦ and w = 1 on ∂Bρ◦/2. Such function w is explicit — it is simply a +truncated and rescaled version of the fundamental solution Φ in (1.17). In +particular, it is immediate to check that w ≥ c2(ρ◦ − |x|) in Bρ◦ for some +c2 > 0. +By using the function c1w(x◦+x) as a subsolution in any ball Bρ◦(x◦) ⊂ +Ω ∩ B3/2, we deduce that u(x) ≥ c1w(x◦ + x) ≥ c1c2(ρ◦ − |x − x◦|) ≥ c1c2d +in Bρ◦(x◦). Setting c◦ = c1c2 and using the previous inequality for every +ball Bρ◦(x◦) ⊂ Ω ∩ B3/2, the result follows. +□ +Mean value property and Liouville theorem. If u is harmonic in Ω +(i.e., ∆u = 0 in Ω), then +(1.19) +u(x) = +� +Br(x) +u(y)dy +for any ball +Br(x) ⊂ Ω. +This is called the mean value property. +Conversely, if u ∈ C2(Ω) satisfies the mean value property, then ∆u = 0 +in Ω. This can be seen for example by using (1.11) above. +In fact, the mean value property (1.19) can be used to give yet another +(weak) definition of harmonic functions that only requires u to be locally +integrable. +Similarly, it is not difficult to deduce the corresponding pro- +perty arising from the definitions of weak super- and subharmonicity (see +Definition 1.9): + +— DRAFT — +1.2. A review on the Laplace equation +17 +From (1.12), if u is weakly superharmonic in Ω (∆u ≤ 0 in Ω in the +weak sense) then for all x ∈ Ω +(1.20) +r �→ +� +Br(x) +u(y) dy +is monotone non-increasing for r ∈ (0, dist(x, ∂Ω)). +(And it is monotone non-decreasing for weakly subharmonic functions.) +Thus, we can define (weak) super- and subharmonicity for L1 +loc functions: +we say that u ∈ L1 +loc(Ω) is superharmonic in Ω if (1.20) holds for all x ∈ Ω. +Similarly, we say that u ∈ L1 +loc(Ω) is subharmonic in Ω if the map in (1.20) +is monotone non-decreasing for all x ∈ Ω and r ∈ (0, dist(x, ∂Ω)). +We now give two lemmas that will be used in Chapter 5. +The first +lemma says that the pointwise limit of a sequence of superharmonic uni- +formly bounded functions is superharmonic. +Lemma 1.16. Let Ω ⊂ Rn, and let {wn}n∈N be a sequence of uniformly +bounded functions wn : Ω → R satisfying (1.20), converging pointwise to +some w : Ω → R. Then w satisfies (1.20). +Proof. Let w∞ := w and let us define for n ∈ N ∪ {∞}, ϕx,n(r) := +� +Br(x) wn. +Notice that ϕx,n(r) is non-increasing in r for all n ∈ N. +In +particular, given 0 < r1 < r2 < Rx, we have that ϕx,n(r1) ≥ ϕx,n(r2) for +n ∈ N. Now we let n → ∞ and use that wn → w pointwise to deduce, by +the dominated convergence theorem (notice that wn are uniformly bounded), +that ϕx,∞(r1) ≥ ϕx,∞(r2). That is, w∞ = w satisfies (1.20). +□ +The second lemma shows that superharmonic functions are lower semi- +continuous. +Lemma 1.17. Let us assume that w is bounded and satisfies (1.20) in +Ω ⊂ Rn. Then, up to changing w in a set of measure 0, w is lower semi- +continuous. +Proof. The proof is standard. If we define w0(x) := limr↓0 +� +Br(x) w (which +is well defined, since the average is monotone non-increasing), then w0(x) = +w(x) if x is a Lebesgue point, and thus w0 = w almost everywhere in Ω. +Let us now consider x◦ ∈ Ω, and let xk → x◦ as k → ∞. Then, by the +dominated convergence theorem we have that +� +Br(x◦) +w = lim +k→∞ +� +Br(xk) +w ≤ lim inf +k→∞ w0(xk) +for 0 < r < 1 +2dist(x◦, ∂Ω). Now, by letting r ↓ 0 on the left-hand side, we +reach that +w0(x◦) ≤ lim inf +k→∞ w0(xk), + +— DRAFT — +18 +1. Overview and Preliminaries +that is, w0 is lower semi-continuous. +□ +On the other hand, a well-known theorem that can be deduced from +the mean value property is the classification of global bounded harmonic +functions. +Theorem 1.18 (Liouville’s theorem). Any bounded solution of ∆u = 0 in +Rn is constant. +Proof. Let u be any global bounded solution of ∆u = 0 in Rn. +Since +u is smooth (by Corollary 1.12), each derivative ∂iu is well-defined and is +harmonic too. Thus, thanks to the mean-value property and the divergence +theorem, for any x ∈ Rn and R ≥ 1 we have +|∂iu(x)| = +����� +cn +Rn +� +BR(x) +∂iu +����� = +����� +cn +Rn +� +∂BR(x) +u(y) yi +|y| dy +����� ≤ C +Rn +� +∂BR(x) +|u|. +Thus, using that |u| ≤ M in Rn, we find +|∂iu(x)| ≤ cn +Rn |∂BR(x)|M += cn +Rn |∂B1|Rn−1M = cnM +R +→ 0, +as +R → ∞. +Therefore, ∂iu(x) = 0 for all x ∈ Rn, and u is constant. +□ +More generally, one can even prove a classification result for functions +with polynomial growth. Here, for γ ∈ R, ⌊γ⌋ denotes the floor function, +that is, the largest integer less or equal to γ. +Proposition 1.19 (Liouville’s theorem with growth). Assume that u is a +solution of ∆u = 0 in Rn satisfying |u(x)| ≤ C(1+|x|γ) for all x ∈ Rn, with +γ > 0. Then, u is a polynomial of degree at most ⌊γ⌋. +Proof. Let us define uR(x) := u(Rx), and notice that ∆uR = 0 in Rn. +From Corollary 1.12 and the growth assumption +Rk∥Dku∥L∞(BR/2) = ∥DkuR∥L∞(B1/2) +≤ Ck∥uR∥L∞(B1) = Ck∥u∥L∞(BR) ≤ CkRγ. +In particular, if k = ⌊γ⌋ + 1, +∥Dku∥L∞(BR/2) ≤ CkRγ−k → 0 +as +R → ∞. +That is, Dku ≡ 0 in Rn, and u is a polynomial of degree k − 1 = ⌊γ⌋. +□ + +— DRAFT — +1.2. A review on the Laplace equation +19 +u +x◦ +y◦ +v +w +Figure 1.1. v touches u from above at x◦, w touches u from above at y◦. +Existence of solutions: comparison principle. We saw that one way to +prove existence of solutions to the Dirichlet problem for the Laplacian is by +using energy methods. With such approach, one proves in fact the existence +of a weak solution u ∈ H1(Ω). +Now, we will see an alternative way to construct solutions: via the com- +parison principle. With this method, one can show the existence of a vis- +cosity solution u ∈ C(Ω). +For the Laplace equation, these solutions (weak or viscosity) can then +be proved to be C∞(Ω), and thus they coincide. +We start by giving the definition of sub- and superharmonicity in the +viscosity sense. It is important to remark that in such definition the function +u is only required to be continuous. +Definition 1.20. A function u ∈ C(Ω) is subharmonic (in the viscosity +sense) if for every function v ∈ C2 such that v touches u from above at +x◦ ∈ Ω (that is, v ≥ u in Ω and v(x◦) = u(x◦)), we have ∆v(x◦) ≥ 0. See +Figure 1.1. +The definition of superharmonicity for u ∈ C(Ω) is analogous (touching +from below and with ∆v(x◦) ≤ 0). +A function u ∈ C(Ω) is harmonic if it is both sub- and superharmonic +in the above viscosity sense. +This definition obviously coincides with the one we know in case u ∈ C2. +However, it allows non-C2 functions u, for example u(x) = |x| is subhar- +monic and −|x| is superharmonic. +A useful property of viscosity sub-/supersolutions is the following. + +— DRAFT — +20 +1. Overview and Preliminaries +u1 +u2 +max{u1, u2} +Figure 1.2. The maximum of two functions u1 and u2. +Proposition 1.21. The maximum of two subharmonic functions is also +subharmonic. That is, if u1, u2 ∈ C(Ω) are subharmonic, then the function +v := max{u1, u2} is subharmonic as well. See Figure 1.2. +Similarly, the minimum of two superharmonic functions is superhar- +monic. +The proof follows easily from Definition 1.20 above, and it is left as an +exercise to the reader. +Moreover, we also have the following: +Proposition 1.22. Let Ω ⊂ Rn be a bounded domain, and assume that +u ∈ C(Ω) satisfies, in the viscosity sense, +� −∆u +≥ +0 +in Ω +u +≥ +0 +on ∂Ω. +Then, u ≥ 0 in Ω. +Proof. After a rescaling, we may assume Ω ⊂ B1. +Assume by contradiction that u has a negative minimum in Ω. Then, +since u ≥ 0 on ∂Ω, we have minΩ u = −δ, with δ > 0, and the minimum is +achieved in Ω. +Let us now consider 0 < ε < δ, and v(x) := −κ + ε(|x|2 − 1), with κ > 0 +(that is, a sufficiently flat paraboloid). +Now, notice that u − v > 0 on ∂Ω, and that we can choose κ > 0 so +that minΩ(u − v) = 0. That is, we can slide the paraboloid from below the +solution u until we touch it, by assumption, at an interior point. Thus, there +exists x◦ ∈ Ω such that u(x◦) − v(x◦) = minΩ(u − v) = 0. Therefore, with +such choice of κ, the function v touches u from below at x◦ ∈ Ω, and hence, +by definition of viscosity solution, we must have +∆v(x◦) ≤ 0. +However, a direct computation gives ∆v ≡ 2nε > 0 in Ω, a contradiction. +□ +Thanks to these two propositions, the existence of a (viscosity) solution +to the Dirichlet problem can be shown as follows. + +— DRAFT — +1.3. Probabilistic interpretation of harmonic functions +21 +Let +Sg := +� +v ∈ C(Ω) : v is subharmonic, and v ≤ g on ∂Ω +� +, +and define the pointwise supremum +u(x) := sup +v∈Sg +v(x). +Then, it can be shown that, if Ω is regular and g is continuous, then u ∈ +C(Ω), and ∆u = 0 in Ω, with u = g on ∂Ω. This is the so-called Perron +method. We refer to [HL97] for a complete description of the method in +case of the Laplace operator. +In Chapter 3 we will study the existence of viscosity solutions in the +more general setting of fully nonlinear elliptic equations. +Short summary on existence of solutions. We have two completely dif- +ferent ways to construct solutions: by energy methods; or by the maximum +(or comparison) principle. +In the first case, the constructed solution belongs to H1(Ω), in the second +case to C(Ω). In any case, one can then prove that u ∈ C∞(Ω)∩C(Ω) — as +long as Ω and g are regular enough — and therefore u solves the Dirichlet +problem in the usual sense. +1.3. Probabilistic interpretation of harmonic functions +To end this introductory chapter, we give a well-known probabilistic inter- +pretation of harmonic functions. The discussion will be mostly heuristic, +just to give an intuition on the Laplace equation in terms of stochastic pro- +cesses. We refer to Appendix C for further probabilistic interpretations for +fully nonlinear equations and for the obstacle problem. +Recall that the Brownian motion is a stochastic process Xt, t ≥ 0, +satisfying the following properties: +(1) X0 = 0 almost surely. +(2) Xt has no memory (is independent of the past, or it has indepen- +dent increments). +(3) Xt has stationary increments: Xt+s − Xs is equal in distribution +to Xt. +(4) Xt has continuous paths (t �→ Xt is continuous) almost surely. +(5) Xt is isotropic, i.e., it is rotationally symmetric in distribution. +The previous properties actually determine the stochastic process Xt up to +a multiplicative constant. Another important property of Brownian motion +is that it is scale invariant, i.e., + +— DRAFT — +22 +1. Overview and Preliminaries +x +z +∂Ω +Figure 1.3. A stochastic process Xx +t defined in Ω starting at x until it +hits the first point on the boundary z ∈ ∂Ω. +(6) r−1Xr2t equals Xt in distribution, for any r > 0. +As we will see next, there is a strong connection between the Brownian +motion and the Laplace operator. +Expected payoff. Given a regular domain Ω ⊂ Rn, and a Brownian motion +Xx +t starting at x (i.e., Xx +t := x+Xt), we play the following stochastic game: +When the process Xx +t hits the boundary ∂Ω for the first time we get a payoff +g(z), depending on the hitting point z ∈ ∂Ω. (See Figure 1.3.) +We then ask ourselves: +What is the expected payoff? +To answer this question, we define +τ := first hitting time of Xx +t , +u(x) := E [g(Xx +τ )] +(value function). +The value of u(x) is, by definition, the answer to the question above. Namely, +it is the expected value of g at the first point where Xx +t hits the boundary +∂Ω. +To find u(x), we try to relate it with values of u(y) for y ̸= x. Then, we +will see that this yields a PDE for u, and by solving it we can find u(x). +Indeed, let us consider a ball Br(x) ⊂ Ω, with r > 0. For any such ball, +we know that the process Xx +t will hit (before reaching ∂Ω, by property (4)) + +— DRAFT — +1.3. Probabilistic interpretation of harmonic functions +23 +some point on ∂Br(x), and moreover any point on ∂Br(x) will be hit with +the same probability. This is because the process is rotationally symmetric +in distribution, (5). +Since the process has no memory, (2), and stationary increments, (3), +this means that +(1.21) +u(x) = +� +∂Br(x) +u(y)dy. +Heuristically, this is because when the process hits the boundary ∂Br(x) at +a point y, it simply starts again the game from such point y. But because all +points y ∈ ∂Br(x) are reached for the first time with the same probability, +then (1.21) holds. +Now, since this can be done for every x ∈ Ω and r > 0, we deduce that +u(x) satisfies the mean value property, and therefore it is harmonic, ∆u = 0 +in Ω, (1.11). +Moreover, since we also know that u = g on ∂Ω (since when we hit the +boundary we get the payoff g surely), then u must be the unique solution of +� ∆u += +0 +in Ω +u += +g +on ∂Ω. +We refer to [Law10] for a nice introduction to this topic. +Expected hitting time. A similar stochastic problem is the following. +Given a smooth domain Ω ⊂ Rn, and a Brownian motion Xx +t , we ask: +What is the expected first time at which Xx +t will hit ∂Ω ? +To answer this question, we argue as before, using that the process must +first hit the boundary of balls Br(x) ⊂ Ω. Indeed, we first denote by u(x) +the expected hitting time that we are looking for. Then, for any such ball we +have that the process Xx +t will hit (before reaching ∂Ω) some point on ∂Br(x), +and moreover any point y ∈ ∂Br(x) will be hit with the same probability. +Thus, the total expected time u(x) will be the expected time it takes to hit +∂Br(x) for the first time, plus the expected time when we start from the +corresponding point y ∈ ∂Br(x), which is u(y). In other words, we have +u(x) = T(r) + +� +∂Br(x) +u(y)dy. +Here, T(r) is the expected first time at which Xx +t hits ∂Br(x) — which +clearly depends only on r and n. +Now, using the scale-invariance property of the Brownian motion, i.e. +r−1Xr2t ∼ Xt, we see that T(r) = T(1)r2 = c1r2 for some constant c1 > 0. + +— DRAFT — +24 +1. Overview and Preliminaries +Thus, we have +u(x) = c1r2 + +� +∂Br(x) +u(y)dy, +and by rearranging terms we find +− 1 +r2 +� � +Br(x) +u(y)dy − u(x) +� += c1. +Finally, taking r → 0 and using (1.11), we deduce that −∆u = c2, for some +constant c2 > 0. Since we clearly have u = 0 on ∂Ω, the expected hitting +time u(x) is the unique solution of the problem +� −∆u += +c2 +in Ω +u += +0 +on ∂Ω. +By considering a non-homogeneous medium (in which it takes more time +to move in some regions than others), the same argument leads to the prob- +lem with a right-hand side +� −∆u += +f(x) +in Ω +u += +0 +on ∂Ω, +with f ≥ 0. + +— DRAFT — +Chapter 2 +Linear elliptic PDE +In this chapter we will study linear elliptic PDEs of the type +(2.1) +tr +� +A(x)D2u(x) +� += +n +� +i,j=1 +aij(x)∂iju = f(x) +in +Ω ⊂ Rn, +as well as +(2.2) +div +� +A(x)∇u(x) +� += +n +� +i,j=1 +∂i +� +aij(x)∂ju(x) +� += f(x) +in +Ω ⊂ Rn. +These are elliptic PDEs in non-divergence and divergence form, respectively. +The coefficients (aij(x))ij and the right-hand side f(x) satisfy appropri- +ate regularity assumptions. In addition, we will assume that the coefficient +matrix A(x) = (aij(x))ij satisfies the uniform ellipticity condition +0 < λ Id ≤ (aij(x))ij ≤ Λ Id, +for some ellipticity constants 0 < λ ≤ Λ < ∞. (For two matrices A, B ∈ +Mn, we say A ≥ B if the matrix A − B is positive semi-definite.) +We will show that, under appropriate regularity assumptions on A(x), +solutions u to (2.1) “gain two derivatives” with respect to f and the coef- +ficients A(x). On the other hand, for the divergence-form equation, (2.2), +we expect solutions to “gain one derivative” with respect to the coefficients +A(x). +In order to do that, we will use perturbative methods, by “freezing” +the coefficients around a certain point and studying the constant coefficient +equation first. After a change of variables, one can transform the constant +coefficient equation into the most ubiquitous and simple elliptic equation: +25 + +— DRAFT — +26 +2. Linear elliptic PDE +Laplace’s equation, where (aij(x))ij is the identity. +Thus, we will begin +the chapter by studying properties of Laplace’s equation such as Harnack’s +inequality and the H¨older regularity with bounded right-hand side. After +that, we proceed by showing Schauder estimates for the Laplacian to con- +tinue with the main theorems of the current chapter: Schauder estimates +for (2.1) and (2.2). +We finish the chapter by studying equations of the type (2.1) and (2.2) +with continuous coefficients. In this case we do not gain two (resp. one) +derivatives, and instead we lose an arbitrarily small H¨older exponent of +regularity. +Equations in non-divergence and divergence form will become particu- +larly useful in Chapters 3 and 4 in the context of nonlinear variational PDEs +and fully nonlinear elliptic PDEs. +For both equations in non-divergence and divergence form, we establish +a priori estimates. That is, rather than proving that the solution is regular, +we show that if the solution is regular, then one can actually estimate the +norm of respectively two and one derivative higher in terms of the H¨older +norms of the coefficients (aij(x))ij and the right-hand side f. This is enough +for the application to nonlinear equations in Chapters 3 and 4. +When the operator is the Laplacian, thanks to the a priori estimates, +and by means of an approximation argument, we show that weak solutions +are in fact smooth. For more general elliptic operators, a priori estimates +together with the continuity method yield the existence of regular solutions. +We refer the reader to [GT77] for such an approach. +2.1. Harnack’s inequality +We start this chapter with one of the most basic estimates for harmonic +functions. It essentially gives a kind of “maximum principle in quantitative +form”. +We will usually write that u ∈ H1 is harmonic, meaning in the weak +sense. Recall from the introduction, however, that as soon as a function is +harmonic, it is immediately C∞. +Theorem 2.1 (Harnack’s inequality). Assume u ∈ H1(B1) is a non-negative, +harmonic function in B1. Then the infimum and the supremum of u are +comparable in B1/2. That is, +� ∆u += +0 +in B1 +u +≥ +0 +in B1 +⇒ +sup +B1/2 +u ≤ C inf +B1/2 +u, +for some constant C depending only on n. + +— DRAFT — +2.1. Harnack’s inequality +27 +∂B1 +∂B1 +∂B1/2 +∂B1/2 +O +1 +C +1 +u +Figure 2.1. Graphic representation of Harnack’s inequality for a har- +monic function u > 0 such that supB1 u = 1. +Proof. This can be proved by the mean value property. Alternatively, we +can also use the Poisson kernel representation, +u(x) = cn +� +∂B1 +(1 − |x|2)u(z) +|x − z|n +dz. +Notice that, for any x ∈ B1/2 and z ∈ ∂B1, we have 2−n ≤ |x−z|n ≤ (3/2)n +and 3/4 ≤ 1 − |x|2 ≤ 1. Thus, since u ≥ 0 in B1, +C−1 +� +∂B1 +u(z) dz ≤ u(x) ≤ C +� +∂B1 +u(z) dz, +for all +x ∈ B1/2, +for some dimensional constant C. In particular, for any x1, x2 ∈ B1/2 we +have that u(x1) ≤ C2u(x2). Taking the infimum for x2 ∈ B1/2 and the +supremum for x1 ∈ B1/2, we reach that supB1/2 u ≤ ˜C infB1/2 u, for some +dimensional constant ˜C, as desired. +□ +Remark 2.2. This inequality says that, if u ≥ 0 in B1, then not only +u > 0 in B1/2 (strong maximum principle), but also we get quantitative +information: u ≥ C−1 supB1/2 u in B1/2, for some constant C depending +only on n. See Figure 2.1. +Notice that there is nothing special about B1/2. We can obtain a similar +inequality in Bρ with ρ < 1, but the constant C would depend on ρ as well. +Indeed, repeating the previous argument, one gets that if ∆u = 0 and u ≥ 0 +in B1, then +(2.3) +sup +Bρ +u ≤ +C +(1 − ρ)n inf +Bρ u, +for some C depending only on n, and where ρ ∈ (0, 1). + +— DRAFT — +28 +2. Linear elliptic PDE +From Harnack’s inequality, we deduce the oscillation decay for harmonic +functions. That is, the oscillation of a harmonic function is reduced (quan- +titatively) in smaller domains. +The oscillation in a domain Ω is defined +as +osc +Ω u := sup +Ω +u − inf +Ω u. +We remark that the following lemma is valid for all harmonic functions, not +necessarily positive. +Corollary 2.3 (Oscillation decay). Let u ∈ H1(B1) be a harmonic function +in B1, i.e. ∆u = 0 in B1. Then +osc +B1/2 +u ≤ (1 − θ) osc +B1 u +for some small θ > 0 depending only on n. +Proof. Let +w(x) := u(x) − inf +B1 u, +which satisfies w ≥ 0 in B1 and oscB1/2 w = oscB1/2 u. Since ∆w = 0 in B1, +we get by Harnack’s inequality +sup +B1/2 +w ≤ C inf +B1/2 +w, +so that +osc +B1/2 +u = osc +B1/2 +w = sup +B1/2 +w − inf +B1/2 +w ≤ +� +1 − 1 +C +� +sup +B1/2 +w ≤ +� +1 − 1 +C +� +sup +B1 +w. +Now notice that supB1 w = oscB1 u, and we are done. +□ +Remark 2.4 (Alternative proof of Corollary 2.3). Alternatively, we can +rewrite the previous proof of Corollary 2.3 by taking advantage of the in- +variance of the estimate. +Indeed, the function u−infB1 u is non-negative and harmonic. Since the +estimate we want to prove is invariant under addition and multiplication by +constants, we may assume that infB1 u = 0 and supB1 u = 1. Let θ := +1 +C+1, +where C is the constant in Harnack’s inequality, Theorem 2.1. Now we have +two options: +• If supB1/2 u ≤ 1 − θ, we are done, +• If supB1/2 u ≥ 1 − θ we use Harnack’s inequality to get +inf +B1/2 +u ≥ 1 +C (1 − θ) ≥ θ. +In any case, we get oscB1/2 u ≤ 1 − θ, so we are done. + +— DRAFT — +2.1. Harnack’s inequality +29 +Remark 2.5. We have proved that Harnack’s inequality implies the oscil- +lation decay. This is always true, we did not use the fact that we are dealing +with harmonic functions. In general, we have +� Harnack’s +inequality +� +=⇒ +� Oscillation +decay +� +=⇒ +� +H¨older +regularity +� +Harnack’s inequality and the oscillation decay are scale invariant. That +is, the following corollary holds: +Corollary 2.6 (Rescaled versions). Let u ∈ H1(Br) be such that ∆u = 0 +in Br. Then +• (Harnack’s inequality) If u ≥ 0 in Br, then +sup +Br/2 +u ≤ C inf +Br/2 +u, +for some C depending only on n. +• (Oscillation decay) One has +osc +Br/2 +u ≤ (1 − θ) osc +Br u, +for some small θ > 0 depending only on n. +Proof. Define ˜u(x) := u(rx), which fulfills ∆˜u = 0 in B1 and therefore +sup +Br/2 +u = sup +B1/2 +˜u ≤ C inf +B1/2 +˜u = C inf +Br/2 +u, +by Theorem 2.1. Similarly, +osc +Br/2 +u = osc +B1/2 +˜u ≤ (1 − θ) osc +B1 ˜u = (1 − θ) osc +Br u +by Corollary 2.3. +□ +A standard consequence of the quantitative oscillation decay proved +above is the H¨older regularity of solutions. +Corollary 2.7 (H¨older regularity). Let u ∈ H1(B1) ∩ L∞(B1) be such that +∆u = 0 in B1. Then +∥u∥C0,α(B1/2) ≤ C∥u∥L∞(B1) +for some constants α > 0 and C depending only on n. +Proof. If we denote ˜u := (2∥u∥L∞(B1))−1u, then ˜u ∈ H1 ∩ L∞(B1) fulfills +∆˜u = 0 in B1 and ∥˜u∥L∞(B1) ≤ 1 +2. If we show ∥˜u∥C0,α(B1/2) ≤ C, then the +result will follow. + +— DRAFT — +30 +2. Linear elliptic PDE +O +0 +1 +B1 +B 1 +2 +B 1 +4 +B 1 +8 +� Oscillation +decay +� +⇓ +� +H¨older +regularity +� +Figure 2.2. Graphical representation of the fact that oscillation decay- +type lemmas imply H¨older regularity. +Thus, dividing u by a constant if necessary, we may assume that ∥u∥L∞(B1) ≤ +1 +2. We need to prove that +|u(x) − u(y)| ≤ C|x − y|α +for all +x, y ∈ B1/2, +for some small α > 0. We do it at y = 0 for simplicity. +Let x ∈ B1/2 and let k ∈ N be such that x ∈ B2−k \ B2−k−1. Then, +|u(x) − u(0)| ≤ osc +B2−k u ≤ (1 − θ)k osc +B1 u ≤ (1 − θ)k = 2−αk, +with α = − log2(1 − θ). (Notice that we are using Corollary 2.6 k-times, +where the constant θ is independent from the radius of the oscillation decay.) +Now, since 2−k ≤ 2|x|, we find +|u(x) − u(0)| ≤ (2|x|)α ≤ C|x|α, +as desired. See Figure 2.2 for a graphical representation of this proof. +□ +Finally, another important consequence of Harnack’s inequality is the +Liouville theorem for non-negative harmonic functions. +Corollary 2.8. Let u be a non-negative harmonic function, that is, u ≥ 0 +and ∆u = 0 in Rn. Then, u is constant. +Proof. Let +v = u − inf +Rn u, +where infRn u is well-defined and finite since u ≥ 0. Then, thanks to Har- +nack’s inequality in arbitrary balls from Corollary 2.6, we get that for any + +— DRAFT — +2.1. Harnack’s inequality +31 +R > 0, +sup +BR +v ≤ C inf +BR v = C +� +inf +BR u − inf +Rn u +� +→ 0, +as R → ∞. That is, supRn u = infRn u and therefore u is constant in Rn. +□ +Of course, the previous result also holds if u ≥ −M in Rn, for some +constant M, since then u + M is non-negative and harmonic. +Harnack’s inequality with a right-hand side. We can prove a Harnack +inequality for equations with a right-hand side, that is, when the Laplacian +is not necessarily zero, ∆u = f. Again, we will be dealing with functions +u ∈ H1, so that we have to understand the equation ∆u = f in the weak +sense. +Theorem 2.9. Let f ∈ L∞(B1), and u ∈ H1(B1). Then, +� ∆u += +f +in B1 +u +≥ +0 +in B1 +⇒ +sup +B1/2 +u ≤ C +� +inf +B1/2 +u + ∥f∥L∞(B1) +� +, +for some C depending only on n. +Proof. We express u as u = v + w with +� ∆v += +0 +in B1 +v += +u +on ∂B1 +� ∆w += +f +in B1 +w += +0 +on ∂B1. +Then, we have +sup +B1/2 +v ≤ C inf +B1/2 +v +and +sup +B1 +w ≤ C∥f∥L∞(B1) +by Theorem 2.1 and Lemma 1.14. Thus, +sup +B1/2 +u ≤ sup +B1/2 +v + C∥f∥L∞(B1) +≤ C inf +B1/2 +v + C∥f∥L∞(B1) ≤ C +� +inf +B1/2 +u + ∥f∥L∞(B1) +� +, +where we are taking a larger constant if necessary. Notice that we have also +used here that v ≤ u + C∥f∥L∞(B1). +□ +Thus, as before, we also get an oscillation decay, but now involving an +error term of size ∥f∥L∞. +Corollary 2.10. Let f ∈ L∞(B1) and u ∈ H1(B1). If ∆u = f in B1 and +f ∈ L∞(B1), then +osc +B1/2 +u ≤ (1 − θ) osc +B1 u + 2∥f∥L∞(B1), +for some θ > 0 depending only on n. + +— DRAFT — +32 +2. Linear elliptic PDE +Proof. The proof is the same as in the case f ≡ 0, see the proof of Corol- +lary 2.3. +□ +Remark 2.11. Now, with the right-hand side f = f(x), the equation ∆u = +f and Harnack’s inequality are not invariant under rescalings in x. In fact, +as we zoom-in, the right-hand side gets smaller! +Namely, if ∆u = f in Br, then ˜u(x) := u(rx) satisfies ∆˜u(x) = r2f(rx) +in B1 so that +sup +B1/2 +˜u ≤ C +� +inf +B1/2 +˜u + 2r2∥f∥L∞(Br) +� +, +and therefore +sup +Br/2 +u ≤ C +� +inf +Br/2 +u + 2r2∥f∥L∞(Br) +� +for some constant C depending only on n. +Even if the previous oscillation decay contains an error depending on f, +it is enough to show H¨older regularity of the solution. +Corollary 2.12 (H¨older regularity). Let f ∈ L∞(B1) and u ∈ H1∩L∞(B1). +If ∆u = f in B1, then +∥u∥C0,α(B1/2) ≤ C +� +∥u∥L∞(B1) + ∥f∥L∞(B1) +� +, +for some constants α > 0 and C depending only on n. +Proof. If we denote ˜u := (2∥u∥L∞(B1)+2∥f∥L∞(B1))−1u, then ˜u ∈ H1(B1)∩ +L∞(B1) fulfills ∆˜u = ˜f in B1 with ∥˜u∥L∞(B1) ≤ 1 +2 and ∥ ˜f∥L∞(B1) ≤ 1 +2. If +we show that ∥˜u∥C0,α(B1/2) ≤ C, then the result will follow. +Thus, after dividing u by a constant if necessary, we may assume that +∥u∥L∞(B1) ≤ 1 +2 and ∥f∥L∞(B1) ≤ 1 +2. +As in Corollary 2.7 we want to prove that |u(x◦)−u(0)| ≤ C|x◦|α for all +x◦ ∈ B1/2 and for some constant C depending only on n. +Let us show that it is enough to prove that +(2.4) +osc +B2−k u ≤ C2−αk +for all k ∈ N, k ≥ k◦, +for some α > 0, C, and for some fixed k◦, all three depending only on +n. Indeed, let k ∈ N be such that x ∈ B2−k \ B2−k−1. If k < k◦, then +|x| ≥ 2−k◦−1 and +|u(x) − u(0)| ≤ osc +B1 u ≤ 1 ≤ 2α(k◦+1)|x|α. +On the other hand, if k ≥ k◦, by (2.4) +|u(x) − u(0)| ≤ osc +B2−k u ≤ C2−αk ≤ C(2|x|)α, + +— DRAFT — +2.2. Schauder estimates for the Laplacian +33 +where in the last inequality we used that |x| ≥ 2−k−1. +Thus, it will be +enough to show (2.4). +Let k ∈ N and k ≥ k◦ for some k◦ to be chosen, and define +˜u(x) := u(rx), +r = 2−k+1. +Then ˜u satisfies ∆˜u = r2f(rx) in B1 (in fact, in B2k−1), and thus, by +Corollary 2.10 +osc +B1/2 +˜u ≤ (1 − θ) osc +B1 ˜u + 2r2∥f∥L∞(Br). +Since oscB1 ˜u = oscB2−k+1 u and ∥f∥L∞(Br) ≤ 1 +2, we find +osc +B2−k u ≤ (1 − θ) +osc +B2−k+1 u + 4−k+2. +Now, take k◦ ∈ N large enough so that 4−k◦+1 ≤ θ +2. Then, +osc +B2−k u ≤ (1 − θ) +osc +B2−k+1 u + θ +24k◦−k. +It is immediate to check by induction that this yields +osc +B2−k+1 u ≤ 2α(k◦−k), +for all k ∈ N. +Indeed, the induction step follows as +osc +B2−k u ≤ (1 − θ)2α(k◦−k) + θ +24k◦−k ≤ (1 − θ)2α(k◦−k) + θ +22α(k◦−k) += +� +1 − θ +2 +� +2α(k◦−k) = 2α(k◦−k−1) +if +1 − θ +2 = 2−α. +Thus, (2.4) holds with C = 2αk◦. +□ +Summarizing, we have checked that Harnack’s inequality for harmonic +functions yields the H¨older regularity of solutions, even with a right-hand +side f ∈ L∞. +This is a general fact, and holds for other types of elliptic equations, too. +2.2. Schauder estimates for the Laplacian +We now want to establish sharp results for the equation +∆u = f(x) +in +B1 +(or in Ω ⊂ Rn) . +This will serve as an introduction for the more general case of equations in +non-divergence and divergence form. +The philosophy is that the sharp results should state that “u is two +derivatives more regular than f(x)”. +The main known results in that directions are the following: + +— DRAFT — +34 +2. Linear elliptic PDE +(a) Schauder estimates. If f ∈ C0,α then u ∈ C2,α, for α ∈ (0, 1). +(b) Calder´on–Zygmund estimates. If f ∈ Lp then u ∈ W 2,p, for p ∈ +(1, ∞). +(c) When α is an integer, or when p ∈ {1, ∞}, the above results do not +hold. For example, if f ∈ C0, it is not true in general that u ∈ C2, +not even C1,1. (In that case, u ∈ C1,1−ε for all ε > 0, and u ∈ W 2,p +for all p < ∞.) +Two counterexamples. Let us provide two counterexamples to show that +Schauder and Calder´on–Zygmund estimates in general do not hold for the +limiting values, α = 0 and p = 1 or p = ∞. +We start with an example of a function u whose Laplacian is bounded +(∆u ∈ L∞), but whose second derivatives are not bounded (u /∈ W 2,∞). +Thus, we give a counterexample to Calder´on–Zygmund estimates for p = ∞. +Let +u(x, y) = (x2 − y2) log(x2 + y2) +in +R2. +Then, +∂xxu = 2 log(x2 + y2) + +8x2 +x2 + y2 − 2 +�x2 − y2 +x2 + y2 +�2 +, +∂yyu = −2 log(x2 + y2) − +8y2 +x2 + y2 + 2 +�x2 − y2 +x2 + y2 +�2 +, +that is, both ∂xxu and ∂yyu are unbounded, and u /∈ W 2,∞. However, +∆u = ∂xxu + ∂yyu = 8x2 − y2 +x2 + y2 ∈ L∞(R2). +One can modify such construction in order to make ∆u continuous and +u /∈ C1,1, thus giving a counterexample to Schauder estimates for α = 0, by +taking u(x, y) = (x2 − y2) log | log(x2 + y2)|. (However, recall that Schauder +estimates tell us that this is not possible if ∆u is H¨older continuous (C0,α).) +Let us now provide a counterexample for Calder´on–Zygmund estimates +when p = 1. The fact that the estimate does not hold can be seen by taking +smooth approximations of the Dirac delta (with constant integral) as right- +hand side, so that the solution converges to the fundamental solution, which +is not in W 2,1. +Let us, however, give a specific example of a function u whose Laplacian +is integrable (∆u ∈ L1) but whose second derivatives are not (u /∈ W 2,1). +Let +u(x, y) = log log +1 +x2 + y2 = log log r−2 +in +R2, + +— DRAFT — +2.2. Schauder estimates for the Laplacian +35 +where we are using polar coordinates and denote r2 := x2 + y2. +Since +u = u(r) and ur = +1 +r log r we have that +∆u = urr + 1 +rur = − log r + 1 +r2(log r)2 + +1 +r2 log r = − +1 +r2(log r)2 ∈ L1(B1/2), +since +� +B1/2 ∆u = −2π +� 1/2 +0 +dr +r(log r)2 < ∞. On the other hand, a direct com- +putation gives that ∂xxu (and ∂yyu) are not absolutely integrable around +the origin, and thus u /∈ W 2,1. (Alternatively, since one has the embed- +ding W 2,1(R2) ⊂ L∞(R2) [Bre11, Corollary 9.13] and u /∈ L∞, we deduce +u /∈ W 2,1). +A similar counterexample can be built in any dimension n ≥ 2, by taking +as function u an appropriate primitive of r1−n +log r . +In this book we focus our attention on proving (a) Schauder estimates, +but not (b) Calder´on–Zygmund estimates. Later in the book we will see +applications of Schauder-type estimates to nonlinear equations. +Remark 2.13 (Calder´on-Zygmund estimates for p = 2). In the case p = 2, +one can prove a priori Calder´on-Zygmund estimates with a simple compu- +tation. That is, let u, f ∈ C∞(B1), be such that +∆u = f +in +B1. +Then, +(2.5) +∥u∥W 2,2(B1/2) ≤ C +� +∥u∥L2(B1) + ∥f∥L2(B1) +� +for some constant C depending only on n. Indeed, let w := ηu for some +fixed test function η ∈ C∞ +c (B1) such that η ≡ 1 in B1/2, η ≡ 0 in B1 \ B3/4 +and η ≥ 0. Then, integrating by parts gives +∥D2u∥L2(B1/2) = +n +� +i,j=1 +� +B1/2 +|D2 +iju|2 ≤ +n +� +i,j=1 +� +B1 +|D2 +ijw|2 += − +n +� +i,j=1 +� +B1 +(Diijw)(Djw) = +n +� +i,j=1 +� +B1/2 +(Diiw)(Djjw) += +� +B1 +(∆w)2 ≤ C +� +B1 +� +u2 + (∆u)2 + |∇η|2|∇u|2� +, +where in the last equality we can take C = C′ supB1 +� +η2 + |∆η|2� +for some +dimensional constant C′. Then, again integrating by parts twice and using +2ab ≤ a2 + b2, we get +� +B1 +|∇η|2|∇u|2 = − +� +B1 +|∇η|2u∆u + +� +B1 +1 +2u2∆|∇η|2 ≤ ˜C +� +B1 +� +u2 + (∆u)2� +, +where ˜C = supB1 +� +|∇η|2 + ∆|∇η|2� +. + +— DRAFT — +36 +2. Linear elliptic PDE +This directly yields the result (2.5) for smooth functions u and f such +that ∆u = f. Arguing as in the proof of Corollary 2.16 below, the same +result also holds as long as u ∈ H1(B1) is a weak solution to ∆u = f in B1 +for f ∈ L2(B1) . +Proofs of Schauder estimates: some comments. There are various +proofs of Schauder estimates, mainly using: +(1) integral representation of solutions (fundamental solutions); +(2) energy considerations; +(3) comparison principle. +The most flexible approaches are (2) and (3). Here, we will see different +proofs of type (3). +The common traits in proofs of type (2)-(3) are their “perturbative char- +acter”, that is, that by zooming in around any point the equation gets closer +and closer to ∆u = constant, and thus (after subtracting a paraboloid) close +to ∆u = 0. Thus, the result can be proved by using the information that +we have on harmonic functions. +Let us start by stating the results we want to prove in this section: +Schauder estimates for the Laplacian. +Theorem 2.14 (Schauder estimates for the Laplacian). Let α ∈ (0, 1), and +let u ∈ C2,α(B1) satisfy +∆u = f +in B1, +with f ∈ C0,α(B1). Then +(2.6) +∥u∥C2,α(B1/2) ≤ C +� +∥u∥L∞(B1) + ∥f∥C0,α(B1) +� +. +The constant C depends only on α and the dimension n. +We will, in general, state our estimates in balls B1/2 and B1. By means +of a covering argument explained below, this allows us to obtain interior +regularity estimates in general domains. +Remark 2.15 (Covering argument). Let us assume that we have an esti- +mate, like the one in (2.6), but in a ball Br1 for some r1 ∈ (0, 1), which will +be typically very close to zero. Namely, we know that if ∆u = f in B1, then +(2.7) +∥u∥C2,α(Br1) ≤ C +� +∥u∥L∞(B1) + ∥f∥C0,α(B1) +� +. +Let us suppose that we are interested in finding an estimate for a bigger +ball, Br2 with r1 < r2 ∈ (0, 1), where r2 will be typically close to one. We +do that via a “covering argument”. (See Figure 2.3.) +That is, let us cover the ball Br2 with smaller balls Br(xi) such that +xi ∈ Br2 and r = (1 − r2)r1. We can do so with a finite number of balls, + +— DRAFT — +2.2. Schauder estimates for the Laplacian +37 +B1/4 +B1/2 +xi +Br(xi) +B4r(xi) +B1 +Figure 2.3. Graphical representation of the “covering argument” in +the case r1 = 1 +4, r2 = 1 +2, and r = 1 +8. +so that i ∈ {1, . . . , N}, for some N depending on r1, r2, and n. Notice that +Br/r1(xi) ⊂ B1. +We apply our estimate (2.7) (translated and rescaled) at each of these +balls Br/r1(xi) (we can do so, because ∆u = f in Br/r1(xi) ⊂ B1). Thus, +we obtain a bound for ∥u∥C2,α(Br(xi)) +∥u∥C2,α(Br(xi)) ≤ C(r1, r2) +� +∥u∥L∞(Br/r1(xi)) + ∥f∥C0,α(Br/r1(xi)) +� +≤ C(r1, r2) +� +∥u∥L∞(B1) + ∥f∥C0,α(B1) +� +. +Now, since Br2 can be covered by a finite number of these balls, we obtain +∥u∥C2,α(Br2) ≤ +n +� +i=1 +∥u∥C2,α(Br(xi)) ≤ NC(r1, r2) +� +∥u∥L∞(B1) + ∥f∥C0,α(B1) +� +. +This is the type of bound we wanted, where the constant now also depends +on r1 and r2. +As a consequence of the “a priori” estimate for the Laplacian we will +show: +Corollary 2.16. Let u be any bounded weak solution to +∆u = f +in B1, + +— DRAFT — +38 +2. Linear elliptic PDE +with f ∈ C0,α(B1) for some α ∈ (0, 1). Then, u is in C2,α inside B1, and +the estimate (2.6) holds. +Furthermore, iterating the previous estimate we will establish the fol- +lowing. +Corollary 2.17 (Higher order regularity estimates). Let u be any bounded +weak solution to +∆u = f +in B1, +with f ∈ Ck,α(B1) for some α ∈ (0, 1), and k ∈ N. Then, u is in Ck+2,α +inside B1 and +∥u∥Ck+2,α(B1/2) ≤ C +� +∥u∥L∞(B1) + ∥f∥Ck,α(B1) +� +, +for some constant C that depends only on k, α, and the dimension n. +In case f ∈ L∞, we will prove the following. +Proposition 2.18. Let u be any solution to +∆u = f +in B1, +with f ∈ L∞(B1). Then, u is in C1,1−ε inside B1, for any ε > 0, with the +estimate +∥u∥C1,1−ε(B1/2) ≤ Cε +� +∥u∥L∞(B1) + ∥f∥L∞(B1) +� +for some constant Cε depending only on ε and n. +We will give two different proofs of Theorem 2.14. The first proof follows +a method introduced by Wang in [Wan06] and shows the a priori estimate +using a very much self-contained approach. For the second proof we use an +approach `a la Caffarelli from [Moo12, Caf89]. +Before doing so, let us observe the following: +• If ∆u = f ∈ L∞ then ˜u(x) := u(rx) solves ∆˜u = r2f(rx). In +other words, if |∆u| ≤ C, then |∆˜u| ≤ Cr2 (and if r is small, the +right-hand side becomes smaller and smaller). +• If ∆u = f ∈ C0,α, ˜u(x) = u(rx) − f(0) +2n |x|2 solves ∆˜u = r2(f(rx) − +f(0)), so that |∆˜u| ≤ Cr2+α in B1. This, by the comparison prin- +ciple, means that ˜u is “very close” to a harmonic function. +Let us now show that Corollary 2.16 holds assuming Theorem 2.14. This +follows by an approximation argument. +Proof of Corollary 2.16. We will deduce the result from Theorem 2.14. +Let u be any solution to ∆u = f in B1, with f ∈ C0,α(B1), and let η ∈ +C∞ +c (B1) be any smooth function with η ≥ 0 and +� +B1 η = 1. Let +ηε(x) := ε−nη +�x +ε +� +, + +— DRAFT — +2.2. Schauder estimates for the Laplacian +39 +which satisfies +� +Bε ηε = 1, ηε ∈ C∞ +c (Bε). Consider the convolution +uε(x) := u ∗ ηε(x) = +� +Bε +u(x − y)ηε(y) dy, +which is C∞ and satisfies +∆uε = f ∗ ηε =: fε +in +B1−ε. +(Notice that for smooth functions, derivatives and convolutions commute; +the same can be done for weak derivatives.) Since uε ∈ C∞, we can use +Theorem 2.14 to get +∥uε∥C2,α(B1/2) ≤ C +� +∥uε∥L∞(B1−ε) + ∥fε∥C0,α(B1−ε) +� +, +where we are also using the covering argument in Remark 2.15 to write it +in a ball B1−ε in the right-hand side. Observe now that for any x, y ∈ B1−ε +|uε(x)| ≤ +� +Bε +|u(x − z)|ηε(z) dy ≤ ∥u∥L∞(B1) +� +Bε +ηε(z) dz = ∥u∥L∞(B1), +and +|fε(x) − fε(y)| ≤ +� +Bε +|f(x − z) − f(y − z)|ηε(z) dz = [f]C0,α(B1)|x − y|α. +From here, we deduce ∥uε∥L∞(B1−ε) ≤ ∥u∥L∞(B1) and ∥fε∥C0,α(B1−ε) ≤ +∥f∥C0,α(B1). Thus, the sequence uε is uniformly bounded in C2,α(B1/2), +∥uε∥C2,α(B1/2) ≤ C +� +∥u∥L∞(B1) + ∥f∥C0,α(B1) +� +. +Moreover, since u is continuous (see Corollary 2.12), arguing as before we +get ∥uε − u∥L∞(B1) → 0 as ε ↓ 0, so that uε → u uniformly. We can use +(H8) from Chapter 1 to deduce that u ∈ C2,α and +∥u∥C2,α(B1/2) ≤ C +� +∥u∥L∞(B1) + ∥f∥C0,α(B1) +� +. +By a covering argument (see Remark 2.15) we can get a similar estimate in +any ball Bρ with ρ < 1, +∥u∥C2,α(Bρ) ≤ Cρ +� +∥u∥L∞(B1) + ∥f∥C0,α(B1) +� +, +where now the constant Cρ depends also on ρ, and in fact, blows up when +ρ ↑ 1. In any case, we have that u ∈ C2,α(Bρ) for any ρ < 1, i.e., u is in +C2,α inside B1. +□ +The previous proof is an example of a recurring phenomenon when prov- +ing regularity estimates for PDEs. If one can get estimates of the kind +∥u∥C2,α ≤ C (∥u∥L∞ + ∥f∥C0,α) , +for all C∞ functions u, and with a constant C that depends only on α and +n (but independent of u and f), then, in general, the estimate holds as well +for all solutions u. Thus, if one wants to prove the higher-order regularity + +— DRAFT — +40 +2. Linear elliptic PDE +estimates from Corollary 2.17, it is enough to get a priori estimates in the +spirit of Theorem 2.14. +As a consequence, assuming that Theorem 2.14 +holds, we can prove Corollary 2.17. +Proof of Corollary 2.17. As mentioned above, we just need to show that +for any u ∈ C∞ such that ∆u = f, one has +(2.8) +∥u∥Ck+2,α(B1/2) ≤ C +� +∥u∥L∞(B1) + ∥f∥Ck,α(B1) +� +. +for some constant C depending only on n, α, and k; and then we are done +by a covering argument (see Remark 2.15). We prove it by induction on k, +and it follows applying the induction hypothesis to derivatives of u. Notice +that (2.8) deals with balls B1/2 and B1, but after a rescaling and covering +argument (see Remark 2.15), it could also be stated in balls B1/2 and B3/4 +(we will use it in this setting). +The base case, k = 0, already holds by Theorem 2.14. Let us now assume +that (2.8) holds for k = m − 1, and we will show it for k = m. +In this case, let us differentiate ∆u = f to get ∆∂iu = ∂if, for i ∈ +{1, . . . , n}. Applying (2.8) for k = m − 1 to ∂iu in balls B1/2 and B3/4, we +get +∥∂iu∥Cm+1,α(B1/2) ≤ C +� +∥∂iu∥L∞(B3/4) + ∥∂if∥Cm−1,α(B3/4) +� +≤ C +� +∥u∥C2,α(B3/4) + ∥f∥Cm,α(B3/4) +� +. +Using now Theorem 2.14 in balls B3/4 and B1, +∥∂iu∥Cm+1,α(B1/2) ≤ C +� +∥u∥L∞(B1) + ∥f∥Cα(B1) + ∥f∥Cm,α(B3/4) +� +This, together with the basic estimate from Theorem 2.14 for ∆u = f, and +used for each i ∈ {1, . . . , n}, directly yields that (2.8) holds for k = m. +□ +Similarly, if one wants to prove regularity estimates in other contexts, it +is often enough to obtain the corresponding a priori estimate. For instance, +using an estimate that we prove later in the chapter (in the more general +context of non-divergence-form equations) we can immediately obtain also +the proof of Proposition 2.18. +Proof of Proposition 2.18. The proof is exactly the same as the proof +of Corollary 2.16 but using Proposition 2.31 from below instead of Theo- +rem 2.14. (Alternatively, see Remark 2.19.) +□ +Let us now provide the first proof of Theorem 2.14. The method used +here was introduced by Wang in [Wan06]. + +— DRAFT — +2.2. Schauder estimates for the Laplacian +41 +First proof of Theorem 2.14. We will prove that +|D2u(z) − D2u(y)| ≤ C|z − y|α � +∥u∥L∞(B1) + [f]C0,α(B1) +� +, +for all y, z ∈ B1/32. After a translation, we assume that y = 0, so that the +proof can be centered around 0. This will prove our theorem with estimates +in a ball B1/32, and the desired result in a ball of radius 1 +2 follows by a +covering argument. Moreover, after dividing the solution u by ∥u∥L∞(B1) + +[f]C0,α(B1) if necessary, we may assume that ∥u∥L∞(B1) ≤ 1 and [f]C0,α(B1) ≤ +1, and we just need to prove that for all z ∈ B1/16, +|D2u(z) − D2u(0)| ≤ C|z|α. +Throughout the proof, we will use the following basic estimates for har- +monic functions: +(2.9) +∆w = 0 +in +Br +⇒ +∥Dκw∥L∞(Br/2) ≤ Cr−κ∥w∥L∞(Br), +where C depends only on n and κ ∈ N. +(In fact, we will only use κ ∈ +{1, 2, 3}.) Such estimate follows by rescaling the estimate (1.16) — which +corresponds to the case r = 1. +We will also use the estimate +(2.10) +∆w = λ +in +Br +⇒ +∥D2w∥L∞(Br/2) ≤ C +� +r−2∥w∥L∞(Br) + |λ| +� +, +for some constant C depending only on n. This estimate follows from (2.9) +after subtracting +λ +2n|x|2. +For k = 0, 1, 2, . . . , let uk be the solution to +� ∆uk += +f(0) +in B2−k +uk += +u +on ∂B2−k. +Then, ∆(uk − u) = f(0) − f, and by the rescaled version of Lemma 1.14 +(2.11) +∥uk − u∥L∞(B2−k) ≤ C(2−k)2∥f(0) − f∥L∞(B2−k) ≤ C2−k(2+α), +where we are using that [f]C0,α(B1) ≤ 1. Hence, the triangle inequality yields +∥uk+1 − uk∥L∞(B2−k−1) ≤ C2−k(2+α). +Since uk+1 − uk is harmonic, we have +(2.12) +∥D2(uk+1 − uk)∥L∞(B2−k−2) ≤ C22(k+1)∥uk+1 − uk∥L∞(B2−k−1) ≤ C2−kα. +Now, notice that +(2.13) +D2u(0) = lim +k→∞ D2uk(0). + +— DRAFT — +42 +2. Linear elliptic PDE +Indeed, let ˜u(x) := u(0) + x · ∇u(0) + 1 +2x · D2u(0)x be the second order +expansion of u at 0. Then, since u ∈ C2,α, we have ∥˜u−u∥L∞(Br) ≤ Cr2+α = +o(r2). Using that ˜u − uk is harmonic together with (2.9) we deduce +|D2uk(0) − D2u(0)| ≤ ∥D2(uk − ˜u)∥L∞(B2−k−1) +≤ C22k∥uk − ˜u∥L∞(B2−k) += C22k∥u − ˜u∥L∞(∂B2−k) → 0 +as +k → ∞. +Now, for any point z near the origin, we have +|D2u(z) − D2u(0)| ≤ |D2uk(0) − D2u(0)| ++ |D2uk(0) − D2uk(z)| + |D2uk(z) − D2u(z)|. +For a given z ∈ B1/16, we choose k ∈ N such that +2−k−4 ≤ |z| ≤ 2−k−3. +Thanks to (2.12)-(2.13), and by the triangle inequality, we get +|D2uk(0) − D2u(0)| ≤ +∞ +� +j=k +|D2uj(0) − D2uj+1(0)| ≤ C +∞ +� +j=k +2−jα = C2−kα, +where we use that α ∈ (0, 1). +In order to estimate |D2u(z) − D2uk(z)|, the same argument can be +repeated around z instead of 0. That is, take solutions of ∆vj = f(z) in +B2−j(z) and vj = u on ∂B2−j(z). Then, +|D2uk(z) − D2u(z)| ≤ |D2uk(z) − D2vk(z)| + |D2vk(z) − D2u(z)|. +The second term above can be bounded by C2−kα arguing as before. For +the first term, we use (2.10) by noticing that ∆(uk − vk) = f(0) − f(z) in +B2−k ∩ B2−k(z) ⊃ B2−k−1(z) (recall |z| ≤ 2−k−3), so that, in B2−2−k(z) we +have +|D2uk(z) − D2vk(z)| ≤ ∥D2(uk − vk)∥L∞(B2−2−k(z)) +≤ C22k∥uk − vk∥L∞(B2−k−1(z)) + C|f(z) − f(0)| +≤ C22k∥uk − vk∥L∞(B2−k−1(z)) + C2−kα, +where we use, again, that |z| ≤ 2−k−3, and [f]C0,α(B1) ≤ 1 +Finally, from (2.11), we know that +∥uk − u∥L∞(B2−k−1(z)) ≤ ∥uk − u∥L∞(B2−k) ≤ C2−k(2+α), +and +∥u − vk∥L∞(B2−k−1(z)) ≤ C2−k(2+α), + +— DRAFT — +2.2. Schauder estimates for the Laplacian +43 +which gives +∥uk − vk∥L∞(B2−k−1(z)) ≤ C2−k(2+α). +Thus, we deduce that +|D2uk(z) − D2u(z)| ≤ C2−kα. +Finally, to estimate |D2uk(z) − D2uk(0)|, we denote hj := uj − uj−1 for +j = 1, 2, . . . , k. Since hj are harmonic, by (2.9) with κ = 3 and using that +B2−k−3 ⊂ B2−j−1, we see that +���� +D2hj(z) − D2hj(0) +|z| +���� ≤ ∥D3hj∥L∞(B2−k−3) +≤ C23j∥hj∥L∞(B2−j ) ≤ C2j(1−α). +Hence, +|D2uk(0) − D2uk(z)| ≤ |D2u0(z) − D2u0(0)| + +k +� +j=1 +|D2hj(z) − D2hj(0)| +≤ C|z|∥u0∥L∞(B1) + C|z| +k +� +j=1 +2j(1−α). +We have also used here that, if we define w := u0 − f(0) +2n |x|2 + f(0) +2n then w is +harmonic, D3w = D3u0, and w = u0 on ∂B1, and +|z|−1|D2u0(z) − D2u0(0)| ≤ ∥D3u0∥L∞(B1/2) = ∥D3w∥L∞(B1/2) +≤ C∥w∥L∞(B1) = C∥u0∥L∞(∂B1) ≤ C∥u0∥L∞(B1), +by higher order regularity estimates for harmonic functions (see (2.9)) and +the maximum principle (Lemma 1.14). Note that here the constant C de- +pends only on n. +Combined with the fact that, from (2.11), +∥u0∥L∞(B1) ≤ C, +(where we also use ∥u∥L∞(B1) ≤ 1) and |z| ≤ 2−k−3, we deduce that +|D2uk(0) − D2uk(z)| ≤ C|z| + C|z|2k(1−α) ≤ C2−kα. +We finish by noticing that |z| ≥ 2−k−4 and combining all the last in- +equalities we reach +|D2u(z) − D2u(0)| ≤ C2−kα ≤ C|z|α +for all z ∈ B1/16. That is, +[D2u]C0,α(B1/16) ≤ C. + +— DRAFT — +44 +2. Linear elliptic PDE +Now, thanks to the interpolation inequalities (see (1.9) with ε = 1), +∥u∥C2,α(B1/16) = ∥u∥C2(B1/16) + [D2u]C0,α(B1/16) +≤ C∥u∥L∞(B1/16) + 2[D2u]C0,α(B1/16) ≤ C. +We finish by recalling that we divided the solution u by ∥u∥L∞(B1)+[f]C0,α(B1), +and we use a covering argument to get the desired result (see Remark 2.15). +□ +For the second proof of Theorem 2.14, we use the methods from [Moo12], +originally from [Caf89]. +Second proof of Theorem 2.14. After subtracting f(0) +2n |x|2 we may as- +sume that f(0) = 0. +After dividing u by ∥u∥L∞(B1) + ε−1∥f∥C0,α(B1) if +necessary, we may also assume that ∥u∥L∞(B1) ≤ 1 and ∥f∥C0,α(B1) ≤ ε, +where ε > 0 is a constant to be chosen depending only on n and α. After +these simplifications, it is enough to show that +(2.14) +∥u∥C2,α(B1/2) ≤ C +for some constant C depending only on n and α. +We will show that, for every x ∈ B1/2, there exist a sequence of quadratic +polynomials, (Pk)k∈N, and a ρ◦ < 1 such that +(2.15) +∥u − Pk∥L∞(Bρk◦ (x)) ≤ C◦ρk(2+α) +◦ +for all k ∈ N, +for some constant C◦. By property (H5) from Chapter 1, this yields that +[D2u]C0,α(B1/2) ≤ CC◦. After using an interpolation inequality (1.9), we get +(2.14). +We will prove (2.15) for x = 0 (after a translation, it follows for all +x ∈ B1/2). We are going to use that ∆u = f, ∥u∥L∞(B1) ≤ 1, f(0) = 0 and +[f]C0,α(B1) ≤ ε. +Notice that ∥∆u∥C0,α(B1) = ∥f∥C0,α(B1) ≤ 2ε, i.e., u is 2ε-close in H¨older +norm to a harmonic function: let w be such that ∆w = 0 and w = u on ∂B1. +Then, ∆(u − w) = f in B1, and u − v = 0 on ∂B1, so that by Lemma 1.14, +(2.16) +∥u − w∥L∞(B1) ≤ C′∥f∥L∞(B1) ≤ Cε, +for some C universal (we are only using ∥f∥L∞(B1) ≤ 2ε, and not using its +Cα norm at this point). The function w is harmonic and |w| ≤ 1 (since +|u| ≤ 1). Therefore, it has a quadratic Taylor polynomial P1 at the origin, +which satisfies ∆P1 ≡ 0 and |P1| ≤ C. Moreover, since w is harmonic (and +in particular w ∈ C3), we have +(2.17) +∥w − P1∥L∞(Br) ≤ Cr3 +for all r ≤ 1, +for some C depending only on n. + +— DRAFT — +2.2. Schauder estimates for the Laplacian +45 +Combining (2.16) and (2.17) we obtain +∥u − P1∥L∞(Br) ≤ C(r3 + ε) +for all r ≤ 1. +Choose now r◦ small enough such that Cr3 +◦ ≤ 1 +2r2+α +◦ +(notice α < 1), and +ε small enough such that Cε < 1 +2r2+α +◦ +. (Notice that both r◦ and ε can be +chosen depending only on n and α.) Then, +∥u − P1∥L∞(Br◦) ≤ r2+α +◦ +. +Let us now define +u2(x) := (u − P1)(r◦x) +r2+α +◦ +. +Notice that ∥u2∥L∞(B1) ≤ 1 and ∆u2(x) = r−α +◦ f(r◦x) =: f2(x). +Then, +f2(0) = 0 and [f2]C0,α(B1) ≤ [f]C0,α(B1) ≤ ε. That is, the same hypotheses as +before are fulfilled. Repeating the same procedure, there exists a polynomial +P2 such that +∥u2 − P2∥L∞(Br◦) ≤ r2+α +◦ +. +That is, substituting back, +��u − P1 − r2+α +◦ +P2(x/r◦) +�� +L∞(Br2◦) ≤ r2(2+α) +◦ +. +Continuing iteratively, for every k ∈ N we can define +uk+1(x) := (uk − Pk)(r◦x) +r2+α +◦ +, +which satisfies +∥uk+1∥L∞(B1) ≤ 1, +∆uk+1(x) = r−α +◦ fk(r◦x) = r−kα +◦ +f(rk +◦x) =: fk+1(x), +and there exists some Pk+1 such that +∥uk+1 − Pk+1∥L∞(Br◦) ≤ r2+α +◦ +. +Substituting back, +��u − P1 − r2+α +◦ +P2(x/r◦) − · · · − rk(2+α) +◦ +Pk+1(x/rk +◦) +�� +L∞(Brk+1 +◦ +) ≤ r(k+1)(2+α) +◦ +. +That is, we have constructed a sequence of quadratic polynomials approxi- +mating u in a decreasing sequence of balls around 0; which shows that (2.15) +holds around 0. After a translation, the same argument can be repeated +around any point x ∈ B1/2, so that, by (H5) we are done. +□ +Remark 2.19. When α = 0, the previous proof implies that if f ∈ L∞(B1) +then, by (2.15), ∇u is in the Zygmund space Λ1(B1); see Remark A.1 in +the Appendix A for more details. +In particular, we also get a proof of +Proposition 2.18. + +— DRAFT — +46 +2. Linear elliptic PDE +Notice that in the previous proof we have not directly used that u is C2. +In fact, the only properties of u (and the Laplacian) we have used are that +the maximum principle holds and that ∆(u(rx)) = r2(∆u)(rx). +In particular, the second proof of Theorem 2.14 is not an a priori esti- +mate, and rather it says that any weak solution to the Laplace equation with +Cα right-hand side is C2,α. That is, we have directly proved Corollary 2.16. +2.3. Schauder estimates for operators in non-divergence form +After proving the Schauder estimates for the Laplacian, we will study now +more general second order linear elliptic operators. We start with operators +in non-divergence form. The type of equation we are interested in is +tr +� +A(x)D2u(x) +� += +n +� +i,j=1 +aij(x)∂iju(x) = f(x) +in +B1 +where the matrix A(x) = (aij(x))ij is uniformly elliptic — in the sense that +(2.18) below holds — and aij(x) ∈ C0,α(B1). We will prove the following a +priori estimates. +Theorem 2.20 (Schauder estimates in non-divergence form). Let α ∈ (0, 1), +and let u ∈ C2,α be any solution to +n +� +i,j=1 +aij(x)∂iju = f(x) +in +B1, +with f ∈ C0,α(B1) and aij(x) ∈ C0,α(B1), and (aij(x))ij fulfilling the ellip- +ticity condition +(2.18) +0 < λ Id ≤ (aij(x))ij ≤ Λ Id +in +B1, +for some 0 < λ ≤ Λ < ∞. Then, +∥u∥C2,α(B1/2) ≤ C +� +∥u∥L∞(B1) + ∥f∥C0,α(B1) +� +for some constant C depending only on α, n, λ, Λ, and ∥aij∥C0,α(B1). +As for the Laplacian, we will provide two different proofs of the previous +result. On the other hand, as a consequence of the previous result, we also +obtain higher order Schauder estimates in non-divergence form. +Corollary 2.21 (Higher order Schauder estimates in non-divergence form). +Let u ∈ Ck+2,α be a solution to +n +� +i,j=1 +aij(x)∂iju = f(x) +in +B1, + +— DRAFT — +2.3. Schauder estimates for operators in non-divergence form +47 +with f ∈ Ck,α(B1) and aij(x) ∈ Ck,α(B1) for some α ∈ (0, 1), k ∈ N, and +(aij(x))ij fulfilling the ellipticity conditions (2.18) for some 0 < λ ≤ Λ < ∞. +Then, +∥u∥Ck+2,α(B1/2) ≤ C +� +∥u∥L∞(B1) + ∥f∥Ck,α(B1) +� +for some constant C depending only on α, k, n, λ, Λ, and ∥aij∥Ck,α(B1). +Remark 2.22 (Ellipticity condition). The uniform ellipticity condition in +B1, (2.18), is a quantification of the fact that the matrix +A(x) := (aij(x))ij +is uniformly positive definite and uniformly bounded as well. Notice that +we can always assume that A(x) is symmetric (from ∂iju = ∂jiu). We recall +that the inequality A1 ≤ A2 for symmetric matrices A1, A2 ∈ Mn has to be +understood in the sense that A2−A1 is positive semi-definite. Alternatively, +(2.18) will hold if +0 < λ|ξ|2 ≤ +n +� +i,j=1 +ξiξjaij(x) ≤ Λ|ξ|2 +for all +x ∈ B1 +for all ξ ∈ Rn. +Remark 2.23 (Constant coefficients). Let us start by understanding the +case of constant coefficients, +n +� +i,j=1 +aij∂iju(x) = 0 +in +B1, +where aij are constants and satisfy the uniform ellipticity assumption, +0 < λId ≤ (aij)ij ≤ ΛId, +for 0 < λ ≤ Λ < ∞. +Let us denote A := (aij)ij ∈ Mn. +Then, A is a symmetric positive +definite matrix, and therefore has a unique positive definite square root +A1/2. After an affine change of variables +z = A1/2x, +the equation +n +� +i,j=1 +aij∂xixju = 0 +becomes +n +� +i=1 +∂ziziu = 0 +or ∆zu = 0. Indeed, +n +� +i,j=1 +aij∂xixju = tr(AD2 +xu) = tr(A1/2D2 +xuA1/2) = tr(D2 +zu) = ∆zu. + +— DRAFT — +48 +2. Linear elliptic PDE +Therefore (and since 0 < λId ≤ A ≤ ΛId), the case of constant coefficients +(uniformly elliptic) can be reduced to the case of harmonic functions. +Thanks to the uniform ellipticity, the change of variables is not degen- +erate, and thus the estimates on ∥u∥C2,α that we get depend only on α, n, +λ, and Λ (but not on A). Similarly, after changing variables, there could +be a shrinking of the domain, say that the C2,α norm of u is bounded in +Bρ instead of B1/2, for some ρ < 1/2. Once again, since the change is non- +degenerate, such ρ depends only on n, λ, and Λ, and one can complete the +proof by a covering argument in B1/2 (see Remark 2.15). +The maximum principle. We state the maximum principle for equations +in non-divergence form, which will be used in this section. +Proposition 2.24 (Maximum Principle in non-divergence form). Let Ω ⊂ +Rn be any bounded open set. Suppose that u ∈ C0(Ω) ∩ C2(Ω) satisfies +n +� +i,j=1 +aij(x)∂iju ≥ 0 +in +Ω, +where (aij(x))ij satisfy +0 < λ Id ≤ (aij(x))ij +in +Ω. +Then, +sup +Ω +u = sup +∂Ω +u. +Proof. Let us begin by showing the maximum principle in the case +(2.19) +n +� +i,j=1 +aij(x)∂iju > 0 +in +B1, +that is, when we have a strict inequality. +We show it by contradiction: +suppose that there exists some x◦ ∈ Ω such that supΩ u = u(x◦). Since it is +an interior maximum, we must have ∇u(x◦) = 0 and D2u(x◦) ≤ 0, that is, +D2u(x◦) is a negative semi-definite symmetric matrix. In particular, all its +eigenvalues are non-positive, and after a change of variables we have that +P T D2u(x◦)P = diag(λ1, . . . , λn) := Dx◦ +for some orthogonal n × n matrix P, and with λi ≤ 0 for all 1 ≤ i ≤ n. +Let A(x) = (aij(x))ij, and let AP (x◦) := P T A(x◦)P. Then, since A(x◦) is +positive definite, so is AP (x◦) = (aP +ij(x◦))ij. In particular, aP +ii(x◦) ≥ 0 for +all 1 ≤ i ≤ n. Then, +tr(A(x◦)D2u(x◦)) = tr(A(x◦)PDx◦P T ) = tr(P T A(x◦)PDx◦) + +— DRAFT — +2.3. Schauder estimates for operators in non-divergence form +49 +and, therefore +0 < tr(A(x◦)D2u(x◦)) = tr(AP (x◦)Dx◦) = +n +� +i=1 +aP +ii(x◦)λi ≤ 0, +a contradiction. Here, we used that aP +ii(x◦) ≥ 0 and λi ≤ 0 for all 1 ≤ i ≤ n. +This shows that the maximum principle holds when the strict inequality +(2.19) is satisfied. +Let us now remove this hypothesis. Let R be large enough such that +BR ⊃ Ω — after a translation, we can take R = 1 +2diam(Ω). Consider now +the function +uε(x) := u(x) + εex1 +for +x ∈ Ω, +for ε > 0. Notice that, +n +� +i,j=1 +aij(x)∂ijuε(x) ≥ λεex1 > 0 +in +Ω. +In particular, we can apply the result for (2.19) to obtain that +sup +Ω +u ≤ sup +Ω +uε = sup +∂Ω +uε ≤ sup +∂Ω +u + εeR. +By letting ε ↓ 0, we obtain the desired result. +□ +As a consequence, we find: +Lemma 2.25. Let Ω ⊂ Rn be a bounded open set, and let u ∈ C0(Ω)∩C2(Ω) +be a function satisfying +� �n +i,j=1 aij(x)∂iju += +f +in Ω +u += +g +on ∂Ω, +where (aij)ij fulfill the ellipticity conditions (2.18) for some 0 < λ ≤ Λ < ∞. +Then, +∥u∥L∞(Ω) ≤ C +� +∥f∥L∞(Ω) + ∥g∥L∞(∂Ω) +� +, +for a constant C depending only on the diameter of Ω, λ, and Λ. +Proof. This follows exactly as in the proof of Lemma 1.14 using Proposi- +tion 2.24. +□ +Proof of Schauder estimates. Let us now proceed with the proof of +Schauder estimates for equations in non-divergence form, Theorem 2.20. We +will first prove (in two ways) the following proposition, which is a weaker +version of the estimate we want to show. +We will later prove that, in fact, such estimate is enough to prove The- +orem 2.20. + +— DRAFT — +50 +2. Linear elliptic PDE +Proposition 2.26. Let u ∈ C2,α be a solution to +n +� +i,j=1 +aij(x)∂iju = f(x) +in +B1, +with f ∈ C0,α(B1) and aij(x) ∈ C0,α(B1) for some α ∈ (0, 1), and (aij(x))ij +fulfilling the ellipticity condition +0 < λId ≤ (aij(x))ij ≤ ΛId +in +B1, +for some 0 < λ ≤ Λ < ∞. Then, for any δ > 0, +[D2u]C0,α(B1/2) ≤ δ[D2u]C0,α(B1) + Cδ +� +∥u∥L∞(B1) + ∥f∥C0,α(B1) +� +, +for some constant Cδ depending only on δ, α, n, λ, Λ, and ∥aij∥C0,α(B1). +Notice that, the previous statement is almost what we want: if we could +let δ ↓ 0 and Cδ remained bounded, Theorem 2.20 would be proved (after +using interpolation inequalities (1.9)). +On the other hand, if the H¨older +norm was in B1/2 instead of B1, choosing δ = 1 +2 would also complete the +proof. As we will see, although it is not so straightforward, Proposition 2.26 +is just one step away from the final result. +Let us provide two different proofs of Proposition 2.26. The first proof +is a sketch that follows the same spirit as the first proof of Theorem 2.20. +The second proof is through a blow-up argument (by contradiction). +First Proof of Proposition 2.26. The proof is very similar to the case of +the Laplacian, the first proof of Theorem 2.14. +We define uk as the solution to +� �n +i,j=1 aij(0)∂ijuk += +f(0) +in B2−k +uk += +u +on ∂B2−k. +(We freeze the coefficients at zero.) Then, +vk := u − uk +satisfies +n +� +i,j=1 +aij(0)∂ijvk = f(x) − f(0) + +n +� +i,j=1 +� +aij(0) − aij(x) +� +∂iju +in +B2−k. +By the maximum principle (Lemma 2.25) we get +∥u − uk∥L∞(B2−k) ≤ C2−2k +� +2−αk∥f∥C0,α(B2−k) ++ 2−αk∥D2u∥L∞(B2−k) +n +� +i,j=1 +∥aij∥C0,α(B2−k) +� +. + +— DRAFT — +2.3. Schauder estimates for operators in non-divergence form +51 +Thus, +∥uk − uk+1∥L∞(2−k−1) ≤ C2−k(2+α) � +∥f∥C0,α(B2−k) + ∥D2u∥L∞(B2−k) +� +, +where the constant C depends only on α, n, λ, Λ, and ∥aij∥C0,α(B1). +Following the exact same proof as in the case of the Laplacian, ∆u = +f(x), we now get +[D2u]C0,α(B1/2) ≤ C +� +∥u∥L∞(B1) + ∥f∥C0,α(B1) + ∥D2u∥L∞(B1) +� +. +This is almost exactly what we wanted to prove. However, we have an +extra term ∥D2u∥L∞(B1) on the right-hand side. This can be dealt with by +means of interpolation inequalities. +We use that, for any ε > 0, there is Cε such that +(2.20) +∥D2u∥L∞(B1) ≤ ε[D2u]C0,α(B1) + Cε∥u∥L∞(B1) +see (1.9) in Chapter 1. +The idea is that, since the ∥D2u∥L∞ term is lower order, we can absorb +it in the left-hand side by paying the price of adding more ∥u∥L∞ norm on +the right-hand side. +Namely, we have +[D2u]C0,α(B1/2) ≤ C +� +∥u∥L∞(B1) + ∥f∥C0,α(B1) + ∥D2u∥L∞(B1) +� +(by interpolation) ≤ C +� +Cε∥u∥L∞(B1) + ∥f∥C0,α(B1) + ε[D2u]C0,α(B1) +� +≤ Cδ +� +∥u∥L∞(B1) + ∥f∥C0,α(B1) +� ++ δ[D2u]C0,α(B1), +where we have used the interpolation inequality, and in the last step we have +chosen ε = δ/C > 0. The constant Cδ depends only on δ, α, n, λ, Λ, and +∥aij∥C0,α(B1). This concludes the proof. +□ +For the second proof of Proposition 2.26 we use a robust blow-up method +due to L. Simon, [Sim97]. +For simplicity, we will first prove it for the +Laplacian case. After proving it for the Laplacian, we explain in detail how +to adapt the method for the more general non-divergence operators. +Second Proof of Proposition 2.26. Assume first that (aij(x))ij = Id, +that is, ∆u = f in B1. We then explain the modifications needed to show +the result in the general case, �n +i,j=1 aij(x)∂iju(x) = f(x) in B1. +Thanks to interpolation inequalities we only need to prove the following +estimate for any δ > 0 sufficiently small, +(2.21) +[D2u]C0,α(B1/2) ≤ δ[D2u]C0,α(B1) + Cδ +� +∥D2u∥L∞(B1) + [f]C0,α(B1) +� + +— DRAFT — +52 +2. Linear elliptic PDE +for all u ∈ C2,α(B1) with ∆u = f in B1. Indeed, if (2.21) holds then, by +interpolation (2.20), with ε = δ/Cδ, +[D2u]C0,α(B1/2) ≤ 2δ[D2u]C0,α(B1) + Cδ +� +∥u∥L∞(B1) + [f]C0,α(B1) +� +for some new Cδ depending only on δ, n and α, which is the desired result. +We will now show that (2.21) holds by contradiction, for some Cδ de- +pending only on δ, n, and α. Indeed, suppose that it does not hold. Then, +there exist sequences uk ∈ C2,α(B1) and fk ∈ C0,α(B1) for k ∈ N such that +∆uk = fk +in +B1, +and for a fixed small constant δ◦ > 0 we have +(2.22) +[D2uk]C0,α(B1/2) > δ◦[D2uk]C0,α(B1) + k +� +∥D2uk∥L∞(B1) + [fk]C0,α(B1) +� +. +We now have to reach a contradiction. +Select xk, yk ∈ B1/2 such that +(2.23) +|D2uk(xk) − D2uk(yk)| +|xk − yk|α +≥ 1 +2[D2uk]C0,α(B1/2) +and let +ρk := |xk − yk|. +Observe that we must necessarily have ρk → 0 as k → ∞, since +1 +2[D2uk]C0,α(B1/2) ≤ |D2uk(xk) − D2uk(yk)| +ρα +k +≤ 2∥D2uk∥L∞(B1) +ρα +k +≤ +2[D2uk]C0,α(B1/2) +kρα +k +, +where we have used (2.22) in the last inequality. Thus, +ρk ≤ Ck− 1 +α → 0 +as +k → ∞ +Now, we rescale and blow up. Define +˜uk(x) := uk(xk + ρkx) − pk(x) +ρ2+α +k +[D2uk]C0,α(B1) +, +˜fk(x) := fk(xk + ρkx) − fk(xk) +ρα +k[D2uk]C0,α(B1) +, +where the quadratic polynomial pk is chosen so that +(2.24) +˜uk(0) = |∇˜uk(0)| = |D2˜uk(0)| = 0. +Namely, +pk(z) := uk(xk) + ρk +n +� +i=1 +∂iuk(xk)zi + 1 +2ρ2 +k +n +� +i,j=1 +∂ijuk(xk)zizj. +It is now a simple computation to check that +(2.25) +∆˜uk = ˜fk +in +B1/(2ρk). + +— DRAFT — +2.3. Schauder estimates for operators in non-divergence form +53 +Let us also denote +ξk := yk − xk +ρk +∈ Sn−1. +Notice that +(2.26) +[D2˜uk]C0,α� +B1/(2ρk) +� ≤ 1, +and +��D2˜uk(ξk) +�� > δ◦ +2 , +where for the second inequality we use (2.22)-(2.23). +Since ˜uk are uniformly bounded in compact subsets, and bounded in the +C2,α norm (see (2.24)-(2.26)), we have by Arzel`a–Ascoli that the sequence +˜uk converges (up to a subsequence and in the C2 norm) to a C2,α function +˜u on compact subsets of Rn. Moreover, again up to a subsequence, we have +that ξk → ξ ∈ Sn−1. +By the properties of ˜uk, we deduce that ˜u satisfies +(2.27) +˜u(0) = |∇˜u(0)| = |D2˜u(0)| = 0, +[D2˜u]C0,α(Rn) ≤ 1, +|D2˜u(ξ)| > δ◦ +2 . +On the other hand, for any R ≥ 1 we have +∥ ˜fk∥L∞(BR) = sup +x∈BR +|fk(xk + ρkx) − fk(xk)| +ρα +k[D2uk]C0,α(B1) +≤ (ρkR)α[fk]C0,α(B1) +ρα +k[D2uk]C0,α(B1) +≤ +Rα[D2uk]C0,α(B1/2) +k[D2uk]C0,α(B1) +≤ Rα +k → 0, as k → ∞. +Thus, ˜fk → 0 uniformly on compact sets of Rn. Together with the fact that +˜uk → ˜u in the C2 norm in compact sets, we deduce (recall (2.25)) +∆˜u = 0 +in +Rn. +That is, ˜u is harmonic and, in particular, so is ∂ij ˜u for any i, j = 1, . . . , n. +Let us now use the three properties in (2.27) to get a contradiction. +First notice that we have [D2˜u]C0,α(Rn) ≤ 1. +Thus, D2˜u has sub-linear +growth at infinity, and by Liouville’s theorem (Proposition 1.19) we find +that D2˜u is constant. +That is, ˜u is a quadratic polynomial, which also +fulfills ˜u(0) = |∇˜u(0)| = |D2˜u(0)| = 0. The only possibility is that ˜u ≡ 0 in +Rn, which is a contradiction with |D2˜u(ξ)| > δ◦ +2 . +Thus, the proposition is proved in the case of the Laplacian. +We now treat the case of variable coefficients, +n +� +i,j=1 +aij(x)∂iju(x) = f(x) +in +B1, +with aij(x) uniformly elliptic in B1 (i.e., 0 < λId ≤ (aij(x))ij ≤ ΛId for +x ∈ B1) and with ∥aij∥C0,α(B1) ≤ M < ∞ for some M. +The proof is + +— DRAFT — +54 +2. Linear elliptic PDE +essentially the same. As before, we proceed by contradiction, by assuming +that there exist sequences uk, fk, and a(k) +ij +such that +n +� +i,j=1 +a(k) +ij (x)∂ijuk(x) = fk(x) +in +B1, +and (2.22) holds. +The only difference with respect to the Laplacian case is the equation +satisfied by ˜uk. Let us define, +˜a(k) +ij (x) := a(k) +ij (xk + ρkx). +Notice that +[˜a(k) +ij ]C0,α(B1/(2ρk)) ≤ ρα +k[a(k) +ij ]C0,α(B1) → 0, +as +k → ∞. +In particular, up to subsequences, ˜a(k) +ij +converges uniformly in compact sets +to some ˜aij with [˜a(k) +ij ]C0,α(Rn) = 0, i.e., ˜aij is constant. Then ˜uk satisfies +n +� +i,j=1 +˜a(k) +ij ∂ij ˜uk = ˜fk(x) − +n +� +i,j=1 +� +a(k) +ij (xk + ρkx) − a(k) +ij (xk) +� +∂ijuk(xk) +ρα +k[D2uk]C0,α(B1) +. +Thus, +������ +n +� +i,j=1 +˜a(k) +ij ∂ij ˜uk − ˜fk(x) +������ +≤ +n +� +i,j=1 +|x|αρα +k[a(k) +ij ]C0,α(B1)∥∂ijuk∥L∞(B1) +ρα +k[D2uk]C0,α(B1) +≤ C|x|α ∥D2uk∥L∞(B1) +[D2uk]C0,α(B1) +≤ C|x|α ∥D2uk∥L∞(B1) +[D2uk]C0,α(B1/2) +. +Using (2.22) we deduce that, for any x ∈ Bσ for some fixed σ ∈ (0, ∞), and +for k large enough, +������ +n +� +i,j=1 +˜a(k) +ij ∂ij ˜uk − ˜fk(x) +������ +≤ C(σ) ∥D2uk∥L∞(B1) +[D2uk]C0,α(B1/2) +≤ C(σ) +k +. +Taking the limit k → ∞ (and recalling that ˜fk → 0 uniformly in compact +sets) we get +n +� +i,j=1 +˜aij∂ij ˜u = 0 +in +Rn, +an equation with constant coefficients, which is equivalent to ∆˜u = 0 in Rn +(see Remark 2.23), and we reach a contradiction as well. +□ + +— DRAFT — +2.3. Schauder estimates for operators in non-divergence form +55 +We can now proceed with the proof of the Schauder estimates in non- +divergence form. Namely, we will show how to go from Proposition 2.26 to +Theorem 2.20. As with the previous results, we will do it in two different +ways. In this case, however, both ways reduce to the same idea. +First Proof of Theorem 2.20. Define the semi-norm +[D2u]∗ +α;B1 := +sup +Bρ(x◦)⊂B1 +ρ2+α[D2u]C0,α(Bρ/2(x◦)). +Notice that this norm measures in a precise way how the C2,α norm of U +blows up as we approach ∂B1. +From the fact that H¨older semi-norms are sub-additive with respect to +unions of convex sets, +(2.28) +[D2u]∗ +α;B1 ≤ C +sup +Bρ(x◦)⊂B1 +ρ2+α[D2u]C0,α(Bρ/4(x◦)) +(and, in fact, they are comparable) for some constant C depending only on +α and n. Indeed, for any fixed ball Bρ(x◦) ⊂ B1, we cover Bρ/2(x◦) with N +smaller balls (Bρ/8(zj))1≤j≤N, which, since Bρ/2(zj) ⊂ B1, gives +�ρ +2 +�2+α +[D2u]C0,α(Bρ/8(zj)) ≤ +sup +Bρ(x◦)⊂B1 +ρ2+α[D2u]C0,α(Bρ/4(x◦)). +Thus, +ρ2+α[D2u]C0,α(Bρ/2(x◦)) ≤ ρ2+α +N +� +j=1 +[D2u]C0,α(Bρ/8(zj)) +≤ 22+αN +sup +Bρ(x◦)⊂B1 +ρ2+α[D2u]C0,α(Bρ/4(x◦)). +Taking the supremum on the left-hand side gives (2.28). +Applying the inequality +[D2u]C0,α(B1/2) ≤ δ[D2u]C0,α(B1) + Cδ +� +∥u∥L∞(B1) + ∥f∥C0,α(B1) +� +from Proposition 2.26 to any ball Bρ/2(x◦) ⊂ Bρ(x◦) ⊂ B1 we get +ρ2+α[D2u]C0,α(Bρ/4) ≤ δρ2+α[D2u]C0,α(Bρ/2) + Cδ +� +∥u∥L∞(B1) + ∥f∥C0,α(B1) +� +≤ δ[D2u]∗ +α;B1 + Cδ +� +∥u∥L∞(B1) + ∥f∥C0,α(B1) +� +. +Taking the supremum and using (2.28) we get +1 +C [D2u]∗ +α;B1 ≤ +sup +Bρ(x◦)⊂B1 +ρ2+α[D2u]C0,α(Bρ/4(x◦)) +≤ δ[D2u]∗ +α;B1 + C +� +∥u∥L∞(B1) + ∥f∥C0,α(B1) +� +. + +— DRAFT — +56 +2. Linear elliptic PDE +Now, if we fix a small enough δ > 0, we can absorb the [D2u]∗ +α;B1 term on +the left-hand side to get +[D2u]C0,α(B1/2) ≤ [D2u]∗ +α;B1 ≤ Cδ +� +∥u∥L∞(B1) + ∥f∥C0,α(B1) +� +, +which, after interpolation (see (1.9)) gives the desired result. +□ +We also give an alternative proof of Theorem 2.20 by directly using the +following abstract lemma. Such lemma constitutes a generalization of the +previous proof. +Lemma 2.27. Let k ∈ R and γ > 0. Let S be a non-negative function on +the class of open convex subsets of B1, and suppose that S is sub-additive. +That is, if A, A1, . . . , AN are open convex subsets of B1 with A ⊂ �N +j=1 Aj, +then S(A) ≤ �N +j=1 S(Aj). +Then, there is δ > 0 small (depending only on n and k) such that, if +ρkS(Bρ/2(x◦)) ≤ δρkS(Bρ(x◦)) + γ +for all Bρ(x◦) ⊂ B1, +then +S(B1/2) ≤ Cγ, +for some C depending only on n and k. +Proof. Let +Q := +sup +Bρ(x◦)⊂B1 +ρkS(Bρ/2(x◦)). +Thanks to the assumption in the Lemma, we get +�ρ +2 +�k +S(Bρ/4(x◦)) ≤ δ +�ρ +2 +�k +S(Bρ/2(x◦))+γ ≤ δQ+γ, +for all Bρ(x◦) ⊂ B1. +Taking now the supremum for all Bρ(x◦) ⊂ B1 we get +˜Q := +sup +Bρ(x◦)⊂B1 +�ρ +2 +�k +S(Bρ/4(x◦)) ≤ δQ + γ. +We now claim that +(2.29) +Q ≤ C ˜Q, +for some C depending only on n and k. This will yield +1 +C Q ≤ ˜Q ≤ δQ + γ ⇒ Q ≤ ˜Cγ +if δ > 0 is small enough depending only on n and k. Thus, we have to show +(2.29). +Take any Bρ(x◦) ⊂ B1, and cover Bρ/2(x◦) with a finite collection of +smaller balls Bρ/8(zj) (j = 1, 2, . . . , N), with zj ∈ Bρ/2(x◦) and N ≤ C + +— DRAFT — +2.3. Schauder estimates for operators in non-divergence form +57 +(universally bounded depending only on the dimension). Since Bρ/2(zj) ⊂ +B1 we then have +�ρ +4 +�k +S(Bρ/8(zj)) ≤ ˜Q. +Adding up over all indices j, and using the sub-additivity of S, we obtain +ρkS(Bρ/2(x◦)) ≤ +N +� +j=1 +ρkS(Bρ/8(zj)) ≤ N4k ˜Q = C ˜Q. +Taking the supremum, we reach (2.29). +□ +Second Proof of Theorem 2.20. We use Lemma 2.27, with k = α and +S(A) := [D2u]C0,α(A), +which is sub-additive on open convex subsets. From the estimate in Propo- +sition 2.26, fixing δ > 0 from Lemma 2.27 (which depends only on α and n) +we know +[D2u]C0,α(B1/2) ≤ C +� +∥u∥L∞(B1) + ∥f∥C0,α(B1) +� ++ δ[D2u]C0,α(B1). +Rescaling1 to Bρ(x◦) with ρ ≤ 1 we obtain +ρ2+α[D2u]C0,α(Bρ/2(x◦)) ≤ +≤ δρ2+α[D2u]C0,α(Bρ(x◦)) ++ C +� +∥u∥L∞(Bρ(x◦)) + ρ2∥f∥L∞(Bρ(x◦)) + ρ2+α[f]C0,α(Bρ(x◦)) +� +≤ δρ2+α[D2u]C0,α(Bρ) + C +� +∥u∥L∞(B1) + ∥f∥C0,α(B1) +� +. +This is exactly +ρkS(Bρ/2(x◦)) ≤ δρkS(Bρ(x◦)) + γ, +with +γ = Cδ +� +∥u∥L∞(B1) + ∥f∥C0,α(B1) +� +. +Thus, thanks to Lemma 2.27, we immediately deduce +S(B1/2) ≤ Cγ, +that is, +[D2u]C0,α(B1/2) ≤ C +� +∥u∥L∞(B1) + ∥f∥C0,α(B1) +� +. +Therefore, after using interpolation inequalities (see (1.9)) we get +∥u∥C2,α(B1/2) ≤ C +� +∥u∥L∞(B1) + ∥f∥C0,α(B1) +� +as desired. +□ +1The rescaling is done by considering the estimate on uρ(x) = u(x◦ + ρx), which ful- +fills � a(ρ) +ij (x)∂ijuρ(x) = ρ2f(x◦ + ρx) =: fρ(x) in B1, with a(ρ) +ij (x) = aij(x◦ + ρx) (notice +that ∥a(ρ) +ij ∥C0,α(B1) ≤ ∥aij∥C0,α(B1)). Then, [D2uρ]C0,α(B1/2) = ρ2+α[D2u]C0,α(Bρ/2(x◦)) and +[fρ]C0,α(B1) = ρ2+α[f]C0,α(Bρ(x◦)). + +— DRAFT — +58 +2. Linear elliptic PDE +We finish this section by proving Corollary 2.21. +Proof of Corollary 2.21. We follow the proof of Corollary 2.17. We will +show by induction on k that +(2.30) +∥u∥Ck+2,α(B1/2) ≤ C +� +∥u∥L∞(B1) + ∥f∥Ck,α(B1) +� +for some constant C depending only on n, α, k, λ, Λ, and ∥aij∥Ck,α(B1). +We apply the induction hypothesis to derivatives of the equation in non- +divergence form. +As in the proof of Corollary 2.17, (2.30) deals with balls B1/2 and B1, +but after a rescaling and covering argument (see Remark 2.15), it could also +be stated in balls B1/2 and B3/4. +The base case, k = 0, already holds by Theorem 2.20. Let us now assume +that (2.30) holds for k = m − 1, and we will show it for k = m. +We differentiate the non-divergence-form equation with respect to ∂e to +get +n +� +i,j=1 +aij(x)∂ij∂eu(x) = ∂ef(x) − +n +� +i,j=1 +∂eaij(x)∂iju(x) +in +B1. +Now, we apply the estimate (2.30) with k = m−1 to ∂eu in the previous +expression, in balls B1/2 and B3/4, to get +∥∂eu∥Cm+1,α(B1/2) ≤ C +� +∥∂eu∥L∞(B3/4) + ∥∂ef∥Cm−1,α(B3/4) ++ +n +� +i,j=1 +∥∂eaij∂iju∥Cm−1,α(B3/4) +� +. +Notice that +∥∂eaij∂iju∥Cm−1,α(B3/4) ≤ ∥∂eaij∥Cm−1,α(B3/4)∥∂iju∥Cm−1,α(B3/4) += C∥∂iju∥Cm−1,α(B3/4) +≤ C +� +∥u∥L∞(B1) + ∥f∥Cm−1,α(B1) +� +, +where in the last inequality we have used the induction hypothesis in balls +B3/4 and B1 (see Remark 2.15). Using that ∥∂eu∥L∞(B3/4) ≤ ∥u∥C2,α(B3/4) +we can use the base case (with balls B3/4 and B1) of (2.30) to bound this +term. In all, we obtain that +∥∂eu∥Cm+1,α(B1/2) ≤ C +� +∥u∥L∞(B1) + ∥f∥Cm−1,α(B1) +� +, +which, combined with the base case, and for every e ∈ Sn−1, yields the +desired estimate. +□ + +— DRAFT — +2.4. Schauder estimates for operators in divergence form +59 +2.4. Schauder estimates for operators in divergence form +We will next prove Schauder estimates for operators in divergence form. In +particular, we will study the equation +(2.31) +div +� +A(x)∇u(x) +� += +n +� +i,j=1 +∂i +� +aij(x)∂ju(x) +� += f(x) +in +B1, +where A(x) := (aij(x))ij is uniformly elliptic, and aij(x) ∈ C0,α. Notice that, +a priori, the expression (2.31) does not make sense even for C∞ functions u: +we are taking derivatives of aij(x), which is only C0,α. +That is why we +need to define a weak notion of solution to (2.31). Thus, we will say that +u ∈ H1(B1) solves (2.31) weakly if +� +B1 +∇φ(y) · A(y)∇u(y) dy = − +� +B1 +φ(y)f(y) dy +for all +φ ∈ C∞ +c (B1). +We will prove the following: +Theorem 2.28 (Schauder estimates in divergence form). Let u ∈ C1,α be +a weak solution to +n +� +i,j=1 +∂i +� +aij(x)∂ju(x) +� += f(x) +in +B1, +with f ∈ Lq(B1) for q ≥ +n +1−α, and aij(x) ∈ C0,α(B1) for some α ∈ (0, 1), +such that (aij(x))ij fulfills the ellipticity condition +(2.32) +0 < λ Id ≤ (aij(x))ij ≤ Λ Id +in +B1, +for some 0 < λ ≤ Λ < ∞. Then, +∥u∥C1,α(B1/2) ≤ C +� +∥u∥L∞(B1) + ∥f∥Lq(B1) +� +for some constant C depending only on α, n, λ, Λ, and ∥aij∥C0,α(B1). +And as a consequence, we also get higher order Schauder estimates for +operators in divergence form. +Corollary 2.29 (Higher order Schauder estimates in divergence form). Let +u ∈ Ck+1,α be a weak solution to +n +� +i,j=1 +∂i +� +aij(x)∂ju(x) +� += f(x) +in +B1, +with f ∈ Ck−1+α(B1) and aij(x) ∈ Ck,α(B1) for some α ∈ (0, 1), k ∈ N, +such that (aij(x))ij fulfills the ellipticity condition (2.32) for some 0 < λ ≤ +Λ < ∞. Then, +∥u∥Ck+1,α(B1/2) ≤ C +� +∥u∥L∞(B1) + ∥f∥Ck−1,α(B1) +� + +— DRAFT — +60 +2. Linear elliptic PDE +for some constant C depending only on α, k, n, λ, Λ, and ∥aij∥Ck,α(B1). +The maximum principle. As in the case of operators in non-divergence +form, we also have a maximum principle for equations in divergence form. +Proposition 2.30 (Maximum Principle in divergence form). Let Ω ⊂ Rn +be a bounded open set. Suppose that u ∈ H1(Ω) satisfies, in the weak sense, +n +� +i,j=1 +∂i +� +aij(x)∂ju(x) +� +≥ 0 +in +Ω, +where (aij(x))ij ∈ L∞(Ω) fulfill the pointwise ellipticity condition, +0 < (aij(x))ij +in +Ω. +Then, +sup +Ω +u = sup +∂Ω +u. +Proof. We know that, denoting A(x) = (aij(x))ij, +� +Ω +∇φ · A(x)∇u dx ≤ 0 +for all +φ ∈ C∞ +c (Ω), φ ≥ 0. +In particular, by approximation (see (S6) in Chapter 1), the previous ex- +pression holds for all φ ∈ H1 +0(Ω) such that φ ≥ 0. We take, as test function, +φ(x) := (u − sup∂Ω u)+ ∈ H1 +0(Ω), where f+ := max{f, 0} denotes the posi- +tive part. Then,� +Ω +∇φ · A(x)∇φ dx = +� +Ω +∇φ · A(x)∇u dx ≤ 0. +Since A(x) > 0, this implies that ∇φ ≡ 0, and φ is constant. Since φ ∈ +H1 +0(Ω), this implies that φ ≡ 0, that is, u ≤ sup∂Ω u in Ω, as wanted. +□ +Proof of Schauder estimates. We proceed with the proof of Theorem 2.28. +We will do so via a blow-up argument, in the spirit of the second proof of +Proposition 2.26. +Proof of Theorem 2.28. As in the (second) proof of Proposition 2.26, we +will show that, for any δ > 0, +(2.33) +[∇u]C0,α(B1/2) ≤ δ[∇u]C0,α(B1) + Cδ +� +∥∇u∥L∞(B1) + ∥f∥Lq(B1) +� +for all u ∈ C1,α(B1) such that +div(A(x)∇u(x)) = +n +� +i,j=1 +∂i (aij(x)∂ju(x)) = f(x), +weakly in +B1. +This yields +∥u∥C1,α(B1/2) ≤ δ[∇u]C0,α(B1) + Cδ +� +∥u∥L∞(B1) + ∥f∥Lq(B1) +� + +— DRAFT — +2.4. Schauder estimates for operators in divergence form +61 +and so, proceeding as in the proof of Theorem 2.20 by using Lemma 2.27, (or, +alternatively, adapting the first proof of Theorem 2.20), we get the desired +result. Let us focus, therefore, on the proof of (2.33): +Suppose that it does not hold. Then, there exist sequences uk ∈ C1,α(B1) +and fk ∈ Lq(B1) for k ∈ N such that +div +� +Ak(x)uk(x) +� += fk(x) +weakly in +B1, +and for a fixed small constant δ◦ > 0 we have +(2.34) [∇uk]C0,α(B1/2) > δ◦[∇uk]C0,α(B1) + k +� +∥∇uk∥L∞(B1) + ∥fk∥Lq(B1) +� +. +We now have to reach a contradiction. +Select xk, yk ∈ B1/2 such that +(2.35) +|∇uk(xk) − ∇uk(yk)| +|xk − yk|α +≥ 1 +2[∇uk]C0,α(B1/2) +and let +ρk := |xk − yk| +2 +, +and +zk := xk + yk +2 +. +Then, as in Proposition 2.26, ρk ≤ Ck− 1 +α → 0 as k → ∞. Define +˜uk(x) := uk(zk + ρkx) + uk(zk − ρkx) − 2uk(zk) +ρ1+α +k +[∇uk]C0,α(B1) +. +Then, +(2.36) +˜uk(0) = |∇˜uk(0)| = 0. +We remark that here, instead of defining ˜uk as in Proposition 2.26 (i.e., +subtracting a quadratic polynomial), we have used second order incremental +quotients. +Let us also denote +ξk := yk − xk +2ρk +∈ Sn−1. +Notice that +(2.37) +[∇˜uk]C0,α(B1/(2ρk)) ≤ 2, +and +|∇˜uk(ξk)| > δ◦ +2 , +where for the second inequality we use (2.34) and (2.35). +Since ˜uk are uniformly bounded in compact subsets, and bounded in the +C1,α norm (due to (2.36) and (2.37)), it follows by Arzel`a–Ascoli that the +sequence ˜uk converges (in the C1 norm) to a C1,α function ˜u on compact +subsets of Rn (up to a subsequence). Moreover, again up to a subsequence, +we have that ξk → ξ ∈ Sn−1. +By the properties of ˜uk, we deduce that ˜u satisfies +(2.38) +˜u(0) = |∇˜u(0)| = 0, +[∇˜u]C0,α(Rn) ≤ 2, +|∇˜u(ξ)| > δ◦ +2 . + +— DRAFT — +62 +2. Linear elliptic PDE +Let us check which equation does ˜uk satisfy. Let +˜a(k) +ij (x) := a(k) +ij (zk + ρkx), +so that, as in Proposition 2.26, ˜a(k) +ij +converges uniformly in compact sets to +some ˜aij constant. For any φ ∈ C∞ +c (B1), we know that +(2.39) +� +B1 +∇φ · Ak(x)∇uk = − +� +B1 +fkφ. +Let ˜Ak(x) := Ak(zk + ρkx) = (˜a(k) +ij (x))ij. Let φ ∈ C∞ +c (Rn), and let k be +large enough so that supp φ ⊂ B1/(2ρk). Let +� +∇φ · ˜Ak(x)∇˜uk = I − II, +where +I = +1 +ρα +k[∇uk]C0,α(B1) +� +∇φ(x) · Ak(zk + ρkx)∇uk(zk + ρkx) dx += +1 +ρα +k[∇uk]C0,α(B1) +� +∇y +� +φ +� +ρ−1 +k (y − zk) +�� +· Ak(y)∇uk(y)ρ−n+1 +k +dy += +−ρ1−α +k +[∇uk]C0,α(B1) +� +φ(x)fk(zk + ρkx) dx, +thanks to (2.39), and +II = +1 +ρα +k[∇uk]C0,α(B1) +� +∇φ(x) · Ak(zk + ρkx)∇uk(zk − ρkx) dx = IIi + IIii. +Here, we have denoted by IIi and IIii the following quantities: +IIi = +1 +ρα +k[∇uk]C0,α(B1) +� +∇φ(x)·(Ak(zk+ρkx)−Ak(zk−ρkx))∇uk(zk−ρkx) dx +and +IIii = +1 +ρα +k[∇uk]C0,α(B1) +� +∇φ(x) · Ak(zk − ρkx)∇uk(zk − ρkx) dx += +ρ1−α +k +[∇uk]C0,α(B1) +� +φ(x)fk(zk − ρkx) dx. +Let us now show that +���� +� +∇φ · ˜Ak∇˜uk +���� → 0, +as +k → ∞ +for all φ ∈ C∞ +c (Rn), by bounding each term separately. + +— DRAFT — +2.4. Schauder estimates for operators in divergence form +63 +Notice that, for 1 +q + 1 +q′ = 1, +���� +� +φ(x)fk(zk + ρkx) dx +���� ≤ +�� +|φ|q′� 1 +q′ �� +|fk(zk + ρkx)|q dx +� 1 +q +≤ C(φ)∥fk∥Lq(B1)ρ +− n +q +k +. +Then, +|I| ≤ C(φ)ρ +1−α− n +q +k +∥fk∥Lq(B1) +[∇uk]C0,α(B1) +≤ C(φ)ρ +1−α− n +q +k +k−1 → 0, +as +k → ∞, +as long as 1 − α − n +q ≥ 0, that is, q ≥ +n +1−α. In the last step we have used +(2.34). Similarly, +|IIii| → 0, +as +k → ∞, +since q ≥ +n +1−α. Finally, +|IIi| ≤ [Ak]C0,α(B1) +[∇uk]C0,α(B1) +� +|∇φ||x|α∥∇u∥L∞(B1) dx +≤ C(φ) ∥∇u∥L∞(B1) +[∇uk]C0,α(B1) +≤ C(φ) +k +→ 0, +as +k → ∞. +Here, we used again (2.34). That is, |IIi| → 0 uniformly in compact sets +of Rn. +Then we conclude that, for any φ ∈ C∞ +c (Rn), +���� +� +∇φ · ˜Ak(x)∇˜uk +���� → 0, +as +k → ∞. +By taking limits, up to a subsequence we will have that ˜Ak → ˜A uniformly +in compact sets, where ˜A is a constant coefficient matrix. Thus, we deduce +that +� +∇φ · ˜A∇˜u = 0 +for all +φ ∈ C∞ +c (Rn). +This means that, after a change of variables, ˜u is harmonic (recall Re- +mark 2.23). By Liouville’s theorem (Proposition 1.19) we obtain that ∇˜u +must be constant (since it is harmonic, and [∇˜u]C0,α(Rn) ≤ 2). However, +∇˜u(0) = 0 and ∇˜u(ξ) ̸= 0 (see (2.38)), a contradiction. +□ +Proof of Corollary 2.29. We proceed by induction on k. The case k = 0 +is due to Theorem 2.28. Then, let us assume that +(2.40) +∥u∥Ck+1,α(B1/2) ≤ C +� +∥u∥L∞(B1) + ∥f∥Ck−1,α(B1) +� +holds for all k ≤ m − 1, and let us show it for k = m. + +— DRAFT — +64 +2. Linear elliptic PDE +To do so, notice that, since aij(x) ∈ Cm,α, and m ≥ 1, we can compute +the derivatives in the divergence-form equation, to get +n +� +i,j=1 +aij(x)∂iju = f(x) − +n +� +i,j=1 +∂iaij(x)∂ju +in +B1, +that is, a non-divergence-form equation, where the right-hand side is in +Cm−1,α. +Applying the higher order Schauder estimates for equations in +non-divergence form, Corollary 2.21 (in balls B1/2 and B3/4), we get that +∥u∥Cm+1,α(B1/2) ≤ C +� +∥u∥L∞(B3/4) + ∥f∥Cm−1,α(B3/4)+ ++ +n +� +i,j=1 +∥∂i(aij)∥Cm−1,α(B3/4)∥∂ju∥Cm−1,α(B3/4) +� +, +that is, +∥u∥Cm+1,α(B1/2) ≤ C +� +∥u∥L∞(B3/4) + ∥f∥Cm−1,α(B3/4) + ∥u∥Cm,α(B3/4) +� +, +where the constant C depends only on n, α, λ, Λ, and ∥aij∥Cm,α(B1). Using +now the hypothesis induction, (2.40) for k = m − 1, in balls B3/4 and B1, +completes the proof. +□ +2.5. The case of continuous coefficients +Let us finish this chapter by studying equations in divergence and non- +divergence form with continuous coefficients. +In this section we establish a priori Schauder estimates for (2.1) and +(2.2) whenever aij ∈ C0(B1) (and the right-hand side is bounded or in Ln +respectively). This kind of estimates will be useful in the next chapters. +In this limiting case (when α ↓ 0), one could extrapolate from the pre- +vious results that the solution has respectively bounded C2 and C1 norm. +However, this is not true. +We will show, instead, that we gain almost two derivatives. Namely, for +any ε > 0, the solution has bounded C2−ε and C1−ε norm. More precisely, +we prove below the following results: +Proposition 2.31. Let u ∈ C2 be any solution to +n +� +i,j=1 +aij(x)∂iju = f(x) +in +B1, +with f ∈ L∞(B1) and aij ∈ C0(B1) for some (aij(x))ij satisfying (2.18) for +some 0 < λ ≤ Λ. Then, for any ε > 0, +∥u∥C1,1−ε(B1/2) ≤ Cε +� +∥u∥L∞(B1) + ∥f∥L∞(B1) +� + +— DRAFT — +2.5. The case of continuous coefficients +65 +for some constant Cε depending only on ε, n, λ, Λ, and (aij)ij. +That is, we are not gaining two full derivatives, but instead we are losing +an arbitrarily small factor. This loss is paired with the fact that the constant +Cε diverges when ε ↓ 0; see [JMV09, EM17] for counterexamples in the +case ε = 0. This is also consistent with what occurs with the Laplacian (see +the counterexample at the beginning of Section 2.2). +We remark that the dependence of C on (aij)i,j in the previous propo- +sition is a dependence on the modulus of continuity of (aij)i,j. That is, if +ω : [0, ∞) → [0, ∞) is a continuous monotone function with ω(0) = 0 and +such that +|aij(x) − aij(y)| ≤ ω(|x − y|), +for all +x, y ∈ B1, +then the constant in the previous proposition depends on ω rather than +on (aij)i,j. +For divergence-form equations we have the following: +Proposition 2.32. Let u ∈ C1 be a weak solution to +n +� +i,j=1 +∂i +� +aij(x)∂ju +� += f(x) +in +B1, +with f ∈ Ln(B1) and aij(x) ∈ C0(B1) satisfying the ellipticity conditions +(2.32) for some 0 < λ ≤ Λ. Then, for any ε > 0, +∥u∥C1−ε(B1/2) ≤ Cε +� +∥u∥L∞(B1) + ∥f∥L∞(B1) +� +for some constant Cε depending only on ε, n, λ, Λ, and (aij)ij. +The proofs of the previous two propositions are analogous to those of +the Schauder estimates for operators in non-divergence and divergence form +respectively. +We give short sketches of the proofs of Propositions 2.31 and 2.32 that +contain all the essential information regarding the steps to take. +Sketch of the proof of Proposition 2.31. We give a short sketch of the +proof in the case (aij(x))ij = Id, and leave the details to the reader. The +proof sketched follows the same steps and arguments as the second proof of +Proposition 2.26. +Proceeding analogously, and after using Lemma 2.27, (cf. first or second +proof of Theorem 2.20), we just need to show that for any δ > 0 +[∇u]C1−ε(B1/2) ≤ δ[∇u]C1−ε(B1) + Cδ(∥∇u∥L∞(B1) + ∥f∥L∞(B1)), +for some Cδ. + +— DRAFT — +66 +2. Linear elliptic PDE +By contradiction, suppose that we have a sequence fk ∈ L∞(B1), uk ∈ +C2(B1), and coefficients (a(k) +ij )ij with a common modulus of continuity, such +that �n +i,j=1 a(k) +ij ∂ijuk = fk and +(2.41) [∇uk]C1−ε(B1/2) > δ◦[∇uk]C1−ε(B1) + k(∥∇uk∥L∞(B1) + ∥fk∥L∞(B1)), +for some δ◦ > 0. +Select xk, yk ∈ B1/2 such that +(2.42) +|∇uk(xk) − ∇uk(yk)| +|xk − yk|1−ε +≥ 1 +2[∇uk]C1−ε(B1/2) +and let ρk := |xk − yk|, so that as in the second proof of Proposition 2.26 +ρk ↓ 0. Define +˜uk(x) := uk(xk + ρkx) − uk(xk) − ρk∇uk(xk) · x +ρ2−ε +k +[∇uk]C1−ε(B1) +and +˜fk(x) := ρε +k +fk(xk + ρkx) − fk(xk) +[∇uk]C1−ε(B1) +, +so that +(2.43) +˜uk(0) = |∇˜uk(0)| = 0, +n +� +i,j=1 +˜a(k) +ij ∂ij ˜uk = ˜fk +in +B1/(2ρk). +where +˜a(k) +ij (x) := a(k) +ij (zk + ρkx). +Denoting ξk := yk−xk +ρk +∈ Sn−1, we have +[∇˜uk]C1−ε� +B +1 +2ρk +� ≤ 1, +and +��∇˜uk(ξk) +�� > δ◦ +2 , +by means of (2.41) and (2.42). +As in Proposition 2.26, ˜uk converges (up to a subsequence and in the C1 +norm) to a C1,1−ε function ˜u on compact subsets of Rn, and ξk → ξ ∈ Sn−1. +Furthermore, +˜u(0) = |∇˜u(0)| = 0, +[∇˜u]C1−ε(Rn) ≤ 1, +|∇˜u(ξ)| > δ◦ +2 . +On the other hand, for any R ≥ 1 we have +∥ ˜fk∥L∞(BR) ≤ ρε +k +k → 0, as k → ∞, +and that, from the uniform modulus of continuity of a(k) +ij , ˜a(k) +ij (x) → ˜aij +locally uniformly in Rn, where the limiting coefficients ˜aij are constant. At +this point, in the equation (2.43) the coefficients converge locally uniformly +to constant coefficients, and the solutions ˜uk converge simply in C1. The + +— DRAFT — +2.5. The case of continuous coefficients +67 +passage to the limit is now more involved than before: in order to do it, we +need the notion of viscosity solutions (see Definition 1.20 and Section 4.3) +and the fact that they are stable under uniform limits (see Proposition 4.20). +In all, we can show that the limiting ˜u satisfies +n +� +i,j=1 +˜aij∂ij ˜u = 0 +in +Rn +(in the viscosity sense). Hence, the limiting solution ˜u is harmonic (after +changing variables) and we reach a contradiction as in the second proof of +Proposition 2.26. +□ +In order to prove the convergence of the sequence in the proof of Propo- +sition 2.32 we will need the following lemma: +Lemma 2.33. Let u ∈ H1(B1) satisfy +(2.44) +div(A(x)∇u(x)) = f(x) +in +B1, +in the weak sense, for some f ∈ L2(B1) and A(x) = (aij(x))ij uniformly +elliptic with ellipticity constants λ and Λ (see (2.18)). Then +∥∇u∥L2(B1/2) ≤ C(∥u∥L2(B1) + ∥f∥L2(B1)) +for some C depending only on λ, Λ, and n. +Proof. Let us prove the lemma in the case A(x) is symmetric for all x ∈ B1. +Let η ∈ C∞ +c (B1) be arbitrary with η ≡ 1 in B1/2, and observe that +� +B1/2 +|∇u|2 ≤ C +� +B1 +∇(uη) · A(x)∇(uη) dx +by ellipticity. In particular, since A(x) is symmetric for all x ∈ B1 we can +use that ∇(uη) · A∇(uη) = u2∇η · A∇η + ∇(uη2) · A∇u and the equation +(2.44) to get +� +B1/2 +|∇u|2 ≤ C +� +B1 +u2|∇η|2 + C +� +B1 +|fu|η2. +By H¨older’s inequality, we get the desired estimate. We refer to the proof +of Lemma 3.8 for more details on the proof and on the non-symmetric case +in a very similar situation. +□ +Let us now give the proof of Proposition 2.32. +Proof of Proposition 2.32. The proof is by contradiction and proceeds +as the proof of Theorem 2.28, with the analogous modifications introduced +in the Sketch of the proof of Proposition 2.31 with respect to the proof of +Proposition 2.26. + +— DRAFT — +68 +2. Linear elliptic PDE +Observe that, in this case, we should define ˜uk(x) as first order incre- +mental quotients: +˜uk(x) = uk(xk + ρkx) − uk(xk) +ρ1−ε +k +[uk]C1−ε(B1) +so that we directly have (differently from the proof of Theorem 2.28) that +˜uk satisfies: +(2.45) +div( ˜Ak(x)∇˜uk(x)) = ˜fk(x) +in +B1/(2ρk), +˜fk(x) = ρ1+ε +k +fk(xk + ρkx) +[uk]C1−ε(B1) +, +in the weak sense, where ˜Ak(x) := Ak(xk + ρkx) and ∥ ˜fk∥Ln(B1/(2ρk)) ↓ 0 as +k → ∞. In particular, ˜Ak(x) converges to some constant matrix ˜A∞ locally +uniformly by uniform continuity of Ak. +On the other hand, observe that each ˜uk is in H1 (since they are C1 by +assumption), and they are locally uniformly in L2 (since they are uniformly +locally bounded). Hence, we can apply Lemma 2.33 to get that ˜uk are locally +uniformly bounded in H1. In particular, by (S4) from Chapter 1 (see (1.3)) +∇˜uk converges weakly to ∇˜u∞. Thus: +� +Rn ∇φ · ˜Ak∇˜uk → +� +Rn ∇φ · ˜A∞∇˜u∞ +as +k → ∞, +for all φ ∈ C∞ +c (Rn), +and from (2.45) we have that u∞ is harmonic (after changing variables) +in Rn. The contradiction is now reached, again, by the Liouville theorem, +Proposition 1.19. +□ +Remark 2.34. The blow-up technique is a common tool in analysis that +has great versatility. In particular, the technique presented in this section +is due to L. Simon, [Sim97], and can be applied in a similar fashion to +many different situations. We have seen the technique applied in interior +a priori estimates for linear second-order equations, both in divergence and +non-divergence form, and blow-up arguments like the one presented above +can be adapted also to boundary estimates, parabolic equations, nonlinear +equations, and even integro-differential equations. +2.6. Boundary regularity +We finish the chapter by stating the corresponding results to Corollaries 2.16 +and 2.17 for the global (up to the boundary) estimates, for a sufficiently +smooth domain. +For the sake of readability we state the result for the Laplacian, but there +exists an analogous result for uniformly elliptic equations in non-divergence +form (with the corresponding regularity on the coefficients). + +— DRAFT — +2.6. Boundary regularity +69 +Theorem 2.35 (Boundary regularity). Let α ∈ (0, 1) and k ∈ N with k ≥ 2, +and let Ω be a bounded Ck,α domain of Rn. Let u ∈ H1(Ω) be a weak solution +to +(2.46) +� ∆u += +f +in Ω +u += +g +on ∂Ω, +for some f ∈ Ck−2,α(Ω), g ∈ Ck,α(∂Ω). +Then, u ∈ Ck,α(Ω) and +∥u∥Ck,α(Ω) ≤ C +� +∥f∥Ck−2,α(Ω) + ∥g∥Ck,α(∂Ω) +� +, +for some constant C depending only on α, n, k, and Ω. +Remark 2.36. Notice that in this case we do not need a term ∥u∥L∞(Ω) on +the right-hand side because, thanks to the maximum principle (Lemma 2.25), +max +Ω +u ≤ C +� +max +∂Ω g + ∥f∥L∞(Ω) +� +for some C depending only on Ω, λ, Λ, and M. +Theorem 2.35 can be proved using similar techniques (correspondingly +adapted) to the ones in the previous sections: after a blow-up, points near +the boundary behave like in a local problem in the half-space (that is, the +blow-up flattens ∂Ω), and we can reach a contradiction with Liouville’s +theorem in the half-space. +One might wonder what happens under lower regularity assumptions +on the domain (we refer to [Ken94, Kry96] for further reading in this +direction). In such case, similar regularity results hold in C1,α (and even +C1) domains, but when Ω is merely Lipschitz, almost all regularity is lost. +Namely, assume that u solves (2.46), with f and g smooth enough. Then, +• If Ω is a C1,α domain, then solutions are C1,α(Ω). +• If Ω is a C1 domain, then solutions are C1−ε(Ω) for all ε > 0, but +not C0,1(Ω) in general. +• If Ω is a Lipschitz domain, then solutions are Cγ(Ω) for some small +γ > 0 that depends on the Lipschitz norm of the domain, and this +is optimal. +We see that, if Ω is a Lipschitz domain, then essentially all regularity is +lost. If one thinks on the blow-up and compactness method, it is clear that +Lipschitz domains are quite different from C1. Indeed, Lipschitz domains +do not get flatter by doing a blow-up (they remain Lipschitz, with the +same Lipschitz norm). +Thus, one cannot improve regularity by blowing +up. Solutions turn out to be Cγ for some small γ > 0 and, in general, not +better. + +— DRAFT — + +— DRAFT — +Chapter 3 +Nonlinear variational +PDE & Hilbert’s +XIXth problem +Eine der begrifflich merkw¨urdigsten Thatsachen in den Elementen der Theorie der +analytischen Functionen erblicke ich darin, daß es partielle Differentialgleichungen +giebt, deren Integrale s¨amtlich notwendig analytische Funktionen der unabh¨angigen +Variabeln sind, die also, kurz gesagt, nur analytischer L¨osungen f¨ahig sind. +— David Hilbert (1900). +Up until this point, we have studied linear elliptic PDEs. In this chapter +we start the study of nonlinear elliptic PDEs. +More precisely, we study variational nonlinear PDEs, that is, those that +appear in the Calculus of Variations (minimizing an energy functional). In +particular, our main goal is to introduce and solve Hilbert’s XIXth problem1. +• Hilbert’s XIXth problem (1900): Consider any local min- +imizer of energy functionals of the form +E(w) := +� +Ω +L(∇w) dx, +where L : Rn → R is smooth and uniformly convex, and Ω ⊂ Rn. +Is is true that all local minimizers to this type of problems are +smooth? +1The original statement by Hilbert says that “there exist partial differential equations whose +integrals are all of necessity analytic functions of the independent variables, that is, in short, +equations susceptible of none but analytic solutions”, and refers to solutions to what he calls +“regular variational problems”, involving convex (in ∇w) and analytic operators of the form +L(∇w, w, x). We deal here with L(∇w) for simplicity. +71 + +— DRAFT — +72 +3. Nonlinear variational PDE & Hilbert’s XIXth problem +Notice that, given a boundary condition +u = g +on +∂Ω, +one can show that there is a unique minimizer to this problem, u ∈ H1(Ω), +with u|∂Ω = g. +That is, there exists a unique u ∈ H1(Ω) such that u +minimizes the functional E(w) := +� +Ω L(∇w) dx, among all functions w ∈ +H1(Ω) such that w|∂Ω = g. We will be more precise about this in the first +two sections of this chapter. +The question in Hilbert’s XIXth problem is that of regularity: Is such +minimizer u smooth? +Remark 3.1 (On the convexity assumption). The uniform convexity of +the function is what gives us existence and uniqueness of a minimizer (see +Theorem 3.3 below). Moreover, from the point of view of regularity, if L is +not convex and reaches its minimum at two different points, then even in +dimension n = 1 there exist counterexamples to regularity. +If n = 1 and L has a minimum at two points p1 < p2, then we can +construct Lipschitz only minimizers zigzagging with slopes p1 and p2 (e.g., +if p1 = −1 and p2 = 1, then u(x) = |x| would be a minimizer). +Thus, the convexity assumption is needed. +3.1. Overview +Hilbert’s XIXth problem as posed above is a generalization of the minimiza- +tion of the Dirichlet integral, +� +Ω +|∇w|2 dx. +Local minimizers of the Dirichlet integral verify the corresponding Euler– +Lagrange equation, which in this case is the Laplace equation +∆w = 0 +in +Ω. +Solutions to this PDE, as seen in Chapter 2, are known to be C∞ in the +interior of Ω. +Thus, the Dirichlet integral case L(p) = |p|2 is extremely simple. Sur- +prisingly, the general case is far more difficult, and its resolution took more +than 50 years. +First, let us be more precise about the problem: by a local minimizer of +E(w) = +� +Ω L(∇w) dx, we mean a function u ∈ H1(Ω) such that +E(u) ≤ E(u + φ) +for all +φ ∈ C∞ +c (Ω). +The uniform convexity of the functional is equivalent to +(3.1) +0 < λId ≤ D2L(p) ≤ ΛId +for all +p ∈ Rn, + +— DRAFT — +3.1. Overview +73 +(i.e., uniform convexity of L). Notice the analogy with the uniform ellipticity +from the previous chapter. +Now, what is the PDE satisfied by minimizers of E(u)? (Namely, the +Euler–Lagrange equation of the problem.) If u ∈ H1(Ω) is a local minimizer, +then +E(u) ≤ E(u + εφ) +for all φ ∈ C∞ +c (Ω), +and all ε ∈ R. +Hence, +� +Ω +L(∇u) dx ≤ +� +Ω +L(∇u + ε∇φ) dx +for all φ ∈ C∞ +c (Ω), +and all ε ∈ R, +and thus, as a function of ε, it has a minimum at ε = 0. Taking derivatives +in ε we reach +0 = d +dε +���� +ε=0 +� +Ω +L(∇u + ε∇φ) dx = +� +Ω +DL(∇u)∇φ dx. +The weak formulation of the Euler–Lagrange equation is then +(3.2) +� +Ω +DL(∇u)∇φ dx = 0 +for all φ ∈ C∞ +c (Ω). +That is, u solves in the weak sense the PDE +(3.3) +div (DL(∇u)) = 0 +in +Ω. +(This derivation will be properly justified in Theorem 3.3 below.) +If u is C2, (3.3) is equivalent to +(3.4) +n +� +i,j=1 +(∂ijL)(∇u)∂iju = 0 +in +Ω. +By uniform convexity of L, this is a (nonlinear) uniformly elliptic PDE. +What can we say about the regularity of u? +Regularity of local minimizers: First approach. Let us assume that +u is smooth enough so that it solves (3.4). We can regard (3.4) as a linear +equation with variable coefficients, by denoting +aij(x) := (∂ijL)(∇u(x)), +and we notice that, by uniform convexity of L, we have +0 < λ Id ≤ (aij(x))ij ≤ Λ Id. +Moreover, if ∇u ∈ C0,α, then aij ∈ C0,α. In particular, using Schauder +estimates (see Theorem 2.20), we have +(3.5) +u ∈ C1,α ⇒ aij ∈ C0,α ⇒ u ∈ C2,α. + +— DRAFT — +74 +3. Nonlinear variational PDE & Hilbert’s XIXth problem +We can then bootstrap the regularity and get C∞: +u ∈ C2,α ⇒ ∇u ∈ C1,α ⇒ aij ∈ C1,α ⇒ u ∈ C3,α ⇒ · · · ⇒ u ∈ C∞. +In fact, using the linear estimates for continuous coefficients, one can +actually get u ∈ C1 ⇒ aij ∈ C0 ⇒ u ∈ C1,α. We remark that while the +previous implications are true at a formal level, we did not properly argue the +use of Schauder estimates. Indeed, our results for Schauder estimates in both +non-divergence form (Theorem 2.20) and divergence form (Theorem 2.28) +are a priori, i.e., they already assume regularity on u. We show how to use +them in Theorem 3.5 below to prove the results we want and expect. +Equations with bounded measurable coefficients. We have argued +that using perturbative results for linear equations (Schauder estimates), +one expects to prove that +u ∈ C1 +=⇒ +u ∈ C∞. +However, this approach does not allow us to prove any regularity if we +do not know a priori that u ∈ C1. The main open question in Hilbert’s +XIXth problem was then +is it true that u ∈ H1 ⇒ u ∈ C1 ? +This problem was open for many years, and it was finally solved (inde- +pendently and almost at the same time) by De Giorgi [DeG57] and Nash +[Nas57, Nas58]. +Theorem 3.2 (De Giorgi–Nash). Let u be a local minimizer of +E(w) = +� +Ω +L(∇w) dx, +with L uniformly convex and smooth. Then, u ∈ C1,α for some α > 0. +This theorem solved Hilbert’s XIXth problem. +In order to show regularity of local minimizers u of E(w) = +� +Ω L(∇w) dx, +with w ∈ H1(Ω), we first notice that they solve (in the weak sense) the +nonlinear elliptic equation +div (DL(∇u)) = 0 +in +Ω. +The first idea in the proof is to consider derivatives of u, v = ∂eu, and +to show that they solve an elliptic PDE as well. +If we differentiate the equation div (DL(∇u)) = 0 with respect to e ∈ +Sn−1, we get +div +� +D2L(∇u)∇∂eu +� += 0 +in +Ω. + +— DRAFT — +3.2. Existence and basic estimates +75 +Denoting (as before) v := ∂eu, aij(x) := ∂ijL(∇u(x)) and A(x) := (aij(x))ij, +we can write this equation as +div (A(x)∇v) = 0 +in +Ω. +This is a linear, uniformly elliptic equation in divergence form, but we +do not have any regularity of A(x) in the x-variable. We only know that +the equation is uniformly elliptic. +This is called a (uniformly elliptic) equation in divergence form with +bounded measurable coefficients. (Recall that the uniform convexity of L +yields 0 < λId ≤ A(x) ≤ ΛId.) +De Giorgi and Nash established a new regularity result for such type of +equations, see Theorem 3.7. +The aim of this Chapter is to provide a complete and detailed proof of the +solution to Hilbert’s XIXth problem. We will follow De Giorgi’s approach. +3.2. Existence and basic estimates +We start by showing the existence and uniqueness of minimizers of E among +the class of H1(Ω) functions with prescribed boundary data. That is, we +want a statement analogous to Theorem 1.10, but with the functional in- +volving L instead. We recall that we denote by u|∂Ω the trace of u on ∂Ω; +see (S5) in Chapter 1. +Theorem 3.3 (Existence and uniqueness of minimizers). Assume that Ω ⊂ +Rn is any bounded Lipschitz domain, and that +(3.6) +� +w ∈ H1(Ω) : w|∂Ω = g +� +̸= ∅. +Let L : Rn → R be smooth and uniformly convex, see (3.1). Let +E(w) := +� +Ω +L(∇w) dx. +Then, there exists a unique minimizer u ∈ H1(Ω) with u|∂Ω = g. Moreover, +u solves (3.3) in the weak sense. +In order to prove the existence and uniqueness theorem for minimizers, +we need first to show the following result on the lower semi-continuity of the +energy in this context. We provide two different proofs. +Lemma 3.4 (Lower semi-continuity of the functional). Let Ω ⊂ Rn be a +bounded domain. Let L : Rn → R be smooth and uniformly convex, see +(3.1); and let +E(w) := +� +Ω +L(∇w) dx. + +— DRAFT — +76 +3. Nonlinear variational PDE & Hilbert’s XIXth problem +Then, E is weakly lower semi-continuous in H1(Ω). That is, if H1(Ω) ∋ +wk ⇀ w ∈ H1(Ω) weakly in H1(Ω), then +E(w) ≤ lim inf +k→∞ E(wk). +First proof. Let us define the set +A(t) := +� +v ∈ H1(Ω) : E(v) ≤ t +� +. +Notice that, by convexity of E, A(t) is convex as well. Let us show that it +is closed, i.e., if A(t) ∋ wk → w strongly in H1(Ω), then w ∈ A(t). This +simply follows by noticing that, up to a subsequence, ∇wk → ∇w almost +everywhere, so that, by Fatou’s lemma, +E(w) = +� +Ω +L(∇w) ≤ lim inf +k→∞ +� +Ω +L(∇wk) ≤ t, +that is, w ∈ A(t). Therefore, A(t) is closed (with respect to the H1(Ω) +convergence), and it is convex. By a standard result in functional analysis +(closed and convex sets are weakly closed; see, for example, [Bre11, The- +orem 3.7]), A(t) is also closed under weak convergence; namely, if A(t) ∋ +wk ⇀ w weakly in H1(Ω) then w ∈ A(t). +Let us now consider a sequence weakly converging in H1(Ω), wk ⇀ w, +and let us denote t∗ := lim infk→∞ E(wk). For any ε > 0, there exists some +subsequence kj,ε such that wkj,ε ⇀ w weakly in H1(Ω) and E(wkj,ε) ≤ t∗ +ε. +That is, wkj,ε ∈ A(t∗ + ε), and therefore, since A(t) is weakly closed (in +H1(Ω)) for all t, we have w ∈ A(t∗ +ε) and E(w) ≤ t∗ +ε. Since this can be +done for any ε > 0, we reach that E(w) ≤ t∗, and therefore, we have shown +the weak lower semi-continuity of E in H1(Ω). +□ +Second proof. Let us prove the lower semi-continuity of the functional by +means of a different proof, from [Mag11]. We will actually show that if +uk, u ∈ W 1,1(Ω) and uk → u in L1 +loc(Ω), then +� +Ω +L(∇u) ≤ lim inf +k→∞ +� +Ω +L(∇uk). +In particular, since Ω is bounded, we can apply this result to the sequences +in H1(Ω) converging weakly in H1(Ω) (by (S2) from Chapter 1). Let η ∈ +C∞ +c (B1) be a smooth function with η ≥ 0 and +� +B1 η = 1, and let ηε(x) = +ε−nη(x/ε), so that we can consider the mollifications +(uk)ε(x) := (uk ∗ ηε)(x) = +� +Bε +u(x − y)ηε(y) dy, +uε(x) := (u ∗ ηε)(x). +Let Ω′ ⊂ Ω be such that for all x ∈ Ω′, Bε(x) ⊂ Ω. In particular, since +uk → u in L1 +loc(Ω), we have ∇(uk)ε(x) → ∇uε(x) for every x ∈ Ω′. From + +— DRAFT — +3.2. Existence and basic estimates +77 +the smoothness of L we also have that L(∇(uk)ε(x)) → L(∇uε(x)) and by +Fatou’s lemma (recall that we may assume L ≥ 0) +(3.7) +� +Ω′ L(∇uε) ≤ lim inf +k→∞ +� +Ω′ L(∇(uk)ε). +Noticing now that ∇(uk)ε = (∇uk)ε and using Jensen’s inequality (since L +is convex and +� +ηε = 1) we have +L(∇(uk)ε) = L +�� +Bε(x) +ηε(x − y)∇uk(y) dy +� +≤ +� +Bε(x) +ηε(x−y)L(∇uk(y)) dy +which leads to +� +Ω′ L(∇(uk)ε) ≤ +� +Ω′ +�� +Bε(x) +ηε(x − y)L(∇uk(y)) dy +� +dx +≤ +� +Iε(Ω′) +L(∇uk(y)) +� +Bε(y)∩Ω′ ηε(x − y) dx dy ≤ +� +Ω +L(∇uk), +where Iε(Ω′) ⊂ Ω denotes an ε-neighborhood of Ω′. Combined with (3.7), +this yields +� +Ω′ L(∇uε) ≤ lim inf +k→∞ +� +Ω +L(∇uk). +Now, since u ∈ W 1,1(Ω), we have ∇uε → ∇u as ε ↓ 0 almost everywhere +in Ω′ and so, again by Fatou’s Lemma, we can let ε ↓ 0 to deduce +� +Ω′ L(∇u) ≤ lim inf +k→∞ +� +Ω +L(∇uk). +By taking an increasing sequence of sets Ω′ whose union is Ω we reach the +desired result. +□ +We can now prove Theorem 3.3. +Proof of Theorem 3.3. We divide the proof into three different parts. +Step 1. If u is a local minimizer, then it solves (3.3) in the weak sense. This +follows from the fact that +� +Ω +L(∇u) dx ≤ +� +Ω +L(∇u + ε∇φ) dx +for all ε, +for all φ ∈ C∞ +c (Ω). +Indeed, notice that the integrals are bounded (L being uniformly convex, +i.e., at most quadratic at infinity, and ∇u ∈ L2). Since L is smooth, we can +take a Taylor expansion +L(∇u + ε∇φ) ≤ L(∇u) + εDL(∇u)∇φ + ε2 +2 |∇φ|2 sup +p∈Rn +��D2L(p) +�� . + +— DRAFT — +78 +3. Nonlinear variational PDE & Hilbert’s XIXth problem +Recalling from (3.1) that +��D2L +�� is bounded by Λ, and plugging it back into +the integral we obtain +−Λε +2|∇φ|2 ≤ +� +Ω +DL(∇u)∇φ dx +for all ε > 0, +for all φ ∈ C∞ +c (Ω). +Letting ε go to zero, we reach that +� +Ω +DL(∇u)∇φ dx ≥ 0 +for all φ ∈ C∞ +c (Ω). +On the other hand, taking −φ instead of φ, we reach the equality (3.2), as +we wanted to see. +Step 2. Let us now show the existence of a solution. +Since L is uniformly convex (see (3.1)) it has a unique minimum. That +is, there exists pL ∈ Rn such that L(p) ≥ L(pL) for all p ∈ Rn. In particular, +since L is smooth, ∇L(pL) = 0 and thus, from the uniform convexity (3.1) +we have that +0 < λ|p|2 ≤ L(p − pL) − L(pL) ≤ Λ|p|2, +for all +p ∈ Rn. +Without loss of generality, by taking ˜L(p) = L(p − pL) − L(pL) if neces- +sary, we may assume that L(0) = 0 and ∇L(0) = 0, so that we have +(3.8) +0 < λ|p|2 ≤ L(p) ≤ Λ|p|2, +for all +p ∈ Rn. +(Notice that we may assume that because if u is a minimizer for L, then +u + ⟨pL, x⟩ is a minimizer for ˜L, since the domain is bounded and therefore +the integral of L(pL) is finite.) +Let +E◦ = inf +�� +Ω +L(∇w) dx : w ∈ H1(Ω), w|∂Ω = g +� +, +that is, the infimum value of E(w) among all admissible functions w. Notice +that, by assumption (3.6), such infimum exists. Indeed, if w ∈ H1(Ω), by +(3.8) we have that +E(w) = +� +Ω +L(∇w) ≤ Λ +� +Ω +|∇w|2 = Λ∥∇w∥2 +L2(Ω) < ∞ +that is, the energy functional is bounded for functions in H1(Ω). +Let us take a minimizing sequence of functions. That is, we take {uk} +such that uk ∈ H1(Ω), uk|∂Ω = g, and E(uk) → E◦ as k → ∞. We begin +by showing that E(uk) are bounded, and that uk is a sequence bounded in +H1(Ω). By (3.8), +λ∥∇uk∥2 +L2(Ω) ≤ λ +� +Ω +|∇uk|2 ≤ +� +Ω +L(∇uk) ≤ E(uk) < ∞, + +— DRAFT — +3.2. Existence and basic estimates +79 +That is, since E(uk) is uniformly bounded (being a convergent sequence +with non-infinite elements), we reach that ∥∇uk∥2 +L2(Ω) is uniformly bounded. +Thus, by the Poincar´e inequality (see Theorem 1.6) the sequence uk is uni- +formly bounded in H1(Ω). +In particular, there exists a subsequence ukj converging strongly in L2(Ω) +and weakly in H1(Ω) to some u ∈ H1(Ω), uk ⇀ u weakly in H1(Ω). By the +weak lower semi-continuity (Lemma 3.4) we reach that +E(u) ≤ lim inf +k→∞ E(uk) = E◦, +so that E(u) = E◦ (by minimality) and therefore u is a minimizer. +Step 3. We finish the proof by showing the uniqueness of such minimizer. +This follows from the uniform convexity. Indeed, since L is uniformly +convex, if p ̸= q, then +L(p) + L(q) +2 +> L +�p + q +2 +� +. +Let u, v ∈ H1(Ω) be two distinct minimizers with the same boundary data +(E(u) = E(v) = E◦). In particular, ∇u ̸≡ ∇v in Ω, so that U := {x ∈ Ω : +∇u ̸= ∇v} ⊂ Ω has positive measure. Thus, +L(∇u) + L(∇v) +2 +> L +�∇u + ∇v +2 +� +in +U, +so that, since |U| > 0, +1 +2 +� +U +� +L(∇u) + L(∇v) +� +> +� +U +L +�∇u + ∇v +2 +� +. +Since the integrals are equal in Ω \ U, we reach +E◦ = 1 +2 +� +Ω +� +L(∇u) + L(∇v) +� +> +� +Ω +L +�∇u + ∇v +2 +� +≥ E◦, +where the last inequality comes from the minimality of E◦. We have reached +a contradiction, and thus, the minimizer is unique. +□ +We next give a complete and rigorous proof of the formal argumentation +from the previous section, where we explained that C1 solutions are C∞. +Theorem 3.5. Let u ∈ H1(Ω) be a local minimizer of +E(w) = +� +Ω +L(∇w) dx, +with L uniformly convex and smooth. Assume that u ∈ C1. Then u ∈ C∞. + +— DRAFT — +80 +3. Nonlinear variational PDE & Hilbert’s XIXth problem +Proof. We know that if u ∈ C1 and u is a minimizer of E(w), then +� +Ω +DL(∇u(x))∇φ(x) dx = 0 +for all φ ∈ C∞ +c (Ω). +Let h ∈ Rn, and assume that |h| is small. We have, in particular, that +� +Ω +� +DL(∇u(x + h)) − DL(∇u(x)) +� +∇φ(x) dx = 0 +for all φ ∈ C∞ +c (Ωh), +where Ωh := {x ∈ Ω : dist(x, ∂Ω) > |h|}. Notice that, by the fundamental +theorem of calculus for line integrals, we can write +DL(∇u(x + h)) − DL(∇u(x)) = += +� 1 +0 +D2L +� +t∇u(x + h) + (1 − t)∇u(x) +�� +∇u(x + h) − ∇u(x) +� +dt. +If we define +˜A(x) := +� 1 +0 +D2L +� +t∇u(x + h) + (1 − t)∇u(x) +� +dt, +then ˜A(x) is uniformly elliptic (since L is uniformly convex), and continuous +(since L is smooth and ∇u is continuous). Then, by the previous argumen- +tation, +� +Ω +∇ +�u(x + h) − u(x) +|h| +� +· ˜A(x)∇φ(x) dx = 0 +for all φ ∈ C∞ +c (Ωh), +that is, u(·+h)−u +|h| +solves weakly +div +� +˜A(x)∇ +�u(x + h) − u(x) +|h| +�� += 0 +for x ∈ Ωh. +Moreover, notice that u(·+h)−u +|h| +is C1 for all h ̸= 0, since u is C1. Thus, by +the Schauder-type estimates for operators in divergence form and continuous +coefficients(Proposition 2.32), +���� +u(· + h) − u +|h| +���� +Cβ(Bρ/2(x◦)) +≤ C(ρ) +���� +u(· + h) − u +|h| +���� +L∞(Bρ(x◦)) +≤ C, +for all Bρ(x◦) ⊂ Ωh and β ∈ (0, 1). In the last inequality we used that +∇u is continuous (and thus, bounded). Notice that the constant C(ρ) is +independent of h (but might depend on β). In particular, from (H7) in +Chapter 1, namely (1.7) with α = 1, we obtain that u ∈ C1,β(Ωh) for all +h ∈ Rn. Letting |h| ↓ 0 we get that u ∈ C1,β inside Ω. +We want to repeat the previous reasoning, noticing now that ˜A(x) is +C0,β (since ∇u ∈ C0,β and L is smooth). That is, u(·+h)−u +|h| +∈ C1,β(Ω) and + +— DRAFT — +3.3. De Giorgi’s proof +81 +fulfills +div +� +˜A(x)∇ +�u(x + h) − u(x) +|h| +�� += 0 +for x ∈ Ωh, +in the weak sense, with ˜A ∈ Cβ and uniformly elliptic. By Theorem 2.28, +���� +u(· + h) − u +|h| +���� +C1,β(Bρ/2) +≤ C(ρ) +���� +u(· + h) − u +|h| +���� +L∞(Bρ) +≤ C, +for all Bρ ⊂ Ωh, and again, thanks to (H7), (1.7), we obtain that u ∈ +C2,β(Ω). +We can now proceed iteratively using the higher order interior +Schauder estimates in divergence form (Corollary 2.29) to obtain that u ∈ +Ck(Ω) for all k ∈ N, i.e, u ∈ C∞ inside Ω. +□ +Remark 3.6. Notice that in the formal proof (3.5) we were using Schauder +estimates in non-divergence form, since we were already assuming that the +solution u was C2. Here, in the proof of Theorem 3.5, we need to differentiate +the equation (in incremental quotients) and then we obtain an equation in +divergence form whose coefficients have the right regularity. Thus, in the +actual proof we are using Schauder estimates for equations in divergence +form instead. +3.3. De Giorgi’s proof +The result of De Giorgi and Nash regarding the regularity of solutions to +equations with bounded measurable coefficients is the following (see the +discussion in Section 3.1). +Theorem 3.7 (De Giorgi–Nash). Let v ∈ H1(Ω) be any weak solution to +(3.9) +div (A(x)∇v) = 0 +in +Ω, +with 0 < λId ≤ A(x) ≤ ΛId. Then, there exists some α > 0 such that +v ∈ C0,α(˜Ω) for any ˜Ω ⊂⊂ Ω, with +∥v∥C0,α(˜Ω) ≤ C∥v∥L2(Ω). +The constant C depends only on n, λ, Λ, Ω, and ˜Ω. The constant α > 0 +depends only on n, λ, and Λ. +This theorem yields Theorem 3.2, and combined with previous discus- +sions, solved Hilbert’s XIXth problem. Indeed, if u ∈ H1(Ω) is any local +minimizer of E(w) = +� +Ω L(∇w) dx, then any derivative of u, v = ∂eu, solves +(3.9). +Thanks to Theorem 3.7 we will have +u ∈ H1(Ω) ⇒ v ∈ L2(Ω) ======⇒ +DeGiorgi +−Nash +v ∈ C0,α(˜Ω) ⇒ u ∈ C1,α ======⇒ +Schauder +u ∈ C∞. + +— DRAFT — +82 +3. Nonlinear variational PDE & Hilbert’s XIXth problem +This will be proved in detail in Section 3.4. +Theorem 3.7 is significantly different in spirit than all the results on +elliptic regularity which existed before. +Most of the previous results can be seen as perturbation of the Laplace +equation (they are perturbative results). In Schauder-type estimates, we +always use that, when zooming in a solution at a point, the operator gets +closer and closer to the Laplacian. +In De Giorgi’s theorem, this is not true anymore. The uniform ellipticity +is preserved by scaling, but the equation is not better, nor closer to the +Laplace equation. +General ideas of the proof. We will follow the approach of De Giorgi. +From now on, we denote L any operator of the form +(3.10) +Lv := −div(A(x)∇v), +where A(x) is uniformly elliptic +with ellipticity constants 0 < λ ≤ Λ. +By a standard covering argument (cf. Remark 2.15), we only need to +prove the estimate for Ω = B1 and ˜Ω = B1/2. +Throughout the proof, we will use that, if v solves Lv = 0, then ˜v(x) := +Cv(x◦+rx) solves an equation of the same kind, ˜L˜v = 0, for some operator ˜L +with the same ellipticity constants as L — given by ˜L˜v = div +� +A(x◦+rx)∇˜v +� +. +De Giorgi’s proof is split into two steps: +First step: Show that ∥v∥L∞ ≤ C∥v∥L2 +Second step: Show that ∥v∥C0,α ≤ C∥v∥L∞. +In the first step, we work on the family of balls (see Figure 3.1) +˜Bk := +� +x : |x| ≤ 1 +2 + 2−k−1 +� +. +Note that ˜B0 = B1, and ˜Bk converges to B1/2 as k → ∞. +We assume ∥v∥L2(B1) ≤ δ ≪ 1 and then consider the truncated functions +vk := (v − Ck)+ +with +Ck := 1 − 2−k, +and the numbers +Vk ≈ +� +˜Bk +|vk|2 dx. +Then, the main point is to derive an estimate of the form +(3.11) +Vk ≤ CkV β +k−1 +for some +β > 1, +for some constant C depending only on n, λ, and Λ. This previous inequality +implies that Vk → 0 as k → ∞ if V0 is small enough. In particular, v∞ = +(v − 1)+ is equal to zero in B1/2, and so v ≤ 1 in B1/2. + +— DRAFT — +3.3. De Giorgi’s proof +83 +B1/2 +B1 +Figure 3.1. Representation of the family of balls ˜Bk. +Notice that our equation Lv = 0 in B1 is linear, while the bound (3.11) +is nonlinear. The “game” consists in using the Sobolev inequality (which +gives control of Lp norms of vk in terms of L2 norms of ∇vk), combined +with an energy inequality, which gives a “reversed” Poincar´e inequality, i.e., +a control of ∥∇vk∥L2 in terms of ∥vk∥L2. +Once we have the first step v ∈ L2 ⇒ v ∈ L∞, the second step consists +of showing an oscillation-decay lemma +Lv = 0 +in +B1 +=⇒ +osc +B1/2 +v ≤ (1 − θ) osc +B1 v. +This implies the C0,α regularity of v (as we saw in Corollary 2.7). +In the next proofs we follow [CV10, Vas16]. +De Giorgi’s first step: from L2 to L∞. The two main ingredients are +the Sobolev inequality +∥v∥Lp(Rn) ≤ C∥∇v∥L2(Rn), +p = +2n +n − 2, +(see Theorem 1.4) and the following energy inequality (the Caccioppoli in- +equality): + +— DRAFT — +84 +3. Nonlinear variational PDE & Hilbert’s XIXth problem +Lemma 3.8 (Energy inequality). Let v ∈ H1(B1) with v ≥ 0 such that +Lv ≤ 0 in B1, for some L of the form (3.10). Then, for any ϕ ∈ C∞ +c (B1) +we have +� +B1 +|∇(ϕv)|2 dx ≤ C∥∇ϕ∥2 +L∞(B1) +� +B1∩supp ϕ +v2 dx, +where C depends only on n, λ, and Λ. +Proof. Notice that the weak formulation of −div(A(x)∇v) ≤ 0 in B1 is +� +B1 +∇η · A(x)∇v dx ≤ 0 +for all +η ∈ H1 +0(B1), η ≥ 0. +Take η = ϕ2v, to get +� +B1 +∇(ϕ2v) · A(x)∇v dx ≤ 0. +Now, we want to “bring one of the ϕ from the first gradient to the second +gradient”. Indeed, using +∇(ϕ2v) = ϕ∇(ϕv) + (ϕv)∇ϕ, +∇(ϕv) = ϕ∇v + v∇ϕ, +we get +0 ≥ +� +B1 +∇(ϕ2v) · A(x)∇v dx += +� +B1 +ϕ∇(ϕv) · A(x)∇v dx + +� +B1 +ϕv ∇ϕ · A(x)∇v dx += +� +B1 +∇(ϕv) · A(x)∇(ϕv) dx − +� +B1 +v∇(ϕv) · A(x)∇ϕ dx ++ +� +B1 +ϕv ∇ϕ · A(x)∇v dx += +� +B1 +∇(ϕv) · A(x)∇(ϕv) dx − +� +B1 +v∇(ϕv) · (A(x) − AT (x))∇ϕ dx +− +� +B1 +v2∇ϕ · A(x)∇ϕ dx. + +— DRAFT — +3.3. De Giorgi’s proof +85 +Let us first bound the term involving (A − AT ). By H¨older’s inequality, +using the uniform ellipticity of A and that (A − AT )2 ≤ 4Λ2Id, we get +� +B1 +v∇(ϕv) · (A(x) − AT (x))∇ϕ dx +≤ +�� +B1 +|v (A(x) − AT (x))∇ϕ|2 dx +� 1 +2 �� +B1 +|∇(ϕv)|2 dx +� 1 +2 +≤ 2 Λ +λ +1 +2 +�� +B1 +|v∇ϕ|2 dx +� 1 +2 �� +B1 +∇(ϕv)A(x)∇(ϕv) dx +� 1 +2 +≤ 1 +2 +� +B1 +∇(ϕv)A(x)∇(ϕv) dx + 2Λ2 +λ +� +B1 +|v∇ϕ|2 dx, +where in the last inequality we are using that 2ab ≤ a2 + b2. Combining the +previous inequalities, we obtain that +2Λ2 +λ +� +B1 +|v∇ϕ|2 dx ≥ 1 +2 +� +B1 +∇(ϕv) · A(x)∇(ϕv) dx − +� +B1 +v2∇ϕ · A(x)∇ϕ dx. +Therefore, we deduce +λ +� +B1 +|∇(ϕv)|2 dx ≤ +� +B1 +∇(ϕv) · A(x)∇(ϕv) dx +≤ 2 +� +B1 +v2 ∇ϕ · A(x)∇ϕ dx + 4Λ2 +λ +� +B1 +|v∇ϕ|2 dx +≤ +� +2Λ + 4Λ2 +λ +� +∥∇ϕ∥2 +L∞(B1) +� +B1∩supp ϕ +v2 dx, +and the lemma is proved. +□ +We will use the energy inequality (from the previous lemma) applied to +the function +v+ := max{v, 0}. +Before doing so, let us show that if Lv ≤ 0 (i.e., v is a subsolution), then +Lv+ ≤ 0 (i.e., v+ is a subsolution as well). (More generally, the maximum +of two subsolutions is always a subsolution.) +Lemma 3.9. Let L be of the form (3.10), let v ∈ H1(B1) be such that +Lv ≤ 0 in B1. Then, Lv+ ≤ 0. +Proof. We proceed by approximation. Let F ∈ C∞(R) be a smooth, non- +decreasing, convex function, with globally bounded first derivatives. +We +start by showing that L(F(v)) ≤ 0 in B1. +Notice that if v ∈ W 1,2(B1), then F(v) ∈ W 1,2(B1) as well. + +— DRAFT — +86 +3. Nonlinear variational PDE & Hilbert’s XIXth problem +We know that L(v) ≤ 0, i.e., +� +B1 +∇η · A∇v dx ≤ 0 +for all +η ∈ H1 +0(B1), η ≥ 0. +Let us now compute, for any fixed η ∈ H1 +0(Ω) satisfying η ≥ 0, L(F(v)). +Notice that the weak formulation still makes sense. +� +B1 +∇η · A∇F(v) dx = +� +B1 +F ′(v)∇η · A∇v dx += +� +B1 +∇(F ′(v)η) · A∇v dx − +� +B1 +ηF ′′(v)∇v · A∇v dx. +The first term is non-positive, since F ′(v)η ∈ H1 +0(B1) and F ′(v) ≥ 0 (F is +non-decreasing), so that F ′(v)η is an admissible test function. The second +term is also non-positive, since ηF ′′(v) ≥ 0 and ∇v · A∇v ≥ 0 by ellipticity +(and the integral is well defined, since ηF ′′(v) can be assumed to be bounded +by approximation, and +� +B1 ∇v · A∇v ≤ Λ∥∇v∥2 +L2(B1)). Therefore, +� +B1 +∇η · A∇F(v) dx ≤ 0, +and the proof is complete. +We finish by taking smooth approximations +of the positive part function, Fε, converging uniformly in compact sets +to F(x) = max{x, 0}. +Notice that this can be done in such a way that +∥Fε(v)∥W 1,2(B1) ≤ C, for some C independent of ε > 0, which gives the +desired result. +□ +We want to prove the following. +Proposition 3.10 (from L2 to L∞). Let L be of the form (3.10), and let +v ∈ H1(B1) be a solution to +Lv ≤ 0 +in +B1. +Then +∥v+∥L∞(B1/2) ≤ C∥v+∥L2(B1), +for some constant C depending only on n, λ, and Λ. +We will prove, in fact, the following (which is actually equivalent): +Proposition 3.11 (from L2 to L∞). Let L be of the form (3.10). There +exists a constant δ > 0 depending only on n, λ, and Λ, such that if v ∈ +H1(B1) solves +Lv ≤ 0 +in +B1 +and +� +B1 +v2 ++ ≤ δ, +then +v ≤ 1 +in +B1/2. + +— DRAFT — +3.3. De Giorgi’s proof +87 +Proof. Define, as introduced in the general ideas of the proof, for k ≥ 0, +˜Bk := +� +|x| ≤ 1 +2 + 2−k−1 +� +, +vk := (v − Ck)+ +with +Ck = 1 − 2−k, +and let ϕk be a family of shrinking cut-off functions 0 ≤ ϕk ≤ 1 that fulfill +ϕk ∈ C∞ +c (B1), +ϕk = +� 1 +in ˜Bk +0 +in ˜Bc +k−1 +, +and +|∇ϕk| ≤ C2k in +˜Bk−1\ ˜Bk, +where C here depends only on n. +Let +Vk := +� +B1 +ϕ2 +kv2 +k dx. +Now, the Sobolev inequality, and the energy inequality (Lemma 3.8) give +�� +B1 +|ϕk+1vk+1|p dx +� 2 +p +≤ C +�� +B1 +|∇(ϕk+1vk+1)|2 dx +� +≤ C22k +� +˜Bk +|vk+1|2 dx +≤ C22k +� +B1 +(ϕkvk)2 dx = C22kVk, +for p = +2n +n−2 if n ≥ 3. If n = 1 or n = 2, we can take p = 4. +On the other hand, by H¨older’s inequality, +Vk+1 = +� +B1 +ϕ2 +k+1v2 +k+1 dx ≤ +�� +B1 +(ϕk+1vk+1)p dx +� 2 +p ��{ϕk+1vk+1 > 0} +��γ, +where γ := 2 +n (if n = 1 or n = 2, γ = 1 +2). Here, we are using that +� +A f ≤ +( +� +A |f|p/2)2/p|A|γ. +Now, from Chebyshev’s inequality and the definition of vk and ϕk, +��{ϕk+1vk+1 > 0} +�� ≤ +��{ϕkvk > 2−k−1} +�� += +��{ϕ2 +kv2 +k > 2−2k−2} +�� +≤ 22(k+1) +� +B1 +ϕ2 +kv2 +k dx = 22(k+1)Vk. +Apart from Chebyshev’s inequality, we are using here that if vk+1 > 0 +and ϕk+1 > 0, then vk > 2−k−1 and ϕk = 1. Thus, combining the previous + +— DRAFT — +88 +3. Nonlinear variational PDE & Hilbert’s XIXth problem +inequalities, we get +Vk+1 ≤ +�� +B1 +(ϕk+1vk+1)p dx +� 2 +p ��{ϕk+1vk+1 > 0} +��γ +≤ C22kVk +� +22(k+1)Vk +�γ ≤ Ck+1V 1+γ +k +, +where we recall γ = 2 +n if n ≥ 3, and γ = 1 +2 otherwise; and C depends only +on n, λ, and Λ. +Now, we claim that, if δ > 0 is small enough, then +� +0 +≤ +Vk+1 +≤ +Ck+1V 1+γ +k +0 +≤ +V0 +≤ +δ +=⇒ +Vk → 0 +as +k → ∞. +Indeed, in order to see this it is enough to check by induction that if +V0 ≤ C−1/γ−1/γ2 then +V γ +k ≤ C−k−1 +(2C) +1 +γ +, +which is a simple computation. Alternatively, one could check by induction +that Vk ≤ C +(1+γ)k��k +i=1 +i +(1+γ)i +� +V (1+γ)k +0 +. +Hence, we have proved that +Vk = +� +B1 +(ϕkvk)2 dx → 0 +as +k → ∞. +Passing to the limit, we get� +B1/2 +(v − 1)2 ++ dx = 0, +and thus, v ≤ 1 in B1/2, as wanted. +□ +Proof of Proposition 3.10. To deduce the Proposition 3.10 from Propo- +sition 3.11, just use ˜v := +√ +δ v/∥v+∥L2(B1) (which solves the same equa- +tion). +□ +This proves the first part of the estimate +(3.12) +Lv ≤ 0 +in +B1 +=⇒ +∥v+∥L∞(B1/2) ≤ C∥v+∥L2(B1). +Notice that, as a direct consequence, we have the L2 to L∞ estimate. +Indeed, if Lv = 0 then Lv+ ≤ 0 (see Lemma 3.9) but also Lv− ≤ 0, +where v− := max{0, −v}. Thus, ∥v−∥L∞(B1/2) ≤ C∥v−∥L2(B1), and since +∥v∥L2(B1) = ∥v+∥L2(B1) + ∥v−∥L2(B1), combining the estimate for v+ and v− +we get +(3.13) +Lv = 0 +in +B1 +=⇒ +∥v∥L∞(B1/2) ≤ C∥v∥L2(B1), +as we wanted to see. + +— DRAFT — +3.3. De Giorgi’s proof +89 +Remark 3.12 (Moser’s proof). The proof of (3.12) here presented is the +original proof of De Giorgi. The first ingredient in the proof was to use +ϕ2v+ as a test function in the weak formulation of our PDE to get the +energy inequality from Lemma 3.8, +� +B1 +|∇(ϕv)2| dx ≤ C +� +B1 +v2|∇ϕ|2 dx +for all +ϕ ∈ C∞ +c (B1). +Roughly speaking, this inequality said that v cannot jump too quickly +(the gradient is controlled by v itself). +Moser did something similar, but taking η = ϕ2(v+)β instead, for some +β ≥ 1, to get the inequality +� +B1 +����∇ +� +v +β+1 +2 ϕ +�2���� dx ≤ C +� +B1 +vβ+1|∇ϕ|2 dx +for all +ϕ ∈ C∞ +c (B1). +Combining this with Sobolev’s inequality, one gets +�� +Br1 +vqγ dx +� 1 +qγ +≤ +� +C +|r2 − r1|2 +� +Br2 +vq dx +� 1 +q +, +where γ = 2∗ +2 > 1 and q = β + 1. Taking a sequence of rk ↓ 1 +2 as in De +Giorgi’s proof, one obtains +∥v∥L2γk(Brk) ≤ C∥v∥L2(B1), +and taking k → ∞ we obtain the L∞ bound in B1/2. +We refer the interested reader to [HL97, Chapter 4] for a full proof. +De Giorgi’s second step: L∞ to C0,α. We next prove the second step of +De Giorgi’s estimate. We want to prove: +Proposition 3.13 (Oscillation decay). Let L be of the form (3.10). Let +v ∈ H1(B2) be a solution to +Lv = 0 +in +B2. +Then, +osc +B1/2 +v ≤ (1 − θ) osc +B2 v +for some θ > 0 small depending only on n, λ, and Λ. +As we saw in Chapter 2 (see Corollary 2.7), this proposition immediately +implies C0,α regularity of solutions. +As shown next, Proposition 3.13 follows from the following lemma. + +— DRAFT — +90 +3. Nonlinear variational PDE & Hilbert’s XIXth problem +B2 +1 +v ≤ 1 +B1 +v ≤ 1 − γ +in +B1/2 +|{v ≤ 0} ∩ B1| ≥ µ > 0 +1 − γ +B1/2 +Figure 3.2. Graphical representation of v with Lv ≤ 0 from Lemma 3.14. +Lemma 3.14. Let L be of the form (3.10), and let v ∈ H1(B2) be such that +v ≤ 1 +in +B2, +and +Lv ≤ 0 +in +B2. +Assume that +��{v ≤ 0} ∩ B1 +�� ≥ µ > 0. +Then, +sup +B1/2 +v ≤ 1 − γ, +for some small γ > 0 depending only on n, λ, Λ, and µ. +In other words, if v ≤ 1, and it is “far from 1” in a set of non-zero +measure, then v cannot be close to 1 in B1/2. (See Figure 3.2.) +Let us show how this lemma yields the oscillation decay: +Proof of Proposition 3.13. Consider the function +w(x) := +2 +oscB2 v +� +v(x) − supB2 v + infB2 v +2 +� +and notice that +−1 ≤ w ≤ 1 +in +B2, +(in fact, oscB2 w = 2). Let us assume that +��{w ≤ 0}∩B1 +�� ≥ 1 +2|B1| (otherwise, +we can take −w instead). Then, by Lemma 3.14, we get +w ≤ 1 − γ +in +B1/2, + +— DRAFT — +3.3. De Giorgi’s proof +91 +and thus +osc +B1/2 +w ≤ 2 − γ. +This yields +osc +B1/2 +v ≤ +� +1 − γ +2 +� +osc +B2 v, +and thus the proposition is proved. +□ +To prove Lemma 3.14, we will need the following De Giorgi isoperimetric +inequality. +It is a kind of a quantitative version of the fact that an H1 +function cannot have a jump discontinuity. +Lemma 3.15. Let w ∈ H1(B1) be such that +� +B1 +|∇w|2 dx ≤ C◦. +Let +A := {w ≤ 0} ∩ B1, +D := +� +w ≥ 1 +2 +� +∩ B1, +E := +� +0 < w < 1 +2 +� +∩ B1. +Then, we have +C◦|E| ≥ c|A|2 · |D|2 +for some constant c depending only on n. +Proof. We define ¯w in B1 as ¯w = w in E, ¯w ≡ 0 in A and ¯w = 1 +2 in D. In +this way, ∇ ¯w ≡ 0 in B1 \ E and +� +B1 |∇ ¯w|2 ≤ C◦. +Let us denote the average of ¯w in B1 by ¯wB1 := +� +B1 ¯w(x) dx. Then, +|A| · |D| ≤ 2 +� +A +� +D +| ¯w(x) − ¯w(y)| dx dy +≤ 2 +� +B1 +� +B1 +��� ¯w(x) − ¯wB1 +�� + +�� ¯w(y) − ¯wB1 +��� +dx dy += 4|B1| +� +B1 +�� ¯w(x) − ¯wB1 +�� dx ≤ C +� +E +|∇ ¯w(x)| dx, +where in the last step we have used the Poincar´e inequality (Theorem 1.6 +with p = 1) and the fact that ∇ ¯w ≡ 0 in B1\E. Thus, by H¨older’s inequality +we reach +|A| · |D| ≤ C +� +E +|∇ ¯w| ≤ C +�� +E +|∇ ¯w|2 +�1/2 +|E|1/2 ≤ CC1/2 +◦ +|E|1/2, +as we wanted to see. +□ +Finally, we prove Lemma 3.14: + +— DRAFT — +92 +3. Nonlinear variational PDE & Hilbert’s XIXth problem +Proof of Lemma 3.14. Consider the sequence +wk := 2k � +v − (1 − 2−k) +� ++ . +Notice that wk ≤ 1 in B2 since v ≤ 1 in B2. Moreover, Lwk ≤ 0 in B2. +Using the energy inequality (Lemma 3.8), we easily get that +� +B1 +|∇wk|2 ≤ C +� +B2 +w2 +k ≤ C◦ +(notice 0 ≤ wk ≤ 1 in B2). +We also have +��{wk ≤ 0} ∩ B1 +�� ≥ µ > 0 +(by the assumption on v). We now apply Lemma 3.15 recursively to wk, as +long as +� +B1 +w2 +k+1 ≥ δ2. +We get +���� +� +wk ≥ 1 +2 +� +∩ B1 +���� ≥ +��{wk+1 > 0} ∩ B1 +�� ≥ +� +B1 +w2 +k+1 ≥ δ2. +Thus, from Lemma 3.15, +���� +� +0 < wk < 1 +2 +� +∩ B1 +���� ≥ c +C◦ +δ4µ2 = β > 0, +where β > 0 is independent of k, and depends only on n, δ, and µ. +But notice that the sets +� +0 < wk < 1 +2 +� +are disjoint for all k ∈ N, therefore +we cannot have the previous inequality for every k. This means that, for +some k◦ ∈ N (depending only on n and β) we have +� +B1 +w2 +k◦ < δ2 +and, hence, by the L2 to L∞ estimate from Proposition 3.10 +∥w+∥L∞(B1/2) ≤ C∥w+∥L2(B1). +We get +∥wk◦∥L∞(B1/2) ≤ Cδ ≤ 1 +2, +provided that δ > 0 is small enough, depending only on n, λ, and Λ. This +means that wk◦ ≤ 1 +2 in B1/2, and thus +v ≤ 1 +22−k◦ + +� +1 − 2−k◦� +≤ 1 − 2−k◦−1 = 1 − γ +as desired, where k◦ (and therefore, γ) depends only on n, λ, Λ, and µ. +□ + +— DRAFT — +3.3. De Giorgi’s proof +93 +Summarizing, we have now proved Lemma 3.14 (by using the L2 to L∞ +estimate and Lemma 3.15). Then, Lemma 3.14 implies the oscillation decay, +and the oscillation decay implies the H¨older regularity. +Theorem 3.16. Let L be of the form (3.10), and let v ∈ H1(B1) solve +Lv = 0 +in +B1. +Then, +∥v∥C0,α(B1/2) ≤ C∥v∥L∞(B1) +for some α > 0 and C depending only on n, λ, and Λ. +Proof. The theorem follows from the oscillation decay, in much the same +way as Corollary 2.7). +□ +Combining this last result with the L2 to L∞ estimate, Proposition 3.10, +we finally obtain the theorem of De Giorgi–Nash. +Theorem 3.17. Let v ∈ H1(B1) be a weak solution to div (A(x)∇v) = 0 +in B1, with 0 < λ Id ≤ A(x) ≤ Λ Id. Then, there exists some α > 0 such +that v ∈ C0,α(B1/2) and +∥v∥C0,α(B1/2) ≤ C∥v∥L2(B1). +The constants C and α > 0 depend only on n, λ, and Λ. +Proof. The result follows from Theorem 3.16 combined with Proposition 3.10 +(by (3.13)). +□ +As a consequence of the previous result, we have: +Proof of Theorem 3.7. It follows by Theorem 3.17 by a covering argu- +ment. +□ +In particular, as shown below, Theorem 3.17 solved Hilbert’s XIXth +problem. +This is one of the main results for which De Giorgi got the Wolf Prize +in 1990, and Nash got the Abel Prize in 2015. It has been speculated that +if only one of them had solved Hilbert’s XIXth problem, he would also have +received the Fields Medal for the proof. +Remark 3.18 (Harnack’s inequality). Even though it is not needed to prove +Theorem 3.17, it is interesting to notice that with some more work one can +also prove Harnack’s inequality for operators of the form div (A(x)∇v); see +[LZ17, Mos61]. + +— DRAFT — +94 +3. Nonlinear variational PDE & Hilbert’s XIXth problem +3.4. Solution to Hilbert’s XIXth problem +In this chapter, we have proved the interior regularity result for v ∈ H1(B1) +div +� +A(x)∇v +� += 0 +in +B1 +=⇒ +∥v∥C0,α(B1/2) ≤ C∥v∥L2(B1), +for some small α > 0 depending only on n, λ, and Λ. +For a general domain Ω ⊂ Rn, this gives the estimate for v ∈ H1(Ω) +div +� +A(x)∇v +� += 0 +in +Ω +=⇒ +∥v∥C0,α(˜Ω) ≤ C∥v∥L2(Ω), +for any ˜Ω ⊂⊂ Ω (with a constant C that depends only on n, λ, Λ, Ω, and ˜Ω). +Thanks to this, one can in fact solve Hilbert’s XIXth problem: +Theorem. Let u ∈ H1(Ω) be any local minimizer of +E(w) := +� +Ω +L(∇w) dx, +where L is smooth and uniformly convex, and Ω ⊂ Rn is bounded. +Then, u is C∞ in Ω. +Proof. Any local minimizer u satisfies +� +Ω +DL(∇u)∇φ dx = 0, +for all +φ ∈ C∞ +c (Ω). +(This is the weak formulation of div(DL(∇u)) = 0 in Ω.) +As in the proof of Theorem 3.5, if we define for any h ∈ Rn, +˜A(x) := +� 1 +0 +D2L +� +t∇u(x + h) + (1 − t)∇u(x) +� +dt, +then ˜A(x) is uniformly elliptic (since L is uniformly convex, 0 < λId ≤ +D2L(p) ≤ ΛId). We have that u(·+h)−u +|h| +∈ H1(Ωh) fulfills (again, see Theo- +rem 3.5) +� +Ω +∇ +�u(x + h) − u(x) +|h| +� +· ˜A(x)∇φ(x) dx = 0 +for all φ ∈ C∞ +c (Ωh), +that is, u(·+h)−u +|h| +solves weakly +div +� +˜A∇ +�u(· + h) − u +|h| +�� += 0, +in +Ωh. +(We recall Ωh := {x ∈ Ω : dist(x, ∂Ω) > |h|}.) + +— DRAFT — +3.5. Further results and open problems +95 +By the estimate of De Giorgi and Nash (Theorem 3.7), we find that +���� +u(· + h) − u +|h| +���� +C0,α(˜Ω) +≤ C +���� +u(· + h) − u +|h| +���� +L2(Ωh) +≤ C∥∇u∥L2(Ω) +for any ˜Ω ⊂⊂ Ωh (see (S8) in Chapter 1). By (H7), since the constant C +is independent of h, this yields +∥u∥C1,α(˜Ω) ≤ C∥u∥H1(Ω). +Now, once u is C1,α, we are done by Theorem 3.5. +□ +3.5. Further results and open problems +Let us finish this chapter by mentioning some state-of-the art results and +open problems regarding the minimization of convex energy functionals. +As we have explained, the minimization of a convex functional is a clas- +sical problem in the Calculus of Variations. Namely, +(3.14) +min +w∈W +� +Ω +L(∇w) dx, +with L : Rn → R convex, Ω ⊂ Rn, and some appropriate class of functions +W, say, with prescribed trace on ∂Ω. Hilbert’s XIXth problem deals with +the case in which L is uniformly convex and smooth, to obtain nice reg- +ularity results. In Remark 3.1 we discuss that lack of convexity can yield +non-uniqueness of minimizers, but it is not that clear what occurs if we sim- +ply remove the condition on the uniform convexity, but maintain the strict +convexity. In fact, functionals involving functions L that only involve strict +convexity (that is, D2L could have 0 and ∞ as eigenvalues in some sets) +appear naturally in some applications: anisotropic surface tensions, traffic +flow, and statistical mechanics (see [Moo20] and the references therein). +Minimizers of (3.14) are known to be Lipschitz (under enough smooth- +ness of the domain and boundary data) by the comparison principle. Thus, +the following natural question is to whether first derivatives of minimizers +are continuous: +If L is strictly convex, are minimizers to (3.14) C1? +The answer to that question has been investigated in the last years. The +problem was first addressed by De Silva and Savin in [DS10], where they +studied the case of dimension 2: +Theorem 3.19 ([DS10]). Let u be a Lipschitz minimizer to (3.14) in R2, +and suppose that L is strictly convex. Assume that the set of points where +D2L has some eigenvalue equal to 0 or ∞ is finite. Then, u ∈ C1. + +— DRAFT — +96 +3. Nonlinear variational PDE & Hilbert’s XIXth problem +Later, Mooney in [Moo20] studied the problem in higher dimensions +and showed that the question has a negative answer, in general, in dimen- +sions n ≥ 4: +Theorem 3.20 ([Moo20]). In R4 there exists a Lipschitz minimizer to +(3.14), with L strictly convex, that is not C1. +In the example by Mooney, the minimizer is analytic outside the origin +(having a singularity there), and the corresponding functional has a Hessian +with an eigenvalue going to ∞ in {x2 +1 +x2 +2 = x2 +3 +x2 +4}∩ +√ +2S3, but otherwise, +the eigenvalues are uniformly bounded from below away from zero. +It is currently an open question what happens in dimension n = 3, as +well as what happens for general strictly convex functionals in R2. + +— DRAFT — +Chapter 4 +Fully nonlinear elliptic +PDE +Second order nonlinear elliptic PDEs in their most general form can be +written as +(4.1) +F(D2u, ∇u, u, x) = 0 +in +Ω ⊂ Rn. +Understanding the regularity of solutions to these equations has been a +major research direction since the mid-20th century. +These are called fully nonlinear elliptic equations. +Besides their own +interest in PDE theory, they arise in Probability Theory (stochastic control, +differential games; see Appendix C for a probabilistic interpretation), and +in Geometry. +Thanks to Schauder-type estimates, under natural assumptions on the +dependence on ∇u, u, and x, the regularity for (4.1) can be reduced to +understanding solutions to +(4.2) +F(D2u) = 0 +in +Ω ⊂ Rn. +Indeed, some of the “perturbative” methods that we used in Chapter 2 +to prove Schauder estimates for linear equations � aij(x)∂iju = f(x) in +Ω ⊂ Rn work in such fully nonlinear setting, too. For simplicity, we will +focus here on the study of (4.2). +In the next sections we will discuss the following: +– What is ellipticity for solutions to (4.2)? +– Existence and uniqueness of solutions. +– Regularity of solutions to (4.2). +97 + +— DRAFT — +98 +4. Fully nonlinear elliptic PDE +We will not prove all the main known results of this Chapter, but only +give an overview of what is known. +We refer to the books [CC95] and +[NTV14] for more details about this topic. +4.1. What is ellipticity? +There are (at least) two possible ways to define ellipticity: +– Linearizing the equation. +– “Imposing” that the comparison principle holds. +We will see that they are essentially the same. +Definition 4.1. Let F : Rn×n → R. We say that F is elliptic if for any two +symmetric matrices A, B ∈ Rn×n such that A ≥ B (i.e., A − B is positive +semi-definite) we have +F(A) ≥ F(B), +with strict inequality if A > B (i.e., A − B positive definite). +The Laplace equation ∆u = 0 corresponds to the case F(M) = tr M. +For a linear equation (with constant coefficients) +n +� +i,j=1 +aij∂iju = 0, +F is given by F(M) = tr (AM), where A = (aij)i,j. This equation is elliptic +if and only if the coefficient matrix A is positive definite. +Therefore, it +coincides with our notion of ellipticity for linear equations. +Remark 4.2 (Comparison Principle). If a C2 function v touches u ∈ C2 +from below at a point x◦ (i.e. u ≥ v everywhere, and u(x◦) = v(x◦); see +Figure 4.1), then it follows that +∇u(x◦) = ∇v(x◦), +D2u(x◦) ≥ D2v(x◦). +Therefore, for these functions we would have F(D2u(x◦)) ≥ F(D2v(x◦)) if +F is elliptic. This is essential when proving the comparison principle. +Proposition 4.3 (Comparison Principle). Assume that F is elliptic, and +Ω ⊂ Rn is bounded. Let u, v ∈ C2(Ω) ∩ C0(Ω). Then, +� +u +≥ +v +on ∂Ω +F(D2u) +≤ +F(D2v) +in Ω. +=⇒ +u ≥ v +in +Ω. +Proof. We separate into two cases. +Case 1. +Assume first that F(D2u) < F(D2v) in Ω (with strict inequal- +ity). +If the conclusion is false, then the function u − v would have an +interior minimum inside Ω, say at x◦ ∈ Ω. Then, we would have D2(u − + +— DRAFT — +4.1. What is ellipticity? +99 +v +x◦ +u +Figure 4.1. The function v touches u from below at x◦. +v)(x◦) ≥ 0. Therefore, D2u(x◦) ≥ D2v(x◦) and by ellipticity of F, this yields +F(D2u(x◦)) ≥ F(D2v(x◦)). This is a contradiction with F(D2u) < F(D2v) +in Ω, and hence u ≥ v in Ω. +Case 2. Assume now F(D2u) ≤ F(D2v) in Ω. Then, we can define +¯u(x) := u(x) + ε +� +cΩ − |x|2� +, +where cΩ > 0 is a constant such that cΩ − |x|2 > 0 in Ω (recall that Ω is +bounded). +Then, we have ¯u ≥ u on ∂Ω, and D2¯u = D2u−2εId. Thus, by ellipticity, +F(D2¯u) = F(D2u − 2εId) < F(D2u) ≤ F(D2v) +in +Ω. +By Case 1, +� +¯u +≥ +v +on ∂Ω, +F(D2¯u) +< +F(D2v) +in Ω. +=⇒ +¯u ≥ v +in +Ω. +This gives +u(x) + ε +� +cΩ − |x|2� +≥ v(x) +in +Ω. +Letting ε ↓ 0 we deduce that u ≥ v in Ω. +□ +Thus, we see that ellipticity is exactly what we need in order to prove +the comparison principle. We will see that uniform ellipticity (analogously +to the case of linear equations) implies, in fact, the regularity of solutions. +Definition 4.4. Let F : Rn×n → R. Then F is uniformly elliptic if there +are 0 < λ ≤ Λ (the ellipticity constants), such that for every symmetric +matrices M, N with N ≥ 0 (that is, positive semi-definite), we have +λ∥N∥ ≤ F(M + N) − F(M) ≤ Λ∥N∥, +where ∥N∥ := tr +� +(NT N)1/2� += tr(N) is the sum of the (absolute value of +the) eigenvalues. + +— DRAFT — +100 +4. Fully nonlinear elliptic PDE +We remark that our choice of matrix norm in the previous definition +is not standard. In Rn, all norms are equivalent and thus we could have +chosen any other norm. This definition of norm, however, avoids dealing +with constants in future computations. +Of course, uniform ellipticity implies ellipticity, in a quantitative way. +For linear equations, i.e. F(M) = tr (AM), uniform ellipticity is equiv- +alent to +0 < λId ≤ A ≤ ΛId, +as usual. +The alternative way to see ellipticity is by linearizing the equation: +Assume F ∈ C1 (which is not always the case!). We consider the func- +tions +Fij(M) := +∂F +∂Mij +(M), +i.e., the first derivative of F(M) with respect to the component Mij of the +matrix M. +Then, it is immediate to see that +F is uniformly elliptic ⇐⇒ 0 < λ Id ≤ (Fij(M))i,j ≤ Λ Id, +∀M +⇐⇒ the linearized equation is uniformly elliptic. +Therefore, at least when F is C1, uniform ellipticity can be seen as +uniform ellipticity of the linearized equation. +In general, though, the uniform ellipticity condition implies that F is +Lipschitz, but not always C1. There are, in fact, important examples of +equations F(D2u) = 0 in which the corresponding F is Lipschitz but not C1. +In this case, the previous characterization of ellipticity through the deriva- +tives of F still holds, understanding now that they are defined almost every- +where. +Remark 4.5 (Convex (or concave) equations). An important subclass of +equations F(D2u) = 0 are those for which F is convex (or concave). Namely, +F(M) as a function F : Rn×n → R is convex (or concave). In this case, the +equation can be written as a Bellman equation (see (C.3)), as +F(D2u) = max +α∈A{Lαu} = 0, +where {Lα}α∈A is a family of linear operators of the form +Lαu := +n +� +i,j=1 +aα +ij∂iju + cα, +for a family of coefficients {aα +ij}α∈A uniformly elliptic, with ellipticity con- +stants λ and Λ. + +— DRAFT — +4.1. What is ellipticity? +101 +Notice that if u solves F(D2u) = 0, with F convex, then v = −u solves +G(D2v) = 0, with G(M) = −F(−M), and therefore, G is concave. +Pucci operators. Within the class of fully nonlinear uniformly elliptic +operators with ellipticity constants λ and Λ, the extremal or Pucci operators, +denoted by M+ and M−, are those that attain the extreme values (from +above and below, respectively). Alternatively, every other elliptic operator +with the same ellipticity constants is ordered with respect to them in the +sense of (4.6) below. +We define M± as follows. +Definition 4.6. Given 0 < λ ≤ Λ, the extremal or Pucci operators with +ellipticity constants λ and Λ, M± : Rn×n → R, are defined as +M−(M) := +inf +λId≤(aij)i,j≤ΛId +� +n +� +i,j=1 +aijMij +� += +inf +λId≤A≤ΛId {tr (AM)} +M+(M) := +sup +λId≤(aij)i,j≤ΛId +� +n +� +i,j=1 +aijMij +� += +sup +λId≤A≤ΛId +{tr (AM)} , +(4.3) +for any symmetric matrix M. They are uniformly elliptic operators, with +ellipticity constants λ and Λ. +In particular, from the definition we have +M±(αM) = αM±(M), +for all +α ≥ 0. +Notice that M± = M± +n,λ,Λ. In general, however, the dependence on the +ellipticity constants and the dimension will be clear in the corresponding +context, and thus we will drop it in the notation. +Sometimes, it is easier to define the Pucci operators through the eigenval- +ues of the corresponding matrix, appropriately weighted with the ellipticity +constants, in the following way. +Lemma 4.7. The Pucci operators as defined in (4.3) can be equivalently +defined as +M−(M) = λ +� +µi>0 +µi + Λ +� +µi<0 +µi = λ∥M+∥ − Λ∥M−∥, +M+(M) = Λ +� +µi>0 +µi + λ +� +µi<0 +µi = Λ∥M+∥ − λ∥M−∥, +(4.4) +where µi = µi(M) denote the eigenvalues of the symmetric matrix M, the +matrices M+ and M− are such that M± ≥ 0, M = M+ − M−, and ∥A∥ = +tr +� +(AT A)1/2� +. + +— DRAFT — +102 +4. Fully nonlinear elliptic PDE +Proof. The proof follows directly using the following rearrangement-type +inequalities involving the eigenvalues and the product of two symmetric +matrices A and B: +n +� +i=1 +λi(A)λn−i(B) ≤ tr(AB) ≤ +n +� +i=1 +λi(A)λi(B), +where λ1(A) ≤ · · · ≤ λn(A) denote the ordered eigenvalues of A, and +λ1(B) ≤ · · · ≤ λn(B) denote the ordered eigenvalues of B. +□ +From the definition of uniform ellipticity of F (Definition 4.4) it follows +that, given two symmetric matrices M, N, +λ∥N+∥ − Λ∥N−∥ ≤ F(M + N) − F(M) ≤ Λ∥N+∥ − λ∥N−∥, +where N = N+ − N−, and N± ≥ 0. Thus, by Lemma 4.7, +(4.5) +M−(N) ≤ F(M + N) − F(M) ≤ M+(N). +If we take M = 0, we see that +(4.6) +M−(N) ≤ F(N) − F(0) ≤ M+(N), +so these operators are like the “worse case” from above and below — up +to a constant, F(0). (Recall that M± are fully nonlinear uniformly elliptic +operators with ellipticity constants λ, Λ.) +If we further assume that F(0) = 0, we see that if u solves any equation +of the form F(D2u) = 0 then in particular +(4.7) +M−(D2u) ≤ 0 ≤ M+(D2u). +Remark 4.8. Equation (4.7) is called equation in non-divergence form with +bounded measurable coefficients. Indeed, notice that given some uniformly +elliptic coefficients (aij(x))i,j with no regularity assumption on x, if u ∈ C2 +fulfills � +i,j aij(x)∂iju then in particular (4.7) holds. On the other hand, +if (4.7) holds for some u ∈ C2, one can recover some uniformly elliptic +coefficients (aij(x))i,j such that � +i,j aij(x)∂iju. +4.2. Equations in two variables +Before going into the general theory of existence and regularity for fully non- +linear equations in Rn, let us study a simpler case: fully nonlinear equations +in two variables. +The main regularity estimate in this context is due to Nirenberg [Nir53], +and reads as follows. +Theorem 4.9. Let F : R2×2 → R be uniformly elliptic with ellipticity +constants λ and Λ. Let u ∈ C2(B1) solve +F(D2u) = 0 +in +B1 ⊂ R2. + +— DRAFT — +4.2. Equations in two variables +103 +Then, +∥u∥C2,α(B1/2) ≤ C∥u∥L∞(B1), +for some constants α > 0 and C depending only on λ and Λ. +The idea of the proof is the following: define v := ∂eu, and differentiate +the equation F(D2u) = 0 in the e direction, to get +(4.8) +2 +� +i,j=1 +aij(x)∂ijv(x) = 0 +in +B1 ⊂ R2, +where aij(x) := Fij(D2u(x)) for i, j ∈ {1, 2}. Since F is uniformly elliptic, +we have a22(x) ≥ λ > 0. Thus, we can divide (4.8) by a22(x) to obtain +(4.9) +a(x)∂11v(x) + b(x)∂12v(x) + ∂22v(x) = 0, +for some coefficients +a(x) = a11(x) +a22(x) +and +b(x) = a12(x) + a21(x) +a22(x) += 2a12(x) +a22(x) . +If we write w := ∂1v and differentiate (4.9) with respect to x1, we get +∂1 +� +a(x)∂1w(x) + b(x)∂2w(x) +� ++ ∂22w(x) = div(A(x)∇w) = 0, +where +A(x) := +�a(x) +b(x) +0 +1 +� +. +That is, w solves an equation in divergence form, and A is uniformly elliptic, +with ellipticity constants depending on λ and Λ. Thus, by the De Giorgi– +Nash result (Theorem 3.7) one has ∂1v = w ∈ C0,α(B1/2). Since the roles +of x1 and x2 can be changed, and since v = ∂eu (with e ∈ Sn−1 arbitrary), +we deduce that u ∈ C2,α(B1/2). +Let us now formally prove it. The idea is the one presented in the lines +above, where we used that u ∈ C4. In reality we can only use that u ∈ C2, +so we proceed by means of incremental quotients. +Proof of Theorem 4.9. Let us define +v(x) = u(x + h) − u(x) +|h| +∈ C2(B1−|h|), +with |h| < 1 +4, and proceed similarly to Theorem 3.5. Since F is translation +invariant, we have +F(D2u(x)) = 0, +F(D2u(x + h)) = 0 +in +B1−|h|. +Then, by the fundamental theorem of calculus for line integrals, +0 = F(D2u(x + h)) − F(D2u(x)) = +2 +� +i,j=1 +aij(x)∂ij +� +u(x + h) − u(x) +� +, + +— DRAFT — +104 +4. Fully nonlinear elliptic PDE +where +aij(x) = +� 1 +0 +Fij +� +tD2u(x + h) + (1 − t)D2u(x) +� +dt. +Since F is uniformly elliptic, (aij)i,j is uniformly elliptic (with the same +ellipticity constants). That is, v ∈ C2(B1−|h|) solves an equation in non- +divergence form +a11(x)∂11v(x) + 2a12(x)∂12v(x) + a22(x)∂22v(x) = 0 +in +B1−|h|, +where a12 = a21 and ∂12v = ∂21v because v ∈ C2. +From the ellipticity +conditions, we have λ ≤ a22(x) ≤ Λ, and we can divide by a22(x) to get +a(x)∂11v(x) + b(x)∂12v(x) + ∂22v(x) = 0 +in +B1−|h|. +Let +A(x) := +�a(x) +b(x) +0 +1 +� +. +It is straightforward to check that A is uniformly elliptic, with ellipticity +constants λ/Λ and Λ/λ. Let η ∈ C2 +c (B1−|h|) and notice that, by integration +by parts, +� +B1−|h| +∂2η ∂12v = +� +B1−|h| +∂1η ∂22v. +Thus, +� +B1−|h| +∇η · A(x)∇∂1v = +� +B1−|h| +∇η(x) · +�a(x)∂11v(x) + b(x)∂12v(x) +∂12v(x) +� +dx += +� +B1−|h| +� +∂1η +� +a(x)∂11v + b(x)∂12v +� ++ ∂2η ∂12v +� +dx += +� +B1−|h| +∂1η +� +a(x)∂11v + b(x)∂12v + ∂22v +� +dx += 0. +That is, ∂1v solves an equation with bounded measurable coefficients A(x) in +divergence form. Thus, by the De Giorgi–Nash theorem (see Theorem 3.16), +we know that ∂1v ∈ Cα and +∥∂1v∥C0,α(B1/2) ≤ C∥∂1v∥L∞(B1−|h|) ≤ C∥∂1u∥C0,1(B1), +(notice that we can go from B1 to B1−|h| in Theorem 3.16 by a covering +argument for |h| small), for some constant C depending only on λ and Λ. +By letting |h| → 0, thanks to (H7), we obtain that +∥∇∂1u∥C0,α(B1/2) ≤ C∥∂1v∥L∞(B1−|h|) ≤ C∥∂1u∥C0,1(B1), +for some constant C depending only on λ and Λ. By symmetry, the same +inequality is true for ∂2v (and ∂2u), so that +∥u∥C2,α(B1/2) ≤ C∥u∥C1,1(B1). + +— DRAFT — +4.3. Existence of solutions +105 +Notice that, by interpolation inequalities (see (1.9)), for each ε > 0, +there exists some Cε > 0 such that +∥u∥C1,1(B1/2) ≤ ε∥u∥C2,α(B1) + Cε∥u∥L∞(B1). +Now, the proof can be concluded by means of Lemma 2.27 analogously to +what has been done in the proof of Theorem 2.20. +□ +Thus, as we can see, in the two-dimensional case it is rather easy to +show a priori C2,α estimates for solutions to the fully nonlinear equation. +Thanks to these estimates, by means of the continuity method (see [GT77] +or [HL97]) one can actually show the existence of C2,α solutions for the +Dirichlet problem. +Nonetheless, as we will see, it turns out that in higher dimensions such +an a priori estimate is no longer available, and one needs to prove existence +of solutions in a different way, by introducing a new notion of weak solution +(viscosity solutions). +This is what we do in the next section. +4.3. Existence of solutions +We now turn our attention to fully nonlinear elliptic equations in Rn. +The first question to understand is the existence of solutions: given a +nice domain Ω ⊂ Rn, and a nice boundary data g : ∂Ω → R, can we always +solve the following Dirichlet problem? +� F(D2u) += +0 +in Ω +u += +g +on ∂Ω. +Notice that here we cannot construct the solution by minimizing a func- +tional, since these fully nonlinear equations do not come, in general, from +any energy functional. +To construct the solution, we only have two options: +– Prove “a priori estimates” and then use the continuity method. +– Use the comparison principle and Perron’s method. +The continuity method is reasonably easy to use, but we need C2,α +estimates for solutions up to the boundary. This is a very difficult problem, +and in fact, in general we do not have C2,α estimates for these equations +in Rn. +Therefore, we need to construct some kind of generalized notion of so- +lution: viscosity solutions. +The right concept of solution must be so that we have +• Existence of solutions. + +— DRAFT — +106 +4. Fully nonlinear elliptic PDE +• Comparison principle (and in particular, uniqueness of solutions). +• Stability (so that limits of solutions are solutions). +Notice that if we consider only C2 solutions, then we have the comparison +principle (and it is easy to prove), but we may not be able to prove existence. +On the other hand, if we relax the notion of solution, then we may be +able to easily prove the existence of a solution, but then it will be more +difficult to prove the uniqueness/comparison principle. +The right notion of generalized solution is the one given in Definition 4.10 +below, known as viscosity solutions. For subsolutions in the viscosity sense, +this notion only requires that the function is upper semi-continuous (USC), +while for supersolutions in the viscosity sense, this notion can be checked +on lower semi-continuous (LSC) functions. This is important in the proof +of existence of solutions. +We recall that a function f is said to be upper semi-continuous at x◦ if +lim sup +x→x◦ +f(x) ≤ f(x◦). +Similarly, it is lower semi-continuous at x◦ if +lim inf +x→x◦ f(x) ≥ f(x◦). +We refer to [Sil15] for a nice introduction to viscosity solutions to elliptic +equations. +Definition 4.10 (Viscosity solutions). Let F : Rn×n → R be uniformly +elliptic, and consider the PDE +F(D2u) = 0 +in +Ω. +• +We say that u ∈ USC(Ω) is a subsolution (in the viscosity sense), and +we write F(D2u) ≥ 0, if for any φ ∈ C2(Ω) such that φ ≥ u in Ω and +φ(x◦) = u(x◦), x◦ ∈ Ω, we have F(D2φ(x◦)) ≥ 0. +• We say that u ∈ LSC(Ω) is a supersolution (in the viscosity sense), and +we write F(D2u) ≤ 0, if for any φ ∈ C2(Ω) such that φ ≤ u in Ω and +φ(x◦) = u(x◦), x◦ ∈ Ω, we have F(D2φ(x◦)) ≤ 0. +• We say that u ∈ C(Ω) solves F(D2u) = 0 in Ω in the viscosity sense if it +is both a subsolution and a supersolution. +Notice that there may be points x◦ ∈ Ω at which no function φ ∈ C2 +touches u at x◦ (from above and/or from below). This is allowed by the +previous definition. +Remark 4.11 (Some history). The concept of viscosity solution was in- +troduced in 1983 by Crandall and P.-L. Lions in the study of first-order + +— DRAFT — +4.3. Existence of solutions +107 +equations. During a few years, the work on viscosity solutions focused on +first-order equations, because it was not known whether second-order uni- +formly elliptic PDEs would have a unique viscosity solution (or if the com- +parison principle would hold for these solutions). In 1988 the comparison +principle for viscosity solutions was finally proved by Jensen [Jen88], and in +subsequent years the concept has become prevalent in the analysis of elliptic +PDEs. +In 1994, P.-L. Lions received the Fields Medal for his contributions to +nonlinear PDEs, one of his major contributions being his work on viscosity +solutions [ICM94]. +A key result in the theory of viscosity solutions is the following (see +[Jen88, CC95]). +Theorem 4.12 (Comparison principle for viscosity solutions). Let Ω ⊂ Rn +be any bounded domain, and F : Rn×n → R be uniformly elliptic. Assume +that u ∈ LSC(Ω) and v ∈ USC(Ω) satisfy +(4.10) +u ≥ v +on +∂Ω, +and +(4.11) +F(D2u) ≤ 0 ≤ F(D2v) +in +Ω +in the viscosity sense. +Then, +u ≥ v +in +Ω. +We already proved this for C2 functions u in Proposition 4.3, and the +proof was very simple. For viscosity solutions the proof is more involved. +The main step in the proof of the comparison principle is the following. +Proposition 4.13. Let Ω ⊂ Rn be any bounded domain, and F : Rn×n → R +be uniformly elliptic. Assume that u ∈ LSC(Ω) and v ∈ USC(Ω) are bounded +functions that satisfy (4.10) and (4.11). Then, +M−(D2(u − v)) ≤ 0 +in +Ω. +We refer the reader to [CC95, Theorem 5.3] for a proof of such result, +where it is proved assuming that u, v ∈ C(Ω). The same proof works under +the hypotheses here presented. +The comparison principle follows using Proposition 4.13 and the next +lemma. +Lemma 4.14. Let Ω ⊂ Rn be any bounded domain, and assume that w ∈ +LSC(Ω) satisfies +w ≥ 0 +on +∂Ω, + +— DRAFT — +108 +4. Fully nonlinear elliptic PDE +v +w +x◦ +Figure 4.2. We slide v from below until it touches w at a point x◦. +and +M−(D2w) ≤ 0 +in +Ω. +Then, w ≥ 0 in Ω. +Proof. The proof is similar to that of Proposition 1.22. Indeed, first notice +that after a rescaling we may assume Ω ⊂ B1, and assume by contradiction +that w has a negative minimum in Ω. Then, since w ≥ 0 on ∂Ω, we have +minΩ w = −δ, with δ > 0, and the minimum is achieved in Ω. +Let us now consider 0 < ε < δ, and v(x) := −κ + ε(|x|2 − 1), with κ > 0 +(that is, a sufficiently flat paraboloid). +Now, notice that v < 0 on ∂Ω, and we can choose κ > 0 so that v touches +w from below at a point inside Ω. In other words, there is κ > 0 such that +w ≥ v in Ω, and w(x◦) = v(x◦) for some x◦ ∈ Ω. (See Figure 4.2.) Then, +by definition of viscosity supersolution, we have +M−(D2v)(x◦) ≤ 0. +However, a direct computation gives M−(D2v) = M−(2εId) ≡ 2λnε > 0 +in Ω, a contradiction. +□ +Once we have the comparison principle for viscosity solutions, we can use +Perron’s method to prove existence of solutions. We next do this, following +[Sil15]. +First let us notice that, for any bounded function u in Ω ⊂ Rn, we may +define its upper semi-continuous envelope as +u∗(x) := sup{lim sup +k +u(xk) : xk → x}, +where the supremum is taken among all sequences Ω ∋ xk → x. Notice that +u∗ is the smallest function satisfying u∗ ∈ USC(Ω) and u∗ ≥ u. Similarly, +we define the lower semi-continuous envelope of u as +(4.12) +u∗(x) := inf{lim inf +k +u(xk) : xk → x}. +We will need the following lemma, which is a generalization of the fact +that the maximum of subsolutions is also a subsolution. + +— DRAFT — +4.3. Existence of solutions +109 +Lemma 4.15. Let F : Rn×n → R be uniformly elliptic, and let Ω ⊂ Rn be +any bounded domain. +Let (ua)a∈A be a family of subsolutions: ua ∈ USC(Ω), and F(D2ua) ≥ 0 +in Ω, for all a ∈ A. Let +u(x) := sup +a∈A +ua, +and let +u∗(x) := sup +� +lim sup +k→∞ +u(xk) : xk → x +� +. +Then, u∗ ∈ USC(Ω) is a subsolution: F(D2u∗) ≥ 0 in Ω. +Proof. We divide the proof into two steps. +Step 1. In the first part, we show that if u∗ has a strict local maximum +at x◦, then one can extract sequences of indices ak ∈ A for k ∈ N, and of +points xk ∈ Ω, such that xk → x◦, uak has a local maximum at xk, and +uak(xk) → u∗(x◦). +By definition of u∗(x◦), we can extract a sequence of indices (aj)j∈N, +aj ∈ A, and of points yj → x◦, such that uaj(yj) → u∗(x◦). Now let us +prove that we can extract a further subsequence ak := ajk such that our +desired conclusion holds. +Indeed, let r > 0 be such that u∗(y) < u∗(x◦) for y ∈ Br(x◦) \ {x◦}, and +let ρ > 0 be so small that, if Kρ := Br(x◦) \ Bρ(x◦), then +max +Kρ u∗ ≤ u∗(x◦) − δ, +for some δ > 0. +Now notice that, for j large enough, uaj ≤ u∗(x◦)−δ/2 in Kρ. Otherwise, +there would be jm → ∞ and zm such that uajm(zm) > u∗(x◦) − δ/2 ≥ +maxKρ u∗ + δ/2. Since Kρ is compact, up to a subsequence, zm → z∞ for +some z∞ in Kρ such that +u∗(z∞) ≥ lim sup +m→∞ uajm(zm) > max +Kρ u∗ + δ/2. +A contradiction. Thus, uaj ≤ u∗(x◦) − δ/2 in Kρ for j large enough. +Let now xj ∈ Br(x◦) be the point where the maximum of uaj in Br(x◦) +is attained. In particular, uaj(xj) ≥ uaj(yj) → u∗(x◦), that is, uaj(xj) ≥ +u∗(x◦) − δ/4 for j large enough. Since uaj ≤ u∗(x◦) − δ/2 in Kρ (again, +for j large enough), this implies that xj ∈ Bρ(x◦). That is, ukj attains its +maximum in Br(x◦), inside Bρ(x◦). By repeating this argument choosing +smaller ρ > 0, we can extract a subsequence ak := ajk to get the desired +result. Notice that xj → x◦, and that by construction, uaj(xj) ≥ uaj(yj) → +u∗(x◦), so that uaj(xj) → u∗(x◦). This completes the first part of the proof. +Notice that so far we have not used that ua are subsolutions. + +— DRAFT — +110 +4. Fully nonlinear elliptic PDE +Step 2. Let us now proceed with the second part of the proof, which proves +the lemma. Let φ ∈ C2 be such that φ(x◦) = u∗(x◦) and u ≤ φ around x◦ +(that is, u−φ attains its local maximum at x◦), with x◦ ∈ Ω. By considering +¯φ(x) = φ(x) + |x − x◦|4, we have that u − ¯φ attains a strict local maximum +at x◦. We apply now the first part of the proof with va := ua − ¯φ. That +is, there exist sequences of indices (ak)k∈N, and points xk → x◦ such that +uak − ¯φ attains its local maximum at xk and uak(xk) → u∗(x◦) (since ¯φ is +continuous). In particular, since uak are subsolutions in the viscosity sense, +we have +F +� +D2 ¯φ(xk) +� +≥ 0 +=⇒ +F +� +D2 ¯φ(x◦) +� += F +� +D2φ(x◦) +� +≥ 0, +by continuity of F and D2φ. Thus, u is a viscosity subsolution. +□ +We can now prove the existence of viscosity solutions. To do so, we +assume that we are given a bounded domain Ω ⊂ Rn such that +for every x◦ ∈ ∂Ω, there exists some ψ+ ∈ C2(Ω) such that +ψ+(x◦) = 0, +ψ+|∂Ω\{x◦} > 0, +and M+(D2ψ+) ≤ 0 in Ω, +(4.13) +where we recall that M+ is the Pucci operator defined in (4.3) with ellipticity +constants λ and Λ. +Notice that, if (4.13) holds, then we also have that +for every x◦ ∈ ∂Ω, there exists some ψ− ∈ C2(Ω) such that ψ−(x◦) = 0, +ψ−|∂Ω\{x◦} < 0, and +M−(D2ψ−) ≥ 0 in Ω, +where ψ− is simply given by ψ− = −ψ+. +We will later show that any bounded C2 domain satisfies (4.13), for any +constants 0 < λ ≤ Λ. +Remark 4.16. In the following results, we will often assume that F(0) = 0. +Otherwise, if F(0) ̸= 0, we can consider the uniformly elliptic operator +˜Ft(D2u) := F +� +D2(u + t|x|2/2) +� += F(D2u + tId) instead. Then, ˜Ft(0) = +F(tId), and we can choose t ∈ R such that F(tId) = 0. Indeed, if F(0) > 0, +by (4.6) ˜Ft(0) = F(tId) ≤ M+(tId) + F(0) = tnλ + F(0) < 0 for t < 0 +negative enough. Since ˜F0(0) = F(0) > 0, by continuity of ˜Ft in t, we are +done for some t ∈ +� +− F(0) +nλ , 0 +� +. The case F(0) < 0 follows analogously. +Theorem 4.17 (Existence and uniqueness of viscosity solutions). Let F : +Rn×n → R be uniformly elliptic with ellipticity constants λ and Λ, let Ω ⊂ Rn +be any bounded domain such that (4.13) holds, and let g ∈ C(∂Ω). +Then, there exists a (unique) viscosity solution to the Dirichlet problem +� F(D2u) += +0 +in Ω +u += +g +on ∂Ω. + +— DRAFT — +4.3. Existence of solutions +111 +Proof. The uniqueness follows directly from the comparison principle, The- +orem 4.12. Thanks to Remark 4.16, we will assume F(0) = 0. The proof of +existence follows by means of Perron’s method, as shown next. +Let us define the set of all subsolutions as +A := +� +v ∈ USC(Ω) : F(D2v) ≥ 0 in Ω, v ≤ g on ∂Ω +� +. +Then, we can define the pointwise supremum of all subsolutions in A, +u(x) := sup +v∈A +v(x). +Notice that since the constant function −∥g∥L∞(∂Ω) belongs to A, such set +is non-empty. Notice also that all elements of A must be below the constant +∥g∥L∞(∂Ω) by the comparison principle, and thus u is bounded. +We define the upper semi-continuous envelope +u∗(x) = sup +� +lim sup +k→∞ +u(xk) : xk → x +� +. +Notice that, by Lemma 4.15, we have F(D2u∗) ≥ 0 in Ω. +The strategy of the proof is as follows. We first prove that u∗ = g on ∂Ω. +This implies that u∗ ∈ A, and therefore u∗ = u. Then, once this is done, we +will define u∗ as the lower semi-continuous envelope of u, and show that u∗ +is a supersolution. By the comparison principle, this will imply that u∗ ≥ u, +and thus u∗ = u. This means that u is continuous, and that it is both a +subsolution and a supersolution, as wanted. +Step 1. Let us start by showing that u∗ = g on ∂Ω, and that u∗ is continuous +on ∂Ω. Namely, we show that for every x◦ ∈ ∂Ω, and every xk → x◦ with +xk ∈ Ω, then lim infk→∞ u∗(xk) = lim supk→∞ u∗(xk) = g(x◦). +Let ε > 0, and let us define +w− +ε := g(x◦) − ε + kεψ− = g(x◦) − ε − kεψ+, +where kε > 0 is chosen large enough (depending on ε but also on g and Ω) +such that w− +ε ≤ g on ∂Ω, and ψ− = −ψ+ is the function given by property +(4.13) at x◦. Let us also define +w+ +ε := g(x◦) + ε + kεψ+, +where kε > 0 is such that w+ +ε ≥ g on ∂Ω (without loss of generality, by +taking it larger if necessary, we can assume it is the same as before). +By the properties of the extremal operators (4.3), we have M−(D2w− +ε ) = +kεM−(D2ψ−) ≥ 0 and M+(D2w+ +ε ) = kεM+(D2ψ+) ≤ 0 in Ω. In particu- +lar, by (4.6) (recall F(0) = 0), +F(D2w− +ε ) ≥ 0 +and +F(D2w+ +ε ) ≤ 0 +in +Ω, + +— DRAFT — +112 +4. Fully nonlinear elliptic PDE +and w− +ε ∈ A. Notice that, by continuity of ψ−, for each ε > 0 there exists +some δ > 0 such that w− +ε ≥ g(x◦) − 2ε in Bδ(x◦) ∩ Ω. This yields, u∗ ≥ +w− +ε ≥ g(x◦) − 2ε in Bδ(x◦) ∩ Ω, so that if xk → x◦, then +lim inf +k→∞ u(xk) ≥ g(x◦) − 2ε. +On the other hand, by the comparison principle, all elements in A are +below w+ +ε for any ε > 0. Again, by continuity of ψ+, for each ε > 0 there +exists some δ > 0 such that w+ +ε ≤ g(x◦) + 2ε in Bδ(x◦) ∩ Ω. This yields, +u∗ ≤ w+ +ε ≤ g(x◦) + 2ε in Bδ(x◦) ∩ Ω, so that if xk → x◦, then +lim sup +k→∞ +u(xk) ≤ g(x◦) + 2ε. +Since ε > 0 is arbitrary, we have that if xk → x◦, then +lim +k→∞ u∗(xk) = g(x◦). +Therefore, u∗ = g on ∂Ω and u is continuous on ∂Ω. In particular, we have +u∗ ∈ A and (since u∗ ≥ u) u∗ ≡ u. This means that u ∈ USC(Ω) and +F(D2u) ≥ 0 in Ω. +Step 2. Now, we show that u is a supersolution as well. To do so, we consider +its lower semi-continuous envelope u∗, (4.12), and prove that F(D2u∗) ≤ 0 +in Ω. +We start by noticing that, since u is continuous on the boundary (by +Step 1), then u∗ = g on ∂Ω. +Assume by contradiction that u∗ is not a +supersolution, that is, there exists some x◦ ∈ Ω such that for some φ ∈ C2 +we have φ(x◦) = u∗(x◦), φ ≤ u∗, but F(D2φ(x◦)) > 0. +By taking ¯φ = φ − |x − x◦|4 if necessary, we may assume that φ < u∗ if +x ̸= x◦, and we still have F(D2φ(x◦)) > 0. Notice that, by continuity of F +and D2φ, we have F(D2φ) > 0 in Bρ(x◦) for some small ρ > 0. +On the other hand, consider φ+δ for δ > 0, and define uδ := max{u, φ+ +δ}. Since φ(x) < u∗(x) ≤ u(x) for x ̸= x◦, we have for δ > 0 small enough +that φδ < u outside Bρ(x◦). +Now, notice that uδ is a subsolution, since it coincides with u outside +Bρ(x◦) and it is the maximum of two subsolutions in Bρ(x◦). This means +that uδ ∈ A, and thus uδ ≤ u. +However, this means that φ + δ ≤ u +everywhere in Ω, and thus φ + δ ≤ u∗, a contradiction. Thus, u∗ had to be +a supersolution. +But then, again by the comparison principle, since u is a subsolution +and u = u∗ = g on ∂Ω, we get that u∗ ≥ u in Ω, which means that u = u∗. +Therefore, u is continuous, both a subsolution and a supersolution, and +u = g on ∂Ω. This concludes the proof. +□ + +— DRAFT — +4.3. Existence of solutions +113 +x◦ +zx◦ +Bρ(zx◦) +Ω +Figure 4.3. Representation of the construction from the proof of Corollary 4.18. +As a consequence, we find the following. +Corollary 4.18. Let Ω be any bounded C2 domain, and F : Rn×n → R +be uniformly elliptic. Then, for any continuous g ∈ C(∂Ω), the Dirichlet +problem +� F(D2u) += +0 +in Ω +u += +g +on ∂Ω, +has a unique viscosity solution. +Proof. The result follows from the previous theorem, we just need to check +that any C2 domain fulfils (4.13). To do so, we need to construct an appro- +priate barrier at every boundary point z ∈ ∂Ω. +Notice, that in the very simple case that Ω is strictly convex, such barrier +ψ+ can simply be a hyperplane with zero level set tangent to Ω at a given +boundary point, such that it is positive in Ω. +In general, since Ω is a bounded C2 domain, it satisfies the exterior ball +condition for some uniform radius ρ > 0: that is, for each point x◦ ∈ ∂Ω +there exist some point zx◦ = z(x◦) ∈ Ωc and a ball Bρ(zx◦) such that +Bρ(zx◦) ⊂ Ωc and Bρ(zx◦) ∩ ∂Ω = {x◦}. See Figure 4.3. +Let us construct the barrier ψ+ from (4.13) for C2 domains. We consider +the function ψ in Rn \ Bρ, for ρ > 0 given by the exterior ball condition, +ψ(x) = e−αρ2 − e−α|x|2, +for some α > 0 also to be chosen. + +— DRAFT — +114 +4. Fully nonlinear elliptic PDE +Notice that +eα|x|2D2ψ(x) = −4α2 +� +� +� +� +� +x2 +1 +x1x2 +. . . +x1xn +x2x1 +x2 +2 +. . . +x2xn +... +... +... +... +xnx1 +. . . +. . . +x2 +n +� +� +� +� +� + 2αId += 2αId − 4α2xxT . +Then, for |x| ≥ ρ we have +eα|x|2M+(D2ψ) ≤ 2αM+(Id) − 4α2M−(xxT ) = 2αnΛ − 4α2λ|x|2 +≤ 2α(nΛ − 2αλρ2). +In particular, if we choose α ≥ +nΛ +2λρ2 , we have +M+(D2ψ) ≤ 0 +in +Bc +ρ. +Therefore, translations of ψ are good candidates for the function ψ+ from +(4.13). +Let now x◦ ∈ ∂Ω be any point on the boundary, and take ψ+(x) := +ψ(x − zx◦). +It is clear that ψ+(x◦) = 0, and that ψ+(x) > 0 for any +x ∈ Ω \ {x◦}. On the other hand, from the discussion above we know that +M+(D2ψ+) ≤ 0. Thus, Ω fulfills (4.13). +□ +Remark 4.19 (Lipschitz domains). It is actually possible to show that +(4.13) holds for any bounded Lipschitz domain, too. +In particular, this +yields the existence of viscosity solutions in such class of domains. +Finally, we also have the following: +Proposition 4.20 (Stability of viscosity solutions). Let Fk be a sequence +of uniformly elliptic operators (with ellipticity constants λ and Λ), and let +uk ∈ C(Ω) be such that Fk(D2uk) = 0 in Ω in the viscosity sense. +Assume that Fk converges to F uniformly in compact sets, and uk → u +uniformly in compact sets of Ω. Then, F(D2u) = 0 in Ω in the viscosity +sense. +Proof. The proof uses the same ideas as the proof of Lemma 4.15. +Let x◦ ∈ Ω and φ ∈ C2 be such that φ(x◦) = u(x◦) and φ ≤ u in Ω. +By taking ¯φ(x) = φ(x) + |x − x◦|4 we have that u − ¯φ attains a strict local +maximum at x◦. +Let now vk := uk − ¯φ. Up to a subsequence, by Step 1 in the proof +of Lemma 4.15, there exists a sequence xk → x◦ such that uk − ¯φ attains +a local maximum at xk, and from the uniform convergence of uk to u, we +also have uk(xk) → u(x◦). Since uk are, in particular, subsolutions in the + +— DRAFT — +4.4. Regularity of solutions: an overview +115 +viscosity sense for the operator Fk, we have that Fk(D2 ¯φ(xk)) ≥ 0. Now, +since xk → x◦, and Fk converges uniformly to F, we get that, letting k → ∞, +F(D2 ¯φ(x◦)) = F(D2φ(x◦)) ≥ 0. +In particular, u is a viscosity subsolution for F. Doing the same for −u, +we reach that u is a viscosity solution. +□ +Remark 4.21. We have seen that for fully nonlinear equations F(D2u) = +0 we have existence, uniqueness, and stability of viscosity solutions. The +same can be done for more general equations like F(D2u, x) = f(x), with +continuous coefficients in x, see [CC95]. However, when we want to study +linear equations in non-divergence form +(4.14) +� +aij(x)∂iju(x) = 0 +with bounded measurable coefficients, it turns out that viscosity solutions do +not behave so well; see the counterexample in [Nad97] (see also [CCKS96]). +This is the reason why, instead of defining viscosity solutions for a specific +equation of the type (4.14), what we do is to say that u solves an equation +with bounded measurable coefficients (in non-divergence form) whenever it +satisfies +M−(D2u) ≤ 0 ≤ M+(D2u) +in viscosity sense, where M± are the Pucci extremal operators (recall Def- +inition 4.6). As explained in Remark 4.8, for C2 functions u these two in- +equalities are equivalent to saying that u solves (4.14) for some coefficients +aij(x). +Summarizing: For viscosity solutions we now have all we need in order +to study regularity issues: +– Existence of solutions. +– Comparison principle. +– Stability under uniform limits. +4.4. Regularity of solutions: an overview +In the last section we saw that for any (smooth) domain Ω ⊂ Rn and any +(continuous) boundary data g, one can find a unique viscosity solution u ∈ +C(Ω) to the Dirichlet problem +� F(D2u) += +0 +in Ω +u += +g +on ∂Ω. +Now, the main question is that of regularity: +If u ∈ C(B1) solves F(D2u) = 0 in B1, +what can we say about the regularity of u? + +— DRAFT — +116 +4. Fully nonlinear elliptic PDE +Is the following implication true? +(4.15) +� +� +� +� +� +� +� +F ∈ C∞ and +uniformly elliptic +& +F(D2u) = 0 in B1 +? +=====⇒ +u ∈ C∞(B1/2). +This is in some sense a question analogous to Hilbert’s XIXth problem. +Regularity for fully nonlinear equations: first results. Assume that +u has some initial regularity, and that F is C∞ and uniformly elliptic. Then, +F(D2u) = 0 +−→ +∂e +n +� +i,j=1 +Fij(D2u)∂ij(∂eu) = 0, +where +Fij := +∂F +∂Mij +is the derivative of F(M) with respect to Mij. Therefore, if we denote +aij(x) := Fij(D2u(x)), +we will then have +aij(x) is uniformly elliptic, +0 < λId ≤ (aij(x))i,j ≤ ΛId, +thanks to the uniform ellipticity of F. +Denoting +v = ∂eu, +we have +F(D2u) = 0 +=⇒ +v = ∂eu +solves +n +� +i,j=1 +aij(x)∂ij(∂eu) = 0, +where aij(x) = Fij(D2u(x)). +Now, if u ∈ C2 (or C2,α), then the coefficients aij(x) are continuous (or +C0,α), and therefore we get, by Schauder-type estimates, +u ∈ C2 ⇒ aij ∈ C0 ⇒ v ∈ C1,α ⇒ u ∈ C2,α ⇒ · · · ⇒ u ∈ C∞, +where we use the bootstrap argument +u ∈ C2,α ⇒ aij ∈ C0,α ⇒ u ∈ C3,α ⇒ aij ∈ C1,α ⇒ · · · ⇒ u ∈ C∞. +In other words, this suggests that the following result. +Proposition 4.22. Let F be uniformly elliptic and C∞. +Let u be any +solution of F(D2u) = 0 in B1, and assume that u ∈ C2. Then, u ∈ C∞. + +— DRAFT — +4.4. Regularity of solutions: an overview +117 +Proof. The idea is the one presented in the lines above, but we can only +use that u ∈ C2 (in the previous argumentation, we used that u is C3). To +do so, we make use of incremental quotients, as in Theorem 3.5. +Let u ∈ C2(B1), and let h ∈ Rn with |h| small. +Notice that F is +translation invariant, so +F(D2u(x)) = 0, +F(D2u(x + h)) = 0 +in +B1−|h|. +Then, +0 = F(D2u(x + h)) − F(D2u(x)) = +n +� +i,j=1 +aij(x)∂ij +� +u(x + h) − u(x) +� +, +where +aij(x) = +� 1 +0 +Fij +� +tD2u(x + h) + (1 − t)D2u(x) +� +dt +(cf. the proof of Theorem 3.5 or Theorem 4.9). This is just the fundamental +theorem of calculus. In particular, since F is uniformly elliptic, (aij)i,j is +uniformly elliptic (with the same ellipticity constants). Since u ∈ C2 and +F is smooth, aij are continuous. That is, u(·+h)−u +|h| +solves the equation in +non-divergence form +n +� +i,j=1 +aij(x)∂ij +�u(x + h) − u(x) +|h| +� += 0 +in +B1−|h|, +for some continuous and uniformly elliptic coefficients aij. By the a priori +estimates for equations with continuous coefficients (Proposition 2.31), we +know that for any α ∈ (0, 1) we have +���� +u(· + h) − u +|h| +���� +C1,α(B1/2) +≤ C +���� +u(· + h) − u +|h| +���� +L∞(B3/4) +≤ C∥u∥C0,1(B3/4), +for some constant C that is independent of h. By (H7), (1.7), from Chap- +ter 1, we reach that u ∈ C2,α(B1/2), and by a covering argument u ∈ C2,α +inside B1. +Now, we proceed iteratively. +Since u ∈ C2,α inside B1, we have that u(·+h)−u +|h| +∈ C2,α inside B1−|h| +for all h. +Together with F being smooth, this implies that aij ∈ C0,α +inside B1−|h|. That is, now u(·+h)−u +|h| +solves a non-divergence-form equation +with H¨older continuous coefficients, and from Theorem 2.20 we get uniform +bounds in the C2,α norm for u(·+h)−u +|h| +, thus yielding that u ∈ C3,α inside B1. +We can repeat this argument iteratively, using the higher order estimates +from Corollary 2.21, to reach the desired result. +□ + +— DRAFT — +118 +4. Fully nonlinear elliptic PDE +This is similar to what happened in Hilbert’s XIXth problem: in that +case we proved C1 ⇒ C∞. +Notice, however, that for fully nonlinear equations, the “gap to be filled” +(from C0 to C2) is “bigger” than in Hilbert’s XIXth problem (from H1 to +C1). +Now, the central question to be answered is: +Is it true that solutions are always C2? +In particular, we wonder whether viscosity solutions are always classical +solutions or not, and thus, whether the Dirichlet problem always admits a +classical solution. +Regularity for fully nonlinear equations. An important observation in +the previous argument was the following: +� u solves +F(D2u) = 0 +=⇒ +� v = ∂eu solves �n +i,j=1 aij(x)∂ijv = 0, +with aij(x) = Fij(D2u(x)). +This means that, at least formally, the derivatives of any solution to +any fully nonlinear equation solve an equation with bounded measurable +coefficients. +This can be argued properly by looking at incremental quotients: +Recall from (4.5) the equivalence +F is uniformly elliptic +�� +M−(D2(u − v)) ≤ F(D2u) − F(D2v) ≤ M+(D2(u − v)), +where M± are the Pucci operators. Thus, +M−� +D2(u(x + h) − u(x)) +� +≤ F(D2u(x + h)) − F(D2u(x)) ≤ +≤ M+� +D2(u(x + h) − u(x)) +� +. +Using F(D2u) = 0 and denoting +vh(x) = u(x + h) − u(x) +|h| +, +we then reach +� M+(D2vh) +≥ +0 +M−(D2vh) +≤ +0 +� equation with bounded +measurable coefficients +� +. +The question is now: in case of divergence-form equations we proved +� equation with bounded +meas. coeff. div(A(x)∇v) = 0 +=⇒ v ∈ C0,α +(De Giorgi-Nash). + +— DRAFT — +4.4. Regularity of solutions: an overview +119 +Is there a similar result for equations in non-divergence form? The answer +is Yes. +Theorem 4.23 (Krylov–Safonov, 1979). Let 0 < λ ≤ Λ be the ellipticity +constants, and v ∈ C(B1) be any solution to +(4.16) +� M+(D2v) +≥ +0 +in B1 +M−(D2v) +≤ +0 +in B1, +in the viscosity sense. Then, +∥v∥C0,α(B1/2) ≤ C∥v∥L∞(B1) +for some small α > 0 and C depending only on n, λ, and Λ. +This result was proved in [KS79] (for classical solutions); see also [Moo19] +for a more recent and simplified proof, and [DS21] for an extension of the +result. +Recall that (see the end of Section 4.1), for C2 functions, (4.16) is actu- +ally equivalent to v solving an equation of the type � +i,j aij(x)∂ijv for some +uniformly elliptic coefficients. This is why (4.16) is called an equation in +non-divergence form with bounded measurable coefficients. +As a consequence of this result, we find the following. We assume for +simplicity F(0) = 0, otherwise see Remark 4.16. +Theorem 4.24 (Krylov–Safonov, 1979). Let F be uniformly elliptic, F(0) = +0, and u ∈ C(B1) be any viscosity solution to +F(D2u) = 0 +in +B1. +Then, +∥u∥C1,α(B1/2) ≤ C∥u∥L∞(B1) +for some small α > 0 and C depending only on n, λ, and Λ. +Proof. By Proposition 4.13 (with v ≡ 0), the function u ∈ C(B1) solves +itself an equation with bounded measurable coefficients +� M+(D2u) +≥ +0 +in B1 +M−(D2u) +≤ +0 +in B1. +Therefore, by Theorem 4.23, u ∈ C0,α inside B1. Now, for β ∈ (0, 1] take +vh(x) := u(x + h) − u(x) +|h|β +, +which (again by Proposition 4.13) also solves an equation with bounded +measurable coefficients, +� M+(D2vh) +≥ +0 +in B1−|h| +M−(D2vh) +≤ +0 +in B1−|h|. + +— DRAFT — +120 +4. Fully nonlinear elliptic PDE +Then, again by Theorem 4.23, we have +∥vh∥C0,α(B1/2) ≤ C∥vh∥L∞(B1−|h|) ≤ C∥u∥Cβ(B1). +By (H7), we deduce that +∥u∥Cα+β(B1/2) ≤ C∥u∥Cβ(B1), +provided that α + β is not an integer and β ≤ 1. +Using this estimate with β = α, 2α, ..., kα, one gets C1,α regularity in a +finite number of steps. +□ +Remark 4.25. Observe that: +• The C0,α estimate for bounded measurable coefficients, Theorem 4.23, +is the best one can get in dimensions n ≥ 3; see [Saf87]. +• In a sense, Theorem 4.23 is the analogue of the result of De Giorgi– +Nash for divergence-form equations. However, it is not enough to get C2 +regularity for solutions to fully nonlinear equations. +Summary: We have F(D2u) = 0 ⇒ u ∈ C1,α (for some small α > 0). +Moreover, u ∈ C2 ⇒ u ∈ C∞. However, we have no idea (yet) if +u ∈ C1,α +?=⇒ +u ∈ C2. +In the two-dimensional case, as we have seen in Theorem 4.9 (as an a +priori estimate), it turns out that one can do something better, and all solu- +tions are C2,α. This is because, in R2, solutions to equations with bounded +measurable coefficients are not only C0,α, but C1,α. +As a consequence, we have the following. +Theorem 4.26. Let F : R2×2 → R be uniformly elliptic and smooth. Let +u ∈ C(B1) be any viscosity solution to +F(D2u) = 0 +in +B1 ⊂ R2. +Then u ∈ C∞. +This completely answers question (4.15) in two dimensions. +In higher dimensions, a famous result established (independently) by +Evans [Eva82] and Krylov [Kry82] gives the following. +Theorem 4.27 (Evans–Krylov, 1982). Let F be any convex (or concave) +uniformly elliptic operator, with F(0) = 0. Let u ∈ C(B1) be any viscosity +solution to +F(D2u) = 0 +in +B1. +Then, +∥u∥C2,α(B1/2) ≤ C∥u∥L∞(B1), + +— DRAFT — +4.5. Further results and open problems +121 +for some α > 0 and C depending only on n, λ, and Λ. In particular, if F +is smooth then u ∈ C∞. +We refer to [CS10] for a shorter proof of such result. +Thus, for any solution to (4.2), with F uniformly elliptic and smooth, +we have: +• If u ∈ C2, then u ∈ C∞. +• u ∈ C1,α always (Krylov–Safonov, 1979). +• In two dimensions, u ∈ C∞ (Nirenberg, 1952). +• If F is convex, then u ∈ C∞ (Evans–Krylov, 1982) +Question: What happens in general? +For decades it was an open problem to decide whether all solutions are +C2 or not. The question was finally answered by Nadirashvili and Vladuts +in the 2000s [NV07, NV08, NV13]: +Theorem 4.28 (Nadirashvili–Vladuts, 2007-2013). There are solutions to +(4.2) that are not C2. These counterexamples exist in dimensions n ≥ 5. +Moreover, for every τ > 0, there exists a dimension n and ellipticity +constants λ and Λ, such that there are solutions u to F(D2u) = 0 with +u /∈ C1,τ. +We refer to the monograph [NTV14] for more references and details. +It is not known what happens in R3 and R4. This is one of the most +remarkable open problems in elliptic PDEs. +4.5. Further results and open problems +As explained above, one of the main open questions regarding the problem +(4.17) +F(D2u) = 0 +in +B1 ⊂ Rn +is the following: +Let u be any solution to (4.17) in R3 or R4. Is it true that u ∈ C2? +We have seen that it is in general not true that solutions to fully nonlinear +equations (in dimension n ≥ 5) are C2 under the assumption that F is +simply uniformly elliptic. Convexity, on the other hand, is a strong condition +under which C2 regularity is achieved, which, unfortunately, does not hold +in some important applications. Even with this, it is still unclear what the +optimal regularity of solutions is when F is convex and uniformly elliptic +(not necessarily smooth). Theorem 4.27 only gives, a priori, C2,α regularity +for some small α > 0. + +— DRAFT — +122 +4. Fully nonlinear elliptic PDE +These observations motivate, on the one hand, a more refined study for +the regularity (and size of singularity) of solutions to general fully nonlinear +elliptic equations, and on the other hand, the study of the optimal regularity +under the convexity assumption. +Partial regularity. Recall that the ellipticity requirement for F implies +that F is Lipschitz. Under the slightly more restrictive requirement that F +is also C1, the following partial regularity result was proved by Armstrong, +Silvestre, and Smart in [ASS12]: +Theorem 4.29 ([ASS12]). Let F be uniformly elliptic, and assume in +addition that F ∈ C1. Let u ∈ C0(B1) be any viscosity solution to (4.17). +Then, there exist some ε > 0 depending only on n, λ, Λ, and a closed +subset Σ ⊂ B1 with dimH Σ ≤ n − ε, such that u ∈ C2(B1 \ Σ). +Here, dimH denotes the Hausdorff dimension of a set; see [Mat95]. +Notice that if dimH Σ ≤ n − ε then in particular Σ has zero measure. +This result is the best known partial regularity result for solutions of +(non-convex) fully nonlinear equations in dimensions n ≥ 3. Notice that +the size of the singular set is not known to be optimal (it could be much +smaller!). Moreover, it is an important open problem to decide whether the +same statement holds without the regularity assumption F ∈ C1. +Optimal regularity when F is convex. When F is convex and uniformly +elliptic, solutions to (4.17) are known to be C2,α for some small α > 0. If +F ∈ C∞, a bootstrap argument then yields higher regularity for u, but the +higher regularity of F is needed. What happens if we just require F to be +convex and uniformly elliptic? +Since F is convex, the expression (4.17) can be reformulated as a supre- +mum of linear uniformly elliptic operators as +sup +a∈A +Lau = 0 +in +B1 ⊂ Rn, +also known as Bellman equation (see (C.3) in the Appendix C), where each +of the operators La is a linear uniformly elliptic operator. +The question that remains open here is: +What is the optimal regularity of solutions to Bellman equations? +In the simpler model of just two different operators, the previous equa- +tion is +(4.18) +max{L1u, L2u} = 0 +in +B1 ⊂ Rn. + +— DRAFT — +4.5. Further results and open problems +123 +The best known result in this direction was proved by Caffarelli, De +Silva, and Savin in 2018, and establishes the optimal regularity of solutions +to (4.18) in two dimensions: +Theorem 4.30 ([CDS18]). Let u be any viscosity solution to (4.18) in +B1 ⊂ R2. Then +∥u∥C2,1(B1/2) ≤ C∥u∥L∞(B1), +for some constant C depending only on λ and Λ. +The approach used in [CDS18] to show this result does not work in +higher dimensions n ≥ 3, and thus the following question remains open: +Let u be any solution to (4.18), with n ≥ 3. Is is true that u ∈ C2,1? + +— DRAFT — + +— DRAFT — +Chapter 5 +The obstacle problem +In this last chapter we focus our attention on a third type of nonlinear +elliptic PDE: a free boundary problem. In this kind of problems we are no +longer only interested in the regularity of a solution u, but also in the study +of an a priori unknown interphase Γ (the free boundary). +As explained later, there is a wide variety of problems in physics, indus- +try, biology, finance, and other areas which can be described by PDEs that +exhibit free boundaries. Many of such problems can be written as variational +inequalities, for which the solution is obtained by minimizing a constrained +energy functional. And one of the most important and canonical examples +is the obstacle problem.1 +Given a smooth function ϕ, the obstacle problem is the following: +(5.1) +minimize +1 +2 +� +Ω +|∇v|2dx +among all functions v ≥ ϕ. +Here, the minimization is subject to boundary conditions v|∂Ω = g. +The interpretation of such problem is clear: One looks for the least ener- +gy function v, but the set of admissible functions consists only of functions +that are above a certain “obstacle” ϕ. +In the two-dimensional case, one can think of the solution v as a “mem- +brane” which is elastic and is constrained to be above ϕ (see Figure 5.1). +1Other examples of important free boundary problems include the one-phase or Bernoulli +problem, the thin or fractional obstacle problem, etc. We refer the interested reader to [CS05, +PSU12, Vel23, Fer22] and the references therein. +125 + +— DRAFT — +126 +5. The obstacle problem +v +ϕ +−∆v ≥ 0 everywhere +v ≥ ϕ everywhere +∆v = 0 in {v > ϕ} +Figure 5.1. The function v minimizes the Dirichlet energy among all +functions with the same boundary values situated above the obstacle. +The Euler–Lagrange equation of the minimization problem is the follow- +ing: +(5.2) +� +� +� +v +≥ +ϕ +in Ω +∆v +≤ +0 +in Ω +∆v += +0 +in the set {v > ϕ}, +together with the boundary conditions v|∂Ω = g. +Indeed, notice that if we denote E(v) = 1 +2 +� +Ω |∇v|2dx, then we will have +E(v + εη) ≥ E(v) +for every ε ≥ 0 and η ≥ 0, η ∈ C∞ +c (Ω), +which yields ∆v ≤ 0 in Ω. That is, we can perturb v with nonnegative +functions (εη) and we always get admissible functions (v + εη). However, +due to the constraint v ≥ ϕ, we cannot perturb v with negative functions in +all of Ω, but only in the set {v > ϕ}. This is why we get ∆v ≤ 0 everywhere +in Ω, but ∆v = 0 only in {v > ϕ}. (We will show later that any minimizer +v of (5.1) is continuous, so that {v > ϕ} is open.) +Alternatively, we may consider u := v−ϕ, and the problem is equivalent +to +(5.3) +� +� +� +u +≥ +0 +in Ω +∆u +≤ +f +in Ω +∆u += +f +in the set {u > 0}, +where f := −∆ϕ. +Such solution u can be obtained as follows: +(5.4) +minimize +� +Ω +�1 +2|∇u|2 + fu +� +dx +among all functions u ≥ 0. + +— DRAFT — +5. The obstacle problem +127 +In other words, we can make the obstacle just zero, by adding a right- +hand side f. Here, the minimization is subject to the boundary conditions +u|∂Ω = ˜g, with ˜g := g − ϕ. +On the Euler–Lagrange equations. As said above, the Euler–Lagrange +equations of the minimization problem (5.1) are: +(i) v ≥ ϕ in Ω (v is above the obstacle). +(ii) ∆v ≤ 0 in Ω (v is a supersolution). +(iii) ∆v = 0 in {v > ϕ} (v is harmonic where it does not touch the +obstacle). +These are inequalities, rather than a single PDE. Alternatively, one can +write also the Euler–Lagrange equations in the following way: +min{−∆v, v − ϕ} = 0 +in +Ω. +(Notice that this resembles a fully nonlinear equation min{L1u, L2u} = 0, +but in the present situation one of the two operators is of order zero.) +Of course, the same can be done for the equivalent problem (5.3). In +that case, moreover, the minimization problem (5.4) is equivalent to +(5.5) +minimize +� +Ω +�1 +2|∇u|2 + fu+ +� +dx, +where u+ = max{u, 0}. +In this way, we can see the problem not as a +constrained minimization but as a minimization problem with a non-smooth +term u+ in the functional. The Euler–Lagrange equation for this functional +is then +(5.6) +∆u = fχ{u>0} +in +Ω. +(Here, χA denotes the characteristic function of a set A ⊂ Rn.) We will +show this in detail later. +The free boundary. Let us take a closer look at the obstacle problem (5.3). +One of the most important features of such problem is that it has two +unknowns: the solution u, and the contact set {u = 0}. In other words, +there are two regions in Ω: one in which u = 0; and one in which ∆u = f. +These regions are characterized by the minimization problem (5.4). More- +over, if we denote +Γ := ∂{u > 0} ∩ Ω, +then this is called the free boundary, see Figure 5.2. +The obstacle problem is a free boundary problem, as it involves an un- +known interface Γ as part of the problem. + +— DRAFT — +128 +5. The obstacle problem +{u = 0} +{u > 0} +∆u = f +Figure 5.2. The free boundary could, a priori, be very irregular. +Moreover, it is not difficult to see that the fact that u is a nonnegative +supersolution must imply ∇u = 0 on Γ, that is, we will have that u ≥ 0 +solves +� +� +� +∆u += +f +in {u > 0} +u += +0 +on Γ +∇u += +0 +on Γ. +This is an alternative way to write the Euler–Lagrange equation of the prob- +lem. In this way, the interface Γ appears clearly, and we see that we have +both Dirichlet and Neumann conditions on Γ. +This would usually be an over-determined problem (too many boundary +conditions on Γ), but since Γ is also free, it turns out that the problem has +a unique solution (where Γ is part of the solution, of course). +5.1. Some motivations and applications +Let us briefly comment on some of the main motivations and applications +in the study of the obstacle problem, which are further developed in Appen- +dix D (see also Appendix C). We refer to the books [DL76, KS80, Rod87, +Fri88, PSU12], for more details and further applications of obstacle-type +problems. +Fluid filtration. The so-called Dam problem aims to describe the filtration +of water inside a porous dam. One considers a dam separating two reservoirs +of water at different heights, made of a porous medium (permeable to water). +Then there is some transfer of water across the dam, and the interior of the +dam has a wet part, where water flows, and a dry part. In this setting, an +integral of the pressure (with respect to the height of the column of water at +each point) solves the obstacle problem, and the free boundary corresponds +precisely to the interphase separating the wet and dry parts of the dam. +Phase transitions. The Stefan problem, dating back to the 19th century, is +one of the most classical and important free boundary problems. It describes + +— DRAFT — +5.1. Some motivations and applications +129 +the temperature of a homogeneous medium undergoing a phase change, +typically a body of ice at zero degrees submerged in water. +In this context, it turns out that the integral of the temperature θ(x, t), +namely u(x, t) := +� t +0 θ, solves the parabolic version of the obstacle problem, +� +� +� +ut − ∆u += +χ{u>0} +in +Ω × (0, T) ⊂ R3 × R, +∂tu +≥ +0, +u +≥ +0. +The moving interphase separating the solid and liquid is exactly the free +boundary ∂{u > 0}. +Hele-Shaw flow. This two-dimensional model, dating back to 1898, de- +scribes a fluid flow between two flat parallel plates separated by a very thin +gap. Various problems in fluid mechanics can be approximated to Hele-Shaw +flows, and that is why understanding these flows is important. +A Hele-Shaw cell is an experimental device in which a viscous fluid is +sandwiched in a narrow gap between two parallel plates. In certain regions, +the gap is filled with fluid while in others the gap is filled with air. When +liquid is injected inside the device through some sinks (e.g. through a small +hole on the top plate) the region filled with liquid grows. In this context, an +integral of the pressure solves, for each fixed time t, the obstacle problem. +In a similar way to the Dam problem, the free boundary corresponds to the +interface between the fluid and the air regions. +Optimal stopping, finance. In probability and finance, the obstacle prob- +lem appears when considering optimal stopping problems for stochastic pro- +cesses. +Indeed, consider a random walk (Brownian motion) inside a domain +Ω ⊂ Rn, and a payoff function ϕ defined on the same domain. We can stop +the random walk at any moment, and we get the payoff at that position. +We want to maximize the expected payoff (by choosing appropriately the +stopping strategy). Then, it turns out that the highest expected payoff v(x) +starting at a given position x satisfies the obstacle problem (5.2), where +the contact set {v = ϕ} is the region where we should immediately stop +the random walk and get the payoff, while {v > ϕ} is the region where we +should wait (see Appendix C for more details). +Interacting particle systems. Large systems of interacting particles arise +in physical, biological, or material sciences. +In some models, the particles attract each other when they are far, and +experience a repulsive force when they are close. In other related models +in statistical mechanics, the particles (e.g. electrons) repel with a Coulomb + +— DRAFT — +130 +5. The obstacle problem +force and one wants to understand their behavior in presence of some exter- +nal field that confines them. +In this kind of models, a natural and interesting question is to deter- +mine the “equilibrium configurations”. For instance, in Coulomb systems +the charges accumulate in some region with a well defined boundary. Inter- +estingly, these problems are equivalent to the obstacle problem — namely, +the electric potential u = u(x) generated by the charges solves such prob- +lem — and the contact set {u = 0} corresponds to the region in which the +particles concentrate. +Quasi-Steady Electrochemical Shaping. Consider a metal inside an +electrolyte under the action of an electric potential, in such a way that the +metal shrinks with time due to a chemical reaction. Then, the integral (in +time) of the potential satisfies, for each fixed time, the obstacle problem, +whose free boundary corresponds to the shape of the metal at that moment. +Heat control. Trying to automatically control the temperature of a room +using only heating devices, under suitable conditions, also yields the obsta- +cle problem (in this case, for the temperature). Here, the free boundary +separates the region where the heating devices are active and where they +are not. +Elasticity. Finally, in elasticity theory we probably find the most visual +representation of the obstacle problem. +Given a thin membrane that is +affected only by tension forces (thus tries to minimize area), it approximately +satisfies the obstacle problem, where the contact region is the area where +the membrane touches the obstacle. +5.2. Basic properties of solutions I +We proceed now to study the basic properties of solutions to the obstacle +problem: existence of solutions, optimal regularity, and nondegeneracy. +We will first study all these properties for minimizers v ≥ ϕ of (5.1), +and then in the next section we will study independently minimizers u ≥ 0 +of (5.4) or (5.5). +This is not only for completeness and clarity of presentation, but also to +have both points of view. For instance, the proof of the optimal regularity +of solutions can be done in two completely different ways, one for each of +the settings. +Existence of solutions. Existence and uniqueness of solutions follows eas- +ily from the fact that the functional +� +Ω |∇v|2dx is convex, and that we want +to minimize it in the closed convex set {v ∈ H1(Ω) : v ≥ ϕ}. + +— DRAFT — +5.2. Basic properties of solutions I +131 +Recall that w|∂Ω denotes the trace of w on ∂Ω whenever it is defined. +Proposition 5.1 (Existence and uniqueness). Let Ω ⊂ Rn be any bounded +Lipschitz domain, and let g : ∂Ω → R and ϕ ∈ H1(Ω) be such that +C = +� +w ∈ H1(Ω) : w ≥ ϕ in Ω, w|∂Ω = g +� +̸= ∅. +Then, there exists a unique minimizer of +� +Ω |∇v|2dx among all functions +v ∈ H1(Ω) satisfying v ≥ ϕ in Ω and v|∂Ω = g. +Proof. The proof is quite similar to that of Theorem 1.10. Indeed, let +θ◦ := inf +�1 +2 +� +Ω +|∇w|2dx : w ∈ H1(Ω), w|∂Ω = g, w ≥ ϕ in Ω +� +, +that is, the infimum value of E(w) = 1 +2 +� +Ω |∇w|2dx among all admissible +functions w. +Let us take a sequence of functions {vk} such that +• vk ∈ H1(Ω) +• vk|∂Ω = g and vk ≥ ϕ in Ω. +• E(vk) → θ◦ as k → ∞. +By the Poincar´e inequality (Theorem 1.6), the sequence {vk} is uniformly +bounded in H1(Ω), and therefore a subsequence {vkj} will converge to a +certain function v strongly in L2(Ω) and weakly in H1(Ω). Moreover, by +compactness of the trace operator (see (S5) in Chapter 1), we will have +vkj|∂Ω → v|∂Ω in L2(∂Ω), so that v|∂Ω = g. Furthermore, such function v +will satisfy E(v) ≤ lim infj→∞ E(vkj) (by (1.4)-(1.5) from (S4) in Chapter 1), +and therefore it will be a minimizer of the energy functional. Since vkj ≥ ϕ +in Ω and vkj → v in L2(Ω), we have v ≥ ϕ in Ω. Thus, we have proved the +existence of a minimizer v. +The uniqueness of the minimizer follows from the strict convexity of the +functional E(v), exactly as in Theorem 1.10. +□ +As in the case of harmonic functions, it is easy to show that if a function +v satisfies +� +� +� +v +≥ +ϕ +in Ω +∆v +≤ +0 +in Ω +∆v += +0 +in the set {v > ϕ}, +then it must actually be the minimizer of the functional. +There are two alternative ways to construct the solution to the obstacle +problem: as the “least supersolution above the obstacle”, or with a “penal- +ized problem”. Let us briefly describe them. + +— DRAFT — +132 +5. The obstacle problem +t +βε(t) = e−t/ε +Figure 5.3. The function βε → β0 as ε ↓ 0. +• Least supersolution: This is related to the existence of viscosity solutions +described in Chapter 4. Indeed, we consider +v(x) := inf +� +w(x) : w ∈ C(Ω), −∆w ≥ 0 in Ω, w ≥ ϕ in Ω, w|∂Ω ≥ g +� +. +Here, the inequality −∆w ≥ 0 in Ω has to be understood in the viscosity +sense. +Then, as in Perron’s method (recall Chapters 1 and 4), it turns out that +v is itself a continuous supersolution, it satisfies ∆v = 0 in {v > ϕ}, and +thus it solves the obstacle problem. Therefore, +� +least +supersolution +� +←→ +� minimizer of +the functional +� +. +• Penalized problem: We consider βε : R → R smooth and convex, converg- +ing to +β0(t) := +� 0 +if +t ≥ 0 +∞ +if +t < 0. +We may take for example βε(t) := e−t/ε, see Figure 5.3. +Then, we minimize the functional +Jε(v) := 1 +2 +� +Ω +|∇v|2dx + +� +Ω +βε(v − ϕ)dx, +subject to the appropriate boundary conditions on ∂Ω, and get a solution +vε ∈ C∞ of ∆vε = β′ +ε(vε − ϕ) in Ω. +Since β′ +ε ≤ 0 everywhere, and β′ +ε(t) = 0 for t ≥ 0, we have +� −∆vε +≥ +0 +everywhere in Ω +∆vε += +0 +in {vε > ϕ}. + +— DRAFT — +5.2. Basic properties of solutions I +133 +As ε → 0, we have vε → v, where v is the solution to the obstacle problem. +We refer to [PSU12] for more details. +Basic properties of solutions. Let us next prove that any minimizer v +of (5.1) is actually continuous and solves (5.2). +From now on we will “forget” about the regularity of the obstacle, and +assume that it is as smooth as needed. +This is why we will always be +dealing with obstacles ϕ ∈ C∞(Ω). One gets analogous results under much +weaker regularity assumptions on ϕ, which depend on the type of result to +be proved. The role of the regularity of the obstacle is beyond the scope of +this book, and thus we will always assume ϕ to be smooth. +We start with the following lemma. +Lemma 5.2. Let Ω ⊂ Rn be any bounded Lipschitz domain, ϕ ∈ C∞(Ω), +and v ∈ H1(Ω) be any minimizer of (5.1) subject to the boundary conditions +v|∂Ω = g. +Then, −∆v ≥ 0 in Ω. +Proof. Let +E(v) = 1 +2 +� +Ω +|∇v|2dx. +Then, since v minimizes E among all functions above the obstacle ϕ (and +with fixed boundary conditions on ∂Ω), we have that +E(v + εη) ≥ E(v) +for every ε ≥ 0 and η ≥ 0, η ∈ C∞ +c (Ω). +This yields +ε +� +Ω +∇v · ∇η + ε2 +2 +� +Ω +|∇η|2dx ≥ 0 +for every ε ≥ 0 and η ≥ 0, η ∈ C∞ +c (Ω), +and thus +� +Ω +∇v · ∇η ≥ 0 +for every η ≥ 0, η ∈ C∞ +c (Ω). +This means that −∆v ≥ 0 in Ω in the weak sense, as desired. +□ +From here, by showing first that {v > ϕ} is open, we obtain the Euler– +Lagrange equations for the functional: +Proposition 5.3. Let Ω ⊂ Rn be any bounded Lipschitz domain, ϕ ∈ +C∞(Ω), and v ∈ H1(Ω) be any minimizer of (5.1) subject to the bound- +ary conditions v|∂Ω = g. +Then, v ∈ C(Ω) and it satisfies +(5.7) +� +� +� +v +≥ +ϕ +in Ω +∆v +≤ +0 +in Ω +∆v += +0 +in {v > ϕ} ∩ Ω. + +— DRAFT — +134 +5. The obstacle problem +Proof. By construction, we already know that v ≥ ϕ in Ω and, thanks +to Lemma 5.2, −∆v ≥ 0 in Ω, i.e, v is (weakly) superharmonic. +Up to +replacing v in a set of measure zero, we may also assume that v is lower +semi-continuous (by Lemma 1.17). Thus, we only need to prove that ∆v = 0 +in {v > ϕ} ∩ Ω and that v is, in fact, continuous. +In order to do that, let us show first that {v > ϕ} ∩ Ω is open. Let +x◦ ∈ {v > ϕ} ∩ Ω be such that v(x◦) − ϕ(x◦) > ε◦ > 0. Since v is lower +semi-continuous and ϕ is continuous, there exists some δ > 0 such that +v(x) − ϕ(x) > ε◦/2 for all x ∈ Bδ(x◦), and hence Bδ(x◦) ⊂ {v > ϕ}. Since +x◦ was arbitrary, this means that {v > ϕ} is open. +This implies, also, +that ∆v = 0 weakly in {v > ϕ} ∩ Ω. Indeed, for any x◦ ∈ {v > ϕ} and +η ∈ C∞ +c (Bδ(x◦)) with |η| ≤ 1, we have v ± εη ≥ ϕ in Ω for all |ε| < ε◦/2, +and therefore it is an admissible competitor to the minimization problem. +Thus, we have E(v + εη) ≥ E(v) for all |ε| < ε◦, and differentiating in ε we +deduce that v is harmonic in {v > ϕ} ∩ Ω. +Finally, let us show that v is continuous. +We already know, by the +regularity of harmonic functions (e.g. Corollary 1.12), that v is continuous +in {v > ϕ} ∩ Ω. Let us now show that v is continuous in {v = ϕ} ∩ Ω as +well. +Let y◦ ∈ {v = ϕ} ∩ Ω, and let us argue by contradiction. That is, since +v is lower semi-continuous, let us assume that there is a sequence yk → y◦ +such that v(yk) → v(y◦) + ε◦ = ϕ(y◦) + ε◦ for some ε◦ > 0. Since ϕ is +continuous, we may assume also that yk ∈ {v > ϕ}. Let us denote by zk the +projection of yk towards {v = ϕ}, so that δk := |zk − y◦| ≤ 2|yk − y◦| ↓ 0 +and v(zk) → v(y◦) = ϕ(y◦). Now, since v is superharmonic by (1.20), +v(zk) ≥ +� +B2δk(yk) +v = (1 − 2−n) +� +B2δk(yk)\Bδk(yk) +v + 2−n +� +Bδk(yk) +v = I1 + I2. +Observe that, for the first term, since v is lower semi-continuous and δk ↓ 0, +we can assume that, for k large enough, v ≥ ϕ(y◦) − 2−nε◦ in B2δk, so that +I1 ≥ (1 − 2−n)[ϕ(y◦) − 2−nε◦]. On the other hand, since v is harmonic in +Bδk(yk), we have by the mean-value property that I2 = 2−nv(yk). Combin- +ing everything, we get +v(zk) ≥ (1 − 2−n)[ϕ(y◦) − 2−nε◦] + 2−nv(yk) → ϕ(y◦) + 2−2nε◦ +which contradicts the fact that we had v(zk) → v(y◦) = ϕ(y◦). Hence, v is +continuous in Ω. +□ +We next prove the following result, which says that v can be character- +ized as the least supersolution above the obstacle. + +— DRAFT — +5.2. Basic properties of solutions I +135 +v +ϕ +∆v = 0 +∆v = ∆ϕ +Figure 5.4. Second derivatives are in general discontinuous across the +free boundary. +Proposition 5.4 (Least supersolution). Let Ω ⊂ Rn be any bounded Lips- +chitz domain, ϕ ∈ H1(Ω), and v ∈ H1(Ω) be any minimizer of (5.1) subject +to the boundary conditions v|∂Ω = g. +Then, for any function w satisfying −∆w ≥ 0 in Ω, w ≥ ϕ in Ω, and +w|∂Ω ≥ v|∂Ω, we have w ≥ v in Ω. In other words, if w is any supersolution +above the obstacle ϕ, then w ≥ v. +Proof. If w is any function satisfying −∆w ≥ 0 in Ω, w ≥ ϕ in Ω, +and w|∂Ω ≥ v|∂Ω, it simply follows from the maximum principle (Propo- +sition 1.13) that w ≥ v. Indeed, we have −∆w ≥ −∆v in Ω ∩ {v > ϕ}, +and on the boundary of such set we have w|∂Ω ≥ v|∂Ω and w ≥ ϕ = v on +{v = ϕ}. +□ +Optimal regularity of solutions. Thanks to Proposition 5.3, we know +that any minimizer of (5.1) is continuous and solves (5.7). From now on, +we will actually localize the problem and study it in a ball: +(5.8) +� +� +� +v +≥ +ϕ +in B1 +∆v +≤ +0 +in B1 +∆v += +0 +in {v > ϕ} ∩ B1. +Our next goal is to answer the following question: +Question: +What is the optimal regularity of solutions? +First, a few important considerations. Notice that in the set {v > ϕ} +we have ∆v = 0, while in the interior of {v = ϕ} we have ∆v = ∆ϕ (since +v = ϕ there); see Figure 5.4. +Thus, since ∆ϕ is in general not zero, ∆v is discontinuous across the +free boundary ∂{v > ϕ} in general. In particular, v /∈ C2. + +— DRAFT — +136 +5. The obstacle problem +We will now prove that any minimizer of (5.1) is actually C1,1, which +gives the: +Answer: v ∈ C1,1 (second derivatives are bounded but not continuous) +The precise statement and proof are given next. +Theorem 5.5 (Optimal regularity). Let ϕ ∈ C∞(B1), and v be any solution +to (5.8). Then, v is C1,1 in B1/2, with the estimate +∥v∥C1,1(B1/2) ≤ C +� +∥v∥L∞(B1) + ∥ϕ∥C1,1(B1) +� +. +The constant C depends only on n. +To prove this, the main step is the following. +Lemma 5.6. Let ϕ ∈ C∞(B1), and v be any solution to (5.8). Let x◦ ∈ B1/2 +be any point on {v = ϕ}. +Then, for any r ∈ (0, 1 +4) we have +0 ≤ sup +Br(x◦) +(v − ϕ) ≤ Cr2, +with C depending only on n and ∥ϕ∥C1,1(B1). +Proof. After dividing v by a constant if necessary, we may assume that +∥ϕ∥C1,1(B1) ≤ 1. +Let ℓ(x) := ϕ(x◦) + ∇ϕ(x◦) · (x − x◦) be the linear part of ϕ at x◦. Let +r ∈ (0, 1 +4). Then, by C1,1 regularity of ϕ, in Br(x◦) we have +ℓ(x) − r2 ≤ ϕ(x) ≤ v(x). +We want to show that, in the ball Br(x◦) (see Figure 5.5), we have +v(x) ≤ ℓ(x) + Cr2. +For this, consider +w(x) := v(x) − +� +ℓ(x) − r2� +. +This function w satisfies w ≥ 0 in Br(x◦), and −∆w = −∆v ≥ 0 in Br(x◦). +Let us split w into +w = w1 + w2, +with +� ∆w1 += +0 +in Br(x◦) +w1 += +w +on ∂Br(x◦) +and +� −∆w2 +≥ +0 +in Br(x◦) +w2 += +0 +on ∂Br(x◦). +Notice that +0 ≤ w1 ≤ w +and +0 ≤ w2 ≤ w. + +— DRAFT — +5.2. Basic properties of solutions I +137 +x◦ +{v = ϕ} +∂Br(x◦) +Figure 5.5. The solution v and a free boundary point x◦ +We have that +w1(x◦) ≤ w(x◦) = v(x◦) − +� +ℓ(x◦) − r2� += r2, +and thus by the Harnack inequality +∥w1∥L∞(Br/2(x◦)) ≤ Cr2. +For w2, notice that ∆w2 = ∆v, and in particular ∆w2 = 0 in {v > ϕ}. +This means that w2 attains its maximum on {v = ϕ}. But in the set {v = ϕ} +we have +w2 ≤ w = ϕ − +� +ℓ − r2� +≤ Cr2, +and therefore we deduce that +∥w2∥L∞(Br(x◦)) ≤ Cr2. +Combining the bounds for w1 and w2, we get ∥w∥L∞(Br(x◦)) ≤ Cr2. +Translating this into v, and using that ∥ϕ∥C1,1(B1) ≤ 1, we find v − ϕ ≤ Cr2 +in Br/2(x◦). +□ +Therefore, we have proved that: +At every free boundary point x◦, v separates from ϕ at most quadratically. +As shown next, this easily implies the C1,1 regularity. +Proof of Theorem 5.5. Dividing v by a constant if necessary, we may +assume that ∥v∥L∞(B1) + ∥ϕ∥C1,1(B1) ≤ 1. +We already know that v ∈ C∞ in the set {v > ϕ} (since v is harmonic), +and also in the interior of the set {v = ϕ} (since ϕ ∈ C∞). Moreover, on the +interface Γ = ∂{v > ϕ} we have proved the quadratic growth supBr(x◦)(v − +ϕ) ≤ Cr2. Let us prove that this yields the C1,1 bound we want. + +— DRAFT — +138 +5. The obstacle problem +{v = ϕ} +x◦ +x1 +ρ +Bρ(x1) +{v > ϕ} +∆v = 0 +Γ +Figure 5.6. A solution v satisfying ∆v = 0 in Bρ(x1) ⊂ {v > ϕ}. +Let x1 ∈ {v > ϕ} ∩ B1/2, and let x◦ ∈ Γ be the closest free boundary +point. Denote ρ = |x1 − x◦|. Then, we have ∆v = 0 in Bρ(x1) (see the +setting in Figure 5.6), and thus we have also ∆(v − ℓ) = 0 in Bρ(x1), where +ℓ is the linear part of ϕ at x◦. +By estimates for harmonic functions, we find +∥D2v∥L∞(Bρ/2(x1)) = ∥D2(v − ℓ)∥L∞(Bρ/2(x1)) ≤ C +ρ2 ∥v − ℓ∥L∞(Bρ(x1)). +But by the growth proved in the previous Lemma, we have ∥v−ℓ∥L∞(Bρ(x1)) ≤ +Cρ2, which yields +∥D2v∥L∞(Bρ/2(x1)) ≤ C +ρ2 ρ2 = C. +In particular, |D2v(x1)| ≤ C. We can do this for all x1 ∈ {v > ϕ} ∩ B1/2, +and on ∂{v > ϕ} we have quadratic growth by Lemma 5.6, hence it follows +that ∥v∥C1,1(B1/2) ≤ C, as wanted. +□ +The overall strategy of the proof of optimal regularity is summarized in +Figure 5.7. +Nondegeneracy. We now want to prove that, at all free boundary points, +v separates from ϕ at least quadratically (we already know at most quadrat- +ically). +That is, we want +(5.9) +0 < cr2 ≤ sup +Br(x◦) +(v − ϕ) ≤ Cr2 +for all free boundary points x◦ ∈ ∂{v > ϕ}. +This property is essential in order to study the free boundary later. + +— DRAFT — +5.2. Basic properties of solutions I +139 +{u = 0} +{u = 0} +{u = 0} +∂{u = 0} +∂{u = 0} +∂{u = 0} +quadratic +growth by +Lemma 5.6 +u ∈ C1,1 by +interior +estimates +u +u +u +Cr2 +Cr2 +Figure 5.7. Strategy of the proof of Theorem 5.5. +Remark 5.7. Since −∆v ≥ 0 everywhere, it is clear that if x◦ ∈ ∂{v > +ϕ} is a free boundary point, then necessarily −∆ϕ(x◦) ≥ 0 (otherwise we +would have −∆ϕ(x◦) < 0, and since u touches ϕ from above at x◦, also +−∆v(x◦) < 0, a contradiction). +Moreover it can be proved that, in fact, if ∆ϕ and ∇∆ϕ do not vanish +simultaneously, then −∆ϕ > 0 near all free boundary points [Caf98]. +This motivates the following: +Assumption: The obstacle ϕ satisfies +−∆ϕ ≥ c◦ > 0 +in the ball B1. +In particular, by Remark 5.7, if ∆ϕ and ∇∆ϕ do not vanish simultane- +ously, then we have −∆ϕ > 0 near any free boundary point, and thus by +zooming in if necessary, we will always have that the assumption is satisfied +in B1, for some small c◦ > 0. +Thus, the only real assumption here is that ∆ϕ and ∇∆ϕ do not vanish +simultaneously, which is a very mild assumption. +Moreover, this is in a +sense a necessary assumption: without this, the nondegeneracy (5.9) does +not hold, and no regularity result can be proved for the free boundary. +(Without the assumption, one can actually construct counterexamples in +which the free boundary is a fractal set with infinite perimeter.) +Proposition 5.8 (Nondegeneracy). Let ϕ ∈ C∞(B1), and v be any solution +to (5.8). Assume that ϕ satisfies −∆ϕ ≥ c◦ > 0 in B1. Then, for every free +boundary point x◦ ∈ ∂{v > ϕ} ∩ B1/2, we have +0 < cr2 ≤ sup +Br(x◦) +(v − ϕ) ≤ Cr2 +for all r ∈ (0, 1 +4), +with a constant c > 0 depending only on n and c◦. +Proof. Let x1 ∈ {v > ϕ} be any point close to x◦ (we will then let x1 → x◦ +at the end of the proof). + +— DRAFT — +140 +5. The obstacle problem +Consider the function +w(x) := v(x) − ϕ(x) − c◦ +2n|x − x1|2. +Then, in {v > ϕ} we have +∆w = ∆v − ∆ϕ − c◦ = −∆ϕ − c◦ ≥ 0 +and hence −∆w ≤ 0 in {v > ϕ} ∩ Br(x1). Moreover, w(x1) > 0. +By the maximum principle, w attains a positive maximum on ∂ +� +{v > +ϕ} ∩ Br(x1) +� +. But on the free boundary ∂{v > ϕ} we clearly have w < 0. +Therefore, there is a point on ∂Br(x1) at which w > 0. In other words, +0 < +sup +∂Br(x1) +w = +sup +∂Br(x1) +(v − ϕ) − c◦ +2n r2. +Letting now x1 → x◦, we find sup∂Br(x◦)(v − ϕ) ≥ cr2 > 0, as desired. +□ +Summary of basic properties. Let v be any solution to the obstacle +problem +� +� +� +v +≥ +ϕ +in B1 +∆v +≤ +0 +in B1 +∆v += +0 +in {v > ϕ} ∩ B1. +Then, we have: +• Optimal regularity: +∥v∥C1,1(B1/2) ≤ C +� +∥v∥L∞(B1) + ∥ϕ∥C1,1(B1) +� +• Nondegeneracy: If −∆ϕ ≥ c◦ > 0, then +0 < cr2 ≤ sup +Br(x◦) +(v − ϕ) ≤ Cr2 +for all r ∈ (0, 1 +2) +at all free boundary points x◦ ∈ ∂{v > ϕ} ∩ B1/2. +• Equivalence with zero obstacle: The problem is equivalent to +� +� +� +u +≥ +0 +in B1 +∆u +≤ +f +in B1 +∆u += +f +in {u > 0} ∩ B1, +where f = −∆ϕ ≥ c◦ > 0. +We will next provide an alternative approach to the optimal regularity. + +— DRAFT — +5.3. Basic properties of solutions II +141 +5.3. Basic properties of solutions II +We proceed now to study the basic properties of solutions u ≥ 0 to the +obstacle problem (5.4) or (5.5). As explained before, the main point here is +that we prove optimal regularity independently from the previous Section. +Throughout this section we will always assume +f ≥ 0 +in +Ω. +Existence of solutions. Since problem (5.4) is equivalent to (5.1), ex- +istence and uniqueness of solutions follow easily from Proposition 5.1, as +shown next. +Proposition 5.9 (Existence and uniqueness). Let Ω ⊂ Rn be any bounded +Lipschitz domain, and let g : ∂Ω → R be such that +C = +� +u ∈ H1(Ω) : u ≥ 0 in Ω, u|∂Ω = g +� +̸= ∅. +Then, for any f ∈ L2(Ω) there exists a unique minimizer of +1 +2 +� +Ω +|∇u|2dx + +� +Ω +fu +among all functions u ∈ H1(Ω) satisfying u ≥ 0 in Ω and u|∂Ω = g. +Proof. We follow the proof of Proposition 5.1. Let +θ◦ := inf +�1 +2 +� +Ω +|∇w|2dx + +� +Ω +fw : w ∈ H1(Ω), w|∂Ω = g, w ≥ 0 in Ω +� +, +that is, the infimum value of E(w) = 1 +2 +� +Ω |∇w|2dx+ +� +Ω fw among all admis- +sible functions w. Notice that, by H¨older’s inequality, E(w) < +∞ if w ∈ +H1(Ω). +We take again a sequence of functions {vk} such that vk ∈ H1(Ω), +vk|∂Ω = g, vk ≥ 0 in Ω, and E(vk) → θ◦ as k → ∞. By the Poincar´e inequal- +ity (Theorem 1.6), H¨older’s inequality, and the fact that E(vk) ≤ θ◦ + 1, for +k large enough +∥vk∥2 +H1(Ω) ≤ C +�� +Ω +|∇vk|2 + +� +∂Ω +g2 +� +≤ C +� +θ◦ + 1 + +� +Ω +|fvk| + 1 +2 +� +∂Ω +g2 +� +≤ C +� +θ◦ + 1 + ∥f∥L2(Ω)∥vk∥H1(Ω) + 1 +2 +� +∂Ω +g2 +� +. +In particular, ∥vk∥H1(Ω) ≤ C for some constant C depending only on n, Ω, +g, f, and θ◦ (recall that g ∈ L2(∂Ω) by the trace theorem, (S5) in Chapter +1). Hence, a subsequence {vkj} converges to a certain function v strongly in +L2(Ω) and weakly in H1(Ω). By compactness of the trace operator vkj|∂Ω → +v|∂Ω = g in L2(∂Ω). Furthermore, v satisfies E(v) ≤ lim infj→∞ E(vkj) (by +(1.4)-(1.5) from (S4) and weak convergence), and therefore it will be a + +— DRAFT — +142 +5. The obstacle problem +minimizer of the energy functional. Since vkj ≥ 0 in Ω and vkj → v in +L2(Ω), we have v ≥ 0 in Ω. Thus, there is a minimizer v. +The uniqueness of the minimizer follows from the strict convexity of the +functional E(v), exactly as in Theorem 1.10. +□ +Remark 5.10. Alternatively, we could have denoted v := u+ϕ with ϕ such +that −∆ϕ = f in Ω, and use Proposition 5.1. +Furthermore, we have the following equivalence. (Recall that we denote +u+ = max{u, 0}, and u− = max{−u, 0}, so that u = u+ − u−.) +Proposition 5.11. Let Ω ⊂ Rn be any bounded Lipschitz domain, and let +g : ∂Ω → R be such that +C = +� +u ∈ H1(Ω) : u ≥ 0 in Ω, u|∂Ω = g +� +̸= ∅. +Then, the following are equivalent. +(i) u minimizes 1 +2 +� +Ω |∇u|2+ +� +Ω fu among all functions satisfying u ≥ 0 +in Ω and u|∂Ω = g. +(ii) u minimizes +1 +2 +� +Ω |∇u|2 + +� +Ω fu+ among all functions satisfying +u|∂Ω = g. +Proof. The two functionals coincide whenever u ≥ 0. Thus, the only key +point is to prove that the minimizer in (ii) must be nonnegative, i.e., u = u+. +(Notice that C ̸= ∅ implies that g ≥ 0 on ∂Ω.) To show this, recall that +the positive part of any H1 function is still in H1, and moreover |∇u|2 = +|∇u+|2 + |∇u−|2 (see (S9) in Chapter 1). Thus, we have that (recall that +f ≥ 0 in Ω) +1 +2 +� +Ω +|∇u+|2 + +� +Ω +fu+ ≤ 1 +2 +� +Ω +|∇u|2 + +� +Ω +fu+, +with strict inequality unless u = u+. This means that any minimizer u of +the functional in (ii) must be nonnegative, and thus we are done. +□ +Basic properties of solutions. Let us next prove that any minimizer of +(5.4) is actually a solution to (5.10) below. +We recall that we are always assuming that obstacles are as smooth as +necessary, ϕ ∈ C∞(Ω), and therefore we assume here that f ∈ C∞(Ω) as +well. +Proposition 5.12. Let Ω ⊂ Rn be any bounded Lipschitz domain, f ∈ +C∞(Ω), and u ∈ H1(Ω) be any minimizer of (5.4) subject to the boundary +conditions u|∂Ω = g. +Then, u solves +(5.10) +∆u = fχ{u>0} +in +Ω + +— DRAFT — +5.3. Basic properties of solutions II +143 +in the weak sense. +In particular, u is C1,α inside Ω, for every α ∈ (0, 1). +Proof. Notice that, by Proposition 5.11, u is actually a minimizer of +E(u) = 1 +2 +� +Ω +|∇u|2 + +� +Ω +fu+ +subject to the boundary conditions u|∂Ω = g. +Thus, for any η ∈ H1 +0(Ω) and ε > 0 we have +E(u + εη) ≥ E(u). +In particular, we obtain +0 ≤ lim +ε↓0 +E(u + εη) − E(u) +ε += +� +Ω +∇u · ∇η + lim +ε↓0 +� +Ω +f (u + εη)+ − u+ +ε +. +Notice that +lim +ε↓0 +(u + εη)+ − u+ +ε += +� η +in +{u > 0} +η+ +in +{u = 0}, +so that we have +� +Ω +∇u · ∇η + +� +Ω +fηχ{u>0} + +� +Ω +fη+χ{u=0} ≥ 0 +for all +η ∈ H1 +0(Ω). +Assume first that η ≥ 0, so that +� +Ω +∇u · ∇η + +� +Ω +fη ≥ 0 +for all +η ∈ H1 +0(Ω), +η ≥ 0, +which implies that ∆u ≤ f in the weak sense. On the other hand, if η ≤ 0, +then +� +Ω +∇u · ∇η + +� +Ω +fηχ{u>0} ≥ 0 +for all +η ∈ H1 +0(Ω), +η ≤ 0, +which implies that ∆u ≥ fχ{u>0} in the weak sense. +In all (recall that +f ≥ 0), +fχ{u>0} ≤ ∆u ≤ f +in +Ω. +(In particular, notice that ∆u = f in {u > 0}.) Now, since f is smooth, this +implies that ∆u ∈ L∞ +loc(Ω). By Proposition 2.18 we deduce that u ∈ C1,1−ε +for every ε > 0. Moreover, since ∆u ∈ L∞ +loc(Ω) we have ∆u ∈ L2 +loc(Ω) and +by Calder´on-Zygmund estimates (see, for example, Remark 2.13) we have +u ∈ W 2,2 +loc (Ω). Thus, ∆u = 0 almost everywhere in the level set {u = 0} (see +(S9) in Chapter 1) and we have +∆u = fχ{u>0} +a.e. in +Ω. +From here we deduce that ∆u = fχ{u>0} in Ω in the weak sense. +□ + +— DRAFT — +144 +5. The obstacle problem +Notice that in the previous Section, when dealing with minimizers v of +(5.1), it was not easy to prove that v is continuous (see Proposition 5.3). +Here, instead, thanks to Proposition 5.12 we simply used Schauder-type +estimates for the Laplacian to directly deduce that u is C1,1−ε, which is the +almost-optimal regularity of solutions. +Optimal regularity of solutions. Thanks to the previous results, we +know that any minimizer of (5.4) is continuous and solves (5.10). From now +on, we will localize the problem and study it in a ball: +(5.11) +� +u +≥ +0 +in B1 +∆u += +fχ{u>0} +in B1. +Our next goal is to answer the following question: +Question: +What is the optimal regularity of solutions? +First, a few important considerations. Notice that in the set {u > 0} we +have ∆u = f, while in the interior of {u = 0} we have ∆u = 0 (since u ≡ 0 +there). +Thus, since f is in general not zero, ∆u is discontinuous across the free +boundary ∂{u > 0} in general. In particular, u /∈ C2. +We will now prove that any minimizer of (5.4) is actually C1,1, which +gives the: +Answer: u ∈ C1,1 (second derivatives are bounded but not continuous) +The precise statement and proof are given next. +Theorem 5.13 (Optimal regularity). Let f ∈ C∞(B1), and let u be any +solution to (5.11). Then, u is C1,1 inside B1/2, with the estimate +∥u∥C1,1(B1/2) ≤ C +� +∥u∥L∞(B1) + ∥f∥Lip(B1) +� +. +The constant C depends only on n. +To prove this, the main step is the following. +Lemma 5.14. Let u be any solution to (5.11). Let x◦ ∈ B1/2 be any point +on {u = 0}. Then, for any r ∈ (0, 1 +4) we have +0 ≤ sup +Br(x◦) +u ≤ Cr2, +with C depending only on n and ∥f∥L∞(B1). + +— DRAFT — +5.3. Basic properties of solutions II +145 +Proof. We have that ∆u = fχ{u>0} in B1, with fχ{u>0} ∈ L∞(B1). Thus, +since u ≥ 0, we can use the Harnack inequality (Theorem 2.9) for the equa- +tion ∆u = fχ{u>0} in B2r(x◦), to find +sup +Br(x◦) +u ≤ C +� +inf +Br(x◦) u + r2∥fχ{u>0}∥L∞(B2r(x◦)) +� +. +Since u ≥ 0 and u(x◦) = 0, this yields supBr(x◦) u ≤ C∥f∥L∞(B1)r2, as +wanted. +□ +Notice that this proof is significantly shorter than the one given in the +previous Section (Lemma 5.6). This is an advantage of using the formula- +tion (5.10). +We have proved the following: +At every free boundary point x◦, u grows (at most) quadratically. +As shown next, this easily implies the C1,1 regularity. +Proof of Theorem 5.13. Dividing u by a constant if necessary, we may +assume that ∥u∥L∞(B1) + ∥f∥Lip(B1) ≤ 1. +We already know that u ∈ C∞ in the set {u > 0} (since ∆u = f ∈ C∞ +there), and also inside the set {u = 0} (since u = 0 there). Moreover, on the +interface Γ = ∂{u > 0} we have proved the quadratic growth supBr(x◦) u ≤ +Cr2. Let us prove that this yields the C1,1 bound we want. +Let x1 ∈ {u > 0} ∩ B1/2, and let x◦ ∈ Γ be the closest free boundary +point. Denote ρ = |x1 − x◦|. Then, we have ∆u = f in Bρ(x1). +By Schauder estimates, we find +∥D2u∥L∞(Bρ/2(x1)) ≤ C +� 1 +ρ2 ∥u∥L∞(Bρ(x1)) + ∥f∥Lip(B1) +� +. +But by the growth proved in the previous Lemma, we have ∥u∥L∞(Bρ(x1)) ≤ +Cρ2, which yields +∥D2u∥L∞(Bρ/2(x1)) ≤ C. +In particular, +|D2u(x1)| ≤ C. +We can do this for each x1 ∈ {u > 0}∩B1/2, and therefore ∥u∥C1,1(B1/2) ≤ C, +as wanted. +□ +Also, notice that as a consequence of the previous results, we have that +as soon as the solution to (5.11) has non-empty contact set, then its C1,1 +norm is universally bounded. + +— DRAFT — +146 +5. The obstacle problem +Corollary 5.15. Let u be any solution to (5.11), and let us assume that +u(0) = 0 and ∥f∥Lip(B1) ≤ 1. Then, +∥u∥C1,1(B1/2) ≤ C +for some C depending only on n. +Proof. It is an immediate consequence of Theorem 5.13 combined with +Lemma 5.14. +□ +Nondegeneracy. For completeness, we now state the nondegeneracy in +this setting (analogously to Proposition 5.8). That is, at all free boundary +points, u grows at least quadratically (we already know at most quadrati- +cally). We want: +0 < cr2 ≤ sup +Br(x◦) +u ≤ Cr2 +for all free boundary points x◦ ∈ ∂{u > 0}. +This property is essential in order to study the free boundary later. As +before, for this we need the following natural assumption: +Assumption: The right-hand side f satisfies +f ≥ c◦ > 0 +in the ball B1. +Proposition 5.16 (Nondegeneracy). Let u be any solution to (5.11). As- +sume that f ≥ c◦ > 0 in B1. Then, for every free boundary point x◦ ∈ +∂{u > 0} ∩ B1/2, we have +0 < cr2 ≤ sup +Br(x◦) +u ≤ Cr2 +for all r ∈ (0, 1 +2), +with a constant c > 0 depending only on n and c◦. +Proof. The proof is the one from Proposition 5.8. +□ +Summary of basic properties. Let u be any solution to the obstacle +problem +� +u +≥ +0 +in B1, +∆u += +fχ{u>0} +in B1. +Then, we have: +• Optimal regularity: +∥u∥C1,1(B1/2) ≤ C +� +∥u∥L∞(B1) + ∥f∥Lip(B1) +� +• Nondegeneracy: If f ≥ c◦ > 0, then +0 < cr2 ≤ sup +Br(x◦) +u ≤ Cr2 +for all r ∈ (0, 1 +2) + +— DRAFT — +5.4. Regularity of free boundaries: an overview +147 +{u = 0} +∂B1 +∆u = f +Γ +u ∈ C1,1 +Figure 5.8. A solution to the obstacle problem in B1. +at all free boundary points x◦ ∈ ∂{u > 0} ∩ B1/2. +Using these properties, we can now start the study of the free boundary. +5.4. Regularity of free boundaries: an overview +From now on, we consider any solution to +(5.12) +� +� +� +� +� +u ∈ C1,1(B1), +u ≥ 0 +in B1, +∆u = f +in {u > 0}, +(see Figure 5.8) with +(5.13) +f ≥ c◦ > 0 +and +f ∈ C∞. +Notice that on the interface +Γ = ∂{u > 0} ∩ B1 +we have that +u = 0 +on Γ, +∇u = 0 +on Γ. +The central mathematical challenge in the obstacle problem is to +understand the geometry/regularity of the free boundary Γ. +Notice that, even if we already know the optimal regularity of u (it is +C1,1), we know nothing about the free boundary Γ. A priori Γ could be a +very irregular object, even a fractal set with infinite perimeter. +As we will see, under the natural assumption f ≥ c◦ > 0, it turns out +that free boundaries are always smooth, possibly outside a certain set of + +— DRAFT — +148 +5. The obstacle problem +{u = 0} +{u = 0} +∆u = f in {u > 0} +all regular points +one singular point +(the contact set has zero density) +Figure 5.9. Singular points are those where the contact set has zero density. +singular points. In fact, in our proofs we will assume for simplicity that +f ≡ 1 (or constant). We do that in order to avoid x-dependence and the +technicalities associated to it, which gives cleaner proofs. In this way, the +main ideas behind the regularity of free boundaries are exposed. +Regularity of free boundaries: main results. Assume from now on +that u solves (5.12)-(5.13). Then, the main known results on the free bound- +ary Γ = ∂{u > 0} can be summarized as follows: +• At every free boundary point x◦ ∈ Γ, we have +0 < cr2 ≤ sup +Br(x◦) +u ≤ Cr2 +∀r ∈ (0, r◦) . +• The free boundary Γ splits into regular points and singular points. +• The set of regular points is an open subset of the free boundary, and Γ is +C∞ near these points. +• Singular points are those at which the contact set {u = 0} has zero density, +and these points (if any) are contained in an (n−1)-dimensional C1 manifold. +Summarizing, the free boundary is smooth, possibly outside a certain set +of singular points. See Figure 5.9. +So far, we have not even proved that Γ has finite perimeter, or anything +at all about Γ. Our goal will be to prove that Γ is C∞ near regular points. +This is the main and most important result in the obstacle problem. It was +proved by Caffarelli in 1977, and it is one of the major results for which he +received the Wolf Prize in 2012 and the Shaw Prize in 2018. +Overview of the strategy. To prove these regularity results for the free +boundary, one considers blow-ups. Namely, given any free boundary point x◦ + +— DRAFT — +5.4. Regularity of free boundaries: an overview +149 +u0(x) = 1 +2(x · e)2 ++ +u0(x) = 1 +2x2 +1 +e +Figure 5.10. Possible blow-ups of the solution to the obstacle problem +at free boundary points. +for a solution u of (5.12)-(5.13), one takes the rescalings +ur(x) := u(x◦ + rx) +r2 +, +with r > 0 small. This is like “zooming in” at a free boundary point. +The factor r−2 is chosen so that +∥ur∥L∞(B1) ≈ 1 +as r → 0; recall that 0 < cr2 ≤ supBr(x◦) u ≤ Cr2. +Then, by C1,1 estimates, we will prove that a subsequence of ur converges +to a function u0 locally uniformly in Rn as r → 0. Such function u0 is called +a blow-up of u at x◦. +Any blow-up u0 is a global solution to the obstacle problem, with f ≡ 1 +(or with f ≡ constant > 0). +Then, the main issue is to classify blow-ups: that is, to show that +either +u0(x) = 1 +2(x · e)2 ++ +(this happens at regular points) +or +u0(x) = 1 +2xT Ax +(this happens at singular points). +Here, e ∈ Sn−1 is a unit vector, and A ≥ 0 is a positive semi-definite matrix +satisfying trA = 1. Notice that the contact set {u0 = 0} becomes a half- +space in case of regular points, while it has zero measure in case of singular +points; see Figure 5.10. +Once this is done, one has to “transfer” the information from the blow- +up u0 to the original solution u. Namely, one shows that, in fact, the free +boundary is C1,α near regular points (for some small α > 0). +Finally, once we know that the free boundary is C1,α, we will “boot- +strap” the regularity to C∞. +This is in a somewhat similar spirit as in +Hilbert’s XIXth problem (Chapter 3), where the really difficult point was to + +— DRAFT — +150 +5. The obstacle problem +prove that minimizers are always C1,α. Once this was done, by Schauder +estimates (Chapter 2) and a bootstrap argument we saw that solutions are +actually C∞. +Classifying blow-ups is not easy. Generally speaking, classifying blow- +ups is of similar difficulty to proving regularity estimates — recall the blow- +up arguments in Chapter 2. +Thus, how can we classify blow-ups? Do we get any extra information +on u0 that we did not have for u? (Otherwise it seems hopeless!) +The answer is yes: Convexity. We will prove that all blow-ups are +always convex. This is a huge improvement, since this yields that the contact +set {u0 = 0} is also convex. Prior to that, we will also show that blow-ups +are also homogeneous. +So, before the blow-up we had no information on the set {u = 0}, but +after the blow-up we get that {u0 = 0} is a convex cone. Thanks to this we +will be able to classify blow-ups, and thus to prove the regularity of the free +boundary. +The main steps in the proof of the regularity of the free boundary will +be the following: +(1) 0 < cr2 ≤ supBr(x◦) u ≤ Cr2 +(2) Blow-ups u0 are homogeneous and convex. +(3) If the contact set has positive density at x◦, then u0(x) = 1 +2(x·e)2 ++. +(4) Deduce that the free boundary is C1,α near x◦. +(5) Deduce that the free boundary is C∞ near x◦. +The proof we will present here for the convexity of blow-ups is new, based +on the fact that they are homogeneous. +We refer to [Caf98], [PSU12], +[Wei99], and [KN77], for different proofs of the classification of blow-ups +and/or of the regularity of free boundaries. +5.5. Classification of blow-ups +The aim of this Section is to classify all possible blow-ups u0. For this, we +will first prove that blow-ups are homogeneous, then we will prove that they +are convex, and finally we will establish their complete classification. +Homogeneity of blow-ups. We start by proving that blow-ups are ho- +mogeneous. This is not essential in the proof of the regularity of the free +boundary (see [Caf98]), but it actually simplifies it. + +— DRAFT — +5.5. Classification of blow-ups +151 +0 +{u = 0} +B1 +∆u = 1 +Figure 5.11. A solution u to the obstacle problem with f ≡ 1. +Recall that, for simplicity, from now on we will assume that f ≡ 1 in B1. +This is only to avoid x-dependence in the equation, it simplifies some proofs. +Therefore, from now on we consider a solution u satisfying (see Fig- +ure 5.11): +(5.14) +u ∈ C1,1(B1) +u ≥ 0 +in B1 +∆u = 1 +in {u > 0} +0 is a free boundary point. +We will prove all the results around the origin (without loss of generality). +We will show that, for the original solution u in B1, the closer we look at +a free boundary point x◦, the closer is the solution to being homogeneous. +Proposition 5.17 (Homogeneity of blow-ups). Let u be any solution to +(5.14). Then, any blow-up of u at 0 is homogeneous of degree 2. +It is important to remark that not all global solutions to the obstacle +problem in Rn are homogeneous. There exist global solutions u0 that are +convex, C1,1, and whose contact set {u0 = 0} is an ellipsoid, for example. +However, thanks to the previous result, we find that such non-homogeneous +solutions cannot appear as blow-ups, i.e., that all blow-ups must be homo- +geneous. +We provide two different proofs of Proposition 5.17. The first one uses +a monotonicity formula as introduced by Weiss; while the second one does +not require any monotonicity formula and is due to Spruck. +Homogeneity of blow-ups `a la Weiss. For the first proof of Proposition 5.17, +we need the following monotonicity formula due to Weiss [Wei99]. + +— DRAFT — +152 +5. The obstacle problem +Theorem 5.18 (Weiss’ monotonicity formula). Let u be any solution to +(5.14). Then, the quantity +(5.15) +Wu(r) := +1 +rn+2 +� +Br +� 1 +2|∇u|2 + u +� +− +1 +rn+3 +� +∂Br +u2 +is monotone in r, that is, +d +drWu(r) = +1 +rn+4 +� +∂Br +(x · ∇u − 2u)2dx ≥ 0 +for r ∈ (0, 1). +Proof. Let ur(x) = r−2u(rx), and observe that +Wu(r) = +� +B1 +� 1 +2|∇ur|2 + ur +� +− +� +∂B1 +u2 +r. +Using this, together with +d +dr(∇ur) = ∇ d +drur, +we find +d +drWu(r) = +� +B1 +� +∇ur · ∇ d +drur + d +drur +� +− 2 +� +∂B1 +ur +d +drur. +Now, integrating by parts we get +� +B1 +∇ur · ∇ d +drur = − +� +B1 +∆ur +d +drur + +� +∂B1 +∂ν(ur) d +drur. +Since ∆ur = 1 in {ur > 0} and +d +drur = 0 in {ur = 0}, we have +� +B1 +∇ur · ∇ d +drur = − +� +B1 +d +drur + +� +∂B1 +∂ν(ur) d +drur. +Thus, we deduce +d +drWu(r) = +� +∂B1 +∂ν(ur) d +drur − 2 +� +∂B1 +ur +d +drur. +Using that on ∂B1 we have ∂ν = x · ∇, combined with +d +drur = 1 +r {x · ∇ur − 2ur} +yields +d +drWu(r) = 1 +r +� +∂B1 +(x · ∇ur − 2ur)2 , +which gives the desired result. +□ +We now give the: + +— DRAFT — +5.5. Classification of blow-ups +153 +First proof of Proposition 5.17. Let ur(x) = r−2u(rx), and notice that +we have the scaling property +Wur(ρ) = Wu(ρr), +for any r, ρ > 0. +If u0 is any blow-up of u at 0 then there is a sequence rj → 0 satisfying +urj → u0 in C1 +loc(Rn). Thus, for any ρ > 0 we have +Wu0(ρ) = lim +rj→0 Wurj (ρ) = lim +rj→0 Wu(ρrj) = Wu(0+). +Notice that the limit Wu(0+) := limr→0 Wu(r) exists by monotonicity of W +and since u ∈ C1,1 implies Wu(r) ≥ −C for all r ≥ 0. +Hence, the function Wu0(ρ) is constant in ρ. However, by Theorem 5.18 +this yields that x · ∇u0 − 2u0 ≡ 0 in Rn, and therefore u0 is homogeneous of +degree 2. +□ +Remark 5.19. Here, we used that a C1 function u0 is 2-homogeneous (i.e. +u0(λx) = λ2u0(x) for all λ ∈ R+) if and only if x · ∇u0 ≡ 2u0. This is +because ∂λ|λ=1 +� +λ−2u0(λx) +� += x · ∇u0 − 2u0. +Homogeneity of blow-ups `a la Spruck. We present an alternative (and quite +different) proof of the homogeneity of blow-ups. Such proof is due to Spruck +[Spr83] and is not based on any monotonicity formula. +Second proof of Proposition 5.17. Let u0 be a blow-up given by the +limit along a sequence rk ↓ 0, +u0(x) := lim +k→∞ r−2 +k u(rkx). +By taking polar coordinates (ϱ, θ) ∈ [0, +∞)×Sn−1 with x = ϱθ, and by +denoting ˜u0(ϱ, θ) = u0(ϱθ) = u0(x), we will prove that u0(x) = ϱ2˜u0(1, θ) = +|x|2u0(x/|x|). +Let us define τ := − log ϱ, ˜u(ϱ, θ) = u(x), and ψ = ψ(τ, θ) as +ψ(τ, θ) := ϱ−2˜u(ϱ, θ) = e2τu(e−τθ) +for τ ≥ 0. We observe that, since ∥u∥L∞(Br) ≤ Cr2, ψ is bounded. Moreover, +ψ ∈ C1((0, ∞) × Sn−1) ∩ C2({ψ > 0}) from the regularity of u; and ∂τψ and +∇θψ are not only continuous, but also uniformly bounded in [0, ∞) × Sn−1. +Indeed, +��∇θψ(τ, θ) +�� ≤ eτ��∇u(e−τθ) +�� ≤ C, +since ∥∇u∥L∞(Br) ≤ Cr by C1,1 regularity and the fact that ∇u(0) = 0. For +the same reason we also obtain +��∂τψ(τ, θ) +�� ≤ 2ψ(τ, θ) + eτ��∇u(e−τθ) +�� ≤ C. + +— DRAFT — +154 +5. The obstacle problem +Observe that, by assumption, if we denote τk := − log rk, +(5.16) +ψ(τk, θ) → ˜u0(1, θ) +uniformly on Sn−1, as k → ∞. +Let us now write an equation for ψ. In order to do that, since we know +that ∆u = χ{u>0} and χ{u>0} = χ{ψ>0}, we have +∆ +� +ϱ2ψ(− log ϱ, θ) +� += χ{ψ>0}. +By expanding the Laplacian in polar coordinates, ∆ = ∂ϱϱ + n−1 +ϱ ∂ϱ + +ϱ−2∆Sn−1 (where ∆Sn−1 denotes the spherical Laplacian, i.e. the Laplace– +Beltrami operator on Sn−1) we obtain +(5.17) +2nψ − (n + 2)∂τψ + ∂ττψ + ∆Sn−1ψ = χ{ψ>0}. +We multiply the previous equality by ∂τψ, and integrate in [0, τ]×Sn−1. +We can consider the terms separately, integrating in τ first, +2n +� +Sn−1 +� τ +0 +ψ∂τψ = n +� +Sn−1 +� +ψ2(τ, θ) − ψ2(0, θ) +� +dθ +and +� +Sn−1 +� τ +0 +∂ττψ∂τψ = 1 +2 +� +Sn−1 +� +(∂τψ)2(τ, θ) − (∂τψ)2(0, θ) +� +dθ, +and then integrating by parts in θ first, to integrate in τ afterwards: +� τ +0 +� +Sn−1 ∆Sn−1ψ∂τψ = −1 +2 +� τ +0 +� +Sn−1 ∂τ|∇θψ|2 += 1 +2 +� +Sn−1 +� +|∇θψ|2(0, θ) − |∇θψ|2(τ, θ) +� +dθ. +Finally, since ∂τψ = 0 whenever ψ = 0, we have χ{ψ>0}∂τψ = ∂τψ and +� +Sn−1 +� τ +0 +χ{ψ>0}∂τψ = +� +Sn−1 +� +ψ(τ, θ) − ψ(0, θ) +� +dθ. +In all, plugging back in (5.17) the previous expressions, and using that +∂τψ and ∇θψ are uniformly bounded in [0, ∞) × Sn−1, we deduce that +(5.18) +� ∞ +0 +� +Sn−1(∂τψ)2 = +� ∞ +0 +∥∂τψ∥2 +L2(Sn−1) ≤ C < ∞. +To finish, now observe that for any |s| ≤ C∗ fixed and for a sufficiently +large k (such that τk + s ≥ 0), +∥ψ(τk + s, ·) − ˜u0(1, ·)∥L2(Sn−1) ≤ ∥ψ(τk + s, ·) − ψ(τk, ·)∥L2(Sn−1) ++ ∥ψ(τk, ·) − ˜u0(1, ·)∥L2(Sn−1). + +— DRAFT — +5.5. Classification of blow-ups +155 +The last term goes to zero, by (5.16). On the other hand, for the first term +and by H¨older’s inequality +∥ψ(τk + s, ·) − ψ(τk, ·)∥2 +L2(Sn−1) ≤ +���� +� s +0 +∂τψ(τk + τ, ·) dτ +���� +2 +L2(Sn−1) +≤ C∗ +���� +� τk+s +τk +∥∂τψ∥2 +L2(Sn−1) +���� → 0, +as k → ∞, where we are using (5.18). Hence, ψ(τk + s, ·) → ˜u0(1, ·) in +L2(Sn−1) as k → ∞, for any fixed s ∈ R. On the other hand, +ψ(τk + s, θ) = e2sr−2 +k u(e−2rkθ) → e2su0(e−sθ) = e2s˜u0(e−s, θ). +That is, for any ρ = e−s > 0, +˜u0(1, ·) = ρ−2˜u0(ρ, θ), +as we wanted to see. +□ +Convexity of blow-ups. By taking advantage of the fact that we know +that blow-ups are 2-homogeneous, we can now give a short (and new) proof +of the fact that they are also convex. More precisely, we will prove that +2-homogeneous global solutions to the obstacle problem are convex (and in +particular, by Proposition 5.17, blow-ups are convex). +Theorem 5.20. Let u0 ∈ C1,1 be any 2-homogeneous global solution to +� +� +� +� +� +u0 +≥ +0 +in Rn +∆u0 += +1 +in {u0 > 0} +0 is a free boundary point. +Then, u0 is convex. +The heuristic idea behind the proof of the previous result is the following: +second derivatives D2u0 are harmonic in {u0 > 0} and satisfy that D2u0 ≥ 0 +on ∂{u0 > 0} (since u0 ≥ 0, it is “convex at the free boundary”). Since D2u0 +is also 0-homogeneous, we can apply the maximum principle and conclude +that D2u0 ≥ 0 everywhere. That is, u0 is convex. Let us formalize the +previous heuristic idea into an actual proof. +We state a short lemma before providing the proof, which says that if +w ≥ 0 is superharmonic in {w > 0}, then it is superharmonic everywhere. +For the sake of generality, we state the lemma for general H1 functions, but +we will use it only for functions that are also continuous. +Lemma 5.21. Let Λ ⊂ B1 be closed. Let w ∈ H1(B1) be such that w ≥ 0 +on Λ and such that w is superharmonic in the weak sense in B1 \ Λ. Then +min{w, 0} is superharmonic in the weak sense in B1. + +— DRAFT — +156 +5. The obstacle problem +Proof. Let us start by assuming that w is, furthermore, continuous. In +this case, we define wε = min{w, −ε} ∈ H1(B1). +Then notice that (by +continuity) in a neighborhood of {w = −ε}, w is superharmonic (∆w ≤ +0). By Lemma 3.9 (we apply the lemma with v = −w − ε) we have that +∆wε ≤ 0 in the weak sense, namely, wε is superharmonic. Moreover, they are +uniformly in H1, so up to subsequences they converge weakly to min{w, 0}. +Since the weak limit of weakly superharmonic functions is superharmonic, +we deduce the desired result. +Finally, to remove the continuity assumption on w ∈ H1(B1), we repeat +the proof of Lemma 3.9. The only thing we need to check is that F ′(v)η ∈ +H1 +0(B1 \Λ), which follows from the fact that such function is in H1(B1) and +vanishes in Λ; see for example [AH96, Theorem 9.1.3]. +□ +We now give the: +Proof of Theorem 5.20. Let e ∈ Sn−1 and consider the second derivatives +∂eeu0. We define +w0 := min{∂eeu0, 0} +and we claim that w0 is superharmonic in Rn, in the sense (1.20). +Indeed, let δ2 +t u0(x) for t > 0 be defined by +δ2 +t u0(x) := u0(x + te) + u0(x − te) − 2u0(x) +t2 +. +Now, since ∆u0 = χ{u0>0}, we have that +∆δ2 +t u0 = 1 +t2 +� +χ{u0( · +te)} + χ{u0( · −te)} − 2 +� +≤ 0 +in +{u0 > 0} +in the weak sense. On the other hand, δ2 +t u0 ≥ 0 in {u0 = 0} and δ2 +t u0 ∈ C1,1. +Thus, by Lemma 5.21, wt := min{δ2 +t u0, 0} is weakly superharmonic, and +hence it satisfies (1.20). +Also notice that δ2 +t u0(x) is uniformly bounded +independently of t, since u0 ∈ C1,1, and therefore wt is uniformly bounded +in t and converges pointwise to w0 as t ↓ 0. In particular, by Lemma 1.16 +we have that w0 is superharmonic in the sense of (1.20), as claimed. +Up to changing it in a set of measure 0, w0 is lower semi-continuous +by Lemma 1.17. In particular, since w0 is 0-homogeneous, it must attain +its minimum at a point y◦ ∈ B1. But since +� +Br(y◦) w0 is non-increasing +for r > 0, we must have that w0 is constant. Since it vanishes on the free +boundary, we have w0 ≡ 0. That is, for any e ∈ Sn−1 we have that ∂eeu0 ≥ 0 +and therefore u0 is convex. +□ +Remark 5.22 (Convexity of blow-ups `a la Caffarelli). The original proof +by Caffarelli on the convexity of blow-ups, [Caf77, Caf98], is more involved +than the previous one, but obtains a quantitative estimate on the convexity + +— DRAFT — +5.5. Classification of blow-ups +157 +without using the homogeneity assumption (in particular, it is valid for any +global solution). +More precisely, for any solution u to (5.14) in B1 +∂eeu(x) ≥ − +C +�� log |x| +��ε +for all +e ∈ Sn−1, x ∈ B1/2, +for some ε > 0. Notice that C +�� log |x| +��−ε → 0 as x → 0. Thus, u becomes +closer and closer to being convex as we approach to the free boundary. +Rescaling this result to BR, and letting R → ∞, this implies that any global +solution is convex. +Finally, we refer to [PSU12, Theorem 5.1] for yet another different proof +of the convexity of blow-ups. +Classification of blow-ups. We next want to classify all possible blow- +ups for solutions to the obstacle problem (5.14). First, we will prove the +following. +Proposition 5.23. Let u be any solution to (5.14), and let +ur(x) := u(rx) +r2 +. +Then, for any sequence rk → 0 there is a subsequence rkj → 0 such that +urkj −→ u0 +in C1 +loc(Rn) +as kj → ∞, for some function u0 satisfying +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +u0 ∈ C1,1 +loc(Rn) +u0 ≥ 0 +in B1 +∆u0 = 1 +in {u0 > 0} +0 is a free boundary point +u0 is convex +u0 is homogeneous of degree 2. +Proof. By C1,1 regularity of u, and by nondegeneracy, we have that +1 +C ≤ sup +B1 +ur ≤ C +for some C > 0. Moreover, again by C1,1 regularity of u, we have +∥D2ur∥L∞(B1/(2r)) ≤ C. +Since the sequence {urk}, for rk → 0, is uniformly bounded in C1,1(K) +for each compact set K ⊂ Rn, there is a subsequence rkj → 0 such that +urkj −→ u0 +in C1 +loc(Rn) + +— DRAFT — +158 +5. The obstacle problem +for some u0 ∈ C1,1(K). Moreover, such function u0 satisfies ∥D2u0∥L∞(K) ≤ +C, with C independent of K, and clearly u0 ≥ 0 in K. +The fact that ∆u0 = 1 in {u0 > 0} ∩ K can be checked as follows. For +any smooth function η ∈ C∞ +c ({u0 > 0} ∩ K) we will have that, for kj large +enough, urkj > 0 in the support of η, and thus +� +Rn ∇urkj · ∇η dx = − +� +Rn η dx. +Since urkj → u0 in C1(K), we can take the limit kj → ∞ to get +� +Rn ∇u0 · ∇η dx = − +� +Rn η dx. +Since this can be done for any η ∈ C∞ +c ({u > 0}∩K), and for every K ⊂ Rn, +it follows that ∆u0 = 1 in {u0 > 0}. +The fact that 0 is a free boundary point for u0 follows simply by taking +limits to urkj (0) = 0 and ∥urkj ∥L∞(Bρ) ≈ ρ2 for all ρ ∈ (0, 1). Finally, the +homogeneity and convexity of u0 follow from Proposition 5.17 and Theo- +rem 5.20. +□ +Our next goal is to prove the following. +Theorem 5.24 (Classification of blow-ups). Let u be any solution to (5.14), +and let u0 be any blow-up of u at 0. Then, +(a) either +u0(x) = 1 +2(x · e)2 ++ +for some e ∈ Sn−1. +(b) or +u0(x) = 1 +2xT Ax +for some matrix A ≥ 0 with tr A = 1. +It is important to remark here that, a priori, different subsequences could +lead to different blow-ups u0. +In order to establish Theorem 5.24, we will need the following. +Lemma 5.25. Let Σ ⊂ Rn be any closed convex cone with nonempty inte- +rior, and with vertex at the origin. Let w ∈ C(Rn) be a function satisfying +∆w = 0 in Σc, w > 0 in Σc, and w = 0 in Σ. +Assume in addition that w is homogeneous of degree 1. Then, Σ must +be a half-space. + +— DRAFT — +5.5. Classification of blow-ups +159 +Proof. By convexity of Σ, there exists a half-space H = {x · e > 0}, with +e ∈ Sn−1, such that H ⊂ Σc. +Let v(x) = (x · e)+, which is harmonic and positive in H, and vanishes +in Hc. By the Hopf Lemma (see Lemma 1.15), we have that w ≥ c◦dΣ in +Σc ∩ B1, where dΣ(x) = dist(x, Σ) and c◦ is a small positive constant. In +particular, since both w and dΣ are homogeneous of degree 1, we deduce +that w ≥ c◦dΣ in all of Σc. Notice that, in order to apply the Hopf Lemma, +we used that — by convexity of Σ — the domain Σc satisfies the interior +ball condition. +Thus, since dΣ ≥ dHc = v, we deduce that w ≥ c◦v, for some c◦ > 0. +The idea is now to consider the functions w and cv, and let c > 0 increase +until the two functions touch at one point, which will give us a contradiction +(recall that two harmonic functions cannot touch at an interior point). To +do this rigorously, define +c∗ := sup{c > 0 : w ≥ cv +in +Σc}. +Notice that c∗ ≥ c◦ > 0. Then, we consider the function w−c∗v ≥ 0. Assume +that w − c∗v is not identically zero. Such function is harmonic in H and +hence, by the strict maximum principle, w − c∗v > 0 in H. Then, using the +Hopf Lemma in H (see Lemma 1.15) we deduce that w−c∗v ≥ c◦dHc = c◦v, +since v is exactly the distance to Hc. But then we get that w−(c∗+c◦)v ≥ 0, +a contradiction with the definition of c∗. +Therefore, it must be w − c∗v ≡ 0. This means that w is a multiple of +v, and therefore Σ = Hc, a half-space. +□ +Remark 5.26 (Alternative proof). An alternative way to argue in the pre- +vious lemma could be the following. Any function w which is harmonic in +a cone Σc and homogeneous of degree α can be written as a function on +the sphere, satisfying ∆Sn−1w = µw on Sn−1 ∩ Σc with µ = α(n + α − 2) +— in our case α = 1. (Here, ∆Sn−1 denotes the spherical Laplacian, i.e. +the Laplace–Beltrami operator on Sn−1.) In other words, homogeneous har- +monic functions solve an eigenvalue problem on the sphere. +Using this, we notice that w > 0 in Σc and w = 0 in Σ imply that w is +the first eigenfunction of Sn−1∩Σc, and that the first eigenvalue is µ = n−1. +But, on the other hand, the same happens for the domain H = {x · e > 0}, +since v(x) = (x · e)+ is a positive harmonic function in H. This means that +both domains Sn−1 ∩Σc and Sn−1 ∩H have the same first eigenvalue µ. But +then, by strict monotonicity of the first eigenvalue with respect to domain +inclusions, we deduce that H ⊂ Σc implies H = Σc, as desired. +We will also need the following. + +— DRAFT — +160 +5. The obstacle problem +Lemma 5.27. Assume that ∆u = 1 in Rn \∂H, where ∂H is a hyperplane. +If u ∈ C1(Rn), then ∆u = 1 in Rn. +Proof. Assume ∂H = {x1 = 0}. For any ball BR ⊂ Rn, we consider the +solution to ∆w = 1 in BR, w = u on ∂BR, and define v = u − w. Then, +we have ∆v = 0 in BR \ ∂H, and v = 0 on ∂BR. We want to show that u +coincides with w, that is, v ≡ 0 in BR. +For this, notice that since v is bounded, for κ > 0 large enough we have +v(x) ≤ κ(2R − |x1|) +in +BR, +where 2R − |x1| is positive in BR and harmonic in BR \ {x1 = 0}. Thus, +we may consider κ∗ := inf{κ ≥ 0 : v(x) ≤ κ(2R − |x1|) +in +BR}. Assume +κ∗ > 0. Since v and 2R − |x1| are continuous in BR, and v = 0 on ∂BR, we +must have a point p ∈ BR at which v(p) = κ∗(2R − |p1|). Moreover, since v +is C1, and the function 2R − |x1| has a wedge on ∂H = {x1 = 0}, we must +have p ∈ BR \ ∂H. However, this is not possible, as two harmonic functions +cannot touch tangentially at an interior point p. This means that κ∗ = 0, +and hence v ≤ 0 in BR. Repeating the same argument with −v instead of +v, we deduce that v ≡ 0 in BR, and thus the lemma is proved. +□ +Finally, we will use the following basic property of convex functions. +Lemma 5.28. Let u : Rn → R be a convex function such that the set {u = 0} +contains the straight line {te′ : t ∈ R}, e′ ∈ Sn−1. Then, u(x + te′) = u(x) +for all x ∈ Rn and all t ∈ R. +Proof. After a rotation, we may assume e′ = en. +Then, writing x = +(x′, xn) ∈ Rn−1 × R, we have that u(0, xn) = 0 for all xn ∈ R, and we +want to prove that u(x′, xn) = u(x′, 0) for all x′ ∈ Rn−1 and all xn ∈ R. +Now, by convexity, given x′ and xn, for every ε > 0 and M ∈ R we have +(1 − ε)u(x′, xn) + εu(0, xn + M) ≥ u((1 − ε)x′, xn + εM). +Since u(0, xn + M) = 0, choosing M = λ/ε and letting ε → 0 we deduce +that +u(x′, xn) ≥ u(x′, xn + λ). +Since this can be done for any λ ∈ R and xn ∈ R, the result follows. +□ +We finally establish the classification of blow-ups at regular points. +Proof of Theorem 5.24. Let u0 be any blow-up of u at 0. We already +proved that u0 is convex and homogeneous of degree 2. We divide the proof +into two cases. +Case 1. +Assume that {u0 = 0} has nonempty interior. +Then, we have +{u0 = 0} = Σ, a closed convex cone with nonempty interior. + +— DRAFT — +5.6. Regularity of the free boundary +161 +For any direction τ ∈ Sn−1 such that −τ ∈ ˚Σ, we claim that +∂τu0 ≥ 0 +in +Rn. +Indeed, for every x ∈ Rn we have that u0(x + τt) is zero for t ≪ −1, and +therefore by convexity of u0 we get that ∂tu0(x + τt) is monotone non- +decreasing in t, and zero for t ≪ −1. This means that ∂tu0 ≥ 0, and thus +∂τu0 ≥ 0 in Rn, as claimed. +Now, for any such τ, we define w := ∂τu0 ≥ 0. Notice that, at least +for some τ ∈ Sn−1 with −τ ∈ ˚Σ, the function w is not identically zero. +Moreover, since it is harmonic in Σc — recall that ∆u0 = 1 in Σc — then +w > 0 in Σc. +But then, since w is homogeneous of degree 1, we can apply Lemma 5.25 +to deduce that we must necessarily have that Σ is a half-space. +By convexity of u0 and Lemma 5.28, this means that u0 is a one- +dimensional function, i.e., u0(x) = U(x · e) for some U : R → R and some +e ∈ Sn−1. +Thus, we have that U ∈ C1,1 solves U ′′(t) = 1 for t > 0, +with U(t) = 0 for t ≤ 0. +We deduce that U(t) = +1 +2t2 ++, and therefore +u0(x) = 1 +2(x · e)2 ++. +Case 2. Assume now that {u0 = 0} has empty interior. Then, by convexity, +{u0 = 0} is contained in a hyperplane ∂H. Hence, ∆u0 = 1 in Rn\∂H, with +∂H being a hyperplane, and u0 ∈ C1,1. It follows from Lemma 5.27 that +∆u0 = 1 in all of Rn. But then all second derivatives of u0 are harmonic +and globally bounded in Rn, so they must be constant. +Hence, u0 is a +quadratic polynomial. Finally, since u0(0) = 0, ∇u0(0) = 0, and u0 ≥ 0, we +deduce that u0(x) = 1 +2xT Ax for some A ≥ 0, and since ∆u0 = 1, we have +tr A = 1. +□ +5.6. Regularity of the free boundary +The aim of this Section is to prove Theorem 5.38 below, i.e., that if u is any +solution to (5.14) satisfying +(5.19) +lim sup +r→0 +��{u = 0} ∩ Br +�� +|Br| +> 0 +(i.e., the contact set has positive density at the origin), then the free bound- +ary ∂{u > 0} is C∞ in a neighborhood of the origin. +For this, we will use the classification of blow-ups established in the +previous Section. +C1,α regularity of the free boundary. The first step here is to transfer +the local information on u given by (5.19) into a blow-up u0. More precisely, + +— DRAFT — +162 +5. The obstacle problem +we next show that +(5.19) +=⇒ +The contact set of a blow-up u0 +has nonempty interior. +Lemma 5.29. Let u be any solution to (5.14), and assume that (5.19) holds. +Then, there is at least one blow-up u0 of u at 0 such that the contact set +{u0 = 0} has nonempty interior. +Proof. Let rk → 0 be a sequence along which +lim +rk→0 +��{u = 0} ∩ Brk +�� +|Brk| +≥ θ > 0. +Such sequence exists (with θ > 0 small enough) by assumption (5.19). +Recall that, thanks to Proposition 5.23, there exists a subsequence rkj ↓ +0 along which urkj → u0 uniformly on compact sets of Rn, where ur(x) = +r−2u(rx) and u0 is convex. +Assume by contradiction that {u0 = 0} has empty interior. Then, by +convexity, we have that {u0 = 0} is contained in a hyperplane, say {u0 = +0} ⊂ {x1 = 0}. +Since u0 > 0 in {x1 ̸= 0} and u0 is continuous, we have that for each +δ > 0 +u0 ≥ ε > 0 +in {|x1| > δ} ∩ B1 +for some ε > 0. +Therefore, by uniform convergence of urkj to u0 in B1, there is rkj > 0 +small enough such that +urkj ≥ ε +2 > 0 +in {|x1| > δ} ∩ B1. +In particular, the contact set of urkj is contained in {|x1| ≤ δ} ∩ B1, so +��{urkj = 0} ∩ B1 +�� +|B1| +≤ +��{|x1| ≤ δ} ∩ B1 +�� +|B1| +≤ Cδ. +Rescaling back to u, we find +��{u = 0} ∩ Brkj +�� +|Brkj | += +��{urkj = 0} ∩ B1 +�� +|B1| +< Cδ. +Since we can do this for every δ > 0, we find that limrkj →0 +|{u=0}∩Brkj | +|Brkj | += 0, +a contradiction. Thus, the lemma is proved. +□ +Combining the previous lemma with the classification of blow-ups from +the previous Section, we deduce: + +— DRAFT — +5.6. Regularity of the free boundary +163 +τ +e +{x · e > 0} +∂τu0 > 0 +Figure 5.12. Derivatives ∂τu0 are nonnegative if τ · e ≥ 1 +2. +Corollary 5.30. Let u be any solution to (5.14), and assume that (5.19) +holds. Then, there is at least one blow-up of u at 0 of the form +u0(x) = 1 +2(x · e)2 ++, +e ∈ Sn−1. +Proof. The result follows from Lemma 5.29 and Theorem 5.24. +□ +We now want to use this information to show that the free boundary +must be smooth in a neighborhood of 0. For this, we start with the following. +Proposition 5.31. Let u be any solution to (5.14), and assume that (5.19) +holds. Fix any ε > 0. Then, there exist e ∈ Sn−1 and r◦ > 0 such that +��ur◦(x) − 1 +2(x · e)2 ++ +�� ≤ ε +in +B1, +and +��∂τur◦(x) − (x · e)+(τ · e) +�� ≤ ε +in +B1 +for all τ ∈ Sn−1. +Proof. By Corollary 5.30 and Proposition 5.23, we know that there is a +subsequence rj → 0 for which urj → 1 +2(x · e)2 ++ in C1 +loc(Rn), for some e ∈ +Sn−1. +In particular, for every τ ∈ Sn−1 we have urj → +1 +2(x · e)2 ++ and +∂τurj → ∂τ +� 1 +2(x · e)2 ++ +� +uniformly in B1. +This means that, given ε > 0, there exists j◦ such that +��urj◦(x) − 1 +2(x · e)2 ++ +�� ≤ ε +in +B1, +and +��∂τurj◦(x) − ∂τ +� 1 +2(x · e)2 ++ +��� ≤ ε +in +B1. +Since ∂τ +� 1 +2(x · e)2 ++ +� += (x · e)+(τ · e), the proposition is proved. +□ +Now, notice that if (τ · e) > 0, then the derivatives ∂τu0 = (x · e)+(τ · e) +are nonnegative, and strictly positive in {x · e > 0} (see Figure 5.12). + +— DRAFT — +164 +5. The obstacle problem +We want to transfer this information to ur◦, and prove that ∂τur◦ ≥ 0 +in B1 for all τ ∈ Sn−1 satisfying τ · e ≥ 1 +2. For this, we need a lemma. +Lemma 5.32. Let u be any solution to (5.14), and consider ur◦(x) = +r−2 +◦ u(r◦x) and Ω = {ur◦ > 0}. +Assume that a function w ∈ C(B1) satisfies: +(a) w is bounded and harmonic in Ω ∩ B1. +(b) w = 0 on ∂Ω ∩ B1. +(c) Denoting Nδ := {x ∈ B1 : dist(x, ∂Ω) < δ}, we have +w ≥ −c1 +in +Nδ +and +w ≥ C2 > 0 +in +Ω \ Nδ. +If c1/C2 is small enough, and δ > 0 is small enough, then w ≥ 0 in B1/2∩Ω. +Proof. Notice that in Ω \ Nδ we already know that w > 0. +Let y◦ ∈ +Nδ ∩ Ω ∩ B1/2, and assume by contradiction that w(y0) < 0. +Consider, in B1/4(y◦), the function +v(x) = w(x) − γ +� +ur◦(x) − 1 +2n|x − y◦|2 +� +. +Then, ∆v = 0 in B1/4(y◦) ∩ Ω, and v(y◦) < 0. Thus, v must have a negative +minimum in ∂ +� +B1/4(y◦) ∩ Ω +� +. +However, if c1/C2 and δ are small enough, then we reach a contradiction +as follows: +On ∂Ω we have v ≥ 0. On ∂B1/4(y◦) ∩ Nδ we have +v ≥ −c1 − C◦γδ2 + γ +2n +�1 +4 +�2 +≥ 0 +on +∂B1/4(y◦) ∩ Nδ. +On ∂B1/4(y◦) ∩ +� +Ω \ Nδ +� +we have +v ≥ C2 − C◦γ ≥ 0 +on +∂B1/4(y◦) ∩ +� +Ω \ Nδ +� +. +Here, we used that ∥ur◦∥C1,1(B1) ≤ C◦, and chose C◦c1 ≤ γ ≤ C2/C◦. +□ +Using the previous lemma, we can now show that there is a cone of +directions τ in which the solution is monotone near the origin. +Proposition 5.33. Let u be any solution to (5.14), and assume that (5.19) +holds. Let ur(x) = r−2u(rx). Then, there exist r◦ > 0 and e ∈ Sn−1 such +that +∂τur◦ ≥ 0 +in +B1/2 +for every τ ∈ Sn−1 satisfying τ · e ≥ 1 +2. + +— DRAFT — +5.6. Regularity of the free boundary +165 +Proof. By Proposition 5.31, for any ε > 0 there exist e ∈ Sn−1 and r◦ > 0 +such that +(5.20) +��ur◦(x) − 1 +2(x · e)2 ++ +�� ≤ ε +in +B1 +and +(5.21) +��∂τur◦(x) − (x · e)+(τ · e) +�� ≤ ε +in +B1 +for all τ ∈ Sn−1. +We now want to use Lemma 5.32 to deduce that ∂τur◦ ≥ 0 if τ · e ≥ 1 +2. +First, we claim that +ur◦ > 0 +in +{x · e > C◦ +√ε}, +(5.22) +ur◦ = 0 +in +{x · e < −C◦ +√ε}, +and therefore the free boundary ∂Ω = ∂{ur◦ > 0} is contained in the strip +{|x · e| ≤ C◦ +√ε}, for some C◦ depending only on n (see Figure 5.13). To +prove this, notice that if x · e > C◦ +√ε then +ur◦ > 1 +2(C◦ +√ε)2 − ε > 0, +while if there was a free boundary point x◦ in {x · e < −C◦ε} then by +nondegeneracy we would get +sup +BC◦ +√ε(x◦) +ur◦ ≥ c(C◦ +√ε)2 > 2ε, +a contradiction with (5.20). +Therefore, we have +∂Ω ⊂ {|x · e| ≤ C◦ +√ε}. +Now, for each τ ∈ Sn−1 satisfying τ · e ≥ 1 +2 we define +w := ∂τur◦. +In order to use Lemma 5.32, we notice: +(a) w is bounded and harmonic in Ω ∩ B1. +(b) w = 0 on ∂Ω ∩ B1. +(c) Thanks to (5.21), if δ ≫ √ε then w satisfies +w ≥ −ε +in +Nδ +and +w ≥ δ/4 > 0 +in +(Ω \ Nδ) ∩ B1. + +— DRAFT — +166 +5. The obstacle problem +Ω +0 +2C◦ +√ε +Nδ ∩ Ω +∂Ω +Figure 5.13. The setting in which we use Lemma 5.32. +(We recall Nδ := {x ∈ B1 : dist(x, ∂Ω) < δ}.) +Indeed, to check the last inequality we use that, by (5.22), we have +{x · e < δ − C◦ +√ε} ∩ Ω ⊂ Nδ. Thus, by (5.21), we get that for all x ∈ +(Ω \ Nδ) ∩ B1 +w ≥ 1 +2(x · e)+ − ε ≥ 1 +2δ − 1 +2C◦ +√ε − ε ≥ 1 +4δ, +provided that δ ≫ √ε. +Using (a)-(b)-(c), we deduce from Lemma 5.32 that +w ≥ 0 +in +B1/2. +Since we can do this for every τ ∈ Sn−1 with τ · e ≥ 1 +2, the proposition is +proved. +□ +As a consequence of the previous proposition, we find: +Corollary 5.34. Let u be any solution to (5.14), and assume that (5.19) +holds. Then, there exists r◦ > 0 such that the free boundary ∂{ur◦ > 0} +is Lipschitz in B1/2. In particular, the free boundary of u, ∂{u > 0}, is +Lipschitz in Br◦/2. +Proof. This follows from the fact that ∂τur◦ ≥ 0 in B1/2 for all τ ∈ Sn−1 +with τ · e ≥ 1 +2 (by Proposition 5.33), as explained next. + +— DRAFT — +5.6. Regularity of the free boundary +167 +Σ1 +Σ2 +e +x◦ +τ +Figure 5.14. Representation of Σ1 and Σ2. +Let x◦ ∈ B1/2 ∩ ∂{ur◦ > 0} be any free boundary point in B1/2, and let +Θ := +� +τ ∈ Sn−1 : τ · e > 1 +2 +� +, +Σ1 := +� +x ∈ B1/2 : x = x◦ − tτ, with τ ∈ Θ, t > 0 +� +, +and +Σ2 := +� +x ∈ B1/2 : x = x◦ + tτ, with τ ∈ Θ, t > 0 +� +, +see Figure 5.14. +We claim that +(5.23) +� ur◦ += +0 +in +Σ1, +ur◦ +> +0 +in +Σ2. +Indeed, since ur◦(x◦) = 0, it follows from the monotonicity property ∂τur◦ ≥ +0 — and the nonnegativity of ur◦ — that ur◦(x◦ − tτ) = 0 for all t > 0 and +τ ∈ Θ. In particular, there cannot be any free boundary point in Σ1. +On the other hand, by the same argument, if ur◦(x1) = 0 for some +x1 ∈ Σ2 then we would have ur◦ = 0 in +� +x ∈ B1/2 : x = x1 − tτ, with τ ∈ +Θ, t > 0 +� +∋ x◦, and in particular x◦ would not be a free boundary point. +Thus, ur◦(x1) > 0 for all x1 ∈ Σ2, and (5.23) is proved. +Finally, notice that (5.23) yields that the free boundary ∂{ur◦ > 0} ∩ +B1/2 satisfies both the interior and exterior cone condition, and thus it is +Lipschitz. +□ +Once we know that the free boundary is Lipschitz, we may assume with- +out loss of generality that e = en and that +∂{ur◦ > 0} ∩ B1/2 = {xn = g(x′)} ∩ B1/2 + +— DRAFT — +168 +5. The obstacle problem +wi > 0 +∆wi = 0 +wi = 0 +B1 +Ω +Figure 5.15. Setting of the boundary Harnack. +for a Lipschitz function g : Rn−1 → R. Here, x = (x′, xn), with x′ ∈ Rn−1 +and xn ∈ R. +Now, we want to prove that Lipschitz free boundaries are C1,α. A key +ingredient for this will be the following basic property of harmonic functions +(see Figure 5.15 for a representation of the setting). +Theorem 5.35 (Boundary Harnack). Let w1 and w2 be positive harmonic +functions in B1 ∩ Ω, where Ω ⊂ Rn is any Lipschitz domain. +Assume that w1 and w2 vanish on ∂Ω ∩ B1, and C−1 +◦ +≤ ∥wi∥L∞(B1/2) ≤ +C◦ for i = 1, 2. Then, +1 +C w2 ≤ w1 ≤ Cw2 +in +Ω ∩ B1/2. +Moreover, +���� +w1 +w2 +���� +C0,α(Ω∩B1/2) +≤ C +for some small α > 0. The constants α and C depend only on n, C◦, and Ω. +For completeness, we provide in Appendix B a proof of this result. We +refer to [DS20] for the boundary Harnack for more general operators and to +[AS19, RT21] for the boundary Harnack for equations with a right hand +side. +Remark 5.36. The main point in Theorem 5.35 is that Ω is allowed to be +Lipschitz. If Ω is smooth (say, C2 or even C1,α) then it follows from a simple +barrier argument that both w1 and w2 would be comparable to the distance +to ∂Ω, i.e., they vanish at a linear rate from ∂Ω. However, in Lipschitz + +— DRAFT — +5.6. Regularity of the free boundary +169 +domains the result cannot be proved with a simple barrier argument, and it +is much more delicate to establish. +The boundary Harnack is a crucial tool in the study of free boundary +problems, and in particular in the obstacle problem. Here, we use it to prove +that the free boundary is C1,α for some small α > 0. +Proposition 5.37. Let u be any solution to (5.14), and assume that (5.19) +holds. Then, there exists r◦ > 0 such that the free boundary ∂{ur◦ > 0} is +C1,α in B1/4, for some small α > 0. In particular, the free boundary of u, +∂{u > 0}, is C1,α in Br◦/4. +Proof. Let Ω = {ur◦ > 0}. By Corollary 5.34, if r◦ > 0 is small enough +then (possibly after a rotation) we have +Ω ∩ B1/2 = {xn ≥ g(x′)} ∩ B1/2 +and the free boundary is given by +∂Ω ∩ B1/2 = {xn = g(x′)} ∩ B1/2, +where g is Lipschitz. +Let +w2 := ∂enur◦ +and +w1 := ∂eiur◦ + ∂enur◦, +i = 1, ..., n − 1. +Since ∂τur◦ ≥ 0 in B1/2 for all τ ∈ Sn−1 with τ ·en ≥ 1 +2, we have that w2 ≥ 0 +in B1/2 and w1 ≥ 0 in B1/2. +This is because ∂ei + ∂en = ∂ei+en = +√ +2∂τ, with τ · en = 1/ +√ +2 > 1 +2. +Notice that we add the term ∂enur◦ in w1 in order to get a nonnegative +function w2 ≥ 0. +Now since w1 and w2 are positive harmonic functions in Ω ∩ B1/2, and +vanish on ∂Ω ∩ B1/2, we can use the boundary Harnack, Theorem 5.35 (or +Corollary B.2), to get +���� +w1 +w2 +���� +C0,α(Ω∩B1/4) +≤ C +for some small α > 0. +Therefore, since w1/w2 = 1 + ∂eiur◦/∂enur◦, we +deduce +(5.24) +���� +∂eiur◦ +∂enur◦ +���� +C0,α(Ω∩B1/4) +≤ C. +Now, we claim that this implies that the free boundary is C1,α in B1/4. +Indeed, if ur◦(x) = t then the normal vector to the level set {ur◦ = t} is + +— DRAFT — +170 +5. The obstacle problem +given by +νi(x) = ∂eiur◦ +|∇ur◦| = +∂eiur◦/∂enur◦ +� +1 + �n−1 +j=1 +� +∂ejur◦/∂enur◦ +�2 , +i = 1, ..., n. +This is a C0,α function by (5.24), and therefore we can take t → 0 to find +that the free boundary is C1,α (since the normal vector to the free boundary +is given by a C0,α function). +□ +So far we have proved that +� {u = 0} has positive +density at the origin +� +=⇒ +� any blow-up is +u0 = 1 +2(x · e)2 ++ +� +=⇒ +� free boundary +is C1,α near 0 +� +As a last step in this section, we will now prove that C1,α free boundaries +are actually C∞. +Higher regularity of the free boundary. We want to finally prove the +smoothness of free boundaries near regular points. +Theorem 5.38 (Smoothness of the free boundary near regular points). Let +u be any solution to (5.14), and assume that (5.19) holds. Then, the free +boundary ∂{u > 0} is C∞ in a neighborhood of the origin. +For this, we need the following result. +Theorem 5.39 (Higher order boundary Harnack). Let Ω ⊂ Rn be any Ck,α +domain, with k ≥ 1 and α ∈ (0, 1). Let w1, w2 be two solutions of ∆wi = 0 +in B1 ∩ Ω, wi = 0 on ∂Ω ∩ B1, with w2 > 0 in Ω. +Assume that C−1 +◦ +≤ ∥wi∥L∞(B1/2) ≤ C◦. Then, +���� +w1 +w2 +���� +Ck,α(Ω∩B1/2) +≤ C, +where C depends only on n, k, α, C◦, and Ω. +Contrary to Theorem 5.35, the proof of Theorem 5.39 is a perturba- +tive argument, in the spirit of (but much more delicate than) the Schauder +estimates from Chapter 3. We will not prove the higher order boundary +Harnack here; we refer to [DS16] for the proof of such result. +Using Theorem 5.39, we can finally prove Theorem 5.38: +Proof of Theorem 5.38. Let ur◦(x) = r−2 +◦ u(r◦x). By Proposition 5.37, +we know that if r◦ > 0 is small enough then the free boundary ∂{ur◦ > 0} +is C1,α in B1, and (possibly after a rotation) ∂enur◦ > 0 in {ur◦ > 0} ∩ B1. + +— DRAFT — +5.7. Singular points +171 +Thus, using the higher order boundary Harnack (Theorem 5.39) with w1 = +∂eiur◦ and w2 = ∂enur◦, we find that +���� +∂eiur◦ +∂enur◦ +���� +C1,α(Ω∩B1/2) +≤ C. +Actually, by a simple covering argument we find that +(5.25) +���� +∂eiur◦ +∂enur◦ +���� +C1,α(Ω∩B1−δ) +≤ Cδ +for any δ > 0. +Now, as in the proof of Proposition 5.37, we notice that if ur◦(x) = t +then the normal vector to the level set {ur◦ = t} is given by +νi(x) = ∂eiur◦ +|∇ur◦| = +∂eiur◦/∂enur◦ +� +1 + �n +j=1 +� +∂ejur◦/∂enur◦ +�2 , +i = 1, ..., n. +By (5.25), this is a C1,α function in B1−δ for any δ > 0, and therefore we +can take t → 0 to find that the normal vector to the free boundary is C1,α +inside B1. But this means that the free boundary is actually C2,α. +Repeating now the same argument, and using that the free boundary is +C2,α in B1−δ for any δ > 0, we find that +���� +∂eiur◦ +∂enur◦ +���� +C2,α(Ω∩B1−δ′) +≤ Cδ′, +which yields that the normal vector is C2,α and thus the free boundary is +C3,α. Iterating this argument, we find that the free boundary ∂{ur◦ > 0} +is C∞ inside B1, and hence ∂{u > 0} is C∞ in a neighborhood of the +origin. +□ +This completes the study of regular free boundary points. It remains to +understand what happens at points where the contact set has density zero +(see e.g. Figure 5.9). This is the content of the next section. +5.7. Singular points +We finally study the behavior of the free boundary at singular points, i.e., +when +(5.26) +lim +r→0 +��{u = 0} ∩ Br +�� +|Br| += 0. +For this, we first notice that, as a consequence of the results of the previous +Section, we get the following. +Proposition 5.40. Let u be any solution to (5.14). +Then, we have the +following dichotomy: + +— DRAFT — +172 +5. The obstacle problem +(a) Either (5.19) holds and all blow-ups of u at 0 are of the form +u0(x) = 1 +2(x · e)2 ++, +for some e ∈ Sn−1. +(b) Or (5.26) holds and all blow-ups of u at 0 are of the form +u0(x) = 1 +2xT Ax, +for some matrix A ≥ 0 with tr A = 1. +Points of type (a) were studied in the previous Section; they are called +regular points and the free boundary is C∞ around them (in particular, the +blow-up is unique). Points of type (b) are those at which the contact set +has zero density, and are called singular points. +To prove the result, we need the following: +Lemma 5.41. Let u be any solution to (5.14), and assume that (5.26) holds. +Then, every blow-up of u at 0 satisfies |{u0 = 0}| = 0. +Proof. Let u0 be a blow-up of u at 0, i.e., urk → u0 in C1 +loc(Rn) along a +sequence rk → 0, where ur(x) = r−2u(rx). +Notice that the functions ur solve +∆ur = χ{ur>0} +in +B1, +in the sense that +(5.27) +� +B1 +∇ur · ∇η dx = +� +B1 +χ{ur>0}η dx +for all η ∈ C∞ +c (B1). +Moreover, by assumption (5.26), we have +��{ur = 0} ∩ B1 +�� −→ 0, and thus +taking limits rk → 0 in (5.27) we deduce that ∆u0 = 1 in B1. Since we +know that u0 is convex, nonnegative, and homogeneous, this implies that +|{u0 = 0}| = 0. +□ +We can now give the: +Proof of Theorem 5.40. By the classification of blow-ups (Theorem 5.24), +the possible blow-ups can only have one of the two forms presented. If (5.19) +holds for at least one blow-up, thanks to the smoothness of the free boundary +(by Proposition 5.37), it holds for all blow-ups, and thus, by Corollary 5.30, +u0(x) = 1 +2(x · e)2 ++ (and in fact, the smoothness of the free boundary yields +uniqueness of the blow-up in this case). +If (5.26) holds, then by Lemma 5.41 the blow-up u0 must satisfy +��{u0 = +0} +�� = 0, and thus we are in case (b) (see the proof of Theorem 5.24). +□ + +— DRAFT — +5.7. Singular points +173 +In the previous Section we proved that the free boundary is C∞ in a +neighborhood of any regular point. A natural question then is to understand +better the solution u near singular points. One of the main results in this +direction is the following. +Theorem 5.42 (Uniqueness of blow-ups at singular points). Let u be any +solution to (5.14), and assume that 0 is a singular free boundary point. +Then, there exists a homogeneous quadratic polynomial p2(x) = 1 +2xT Ax, +with A ≥ 0 and ∆p2 = 1, such that +ur −→ p2 +in +C1 +loc(Rn). +In particular, the blow-up of u at 0 is unique, and u(x) = p2(x) + o(|x|2). +To prove this, we need the following monotonicity formula due to Mon- +neau. +Theorem 5.43 (Monneau’s monotonicity formula). Let u be any solution +to (5.14), and assume that 0 is a singular free boundary point. +Let q be any homogeneous quadratic polynomial with q ≥ 0, q(0) = 0, +and ∆q = 1. Then, the quantity +Mu,q(r) := +1 +rn+3 +� +∂Br +(u − q)2 +is monotone in r, that is, +d +drMu,q(r) ≥ 0. +Proof. We sketch the argument here, and refer to [PSU12, Theorem 7.4] +for more details. +We first notice that +Mu,q(r) = +� +∂B1 +(u − q)2(rx) +r4 +, +and hence a direct computation yields +d +drMu,q(r) = +2 +rn+4 +� +∂Br +(u − q) {x · ∇(u − q) − 2(u − q)} . +On the other hand, it turns out that +1 +rn+3 +� +∂Br +(u − q) {x · ∇(u − q) − 2(u − q)} = Wu(r) − Wu(0+)+ ++ +1 +rn+2 +� +Br +(u − q)∆(u − q), +where Wu(r) (as defined in (5.15)) is monotone increasing in r > 0 thanks +to Theorem 5.18. Thus, we have +d +drMu,q(r) ≥ +2 +rn+3 +� +Br +(u − q)∆(u − q). + +— DRAFT — +174 +5. The obstacle problem +But since ∆u = ∆q = 1 in {u > 0}, and (u−q)∆(u−q) = q ≥ 0 in {u = 0}, +we have +d +drMu,q(r) ≥ +2 +rn+3 +� +Br∩{u=0} +q ≥ 0, +as wanted. +□ +We can now give the: +Proof of Theorem 5.42. By Proposition 5.40 (and Proposition 5.23), we +know that at any singular point we have a subsequence rj → 0 along which +urj → p in C1 +loc(Rn), where p is a 2-homogeneous quadratic polynomial +satisfying p(0) = 0, p ≥ 0, and ∆p = 1. +Thus, we can use Monneau’s +monotonicity formula with such polynomial p to find that +Mu,p(r) := +1 +rn+3 +� +∂Br +(u − p)2 +is monotone increasing in r > 0. In particular, the limit limr→0 Mu,p(r) := +Mu,p(0+) exists. +Now, recall that we have a sequence rj → 0 along which urj → p. In +particular, r−2 +j +{u(rjx) − p(rjx)} −→ 0 locally uniformly in Rn, i.e., +1 +r2 +j +∥u − p∥L∞(Brj ) −→ 0 +as rj → 0. This yields that +Mu,p(rj) ≤ +1 +rn+3 +j +� +∂Brj +∥u − p∥2 +L∞(Brj ) −→ 0 +along the subsequence rj → 0, and therefore Mu,p(0+) = 0. +Let us show that this implies the uniqueness of blow-ups. Indeed, if there +was another subsequence rℓ → 0 along which urℓ → q in C1 +loc(Rn), for a 2- +homogeneous quadratic polynomial q, then we would repeat the argument +above to find that Mu,q(0+) = 0. But then this yields, by homogeneity of p +and q, +� +∂B1 +(p − q)2 = +1 +rn+3 +� +∂Br +(p − q)2 ≤ 2Mu,p(r) + 2Mu,q(r) −→ 0, +and hence +� +∂B1 +(p − q)2 = 0. +This means that p = q, and thus the blow-up of u at 0 is unique. +Let us finally show that u(x) = p(x)+o(|x|2), i.e., r−2∥u−p∥L∞(Br) → 0 +as r → 0. +Indeed, assume by contradiction that there is a subsequence + +— DRAFT — +5.8. On the size of the singular set +175 +rk → 0 along which +r−2 +k ∥u − p∥L∞(Brk) ≥ c1 > 0. +Then, there would be a subsequence of rki along which urki → u0 in C1 +loc(Rn), +for a certain blow-up u0 satisfying ∥u0 − p∥L∞(B1) ≥ c1 > 0. However, by +uniqueness of blow-ups it must be u0 = p, and hence we reach a contradic- +tion. +□ +We refer to [SY23, Bon01] for an alternative approach to the unique- +ness of blow-ups at singular points, not based on monotonicity formulas. +Summarizing, we have proved the following result: +Theorem 5.44. Let u be any solution to (5.14). Then, we have the following +dichotomy: +(a) Either all blow-ups of u at 0 are of the form +u0(x) = 1 +2(x · e)2 ++ +for some +e ∈ Sn−1, +and the free boundary is C∞ in a neighborhood of the origin. +(b) Or there is a homogeneous quadratic polynomial p, with p(0) = 0, +p ≥ 0, and ∆p = 1, such that +∥u − p∥L∞(Br) = o(r2) +as +r → 0. +In particular, when this happens we have +lim +r→0 +��{u = 0} ∩ Br +�� +|Br| += 0. +The last question that remains to be answered is: How large can the set +of singular points be? This is the topic of the following section. +5.8. On the size of the singular set +We finish this chapter with a discussion of more recent results (as well as +some open problems) about the set of singular points. +Recall that a free boundary point x◦ ∈ ∂{u > 0} is singular whenever +lim +r→0 +��{u = 0} ∩ Br(x◦) +�� +|Br(x◦)| += 0. +The main known result on the size of the singular set reads as follows. +Theorem 5.45 ([Caf98]). Let u be any solution to (5.14). Let Σ ⊂ B1 be +the set of singular points. +Then, Σ∩B1/2 is locally contained in a C1 manifold of dimension n−1. + +— DRAFT — +176 +5. The obstacle problem +This result is sharp, in the sense that it is not difficult to construct +examples in which the singular set is (n − 1)-dimensional; see [Sch77]. +As explained below, such result essentially follows from the uniqueness +of blow-ups at singular points, established in the previous section. +Indeed, given any singular point x◦, let px◦ be the blow-up of u at x◦ +(recall that px◦ is a nonnegative 2-homogeneous polynomial). Let k be the +dimension of the set {px◦ = 0} — notice that this is a proper linear subspace +of Rn, so that k ∈ {0, ..., n − 1} — and define +(5.28) +Σk := +� +x◦ ∈ Σ : dim({px◦ = 0}) = k +� +. +Clearly, Σ = �n−1 +k=0 Σk. +The following result gives a more precise description of the singular set. +Proposition 5.46 ([Caf98]). Let u be any solution to (5.14). Let Σk ⊂ B1 +be defined by (5.28), k = 1, ..., n − 1. Then, Σk is locally contained in a C1 +manifold of dimension k. +The rough heuristic idea of the proof of this result is as follows. Assume +for simplicity that n = 2, so that Σ = Σ1 ∪ Σ0. +Let us take a point x◦ ∈ Σ0. +Then, by Theorem 5.44, we have the +expansion +(5.29) +u(x) = px◦(x − x◦) + o +� +|x − x◦|2� +where px◦ is the blow-up of u at x◦ (recall that this came from the uniqueness +of blow-ups at x◦). By definition of Σ0, the polynomial px◦ must be positive +outside the origin, and thus by homogeneity satisfies px◦(x−x◦) ≥ c|x−x◦|2, +with c > 0. This, combined with (5.29), yields then that u must be positive +in a neighborhood of x◦. In particular, all points in Σ0 are isolated. +On the other hand, let us now take a point x◦ ∈ Σ1. Then, by definition +of Σ1 the blow-up must necessarily be of the form px◦(x) = 1 +2(x · ex◦)2, for +some ex◦ ∈ Sn−1. Again by the expansion (5.29), we find that u is positive +in a region of the form +� +x ∈ Bρ(x◦) : +��(x − x◦) · ex◦ +�� > ω(|x − x◦|) +� +, +where ω is a certain modulus of continuity, and ρ > 0 is small (see Fig- +ure 5.16). +This is roughly saying that the set Σ1 “has a tangent plane” at x◦. +Repeating the same at any other point ˜x◦ ∈ Σ1 we find that the same +happens at every point in Σ1 and, moreover, if ˜x◦ is close to x◦ then e˜x◦ +must be close to ex◦ — otherwise the expansions (5.29) at ˜x◦ and x◦ would +not match. Finally, since the modulus ω can be made independent of the + +— DRAFT — +5.8. On the size of the singular set +177 +x◦ +u > 0 +u > 0 +Bρ(x◦) +ex◦ +Figure 5.16. u is positive in {x ∈ Bρ(x◦) : |(x − x◦) · ex◦| > ω(|x − x◦|)}. +x◦ +˜x◦ +ex◦ +e˜x◦ +Figure 5.17. Singular points x◦, ˜x◦ ∈ Σ1. +point (by a compactness argument), it turns out that the set Σ1 is contained +in a C1 curve (see Figure 5.17). +What we discussed here is just an heuristic argument; the actual proof +uses Whitney’s extension theorem and can be found for example in [PSU12]. +Finally, we refer to [CSV18], [FS19], and [FZ21] (and the expository paper +[Fig18b]) for some recent finer results about the set of singular points. +Generic regularity. In PDE problems in which singularities may appear, +it is very natural and important to understand whether these singularities +appear “often”, or if instead “most” solutions have no singularities. +In the context of the obstacle problem, the key question is to understand +the generic regularity of free boundaries. Explicit examples show that sin- +gular points in the obstacle problem can form a very large set, of dimension + +— DRAFT — +178 +5. The obstacle problem +n − 1 (as large as the regular set). Still, singular points are expected to be +rare (see [Sch74]): +Conjecture (Schaeffer, 1974): Generically, the weak solution of the obsta- +cle problem is also a strong solution, in the sense that the free boundary is +a C∞ manifold. +In other words, the conjecture states that, generically, the free boundary +has no singular points. +The first result in this direction was established by Monneau in 2003, +who proved the following. +Theorem 5.47 ([Mon03]). Schaeffer’s conjecture holds in R2. +More precisely, Monneau considers a 1-parameter family of solutions uλ, +with λ ∈ (0, 1), such that +� ∆uλ += +χ{uλ>0} +in Ω +uλ += +gλ +on ∂Ω, +with gλ = g + λ and g ≥ 0 on ∂Ω. +Then, the first step is to notice that not only each of the singular sets +Σλ ⊂ Ω is contained in a C1 manifold of dimension (n − 1), but actually the +union � +λ∈(0,1) Σλ ⊂ Ω is still contained in an (n − 1)-dimensional manifold. +After that, we look at the free boundary as a set in Ω×(0, 1) ∋ (x, λ), and +notice that it can be written as a graph {λ = h(x)}, for some function h. A +second key step in the proof is to show that h is Lipschitz and, furthermore, +it has zero gradient at any singular point. This, combined with the coarea +formula, yields that in R2 the set of singular points is empty for almost every +λ ∈ (0, 1), which implies Theorem 5.47. +Finally, the best known result in this direction was established very +recently by Figalli, Serra, and the second author. +Theorem 5.48 ([FRS20]). Schaeffer’s conjecture holds in R3 and R4. +The proof of this result is based on a new and very fine understanding +of singular points. For this, [FRS20] combines Geometric Measure The- +ory tools, PDE estimates, several dimension reduction arguments, and even +several new monotonicity formulas. +It remains an open problem to decide whether or not Schaeffer’s conjec- +ture holds in dimensions n ≥ 5 or not. + +— DRAFT — +Appendix A +Some properties of +H¨older spaces +In this appendix, we prove the properties (H1)-(H8) stated in Chapter 1. +Recall that, given α ∈ (0, 1], the H¨older space C0,α(Ω) is the set of +functions u ∈ C(Ω) such that +[u]C0,α(Ω) := sup +x,y∈Ω +x̸=y +��u(x) − u(y) +�� +|x − y|α +< ∞. +The H¨older norm is +∥u∥C0,α(Ω) := ∥u∥L∞(Ω) + [u]C0,α(Ω). +When α = 1, this is the usual space of Lipschitz functions. +More generally, given k ∈ N and α ∈ (0, 1], the space Ck,α(Ω) is the set +of functions u ∈ Ck(Ω) such that the following norm is finite +∥u∥Ck,α(Ω) := +k +� +j=1 +∥Dju∥L∞(Ω) + sup +x,y∈Ω +x̸=y +��Dku(x) − Dku(y) +�� +|x − y|α += ∥u∥Ck(Ω) + [Dku]C0,α(Ω). +Finally, when β > 0 is not an integer, we denote Cβ(Ω) := Ck,α(Ω), +where β = k + α, with k ∈ N, α ∈ (0, 1). +Next, we give the proofs of the properties of H¨older spaces that we +have used throughout the book. Unless stated otherwise, in the following +statements we assume α ∈ (0, 1). +179 + +— DRAFT — +180 +A. Some properties of H¨older spaces +(H1) Assume +oscBr(x)u ≤ C◦rα +for all Br(x) ⊂ B1, +where oscAu := supA u − infA u. +Then, u ∈ C0,α(B1) and [u]C0,α(B1) ≤ CC◦, with C depending only on +n, α. +• Proof of (H1) We want to prove that |u(z) − u(x)| ≤ CC◦|z − x|α for +all z, x ∈ B1. Given z, x ∈ B1, let r = |z − x|. For this, we may assume +r < 1/10 and distinguish two cases: +(i) If Br(x) ⊂ B1, then we simply use the assumption to get +|u(z) − u(x)| ≤ oscBr(x)u ≤ C◦rα = C◦|z − x|α. +(ii) Otherwise, we take ¯x and ¯z on the segments 0x and 0z, respectively, +such that |x − ¯x| = r and |z − ¯z| = r. Then, by assumption we have +|u(x) − u(¯x)| ≤ C◦rα, |u(z) − u(¯z)| ≤ C◦rα, and |u(¯x) − u(¯z)| ≤ C◦rα. +The last inequality holds because |¯x − ¯z| < r, which can be easily +checked by construction of ¯x and ¯z. +Combining the last three inequalities, we deduce that |u(x) − u(z)| ≤ +3C◦rα, as wanted. +□ +We also state and prove the following slight modification of (H1), which +will be useful in later proofs. +Notice that the difference with respect to +the previous statement is that now, given any ball in B1, we control the +oscillation in the ball with half the radius. +(H1’) Assume +oscBr(x)u ≤ C◦rα +for all B2r(x) ⊂ B1, +where oscAu := supA u − infA u. +Then, u ∈ C0,α(B1) and [u]C0,α(B1) ≤ CC◦, with C depending only on +n, α. +• Proof of (H1’). We proceed analogously to the proof of (H1). Let z, x ∈ +B1, and let r = |z − x|. We may assume that r < 1/10. If B2r(x) ⊂ B1, the +result follows by assumption. +Otherwise, let us take ¯x and ¯z on the segments 0x and 0z, respectively, +such that |x− ¯x| = 2r and |z − ¯z| = 2r. Let us define xk = (1−2−k)x+2−k¯x +and zk = (1 − 2−k)z + 2−k¯z. Notice that |xk+1 − xk| = |zk+1 − zk| = 2−kr. +Also, |xk| = |x| − 2−k+1r, so that B2|xk+1−xk|(xk) ⊂ B1. That is, we can use +our assumption on xk and xk+1 to get that +|u(xk) − u(xk+1)| ≤ C◦|xk − xk+1|α = C◦2−kαrα. + +— DRAFT — +A. Some properties of H¨older spaces +181 +(An analogous result holds for zk.) On the other hand, by choice of ¯x and ¯z, +they can also be compared in the oscillation of u as +|u(¯x) − u(¯z)| ≤ C◦|¯x − ¯z|α ≤ C◦rα. +Putting everything together, we reach that +|u(x) − u(z)| ≤ +� +k≥0 +|u(xk+1) − u(xk)| + |u(¯x) − u(¯z)| + +� +k≥0 +|u(zk+1) − u(zk)| +≤ 2 +� +k≥0 +C◦2−kαrα + C◦rα ≤ CC◦rα, +for some constant C depending only on α. +□ +(H2) Let ux,r := +� +Br(x) u. Assume +∥u − ux,r∥L∞(Br(x)) ≤ C◦rα +for all Br(x) ⊂ B1. +Then, u ∈ C0,α(B1) and [u]C0,α(B1) ≤ CC◦, with C depending only on n, α. +• Proof of (H2). By the triangle inequality we have +oscBr(x)u ≤ 2∥u − ux,r∥L∞(Br(x)) ≤ 2C◦rα, +and thus the result follows from (H1). +□ +(H3) Let ux,r := +� +Br(x) u. Assume +� � +Br(x) +|u − ux,r|2 +�1/2 +≤ C◦rα +for all Br(x) ⊂ B1. +Then, u ∈ C0,α(B1) and [u]C0,α(B1) ≤ CC◦, with C depending only on n, α. +• Proof of (H3). Notice that, for every z ∈ B1, +��ux,r − ux, r +2 +��2 ≤ 2|u(z) − ux,r|2 + 2 +��u(z) − ux, r +2 +��2. +Thus, integrating in Br/2(x) and using the assumption we deduce +|ux,r − ux, r +2 |2 ≤ 2 +� +Br/2(x) +|u − ux,r|2 + 2 +� +Br/2(x) +��u − ux, r +2 +��2 +≤ 2n+1 +� +Br(x) +|u − ux,r|2 + 2 +� +Br/2(x) +��u − ux, r +2 +��2 ≤ CC2 +◦r2α. +This means that +��ux,r − ux, r +2 +�� ≤ CC◦rα, +and summing a geometric series we get +|ux,r − u(x)| ≤ +� +k≥0 +��ux, r +2k − ux, +r +2k+1 +�� ≤ +� +k≥0 +CC◦ +� r +2k +�α += 2CC◦rα. + +— DRAFT — +182 +A. Some properties of H¨older spaces +Here we used that, up to redefining u on a set of measure zero, by Lebesgue +differentiation theorem (Theorem 1.1) we have that ux,r → u(x) as r → 0. +Let now x, y ∈ B1, r = 2|x − y|, and assume that Br(x) ⊂ B1. Then, we +have +|ux,r − uy,r|2 ≤ +� +Br/2(x) +|u − ux,r|2 + +� +Br/2(x) +��u − uy,r +��2 +≤ 2n +� +Br(x) +|u − ux,r|2 + 2n +� +Br(y) +��u − uy,r +��2 ≤ CC2 +◦r2α, +and thus +��ux,r − uy,r +�� ≤ CC◦rα. +Combining the previous estimates, we deduce that for every x, y ∈ B1 +such that B2|x−y|(x) ⊂ B1, we have +|u(x) − u(y)| ≤ |u(x) − ux,r| + |ux,r − uy,r| + |uy,r − u(y)| ≤ 3CC◦rα. +Once we have this, by (H1’) we are done. +□ +(H4) Assume that for every x there is a constant Cx such that +∥u − Cx∥L∞(Br(x)) ≤ C◦rα +for all Br(x) ⊂ B1. +Then, u ∈ C0,α(B1) and [u]C0,α(B1) ≤ CC◦, with C depending only on n, α. +Assume that for every x there is a linear function ℓx(y) = ax+bx·(y−x) +such that +∥u − ℓx∥L∞(Br(x)) ≤ C◦r1+α +for all Br(x) ⊂ B1. +Then, u ∈ C1,α(B1) and [Du]C0,α(B1) ≤ CC◦, with C depending only on +n, α. +Assume that for every x there is a quadratic polynomial Px(y) such that +∥u − Px∥L∞(Br(x)) ≤ C◦r2+α +for all Br(x) ⊂ B1. +Then, u ∈ C2,α(B1) and [D2u]C0,α(B1) ≤ CC◦, with C depending only on +n, α. +• Proof of (H4). (i) The first statement — with the C0,α norm — follows +from (H1). +(ii) Let us sketch the proof of the second statement — with the C1,α +norm. Let x, y ∈ B1 with y ∈ Br(x) ⊂ B1. Notice that, dividing by r +and taking r → 0 in the assumption, it follows that u is differentiable at +x and that ℓx must be given by ℓx(y) = u(x) + ∇u(x) · (y − x). Thus, by +assumption, we have +u(y) = u(x) + ∇u(x) · (y − x) + O(r1+α) + +— DRAFT — +A. Some properties of H¨older spaces +183 +and, for every z ∈ Br(x) such that |z − y| ≈ |z − x| ≈ |y − x| ≈ r, +u(z) = u(x) + ∇u(x) · (z − x) + O(r1+α) += u(y) + ∇u(y) · (z − y) + O(r1+α) += u(x) + ∇u(x) · (y − x) + ∇u(y) · (z − y) + O(r1+α). +From this, we deduce that +∇u(x) · (z − y) = ∇u(y) · (z − y) + O(r1+α). +Taking z such that z − y is parallel to ∇u(y) − ∇u(x), we get +∇u(x) = ∇u(y) + O(rα), +as wanted. +(iii) Let us prove the third statement concerning the C2,α norm — the +following proof is more general and works also in case (ii). +Let x, y ∈ Br(x◦) with |x − y| = r and suppose B2r(x◦) ⊂ B1. Let us +rescale u around x◦, i.e., ur(z) := u(x◦ + rz), so that |¯x − ¯y| = 1, where +x◦ + r¯x = x and x◦ + r¯y = y. Let us define also +Px,r(z) := Px(x◦ + rz) +and +Py,r := Py(x◦ + rz). +Then, +∥ur − Px,r∥L∞(B1(¯x)) = ∥u − Px∥L∞(Br(x)) ≤ C◦r2+α, +∥ur − Py,r∥L∞(B1(¯y)) = ∥u − Py∥L∞(Br(y)) ≤ C◦r2+α. +Hence, if we denote ¯w = ¯x+¯y +2 +then B1/2( ¯w) ⊂ B1(¯x) ∩ B1(¯y) and +∥Px,r − Py,r∥L∞(B1/2( ¯w)) ≤ ∥ur − Px,r∥L∞(B1(¯x)) + ∥ur − Py,r∥L∞(B1(¯y)) +≤ CC◦r2+α. +This means that all the coefficients of the polynomial Px,r − Py,r are con- +trolled by ˜CC◦r2+α. +Now, notice that if we denote Px(z) = ax+bx·(z−x)+(z−x)T Mx(z−x) +then Px,r(z) = ax + rbx · (z − ¯x) + r2(z − ¯x)T Mx(z − ¯x), and an analogous +expression holds for Py,r. Hence, we can write +Px,r(z) − Py,r(z) = +� +ax − ay + rbx · (¯y − ¯x) + r2(¯y − ¯x)T Mx(¯y − ¯x) +� ++ r +� +bx − by + 2r(¯y − ¯x)T Mx +� +· (z − ¯y) ++ r2(z − ¯y)T (Mx − My)(z − ¯y). +In particular, by looking at the quadratic and linear coefficients of such +polynomial, we have proved that +|Mx − My| ≤ ˜CC◦rα +and +��bx − by + 2r(¯y − ¯x)T Mx +�� ≤ CC◦r1+α. + +— DRAFT — +184 +A. Some properties of H¨older spaces +Since r(¯y − ¯x) = y − x, this is equivalent to +��by − bx − 2(y − x)T Mx +�� ≤ CC◦r1+α. +Notice, also, that +∥u − ax − bx · (· − x)∥L∞(Br(x)) ≤ C◦r2+α + Cxr2 ≤ 2Cxr2 +if r small enough, so that, in particular, arguing as in (i), u is differentiable +at x and ax = u(x), bx = ∇u(x). +Thus, using that r = 2|x − y|, we have +��∇u(y) − ∇u(x) − 2(y − x)T Mx +�� ≤ CC◦|y − x|1+α, +and letting y → x we deduce that ∇u is differentiable at x, with D2u(x) = +2Mx. An analogous result holds for My, so that we have shown that, for +any x, y ∈ Br(x◦) with |x − y| = r and B2r(x◦) ⊂ B1, +��D2u(x) − D2u(y) +�� ≤ ˜CC◦rα. +The result now follows by (H1’). +□ +Remark. Notice that the converse statement to (H4) also holds. For ex- +ample, when k = 1, if u ∈ C1,α(B1) then we have +∥u − ℓx∥L∞(Br(x)) ≤ C◦r1+α +for all Br(x) ⊂ B1, +where ℓx(y) = u(x) + ∇u(x) · (y − x). Indeed, to show this, we use that +u(y) = u(x) + +� 1 +0 +∇u(ty + (1 − t)x) · (y − x)dt, +combined with +��∇u +� +ty + (1 − t)x +� +− ∇u(x) +�� ≤ C◦|ty + (1 − t)x − x|α ≤ C◦|y − x|α, +to get +��u(y) − u(x) − ∇u(x) · (y − x) +�� ≤ +� 1 +0 +C◦|y − x|α|y − x|dt = C◦|y − x|1+α, +as wanted. +(H5) Let ρ◦ ∈ (0, 1). Assume that, for every x ∈ B1/2, there exists a se- +quence of quadratic polynomials, (Pk)k∈N such that +∥u − Pk∥L∞(Bρk◦ (x)) ≤ C◦ρk(2+α) +◦ +for all k ∈ N. +Then, u ∈ C2,α(B1/2) and [D2u]C0,α(B1/2) ≤ CC◦, with C depending only on +n, α, and ρ◦. + +— DRAFT — +A. Some properties of H¨older spaces +185 +• Proof of (H5). Let us take x = 0. By hypothesis, we have +∥Pk−1 − Pk∥L∞(Bρk◦ ) ≤ ∥u − Pk−1∥L∞(Bρk◦ ) + ∥u − Pk∥L∞(Bρk◦ ) +≤ CC◦ρk(2+α) +◦ +. +Then, we use the following: +Claim. Assume that P is a quadratic polynomial satisfying ∥P∥L∞(Br) ≤ γ. +If we denote P(z) = a + b · z + zT Mz, then we have that +|a| ≤ Cγ, +|b| ≤ Cγ +r , +|M| ≤ Cγ +r2 , +where C is a constant depending only on n. +To prove the claim, notice that, by rescaling, we have Pr(z) := P(rz) = +ar + br · z + zT Mrz, where ar = a, br = rb, Mr = r2M. By assumption, +we have that ∥Pr∥L∞(B1) ≤ γ. Since the coefficients of polynomials on B1 +are controlled by the L∞ norm, we get that |ar| ≤ Cγ, |br| ≤ Cγ, and +|Mr| ≤ Cγ. This proves the claim. +Using the previous claim and the bound on Pk−1 − Pk, we deduce that +|ak−1 − ak| ≤ CC◦ρk(2+α) +◦ +, +|bk−1 − bk| ≤ CC◦ρk(1+α) +◦ +, +and +|Mk−1 − Mk| ≤ CC◦ρkα +◦ , +where Pk(z) = ak + bk · z + zT Mkz. +It follows that Pk converge uniformly to a polynomial P(z) = a + b · z + +zT Mz, and that +∥u − P∥L∞(Bρk◦ ) ≤ ∥u − Pk∥L∞(Bρk◦ ) + |ak − a| + ρk +◦|bk − b| + ρ2k +◦ |Mk − M| +≤ CC◦ρk(2+α) +◦ +for all k ≥ 1. From this, it follows that for every r ∈ (0, 1) we have +∥u − P∥L∞(Br) ≤ CC◦r2+α +(simply use that for any r we have ρk+1 +◦ +≤ r ≤ ρk +◦ for some k). Thus, since we +can do this for every x ∈ B1/2, it follows from (H4) that [D2u]C0,α(B1/2) ≤ +CC◦. +□ +We refer to Remark A.1 below for a generalization of property (H5). +(H6) Assume that α ∈ (0, 1), ∥u∥L∞(B1) ≤ C◦, and +(A.1) +sup +h∈B1 +x∈B1−|h| +��u(x + h) + u(x − h) − 2u(x) +�� +|h|α +≤ C◦. +Then, u ∈ C0,α(B1) and ∥u∥C0,α(B1) ≤ CC◦, with C depending only on n, α. + +— DRAFT — +186 +A. Some properties of H¨older spaces +Assume that α ∈ (0, 1), ∥u∥L∞(B1) ≤ C◦, and +(A.2) +sup +h∈B1 +x∈B1−|h| +��u(x + h) + u(x − h) − 2u(x) +�� +|h|1+α +≤ C◦. +Then, u ∈ C1,α(B1) and ∥u∥C1,α(B1) ≤ CC◦, with C depending only on n, α. +However, such property fails when α = 0. +• Proof of (H6). (i) Let us do the case (A.2) first. +Given h ∈ B1 and x ∈ B1−|h|, let +w(h) := u(x + h) − u(x) +|h| +. +Then, by assumption we have +��w(h) − w(h/2) +�� = |u(x + h) + u(x) − 2u(x + h/2)| +|h| +≤ C◦|h|α. +Thus, for every k ≥ 0, +��w(h/2k) − w(h/2k+1) +�� ≤ C◦|h|α2−kα. +This implies the existence of the limit limt→0 w(th), and by summing a +geometric series we get +��w(h) − lim +t→0 w(th) +�� ≤ CC◦|h|α. +Since +lim +t→0 w(th) = lim +t→0 +u(x + th) − u(x) +t|h| += h +|h| · ∇u(x), +this leads to +��u(x + h) − u(x) − h · ∇u(x) +�� ≤ CC◦|h|1+α. +Using (H4), we see that the last inequality implies that [Du]C0,α(B1) ≤ CC◦. +Finally, using that ∥u∥L∞(B1) ≤ C◦, the result follows. +(ii) Let us do now the case (A.1). +As before, let us define w(h) := u(x+h)−u(x) +|h| +and notice that +��w(h) − w(h/2) +�� = |u(x + h) + u(x) − 2u(x + h/2)| +|h| +≤ C◦|h|α−1. +Then, for every k ≥ 0 we have +��w(2kh) − w(2k+1h) +�� ≤ C◦|h|α−12−k(1−α). + +— DRAFT — +A. Some properties of H¨older spaces +187 +Take k◦ ≥ 0 such that 2k◦|h| ≈ 1 (and so that1 still x + 2k◦h ∈ B1), and add +the previous inequality for all 0 ≤ k < k◦. Then, by summing a geometric +series, we deduce that +��w(2k◦h) − w(h) +�� ≤ CC◦|h|α−1. +Since +��w(2k◦h) +�� ≤ C∥u∥L∞(B1) ≤ CC◦ ≤ CC◦|h|α−1, +we finally get +��w(h) +�� ≤ +��w(2k◦h) +�� + CC◦|h|α−1 ≤ CC◦|h|α−1. +Translating back to u, this gives the desired result. +(iii) Finally, let us prove that the function +u(x) = x log |x|, +x ∈ (−1, 1), +satisfies (A.2) with α = 0, but it is not in C0,1. +Indeed, let us show that +��(x + h) log |x + h| + (x − h) log |x − h| − 2x log |x| +�� +|h| +≤ C◦ +for all x, h ∈ (−1, 1) and for some C◦ > 0. For this, notice that +(x + h) log |x + h| + (x − h) log |x − h| − 2x log |x| +h += += +� +1 + h +x +� +log +��1 + h +x +�� + +� +1 − h +x +� +log +��1 − h +x +�� +h +x += (1 + t) log |1 + t| + (1 − t) log |1 − t| +t +, +with t = h/x. Such function of t is smooth in R\{0} and has finite limits at +t = 0 and at t = ∞. Therefore, it is globally bounded in R by some constant +C◦ (actually, C◦ < 2). +□ +Remark. We refer to [And97, Section 2] for higher order versions of the +characterization (H6). +(H7) Assume that α ∈ (0, 1], ∥u∥L∞(B1) ≤ C◦, and that for every h ∈ B1 we +have +���� +u(x + h) − u(x) +|h|α +���� +Cβ(B1−|h|) +≤ C◦, +with C◦ independent of h. Assume in addition that α + β is not an integer. +Then, u ∈ Cα+β(B1) and ∥u∥Cα+β(B1) ≤ CC◦, with C depending only on +n, α, β. +1Note that this is always possible if x, y ∈ B9/10, for example. +If x, y are close to the +boundary ∂B1, then this is possible for example when (x − y) · +x +|x| > 1 +2 |y − x|. It is easy to see +that we can always reduce to this case. + +— DRAFT — +188 +A. Some properties of H¨older spaces +However, such property fails when α + β is an integer. +• Proof of (H7). We prove it in case β ∈ (0, 1], the proof for β > 1 is +analogous. Let us define +vh(x) = u(x + h) − u(x) +|h|α +. +Then, by assumption we have +sup +x,y∈B1−|h| +|vh(x) − vh(y)| +|x − y|β +≤ C◦. +This is equivalent to +sup +x,y∈B1−|h| +|u(x + h) − u(x) − u(y + h) + u(y)| +|h|α+β +≤ C◦. +Taking y = x − h, this yields +sup +x∈B1−2|h| +|u(x + h) + u(x + h) − 2u(x)| +|h|α+β +≤ C◦. +By (H6), we deduce that ∥u∥Cα+β(B1) ≤ CC◦ — as long as α + β ̸= 1. +□ +(H8) Assume that ui → u uniformly in Ω ⊂ Rn, and that ∥ui∥Ck,α(Ω) ≤ C◦, +with α ∈ (0, 1] and for some C◦ independent of i. Then, u ∈ Ck,α(Ω), and +∥u∥Ck,α(Ω) ≤ C◦. +• Proof of (H8). Assume first k = 0. Then, we have that for every x, y ∈ +Ω, x ̸= y, +∥ui∥L∞(Ω) + |ui(x) − ui(y)| +|x − y|α +≤ C◦. +Taking limits ui → u, we deduce that the same inequality holds for u, and +thus ∥u∥C0,α(Ω) ≤ C◦, as wanted. +Assume now that k ≥ 1. +Then, it follows from Arzel`a–Ascoli that +Dmui → Dmu uniformly in Ω for m ≤ k and thus, as before, taking limits +in the inequality +∥ui∥Ck(Ω) + |Dkui(x) − Dkui(y)| +|x − y|α +≤ C◦, +the result follows. +□ +Remark A.1. In relation with property (H5), one can define L ∞,β as +the set of functions u : B1 → R satisfying that, for each x ∈ R and each +r ∈ (0, 1 − |x|), there exists some polynomial Px,r of degree ⌊β⌋ such that +∥u − Px,r∥L∞(Br(x)) ≤ Crβ + +— DRAFT — +A. Some properties of H¨older spaces +189 +for some C universal, and where ⌊β⌋ denotes the integer part of β. More +generally, one can define2 L p,β for p ∈ [1, ∞] as the set of functions u +satisfying +r− n +p ∥u − Px,r∥Lp(Br) ≤ Crβ. +Then, it turns out that, for any β > 0 and p ≥ 1, L p,β = L ∞,β; see +[JTW83, Theorem 2]. Moreover, similarly to what we did in (H5), one +can prove that if β = k + α, then +L p,k+α = L ∞,k+α = Ck,α, +if +α ∈ (0, 1) and k ∈ N. +On the other hand, when β is an integer these spaces do not coincide with +H¨older spaces. Indeed, for β = 1 we have +L p,1 = L ∞,1 = Λ1, +(see [JW84, Section 1.6]), and for β > 1, +u ∈ L p,β +⇐⇒ +∇u ∈ L p,β−1, +(see [JTW83, Theorem 3].) Here, Λ1 denotes the Zygmund space, i.e. the +set of functions u : B1 → R such that +sup +h∈B1 +x∈B1−|h| +��u(x + h) + u(x − h) − 2u(x) +�� +|h| +≤ C, +for some universal C. Finally, when β = 0 we have +L p,0 = L 1,0 = BMO, +if +p ∈ [1, ∞), +where BMO denotes the space of bounded mean oscillation functions, see +[JN61, JW84]. +Notice also that ∇u ∈ BMO implies u ∈ Λ1, but the +opposite implication does not hold, see [Str80, Theorem 3.4]. +2These spaces are called Morrey-Campanato spaces when p < ∞ and β < 1. + +— DRAFT — + +— DRAFT — +Appendix B +Proof of the boundary +Harnack inequality +The goal of this appendix is to prove the boundary Harnack inequality for +Lipschitz domains, Theorem 5.35. The proof we present here is due to De +Silva and Savin [DS20], and is different to the one given in the book [CS05]. +For simplicity, we consider domains Ω such that +(B.1) +Ω ∩ B1 is given by a Lipschitz graph in the en direction, +with Lipschitz norm ≤ 1, and with 0 ∈ ∂Ω. +In other words, we consider (x′, xn) ∈ Rn−1 × R, and let +(B.2) +g : Rn−1 → R, +[g]C0,1(Rn−1) ≤ 1, +g(0) = 0, +Ω := {x ∈ Rn : xn > g(x′)}. +The boundary Harnack inequality in Lipschitz domains is the following. +(See Figure B.1 for a depiction of the setting in the theorem.) +Theorem B.1 (Boundary Harnack). Let w1 and w2 be positive harmonic +functions in B1 ∩ Ω, where Ω ⊂ Rn is a Lipschitz domain as in (B.1)-(B.2). +Assume that w1 and w2 vanish continuously on ∂Ω ∩ B1, and C−1 +◦ +≤ +∥wi∥L∞(B1/2) ≤ C◦ for i = 1, 2. Then, +C−1w2 ≤ w1 ≤ Cw2 +in +Ω ∩ B1/2. +The constant C depends only on n and C◦. +Moreover, an appropriate iteration of the previous result gives the fol- +lowing. +191 + +— DRAFT — +192 +B. Proof of the boundary Harnack inequality +wi > 0 +∆wi = 0 +wi = 0 +B1 +Ω +xn +x′ +g(x′) +Figure B.1. Depiction of the setting in Theorem B.1 and Corollary B.2. +Corollary B.2. Let w1 and w2 be as in Theorem B.1. Then, +���� +w1 +w2 +���� +C0,α(Ω∩B1/2) +≤ C +for some small α > 0. The constants α and C depend only on n and C◦. +Remark B.3. Notice that, for simplicity, we deal with Lipschitz domains +with Lipschitz constant bounded by 1 and, as a consequence, none of the +constants appearing in Theorem B.1 depend on the domain Ω. The same +proof presented here can be adapted to the case of general Lipschitz domains. +The reasons we consider domains with Lipschitz constant bounded by 1 +are to avoid introducing more notation and so that the domain Ω in B1 +has a single connected component. Note, moreover, that when we apply +the boundary Harnack in Proposition 5.37, we are doing so to a Lipschitz +domain with Lipschitz constant smaller than 1 (therefore, we can directly +apply Corollary B.2). +The following two (well-known) lemmas for sub- and superharmonic +functions will be used. Notice that these are interior regularity properties. +Lemma B.4 (Weak Harnack Inequality for supersolutions). Let u ∈ C(B1). +Then, +� −∆u +≥ +0 +in B1 +u +≥ +0 +in B1 +=⇒ +inf +B1/2 +u ≥ c ∥u∥L1(B1/2), + +— DRAFT — +B. Proof of the boundary Harnack inequality +193 +for some c > 0 depending only on n. +Proof. By the mean value property of the Laplace equation, for any x◦ ∈ +B1/3 we have +u(x◦) ≥ +1 +|B2/3| +� +B2/3(x◦) +u = c∥u∥L1(B2/3)(x◦) ≥ c∥u∥L1(B1/3), +with c a dimensional constant, so that we have proved the property in a ball +of radius 1/3. Take now any ¯x◦ ∈ ∂B1/3 and consider the ball B1/6(¯x◦). +Notice that we can repeat the previous steps to derive +inf +B1/6(¯x◦) u ≥ c∥u∥L1(B1/6)(¯x◦). +Moreover, if we denote B := B1/3 ∩ B1/6(¯x◦), then +∥u∥L1(B1/6)(¯x◦) ≥ +� +B +u ≥ |B| inf +B u ≥ c inf +B1/3 +u. +From the first result in this proof, we can conclude +inf +B1/2 +u ≥ c1 inf +B1/3 +u ≥ c2∥u∥L1(B1/3) ≥ c3∥u∥L1(B1/2) +for some dimensional constant c3. In the last step we have used the mono- +tonicity of averages with respect to the radius for superharmonic functions; +see for example (1.12). +□ +The second lemma reads as follows. +Lemma B.5 (L∞ bound for subsolutions). Let u ∈ C(B1). Then, +−∆u ≤ 0 +in +B1 +⇒ +sup +B1/2 +u ≤ Cε∥u∥Lε(B1), +for any ε > 0, and for some Cε depending only on n and ε. +Proof. Again, by the mean value property we have that, for any r > 0, +∥u∥L∞(Br/2) ≤ C +� +Br +u ≤ C∥u∥1−ε +L∞(Br) +� +Br +|u|ε. +We now want to use an interpolation inequality. Notice that, for any δ > 0, +there exists some Cδ (depending only on δ and ε) such that ξ1−ε ≤ δξ + Cδ +for all ξ ≥ 0. Taking ξ = A +B with +A = ∥u∥L∞(Br), +B = +� +C +� +Br +|u|ε +� 1 +ε +we deduce that, for any δ > 0, there exists some Cδ such that +∥u∥L∞(Br/2) ≤ C∥u∥1−ε +L∞(Br) +� +Br +|u|ε ≤ δ ∥u∥L∞(Br) + Cδ +� +C +� +Br +|u|ε +� 1 +ε +. + +— DRAFT — +194 +B. Proof of the boundary Harnack inequality +Figure B.2. A chain of balls to apply the Harnack inequality sequentially. +In particular, +∥u∥L∞(Br/2) ≤ δ∥u∥L∞(Br) + Cδr− n +ε ∥u∥Lε(B1). +We are now in position to apply Lemma 2.27 with S(A) = ∥u∥L∞(A), +k = n +ε and γ = Cδ∥u∥Lε(B1), to deduce that +∥u∥L∞(B1/2) ≤ C∥u∥Lε(B1), +for some constant C depending only on n and ε, as wanted. +□ +As a consequence of the previous lemmas we obtain the following two +useful results, which are partial steps towards the proof of Theorem B.1. +The first one gives an L∞ bound for u in terms of the value of the function +at an interior point in Ω. +Lemma B.6. Let u ∈ C(B1) be a positive harmonic function in B1∩Ω with +u = 0 on B1 \ Ω, where Ω ⊂ Rn is a Lipschitz domain as in (B.1)-(B.2). +Assume, moreover, that u( 1 +2en) = 1. Then, +∥u∥L∞(B1/2) ≤ C, +for some constant C depending only on n. +Proof. Notice that since u ≥ 0 is harmonic whenever u > 0, and it is +continuous, we have ∆u ≥ 0 in B1 in the viscosity sense. +On the other hand, since g in (B.1) has Lipschitz constant bounded by +1, we have Bϱ( 1 +2en) ⊂ {∆u = 0}, with ϱ = +1 +2 +√ +2. In particular, by Harnack’s +inequality (see (2.3)) we have that u ≤ Cn in B1/4( 1 +2en). That is, u(0, xn) ≤ +Cn for xn ∈ +� 1 +4, 1 +2 +� +. Repeating iteratively, we get u(0, xn) ≤ Ck +n for xn ∈ +� +2−k−1, 2−k� +(see Figure B.2 for a sketch of this chain of inequalities), so +that u(0, t) ≤ t−K for t ∈ +� +0, 1 +2 +� +, for some large dimensional constant K. We + +— DRAFT — +B. Proof of the boundary Harnack inequality +195 +can repeat the same procedure at all points in B1/2 by iterating successive +Harnack inequalities, to deduce that +u ≤ d−K +in +B1/2, +where +d(x) := dist(x, Ωc). +In particular, for ε > 0 small enough we have +� +B1/2 +|u|ε ≤ C. +By Lemma B.5, we deduce that ∥u∥L∞(B1/4) ≤ C, and the result in B1/2 +follows from a simple covering argument. +□ +The second lemma reads as follows. +Lemma B.7. Let δ > 0 be small, let Ω ⊂ Rn be a Lipschitz domain as in +(B.1)-(B.2), and let Ωδ := {x ∈ Ω : dist(x, Ωc) ≥ δ}. Let u ∈ C(B1) satisfy +� ∆u += +0 +in Ω ∩ B1 +u += +0 +on ∂Ω ∩ B1 +and +� u +≥ +1 +in B1 ∩ Ωδ +u +≥ +−δ +in B1. +Then, for all k ∈ N such that kδ ≤ 3 +4, we have +u ≥ −δ(1 − c◦)k +in +B1−kδ +for some constant c◦ depending only on n. +Proof. Let u− = min{u, 0}. Notice that u− is superharmonic (in the vis- +cosity sense) since ∆u− = 0 when u− < 0, and u− ≤ 0, so we have ∆u− ≤ 0. +Let w = u− + δ. By assumption, w ≥ 0 and ∆w ≤ 0. +Let x◦ ∈ ∂Ω ∩ B1−2δ. Let us apply Lemma B.4 to a ball of radius 2δ +around x◦, so that (after scaling) we deduce +inf +Bδ(x◦) w ≥ cδ−n∥w∥L1(Bδ(x◦)). +Notice, now, that since the domain is Lipschitz and w ≥ δ in Ωc, we can +bound ∥w∥L1(Bδ(x◦)) ≥ δ|{w ≥ δ} ∩ Bδ(x◦)| ≥ cδn+1 for some c (see Fig- +ure B.3) depending only on n. Thus, +inf +Bδ(x◦) w ≥ c◦δ. +In particular, since w ≥ δ in B1 ∩ Ωδ we have w ≥ c◦δ in B1−δ and therefore +u ≥ −δ(1 − c◦) in B1−δ. Applying iteratively this inequality for balls of +radius 1 − 2δ, 1 − 3δ, ..., we obtain the desired result. +□ +We can now show the following result, which is a key step in the proof +of Theorem B.1. + +— DRAFT — +196 +B. Proof of the boundary Harnack inequality +∂Ω +u = 0 +u ≥ −δ +w ≥ 0 +w = δ +x◦ +Bδ(x◦) +∥w∥L1(Bδ(x◦)) ≥ cδn+1 +Figure B.3. The fact that the domain is Lipschitz allows us to bound +the L1 norm of w in Bδ(x◦) from below. +Proposition B.8. There exists δ > 0, depending only on n, such that the +following holds. +Let Ω ⊂ Rn be a Lipschitz domain as in (B.1)-(B.2), and let Ωδ := {x ∈ +Ω : dist(x, Ωc) ≥ δ}. Assume that u ∈ C(B1) satisfies +� ∆u += +0 +in Ω ∩ B1 +u += +0 +on ∂Ω ∩ B1 +and +� u +≥ +1 +in B1 ∩ Ωδ +u +≥ +−δ +in B1. +Then, u ≥ 0 in B1/2. +Proof. It is enough to show that, for some a > 0, we have +(B.3) +� u +≥ +a +in B1/2 ∩ Ωδ/2 +u +≥ +−δa +in B1/2. +Indeed, iterating (B.3) at all scales, and at all points z ∈ ∂Ω ∩ B1/2, we +obtain +� u +≥ +ak +in B2−k(z) ∩ Ω2−kδ +u +≥ +−δak +in B2−k(z) +for all k ∈ N. In particular, the first inequality yields that u(z + ten) ≥ 0 +for z ∈ ∂Ω ∩ B1/2 and t > 0, and therefore u ≥ 0 in B1/2. +Let us show (B.3). We start with the first inequality. Let x◦ ∈ B1/2 ∩ +Ωδ/2, and let us suppose that δ +2 ≤ dist(x◦, Ωc) < δ (otherwise, we are done +by assumption). Consider the function w = u + δ, which satisfies w ≥ 0 in +Ω by assumption. +Notice that we can connect the points x◦ and x◦ + 1 +2δen with a sequence +of (three) overlapping balls in Ω, so that we can apply Harnack’s inequality +to w to deduce +w(x◦) ≥ 1 +C w +� +x◦ + 1 +2δen +� +≥ 1 +C , + +— DRAFT — +B. Proof of the boundary Harnack inequality +197 +for some dimensional constant C, where in the last step we are using that +w +� +x◦ + 1 +2δen +� +≥ 1+δ by assumption. In particular, by taking δ > 0 smaller +than +1 +2C , we get +u(x◦) ≥ 1 +C − δ ≥ 1 +2C +for all +x◦ ∈ B1/2 ∩ Ωδ/2. +On the other hand, by Lemma B.7 we know that u ≥ −δ(1 − c◦)k in +B1−kδ as long as kδ ≤ 3 +4. If we take k = 1 +2δ, we deduce +u ≥ −δ(1 − c◦) +1 +2δ +in +B1/2, +and taking δ small enough such that (1 − c◦) +1 +2δ ≤ +1 +2C we are done. +□ +Remark B.9 (Proposition B.8 for small Lipschitz constants). The proofs +of Lemma B.7 and Proposition B.8 can be simplified a lot in the case of a +domain with small Lipschitz constant. +Indeed, let us assume that the hypotheses of Proposition B.8 hold, where +the domain Ω satisfies (B.1)-(B.2) but with Lipschitz constant L < +1 +n−1, and +let us consider the harmonic function +ϕ(x) = x2 +n − +1 +n − 1 +� +x2 +1 + x2 +2 + · · · + x2 +n−1 +� +. +Then, for δ small enough, ϕ ≤ u on ∂B1/2 ∩ Ω, and by assumption on the +Lipschitz constant of the domain we have that ϕ ≤ 0 on ∂Ω∩B1. In all, the +maximum principle gives ϕ ≤ u in B1/2 ∩ Ω, which implies that u(ten) ≥ 0 +for t ∈ +� +0, 1 +2 +� +. By repeating the same argument at all boundary points in +∂Ω ∩ B1/2 we reach that u ≥ 0 in B1/2. +We can now give the proof of Theorem B.1. +Proof of Theorem B.1. Thanks to Lemma B.6, up to a constant depend- +ing on C◦, we may assume w1( 1 +2en) = w2( 1 +2en) = 1. Then, let us define +v = Mw1 − εw2 +for some constants M (large) and ε (small) to be chosen. Let δ > 0 be given +by Proposition B.8. Then, since w2 is bounded, +v ≥ −εw2 ≥ −δ +in +B1/2 +for ε > 0 small enough. On the other hand, by the interior Harnack inequal- +ity, we can take M large enough so that Mw1 ≥ 1 + δ in B1/2 ∩ Ωδ, where +we recall that Ωδ = {x ∈ Ω : dist(x, Ωc) ≥ δ}. That is, +v = Mw1 − εw2 ≥ 1 +in +B1/2 ∩ Ωδ, +for M large enough depending only on n. Thus, the hypotheses of Proposi- +tion B.8 are satisfied, and therefore we deduce that v ≥ 0 in B1/2. + +— DRAFT — +198 +B. Proof of the boundary Harnack inequality +This means that, w2 ≤ Cw1 in B1/4 for some constant C depending +only on n. The inequality in B1/2 follows by a covering argument. Finally, +reversing the roles of w1 and w2, we obtain the desired result. +□ +Finally, we give the: +Proof of Corollary B.2. Let us denote +W := w1 +w2 +, +so that we have to prove H¨older regularity for W in Ω ∩ B1/2. +Notice that, by Theorem B.1, we know that +1 +C ≤ W ≤ C +in +B1/2 ∩ Ω, +for some C depending only on n. We start by claiming that, for some θ > 0 +and all k ∈ N, we have +(B.4) +osc +B2−k−1 W ≤ (1 − θ) osc +B2−k W. +Indeed, let +ak := sup +B2−k +W +and +bk := inf +B2−k W. +If we denote pk = +1 +2k+1 en, then either W(pk) ≥ +1 +2(ak + bk) or W(pk) ≤ +1 +2(ak + bk). +Suppose first that W(pk) ≥ 1 +2(ak + bk), and let us define +v := w1 − bkw2 +ak − bk +. +Notice that, by assumption, +1 +2w2(pk) ≤ v(pk) ≤ w2(pk). +In particular, we can apply Theorem B.1 to the pair of functions v and w2 +in the ball B2−k, to deduce that v ≥ 1 +C w2 in B2−k−1, that is, +w1 − bkw2 +ak − bk +≥ 1 +C w2 +in +B2−k−1 +⇐⇒ +inf +B2−k−1 W ≥ 1 +C (ak − bk) + bk. +Since supB2−k−1 W ≤ supB2−k W ≤ ak, we deduce that +osc +B2−k−1 W ≤ ak − 1 +C (ak − bk) − bk = +� +1 − 1 +C +� +(ak − bk) = (1 − θ) osc +B2−k W, +with θ = 1 +C , as wanted. +If we assume instead that W(pk) ≤ 1 +2(ak + bk), then the argument is +similar taking v := (akw2 − w1)/(ak − bk) instead. In all, (B.4) holds. + +— DRAFT — +B. Proof of the boundary Harnack inequality +199 +In particular, we have shown that, for some small α depending only on +n, we have +(B.5) +osc +Br(x◦) W ≤ Crα +for all +r ∈ (0, 1 +4) +and +x◦ ∈ ∂Ω ∩ B1/2, +(compare with the proof of Corollary 2.7). We now need to combine (B.5) +with interior estimates for harmonic functions to deduce our desired result. +Indeed, letting x, y ∈ Ω ∩ B1/2, we want to show that +(B.6) +|W(x) − W(y)| ≤ C|x − y|α, +for some constant C depending only on n. +Let 2r = dist(x, ∂Ω) = |x − x∗|, with x∗ ∈ ∂Ω. We consider two cases: +• If |x − y| ≥ +r +2, then we apply (B.5) in a ball Bρ(x∗) with radius ρ = +2r + |x − y| to deduce that +|W(x) − W(y)| ≤ +osc +Bρ(x∗) W ≤ C(2r + |x − y|)α ≤ C|x − y|α. +• If |x−y| ≤ r +2, then by (B.5) we know that oscBr(x) W ≤ Crα. In particular, +if we denote c∗ := W(x), then +∥w1 − c∗w2∥L∞(Br(x)) = ∥w2 (W − c∗) ∥L∞(Br(x)) ≤ Crα∥w2∥L∞(Br(x)). +On the other hand, since w1 − c∗w2 is harmonic in Br(x), by Corollary 2.7 +(rescaled) we know that +[w1 − c∗w2]C0,α(Br/2(x)) ≤ C +rα ∥w1 − c∗w2∥L∞(Br(x)) ≤ C∥w2∥L∞(Br(x)). +Hence, +|W(y) − W(x)| = +���� +w1(y) − c∗w2(y) +w2(y) +���� ≤ C|x − y|α ∥w2∥L∞(Br(x)) +w2(y) +. +We finish by noticing that, by Harnack’s inequality applied to w2 in B2r(x), +we have ∥w2∥L∞(Br(x)) ≤ Cw2(y) for some C depending only on n. +With these two cases, we have shown (B.6). This proves the result. +□ +Remark B.10. As said above, the proofs in this Appendix have been car- +ried out in case that Ω is a Lipschitz domain as in (B.1), with Lipschitz +constant bounded by 1. This slightly simplifies the notation, and we have +that Ω ∩ B1 has only one connected component. +In case of general Lipschitz domains (with Lipschitz constant bounded +by L), the same proofs can be carried out, provided that one is slightly more +careful with the underlying geometry. A simple way to do this is to prove +all the results with B1/2 replaced by Bρ, with ρ > 0 small depending on L. + +— DRAFT — +200 +B. Proof of the boundary Harnack inequality +An alternative way to do this is to work with cylinders, rather than balls, +as in [DS20]. + +— DRAFT — +Appendix C +Probabilistic +interpretation of fully +nonlinear equations +In this appendix, we heuristically describe the probabilistic interpretation of +fully nonlinear elliptic PDEs. This extends the discussion from Section 1.3 +in the context of the Laplace operator. +We start by recalling the following probabilistic interpretation of har- +monic functions from Chapter 1: +We have a Brownian motion Xx +t , starting at x ∈ Ω, and a payoff function +g : ∂Ω → R. When we hit the boundary ∂Ω (for the first time) at a point +z ∈ ∂Ω, we get a paid g(z). The question is then: +What is the expected payoff? +It turns out that +u(x) := +� expected +payoff +� += E +� +g (Xx +τ ) +� +satisfies +� ∆u += +0 +in Ω +u += +g +on ∂Ω, +where τ is the first time at which Xx +t hits ∂Ω. +We already saw this in Chapter 1. Now, we will see more general “prob- +abilistic games” that lead to more general elliptic PDEs. +Stochastic processes. A stochastic process Xt is a collection of random +variables indexed by a parameter, that for us is going to be t ≥ 0, taking +values in a state space, that for us is going to be Rn. One can think of them +as simply a “particle” moving randomly in Rn, with t ≥ 0 being the time. +201 + +— DRAFT — +202 +C. Probabilistic interpretation of fully nonlinear equations +The most famous and important stochastic process is the Brownian mo- +tion, that we already introduced in Section 1.3. We recall that it is charac- +terized by the following properties: +(1) X0 = 0 almost surely. +(2) Xt has no memory (is independent of the past, or it has indepen- +dent increments). +(3) Xt has stationary increments: Xt+s − Xs is equal in distribution +to Xt. +(4) Xt has continuous paths (t �→ Xt is continuous) almost surely. +(5) Xt is isotropic, i.e., it is rotationally symmetric in distribution. +A more general class of stochastic processes is obtained by removing the +assumption (5). +Infinitesimal generator. The infinitesimal generator of a stochastic pro- +cess Xt is an operator L defined to act on functions u : Rn → R by +(C.1) +Lu(x) := lim +t↓0 +E [u (x + Xt)] − u(x) +t +. +It takes C2 functions u, and gives Lu. +For the Brownian motion, we have that L is the Laplacian ∆. +More generally, under the assumptions (1)-(2)-(3)-(4), the infinitesimal +generator L will be a second order elliptic operator of the form +Lu = +n +� +i,j=1 +aij∂iju + +n +� +i=1 +bi∂iu + cu. +Why is this infinitesimal generator useful? +The infinitesimal generator of a stochastic process encodes all the infor- +mation of such process. Indeed, it is a classical fact that the definition of L +leads to the formula +(C.2) +E [u(x + Xt)] = u(x) + E +�� t +0 +Lu(x + Xs) ds +� +. +(This is analogous to the fundamental theorem of Calculus!) +We can come back to the “expected payoff” problem: +Let Ω ⊂ Rn be a fixed domain, and consider a stochastic process (x+Xt) +starting at x ∈ Ω, satisfying (2)-(3)-(4) above. Given a payoff function +g : ∂Ω → R, we have the following: when Xx +t hits the boundary ∂Ω for +the first time at z ∈ ∂Ω, we get a payoff g(z). (See Figure C.1.) +What is the expected payoff? + +— DRAFT — +C. Probabilistic interpretation of fully nonlinear equations +203 +x +z +∂Ω +Figure C.1. A stochastic process Xx +t defined in Ω starting at x until +it hits the first point on the boundary z ∈ ∂Ω. +Of course, the expected payoff will depend on x ∈ Ω. For the Brown- +ian motion, we defined u(x) to be the expected payoff when starting at x, +E [g(Xx +τ )], where τ is the first time we hit ∂Ω. Then, we observed that, since +the Brownian motion is isotropic, u must satisfy the mean value property, +and thus u is harmonic: ∆u = 0 in Ω. +Now, for more general stochastic processes, we must use (C.2). Indeed, +we define u as before (expected payoff), and notice that if t > 0 is small +enough, then x+Xt will still be inside Ω, and therefore, the expected payoff +is simply equal to E [u(x + Xt)] (up to a small error), i.e, +u(x) = E [u(x + Xt)] + o(t) +(for t > 0 small enough). +where the term o(t) is due to the fact that x+Xt could potentially lie outside +of Ω, even for arbitrarily small times t > 0. +Now, using the definition of infinitesimal generator, (C.1), we obtain +that +Lu(x) = lim +t↓0 +E [u(x + Xt)] − u(x) +t += lim +t↓0 +o(t) +t += 0. +Therefore, for every x ∈ Ω, we get Lu(x) = 0. We clearly have u = g on +∂Ω, thus, u must be the solution of +� Lu += +0 +in Ω +u += +g +on ∂Ω. + +— DRAFT — +204 +C. Probabilistic interpretation of fully nonlinear equations +Summarizing: +� +Expected +payoff for Xt +� +←→ +� Dirichlet problem for L +(infinitesimal generator) +� +. +Something similar can be done to solve other probabilistic problems +related to Xt: +– What is the expected time it will take to exit Ω if we start at x? +� −Lu += +1 +in Ω +u += +0 +on ∂Ω. +– What is the probability density p(x, t) of Xt in Rn? +� ∂tp − Lp += +0 +in Rn × (0, ∞) +p(·, 0) += +δ{x=0} +on ∂Ω. +We next see what happens when we have a control, or a two-player game. +In that case, we get nonlinear PDEs. +Optimal stopping. We start with the optimal stopping problem. +This +kind of problem appears very often in Mathematical Finance, for example. +Given a process Xt in Rn, we can decide at each instant of time whether +to stop it or not. When we stop, we get a payoff ϕ (which depends on the +point we stopped at). The goal is to discover what is the optimal strategy +so that we maximize the payoff. +Let us consider the process x + Xt (starting at x ∈ Rn), and a payoff +ϕ ∈ C∞ +c (Rn). For any stopping time θ, we get a payoff E [ϕ(x + Xθ)], and +therefore we want to maximize +u(x) := max +θ +E [ϕ(x + Xθ)] +among all possible stopping times θ (notice that a stopping time θ is actually +a random variable; see [Eva13] for more details). +Can we find a PDE for u(x)? +Roughly speaking, the only important thing to decide here is: +If we are at x, is it better to stop and get ϕ(x), or to continue and hope +for a better payoff later? +Let us find the PDE for u: +– First, since we can always stop (take θ = 0), we have u(x) ≥ ϕ(x) +for every x ∈ Rn. +– Second, since we can always continue for some time (take θ ≥ t◦ > +0), we have that u(x) ≥ E [u(x + Xt)] for t ≤ t◦. This, combined +with (C.2) (or with the definition (C.1)), gives +Lu(x) ≤ 0 +for every x ∈ Rn. + +— DRAFT — +C. Probabilistic interpretation of fully nonlinear equations +205 +u +ϕ +−Lu ≥ 0 everywhere +u ≥ ϕ everywhere +Lu = 0 in {u > ϕ} +Figure C.2. The obstacle problem. +– Third, at those points where we have u(x) > ϕ(x), we are clearly +not stopping there, so we have u(x) = E [u(x + Xt)] + o(t) for t +very small, and thus Lu(x) = 0 whenever u(x) > ϕ(x). +The PDE for u is +� +� +� +u +≥ +ϕ +in Rn, +−Lu +≥ +0 +in Rn, +Lu += +0 +in {u > ϕ}, +←→ min{−Lu, u − ϕ} = 0 +in +Rn. +This is the obstacle problem in Rn from Chapter 5. (See Figure C.2.) +Notice that once we know u, we know the sets {u = ϕ} and {u > ϕ}, so +we have the optimal strategy! +Controlled diffusion. Let us now take a different problem, that nonethe- +less is quite similar to the optimal stopping. +Consider two stochastic processes, X(1) +t +and X(2) +t +, with infinitesimal gen- +erators L1 and L2 respectively. Let Ω ⊂ Rn be a domain, and let g : ∂Ω → R +be a payoff. We have the same “game” as before (we get a payoff when we +hit the boundary), but now we have a control: for every x ∈ Ω, we can +choose to move according to X(1) +t +or X(2) +t +. +The question is then: +What is the optimal strategy if we want to maximize the payoff? + +— DRAFT — +206 +C. Probabilistic interpretation of fully nonlinear equations +Notice that now the strategy consists of choosing between X(1) +t +and X(2) +t +for every x ∈ Ω. As before, we define +u(x) := +max +all possible choices +of a : Ω → {1, 2} +E [g(Xa +τ )] +(where τ is the time we hit the boundary ∂Ω). Notice that for every a : +Ω → {1, 2} we have Xa +t , a process which could change from point to point. +Is there any PDE for u? +The optimality conditions are: +– First, when we are at x we can simply decide to continue with X(1) +t +, +and therefore, u(x) ≥ E +� +u(x + X(1) +t +) +� +for every x ∈ Ω. This yields +L1u(x) ≤ 0 for every x ∈ Ω. +– Similarly, we can do the same for X(2) +t +, and get L2u(x) ≤ 0 for +every x ∈ Ω. +– Finally, it turns out that either +u(x) = lim +t↓0 E +� +u(x + X(1) +t +) +� +or +u(x) = lim +t↓0 E +� +u(x + X(2) +t +) +� +, +since close to x we are taking either X(1) +t +or X(2) +t +. This means that +either L1u(x) = 0 or L2u(x) = 0, for every x ∈ Ω. +Therefore, u satisfies +� +� +� +−L1u +≥ +0 +in Ω, +−L2u +≥ +0 +in Ω, +either L1u = 0 or L2u = 0 in Ω +←→ +max{L1u, L2u} = 0 +in +Ω. +More generally, if we have a family of processes Xα +t , with α ∈ A, then +the PDE for u becomes +(C.3) +max +α∈A {Lαu} = 0 +in +Ω. +Even more generally, we could have two players, one that wants to max- +imize the payoff and the other one that wants to minimize the payoff. They +have two parameters, Xαβ +t +, α ∈ A, β ∈ B, and each player controls one +parameter. Then, the optimal payoff solves the PDE +(C.4) +min +β∈B max +α∈A {Lαβu} = 0 +in +Ω. +Equation (C.3) above is called the Bellman equation (stochastic control). +Equation (C.4) above is called the Isaacs equation (differential games). +These two equations are fully nonlinear elliptic equations! + +— DRAFT — +C. Probabilistic interpretation of fully nonlinear equations +207 +Indeed, assume that we have (C.3), and that the infinitesimal generators +Lαu are of the form +Lαu = +n +� +i,j=1 +a(α) +ij ∂iju, +(α ∈ A) +with a(α) +ij +uniformly elliptic: 0 < λId ≤ (a(α) +ij )ij ≤ ΛId. Then, the equation +(C.3) is +max +α∈A +� +� +� +n +� +i,j=1 +a(α) +ij ∂iju +� +� +� = 0 +in +Ω. +This is a nonlinear function of the Hessian D2u: +F(D2u) = 0 +in +Ω, +with +F(M) := max +α∈A +� +� +� +n +� +i,j=1 +a(α) +ij Mij +� +� +� . +The function F : Rn×n → R is the maximum of linear functions. In partic- +ular, F is convex. +Moreover, F is uniformly elliptic: +0 < λ∥N∥ ≤ min +α∈A +� +� +� +n +� +i,j=1 +a(α) +ij Nij +� +� +� ≤ F(M + N) − F(M) ≤ +≤ max +α∈A +� +� +� +n +� +i,j=1 +a(α) +ij Nij +� +� +� ≤ Λ∥N∥ +for any symmetric matrix N ≥ 0 (we are using here that max f + min g ≤ +max(f + g) ≤ max f + max g). +Furthermore, any convex function can be written as the maximum of +linear functions (see Figure C.3), and thus: +Remark C.1. Any F : Rn×n → R which is uniformly elliptic and convex +can be written as +F(M) = max +α∈A +� +tr (A(α)M) + cα +� +(where cα are constants). +(If F is homogeneous of degree 1, then we do not need the cα.) +In particular, every fully nonlinear uniformly elliptic equation +F(D2u) = 0 +in +Ω, +with F being convex, can be written as a Bellman equation +max +α∈A {Lαu} = 0 +in +Ω +with +Lαu = �n +i,j=1 a(α) +ij ∂iju + cα. +Finally, for non-convex F it turns out that: + +— DRAFT — +208 +C. Probabilistic interpretation of fully nonlinear equations +Figure C.3. Convex function as the maximum of linear functions. +Observation. Any F : Rn×n → R which is uniformly elliptic (not neces- +sarily convex), can be written as +F(M) = min +β∈B max +α∈A +� +� +� +n +� +i,j=1 +a(αβ) +ij +Mij + cαβ +� +� +� = min +β∈B max +α∈A +� +tr +� +A(α,β)M +� ++ cαβ +� +. +This is because any Lipschitz function F can be written as the minimum of +convex functions, and convex functions can be written as the maximum of +linear functions. +In particular, every fully nonlinear uniformly elliptic equation +F(D2u) = 0 +in +Ω +can be written as an Isaacs equation +min +β∈B max +α∈A {Lαβu} = 0 +in +Ω +with +Lαβu = �n +i,j=1 a(α,β) +ij +∂iju + cαβ. +Summary: Every fully nonlinear elliptic PDE has an interpretation in +terms of a probabilistic game! + +— DRAFT — +C. Probabilistic interpretation of fully nonlinear equations +209 +Probabilistic interpretation of PDEs. +Expected payoff +←→ +Dirichlet problem +� Lu += +0 +in Ω +u += +g +on ∂Ω. +Expected exit time +(or running costs/ +non-homogeneous environments) +←→ +Dirichlet problem +� −Lu += +f +in Ω +u += +0 +on ∂Ω. +Distribution of the process +←→ +Heat equation +∂tu − Lu = 0. +Optimal stopping +←→ +Obstacle problem +min{−Lu, u − ϕ} = 0. +Controlled diffusion +←→ +Fully nonlinear equation +F(D2u) = 0, +F convex. +Two-player games +←→ +Fully nonlinear equation +F(D2u) = 0. +One could even consider the equations with x-dependence, or with lower +order terms. All equations studied in Chapters 4 and 5 have a probabilistic +interpretation. + +— DRAFT — + +— DRAFT — +Appendix D +Motivations and +applications for the +obstacle problem +Here, we give a brief overview of the motivations and applications for the +obstacle problem listed in Chapter 5. We refer to the books [DL76, KS80, +Rod87, Fri88, PSU12] for more details, as well as for further applications +of obstacle-type problems. +Fluid filtration. Consider two reservoirs of water at different heights sep- +arated by a porous dam. For simplicity, we will assume a flat dam, with +rectangular cross section, which yields a problem in R2. Alternatively, one +could consider variable cross sections, which would yield an analogous ob- +stacle problem in R3 instead. +The dam is permeable to the water, except in the base. Thus, there is +some flow of fluid between the two reservoirs across the dam, and some wet +part of the cross section depending only on the relative distance to each of +the two water sources. +Let us assume one reservoir has water at height 1, and the other has +water at height 0 < h < 1. Let us denote by ϕ(x) the profile of the water +through the dam cross section. See Figure D.1 for a representation of the +situation. +Let us denote by u = u(x, y) : [0, 1] × [0, 1] → R+ the hydraulic piezo- +metric head of the fluid, given by the sum between the pressure p(x, y) and +211 + +— DRAFT — +212 +D. Motivations and applications for the obstacle problem +y +x +1 +1 +h +D +y = ϕ(x) +Figure D.1. Graphic representation of the cross section of a porous dam. +the elevation head (i.e., the potential energy of the fluid): +u(x, y) = y + 1 +γ p(x, y), +where γ is a constant depending on the fluid. The hydraulic head is defined +where there is fluid, namely, in +D := +� +(x, y) ∈ (0, 1) × (0, 1) : y < ϑ(x) +� +, +and is such that u(0, y) = 1 for 0 ≤ y ≤ 1, and u(1, y) = h for 0 ≤ y ≤ h +and u(1, y) = y for h ≤ y ≤ ϑ(1). +Here, u itself is an unknown, but D is also to be determined (and there- +fore, ϑ). In these circumstances we have that u(x, y) ≥ y in D, and if we +define +w(x, y) := +� ϕ(x) +y +� +u(x, ζ) − ζ +� +dζ +for +(x, y) ∈ D, +and w(x, y) ≡ 0 for (x, y) ∈ [0, 1] × [0, 1] \ D, then w fulfils the equation +∆w = χ{w>0} = χD +in +[0, 1] × [0, 1]. +That is, w is a solution to the obstacle problem (see (5.6)) with f ≡ 1. +We refer to [Bai74] and the references therein for more details about +the Dam problem. +Phase transitions. The Stefan problem, dating back to the 19th century, is +the most classical and important free boundary problem. It aims to describe +the temperature distribution in a homogeneous medium undergoing a phase +change, such as ice melting to water. +We denote by θ(x, t) the temperature (at position x and time t), and +assume θ ≥ 0. The function θ satisfies the heat equation ∂tθ − ∆θ = 0 in +the region {θ > 0}, while the evolution of the free boundary ∂{θ > 0} is + +— DRAFT — +D. Motivations and applications for the obstacle problem +213 +dictated by the Stefan condition ∂tθ = |∇xθ|2 on ∂{θ > 0} — where the +gradient is computed from inside {θ > 0}. +After the transformation u(x, t) := +� t +0 θ(x, τ)dτ (see [Duv73, Fig18]), +the problem is locally equivalent to +� +� +� +∂tu − ∆u += +−χ{u>0} +in +B1 × (0, T) ⊂ R3 × R +u +≥ +0 +∂tu +≥ +0. +This is the parabolic version of the obstacle problem ∆u = χ{u>0} in B1. +Hele-Shaw flow. This model, dating back to 1898, describes a fluid flow +between two flat parallel plates separated by a very thin gap. Various prob- +lems in fluid mechanics can be approximated to Hele-Shaw flows, and that +is why understanding these flows is important. +A Hele-Shaw cell is an experimental device in which a viscous fluid is +sandwiched in a narrow gap between two parallel plates. In certain regions, +the gap is filled with fluid while in others the gap is filled with air. When +liquid is injected inside the device through some sinks (e.g. through a small +hole on the top plate) the region filled with liquid grows. +We denote by p(x, t) the pressure of the fluid (at position x and time +t). By definition, {p > 0} is the region filled with liquid, while in {p = 0} +there is just air. The pressure p is harmonic in {p > 0}, and the evolution +of the free boundary ∂{p > 0} is dictated by ∂tp = |∇xp|2 on ∂{p > 0} — +where the gradient is computed from inside {p > 0}. Notice the striking +similarity to the Stefan problem — the only important difference here is +that p is harmonic (and not caloric) in the region where it is positive. +After the transformation u(x, t) = +� t +0 p(x, τ)dτ, it turns out that u solves +locally (i.e., outside the region where liquid is injected) +� +� +� +∆u += +χ{u>0} +in +B1 × (0, T) ⊂ R2 × R +u +≥ +0 +∂tu +≥ +0. +This means that, for each fixed time t, u(·, t) is a solution to the (stationary) +obstacle problem. +Optimal stopping, finance. As explained in Appendix C, the obstacle +problem appears when considering optimal stopping problems for stochastic +processes. +A typical example is the Black–Scholes model for pricing of American +options. An American option is a contract that entitles its owner to buy +some financial asset (typically a share of some company) at some specified +price (the “strike price”) at any time — often before some specified date. + +— DRAFT — +214 +D. Motivations and applications for the obstacle problem +This option has some value, since in case that the always fluctuating market +price of the asset goes higher than the strike price then the option can be +“exercised” to buy the asset at the lower price. The Black-Sholes model +aims to calculate the rational price u = u(x, t) of an option at any time t +prior to the maturity date and depending on the current price x of the +financial asset. Since the option can be exercised at any time, determining +the “exercise region” (i.e. the region in which it is better to exercise the +option) is a part of the problem. Interestingly, this problem leads to an +obstacle problem (often parabolic) posed in Rn, where the dimension n is +the number of assets. +We refer to [LS09] and the references therein for more details about +such kind of models. +Interacting particle systems. Large systems of interacting particles arise +in several models in the natural sciences (one can think of physical particles +in Physics or Biology, for example). In such systems the discrete energy can +be well approximated by the continuum interacting energy. We denote µ +the (probability) measure representing the particle density. +In several models the particles attract each other when they are far, +but experience a repulsive force when they are close [CDM16]. Then, the +interaction energy E associated to the interaction potential W ∈ L1 +loc(R3), +is given by +E[µ] := 1 +2 +� +R3 +� +R3 W(x − y)dµ(x) dµ(y). +In general, the interaction potential can have very different structures. It +is common to assume a repulsive behaviour for particles that are very close +(blowing up at zero distance), and attractive behaviour when they are far. +A typical assumption is to have W(z) ∼ |z|−1 near the origin. +In other models in statistical mechanics, the particles (e.g. electrons) +repel with a Coulomb force and one wants to understand their behaviour +in presence of some external field that confines them [Ser18]. In that case, +the interaction energy associated with the system is given by +E[µ] := 1 +2 +� +R3 +� +R3 +dµ(x) dµ(y) +|x − y| ++ +� +R3 V dµ. +One of the main questions when dealing with these systems is to under- +stand the “equilibrium configurations”, that is, minimizers of the energy E. +It turns out that, in both cases, any minimizer µ◦ is given by µ◦ = −∆u, +with u satisfying (locally) the obstacle problem +min{−∆u, u − ϕ} = 0, + +— DRAFT — +D. Motivations and applications for the obstacle problem +215 +for some obstacle ϕ that depends on W (or on V ). +The free boundary +corresponds to the boundary of the region in which the particles concentrate. +We refer to [CDM16, Ser18] and the references therein for a thorough +study of these problems. +Quasi-Steady Electrochemical Shaping. Electrochemical Machining (ECM) +is an electrochemical method to remove metals (electroconductive) by plac- +ing the material inside an electrolytic call as an anode, surrounded by a +fixed cathode. Then an electric potential is applied between a cathode and +an anode, which is submerged in an appropriate electrolyte, thus producing +a chemical reaction that removes the metal from the anode and gives rise to +a moving boundary. This method is used to shape extremely hard materials, +to produce complicated shapes which are otherwise very difficult to obtain. +Let us suppose we have cylindrical symmetry (that is, both anode and +cathode are long cylindrical materials), so that we can work with the cross +section and thus in two dimensions. A similar approach works in the three- +dimensional case. +Let Ω ⊂ R2 denote the domain enclosed by the cathode, and Λ(0) ⊂ Ω +denote the anode at time t = 0 (an electric potential is applied between ∂Ω +and ∂Λ(0), where the region Ω \ Λ(0) contains the electrolyte). Then, the +metal starts to be removed, so that after a time t ≥ 0, we denote by Λ(t) +the set defining the anode. By this process we have that Λ(t) ⊂ Λ(t′) if +t ≥ t′. The boundary Γ(t) = ∂Λ(t) is unknown, it is a free boundary, which +we assume is represented by a function γ : Ω → R as +Γ(t) = {(x, y) ∈ Ω : γ(x, y) = t}, +for some function γ to be determined. +We assume that γ(x, y) = 0 in +Ω \ Λ(0). If we denote by π = π(t) > 0 the potential difference at time +t > 0 between anode and cathode, then the ECM problem is concerned with +finding a function η(t, x, y) that solves +∆η(t, x, y) = 0 +in +Ω \ Γ(t), +η(t, x, y) = 0 +on +{t > 0} × ∂Ω, +η(t, x, y) = π(t), +∇η(t, x, y) · ∇γ(x, y) = λ +on +{t > 0} × Γ(t) +(with the convention that the gradient and the Laplacian are only taken in +the spatial variables), for some constant λ > 0 (the ECM constant). Notice +that 0 ≤ η(t, x, y) ≤ π(t) in Λ(t) by the maximum principle, and let us +extend η to Ω as η(t, x, y) = π(t) in Λ(t). Now, if we define +u(t, x, y) = +� t +0 +(π(s) − η(s, x, y)) ds, +then u ≥ 0 and in Λ(0), u fulfils +∆u(t, ·, ·) = λχ{u(t,·,·)>0} +for any +t > 0. + +— DRAFT — +216 +D. Motivations and applications for the obstacle problem +That is, u fulfils an obstacle problem (compare with (5.6)) with f ≡ λ, for +each time t > 0. We refer to [Rod87] for more details. +Heat control. Given a domain Ω and a temperature T◦, we have heating +devices evenly distributed on Ω that need to ensure that the temperature +u(x), x ∈ Ω, is as close as possible to T◦, by injecting flux proportional to +the distance between u(x) and T◦. Due to the limited power of the devices, +the heat flux generated by them needs to remain in the interval (−q, 0] for +q ≥ 0. +Thus, the heat flux injected is +Φ(u) = max{C(u − T◦)−, −q} +for some constant C > 0. In equilibrium, the temperature satisfies +∆u = Φ(u) +in +Ω, +In particular, letting C → ∞, the previous equation becomes +∆u = −qχ{u0} +in +Ω, +(see the parallelism to (5.6) with f ≡ q > 0). If w ≥ 0 (that is, u ≤ T◦) +then this is exactly the obstacle problem. This can be obtained by putting +Dirichlet boundary conditions on ∂Ω that are u|∂Ω ≤ T◦ (for example, in a +room with lateral walls without thermal insulation). We refer to [DL76] for +more details. +Elasticity. We finish with probably the most intuitive physical interpre- +tation of the obstacle problem: the deformation of a thin membrane in +elasticity theory. +Let us consider an elastic membrane represented by a function in R2, +u : R2 → R, so that u(x, y) represents the vertical displacement with respect +to the xy-plane. Given a domain Ω ⊂ R2, we suppose that the membrane +has a fixed boundary, that is, we prescribe the value of u on ∂Ω, by some (say +continuous) function g : ∂Ω → R. We assume an homogeneous membrane +equally stretched in all directions, whose shape is determined by the surface +tension. For simplicity we also assume lack of external forces. +In this setting, the shape of the membrane will be such that the total +area is minimized, among all possible configurations with the same boundary + +— DRAFT — +D. Motivations and applications for the obstacle problem +217 +values. Namely, the following functional +� +Ω +� +1 + |∇w|2 dx dy +is minimized among functions w ∈ H1(Ω) such that w|∂Ω = g. This yields +the classical Plateau’s problem. The Dirichlet energy appears as a lower +order approximation of the previous functional. Namely, if we assume that +the vertical displacements are not large (say, the membrane is rather flat), +then a Taylor expansion of the functional yields +� +Ω +� +1 + |∇w|2 dx dy ∼ +� +Ω +� +1 + 1 +2|∇w|2 +� +dx dy, +so that the minimization of the area is roughly a minimization of the Dirichlet +energy. +The obstacle problem is concerned with finding the membrane that min- +imizes the Dirichlet energy (thus, approximately the area) among those with +prescribed boundary, that lie above a given obstacle ϕ : R2 → R. + +— DRAFT — + +— DRAFT — +Notation +Let us introduce some of the notation be used throughout the book. +Matrix notation. +A = (aij)ij +Matrix with (i, j) − th entry denoted by aij. +Mn +Space of matrices of size n × n. +Id +Identity matrix. +tr A +Trace of the matrix A, tr A = a11 + · · · + ann. +det A +Determinant of the matrix A. +AT +Transpose of the matrix A. +Geometric notation. +Rn, Sn +n-dimensional Euclidean space, n-sphere. +ei ∈ Sn−1 +i − th element of the base, ei = (0, . . . , 0, +(i) +1 , 0, . . . 0). +x ∈ Rn +Typical point x = (x1, . . . , xn). +|x| +Modulus of the point x, |x| = +� +x2 +1 + · · · + x2n. +|U| +n-dimensional Lebesgue measure of a set U ⊂ Rn. +Rn ++ +{x = (x1, . . . , xn) ∈ Rn : xn > 0}. +∂U +Boundary of the set U ⊂ Rn. +V ⊂⊂ U +The set V is compactly contained in U, that is V ⊂ U. +Br(x) +Ball of radius r centered at x, Br(x) := {y ∈ Rn : |x − y| < r}. +x · y +For x, y ∈ Rn, scalar product of x and y, x · y = x1y1 + · · · + xnyn. +219 + +— DRAFT — +220 +Notation +Functional notation. +u +In general, u denotes a function u : Rn → R (unless stated other- +wise). +u+, u− +Positive and negative part of a function, u+ = max{u, 0}, u− = +max{−u, 0}. +χE +Characteristic function of the set E, χE(x) = 1 for x ∈ E, and +χE(x) = 0 for x /∈ E. +supp u +Support of u, supp u = {x : u(x) ̸= 0}. +� +A +Average integral over the positive measure set A, +� +A f := +1 +|A| +� +A f. +Function spaces. Let U ⊂ Rn be an open set. +C(U), C0(U) +Space of continuous functions u : U → R. +C(U), C0(U) +Functions u ∈ C(U) continuous up to the boundary. +Ck(U), Ck(U) +Space of functions k times continuously differentiable. +Ck,α(U) +H¨older spaces, see Section 1.1. +C∞(U), C∞(U) +Set of functions in Ck(U) or Ck(U) for all k ≥ 1. +Cc(U), Ck +c (U) +Set of functions with compact support in U. +C0(U), Ck +0 (U) +Set of functions with u = 0 on ∂U. +Lp +Lp space, see Section 1.1. +L∞ +L∞ space, see Section 1.1 (see esssupΩu below). +esssupΩu +Essential supremum of u in Ω: infimum of the essential +upper bounds, esssupΩu := inf{b > 0 : |{u > b}| = 0}. +W 1,p, W 1,p +0 +Sobolev spaces, see Section 1.1 and (S7). +H1, H1 +0 +Sobolev spaces with p = 2, see Section 1.1 and (S7). +∥ · ∥F +Norm in the functional space F ∈ {C0, Ck, Lp, . . . }, de- +fined when used for the first times. + +— DRAFT — +Notation +221 +Differential notation. Let u : U → R be a function. +∂iu, ∂xiu, uxi +Partial derivative in the ei direction, ∂u +∂xi . +∂eu +Derivative in the e ∈ Sn−1 direction. +∇u, Du +Gradient, ∇u = (∂1u, . . . , ∂nu). +∂iju, ∂xixju, uxixj +Second partial derivatives in the directions ei and ej, +∂2u +∂xi∂xj . +D2u +Hessian, D2u = (∂iju)ij ∈ Mn. +Dku +Higher derivatives forms, Dku := (∂i1 . . . ∂iku)i1,...,ik. +|Dku(x)| +Norm of Dku(x) (any equivalent norm). +∥Dku(x)∥F +Norm of Dku, ∥|Dku|∥F. +∆u +Laplacian of u, ∆u = ∂11u + · · · + ∂nnu. +Domains. We say that Ω ⊂ Rn is a domain if it is an open connected set. +A domain Ω is said to be Ck,α (resp. Ck) if ∂Ω can be written locally as +the graph of a Ck,α (resp. Ck) function. + +— DRAFT — + +— DRAFT — +Bibliography +[AH96] D. Adams, +L. Hedberg, +Function Spaces and Potential Theory, +Springer, +Grundlehren der mathematischen Wissenschaften, 1996. +[AS19] M. Allen, H. Shahgholian, A new boundary Harnack principle (equations with right +hand side), Arch. Rat. Mech. Anal. 234 (2019), 1413-1444. +[ACM18] L. Ambrosio, A. Carlotto, A. Massaccesi, Lectures on Elliptic Partial Differential +Equations, Lecture notes Scuola Normale Superiore di Pisa 18, Springer, 2018. +[And97] P. Andersson, Characterization of pointwise H¨older regularity, Appl. Comp. Har- +mon. Anal. 4 (1997), 429-443. +[ASS12] S. N. Armstrong, L. Silvestre, C. K. Smart, Partial regularity of solutions of fully +nonlinear uniformly elliptic equations, Comm. Pure Appl. Math. 65 (2012), 1169-1184. +[Bai74] C. Baiocchi, Free boundary problems in the theory of fluid flow through porous +media, in Proceedings of the ICM 1974. +[Bon01] L. P. Bonorino, Regularity of the free boundary for some elliptic and parabolic +problems. I, Comm. Partial Differential Equations 26 (2001), 175-203. +[Bre11] H. Brezis, Functional Analysis, Sobolev spaces and Partial Differential Equations, +Springer, 2011. +[Caf77] L. 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Press, Somerville, MA, +2016. +[Vel23] B. Velichkov, Regularity of the One-Phase Free Boundaries, Lect. Notes Unione +Mat. Ital., Springer, Cham, 2023. +[Wan06] X.-J. Wang, Schauder estimates for elliptic and parabolic equations, Chinese Ann. +Math. Ser. B 27 (2006), 637-642. +[Wei99] G. S. Weiss, A homogeneity improvement approach to the obstacle problem, Invent. +Math. 138 (1999), 23-50. + +— DRAFT — + +— DRAFT — +Index +Arzel`a-Ascoli Theorem, 9 +Average integral, 3 +Bellman equation, 100, 206 +Boundary Harnack, 168, 191 +Higher order regularity, 170 +Boundary regularity Laplace equation, +69 +Brownian motion, 21 +Classification of blow-ups, 150, 158 +Comparison principle +Fully nonlinear equations (classical), +98 +Fully nonlinear equations (viscosity), +107 +Laplace equation, 14 +Controlled diffusion, 205 +Convex fully nonlinear equations, 100 +Covering argument, 36 +De Giorgi–Nash, 74, 81 +Dirichlet problem +Fully nonlinear equations, 113 +Laplace equation, 9 +Divergence-form PDE, 25 +Elasticity, 216 +Ellipticity condition +Fully nonlinear equations, 98 +Ellipticity constants +Fully nonlinear equations, 99 +Linear equations, 25 +Energy inequality, 84 +Energy method, 10 +Equation in divergence form with +bounded measurable coefficients, +74, 81 +Equation in non-divergence form with +bounded measurable coefficients, +102 +Equations in two variables (fully +nonlinear), 102, 120 +Euler–Lagrange for obstacle problem, +126 +Evans–Krylov Theorem, 120 +Existence and uniqueness +Laplace equation, 12 +Minimizers convex functional, 75 +Obstacle problem, 131, 141 +Viscosity solutions, 110 +Expected exit time, 204 +Expected hitting time, 23 +Expected payoff, 22, 202 +Extremal operators, 101 +Fluid filtration, 211 +Free boundary, 127 +Fully nonlinear equation, 97 +Fundamental solution, 13 +Generic regularity, 177 +H¨older norm, 6 +H¨older semi-norm, 6 +H¨older space, 6 +Harnack inequality, 26 +Harnack’s inequality, 31 +229 + +— DRAFT — +230 +Index +Heat control, 216 +Hele-Shaw flow, 213 +Higher order Schauder estimates +Divergence form, 59 +Laplacian, 38 +Non-divergence form, 46 +Hilbert XIXth problem, 71, 94 +Homogeneity of blow-ups, 151 +Hopf Lemma, 16 +Infinitesimal generator, 202 +Integration by parts, 4 +Interacting particle systems, 214 +Interpolation inequalities, 9 +Isaacs equation, 206 +Krylov–Safonov Theorem, 119 +Least supersolution, 134 +Lebesgue differentiation theorem, 3 +Liouville Theorem, 18, 30 +Maximum principle +Divergence form, 60 +Laplace equation, 14 +Non-divergence form, 48 +Mean value property, 16 +Monneau monotonicity formula, 173 +Morrey inequality, 6 +Nadirashvili–Vladuts counterexamples, +121 +Non-divergence-form PDE, 25 +Nondegeneracy, Obstacle problem, 139, +146 +Obstacle problem, 125 +Optimal regularity, Obstacle problem, +136, 144 +Optimal stopping, 204, 213 +Oscillation decay, 28, 31, 89 +Perron’s method for viscosity solutions, +108 +Phase transitions, 212 +Poincar´e inequality, 6 +Poisson kernel, 13 +Probabilistic interpretation of PDEs, +209 +Pucci operators, 101 +Quasi-Steady Electrochemical Shaping, +215 +Regular points, 148 +Regularity of the free boundary, 169 +Higher regularity, 170 +Schaeffer conjecture, 178 +Schauder estimates +Divergence form, 59 +Laplacian, 36 +Non-divergence form, 46 +Schauder estimates for continuous +coefficients +Divergence form, 65 +Non-divergence form, 64 +Singular points, 148 +Size, 175 +Sobolev inequality, 5 +Sobolev space, 4 +Stability of viscosity solutions, 114 +Subharmonic function, 14 +Superharmonic function, 14 +Two-players game, 206 +Uniform ellipticity condition, 25 +Fully nonlinear equations, 99, 100 +Linear equations, 47 +Uniqueness of blow-ups at singular +points, 173 +Viscosity solution, 19, 106 +Viscosity subsolution, 106 +Viscosity supersolution, 106 +Weak solution, 11 +Weiss monotonicity formula, 152 + diff --git a/XdAzT4oBgHgl3EQfmf14/content/tmp_files/load_file.txt b/XdAzT4oBgHgl3EQfmf14/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..76bb61e6be073183b64997d5a86f82dd1ff12a66 --- /dev/null +++ b/XdAzT4oBgHgl3EQfmf14/content/tmp_files/load_file.txt @@ -0,0 +1,5803 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf,len=5802 +page_content='— DRAFT — Regularity Theory for Elliptic PDE Xavier Fern´andez-Real Xavier Ros-Oton EPFL SB MATH, Institute of Mathematics, Station 8, CH- 1015 Lausanne, Switzerland E-mail address: xavier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='fernandez-real@epfl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='ch Universit¨at Z¨urich, Institut f¨ur Mathematik, Winterthur- erstrasse 190, 8057 Z¨urich, Switzerland, & ICREA, Pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Llu´ıs Companys 23, 08010 Barcelona, Spain, & Universitat de Barcelona, Departament de Matem`atiques i Inform`atica, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain, & Centre de Recerca Matem`atica, Edifici C, Campus Bellaterra, 08193 Bellaterra, Spain E-mail address: xros@icrea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='cat arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='01564v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='AP] 4 Jan 2023 — DRAFT — 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 35J15, 35B65, 35J05, 35J20, 35J60, 35R35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Elliptic PDE, Schauder estimates, Hilbert XIXth problem, nonlinear elliptic equations, obstacle problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — Contents Preface vii Chapter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Overview and Preliminaries 1 §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Preliminaries: Sobolev and H¨older spaces 3 §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A review on the Laplace equation 9 §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Probabilistic interpretation of harmonic functions 21 Chapter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Linear elliptic PDE 25 §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Harnack’s inequality 26 §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Schauder estimates for the Laplacian 33 §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Schauder estimates for operators in non-divergence form 46 §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Schauder estimates for operators in divergence form 59 §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The case of continuous coefficients 64 §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Boundary regularity 68 Chapter 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Nonlinear variational PDE & Hilbert’s XIXth problem 71 §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Overview 72 §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Existence and basic estimates 75 §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' De Giorgi’s proof 81 §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Solution to Hilbert’s XIXth problem 94 §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Further results and open problems 95 Chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Fully nonlinear elliptic PDE 97 §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' What is ellipticity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 98 §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Equations in two variables 102 v — DRAFT — vi Contents §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Existence of solutions 105 §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Regularity of solutions: an overview 115 §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Further results and open problems 121 Chapter 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem 125 §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Some motivations and applications 128 §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Basic properties of solutions I 130 §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Basic properties of solutions II 141 §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Regularity of free boundaries: an overview 147 §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Classification of blow-ups 150 §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Regularity of the free boundary 161 §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Singular points 171 §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the size of the singular set 175 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Some properties of H¨older spaces 179 Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of the boundary Harnack inequality 191 Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Probabilistic interpretation of fully nonlinear equations 201 Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Motivations and applications for the obstacle problem 211 Notation 219 Bibliography 223 Index 229 — DRAFT — Preface One of the most basic and important questions in PDE is that of regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It is also a unifying problem in the field, since it affects all kinds of PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A classical example is Hilbert’s XIXth problem (1900), which roughly speak- ing asked to determine whether all solutions to uniformly elliptic variational PDEs are smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The question was answered positively by De Giorgi and Nash in 1956 and 1957, and it is now one of the most famous and important theorems in the whole field of PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The question of regularity has been a central line of research in elliptic PDE since the mid-20th century, with extremely important contributions by Nirenberg, Caffarelli, Krylov, Evans, Figalli, and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Their works have enormously influenced many areas of Mathematics linked one way or another with PDE, including: Harmonic Analysis, Calculus of Variations, Differential Geometry, Geometric Measure Theory, Continuum and Fluid Mechanics, Probability Theory, Mathematical Physics, and Computational and Applied Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This text emerged from two PhD courses on elliptic PDE given by the second author at the University of Z¨urich in 2017 and 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It aims to pro- vide a self-contained introduction to the regularity theory for elliptic PDE, focusing on the main ideas rather than proving all results in their greatest generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The book can be seen as a bridge between an elementary PDE course and more advanced textbooks such as [GT77] or [CC95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, we believe that the present selection of results and techniques complements nicely other books on elliptic PDE such as [Eva98], [HL97], and [Kry96], as well as the recent book [ACM18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For example, we give a different proof of the Schauder estimates (due to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Simon) which is not contained in other textbooks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' we prove some basic results for fully nonlinear equations that vii — DRAFT — viii Preface are not covered in [CC95];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' and we also include a detailed study of the ob- stacle problem, often left to more specialized textbooks such as [Fri88] or [PSU12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Furthermore, at the end of Chapters 3, 4, and 5 we provide a review of some recent results and open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We would like to thank Alessio Figalli, Thomas Kappeler, Alexis Michelat, Joaquim Serra, and Wei Wang, for several comments and suggestions on this book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, we acknowledge the support received from the following funding agencies: X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' was supported by the European Research Council under the Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 721675 “Regularity and Stability in Partial Differen- tial Equations (RSPDE)”, by the Swiss National Science Foundation (SNF grants 200021 182565 and PZ00P2 208930), and by the Swiss State Secre- tariat for Education, Research and lnnovation (SERI) under contract num- ber M822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='00034;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' was supported by the European Research Council un- der the Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 801867 “Regularity and singularities in elliptic PDE (EllipticPDE)”, by the Swiss National Science Foundation (SNF grant 200021 178795), by AEI project PID2021-125021NA-I00 (Spain), by the grant RED2018-102650-T funded by MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13039/501100011033, and by the Spanish State Research Agency through the Mar´ıa de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Z¨urich, 2020 — DRAFT — Chapter 1 Overview and Preliminaries A beautiful result in Complex Analysis states that because the real part u(x, y) of any holomorphic function satisfies uxx + uyy = 0, it must be real analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, the oscillation of u in any given domain controls all the derivatives in any (compactly contained) subdomain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In higher dimensions, the same phenomenon occurs for solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) ∆u = 0 in Ω ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' These are harmonic functions, and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) is the simplest elliptic partial dif- ferential equation (PDE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Any solution to this equation is smooth (real an- alytic), and satisfies ∥u∥Ck(Q) ≤ Ck,Q∥u∥L∞(Ω) for all k = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' for any compact subdomain Q ⊂⊂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, all derivatives are controlled by the supremum of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Here, and throughout the book, Ω is any bounded domain of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Regularity for Laplace’s equation: ∆u = 0 in Ω ⊂ Rn =⇒ u is C∞ inside Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This kind of regularization property is common in elliptic PDEs and is the topic of the present book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 1 — DRAFT — 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Overview and Preliminaries One can give three different kinds of explanations for this phenomenon: (a) Integral representation of solutions: Poisson kernels, fundamental solutions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (b) Energy considerations: Harmonic functions are local minimizers of the Dirichlet energy E(u) := � Ω |∇u|2 dx (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', if we change u to w in ˜Ω ⊂ Ω, then E(w) ≥ E(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (c) Comparison principle: A harmonic function cannot have any inte- rior maximum point (maximum principle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' These three approaches are extremely useful in different contexts, as well as in the development of the regularity theory for nonlinear elliptic PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The structure of the book is as follows: ⋆ First, in Chapter 2 we will study linear elliptic PDEs n � i,j=1 aij(x)∂iju = f(x) in Ω ⊂ Rn and n � i,j=1 ∂i � aij(x)∂ju � = f(x) in Ω ⊂ Rn, where the coefficients aij(x) and the right-hand side f(x) satisfy appropriate regularity assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In the simplest case, (aij)i,j ≡ Id, we have ∆u = f(x) in Ω ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The type of result we want to prove is: “u is two derivatives more regular than f”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' ⋆ Then, in Chapter 3 we will turn our attention to nonlinear variational PDEs: minimizers of E(u) := � Ω L(∇u)dx, L smooth and uniformly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The regularity for such kind of nonlinear PDEs was Hilbert’s XIXth problem (1900).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' ⋆ In Chapter 4 we will study nonlinear elliptic PDEs in their most general form F(D2u, ∇u, u, x) = 0 in Ω ⊂ Rn, — DRAFT — 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Preliminaries: Sobolev and H¨older spaces 3 or simply F(D2u) = 0 in Ω ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' These are called fully nonlinear elliptic equations, and in general they do not have a variational formulation in terms of an energy functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' ⋆ In Chapter 5 we will study the obstacle problem, a constrained min- imization problem: minimize � Ω |∇u|2dx, among functions u ≥ ϕ in Ω, where ϕ is a given smooth “obstacle”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is the simplest and most impor- tant elliptic free boundary problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, it can be seen as a nonlinear PDE of the type min{−∆u, u − ϕ} = 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As we will see, in each of these contexts we will use mainly: (b) energy considerations, or (c) maximum principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' At the end of the book, we have also included four appendices to com- plement the theory from the main chapters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Preliminaries: Sobolev and H¨older spaces We next give a quick review on Lp, Sobolev, and H¨older spaces, stating the results that will be used later in the book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lp spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Given Ω ⊂ Rn and 1 ≤ p < ∞, the space Lp(Ω) is the set Lp(Ω) := � u measurable in Ω : � Ω |u|pdx < ∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It is a Banach space, with the norm ∥u∥Lp(Ω) := ( � Ω |u|p)1/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' When p = ∞, the space L∞(Ω) is the set of bounded functions (up to sets of measure zero), with the norm ∥u∥L∞(Ω) := esssupΩ|u|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A well-known result in this setting is the Lebesgue differentiation theo- rem (see, for example, [EG92]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If u ∈ L1(Ω), then for almost every x ∈ Ω we have lim r→0 � Br(x) ��u(x) − u(y) ��dy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' When this holds at a point x ∈ Ω, we say that x is a Lebesgue point of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Here, and throughout the book, � A denotes the average 1 |A| � A, where A ⊂ Rn is any set of finite and positive measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A useful consequence of this result is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Overview and Preliminaries Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume u ∈ L1(Ω), and � Ω uv dx = 0 for all v ∈ C∞ c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u = 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A fundamental identity in the study of PDEs is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3 (Integration by parts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume Ω ⊂ Rn is any bounded C1 domain1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, for any u, v ∈ C1(Ω) we have (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2) � Ω ∂iu v dx = − � Ω u ∂iv dx + � ∂Ω uv νi dS, where ν is the unit (outward) normal vector to ∂Ω, and i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that, as an immediate consequence, we find the divergence theo- rem, as well as Green’s first identity � Ω ∇u · ∇v dx = − � Ω u ∆v dx + � ∂Ω u ∂v ∂ν dS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The regularity requirements of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3 can be relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For instance, the domain Ω need only be Lipschitz, while only u, v ∈ H1(Ω) is necessary in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2) — where H1 is a Sobolev space, defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Given any domain Ω ⊂ Rn and 1 ≤ p ≤ ∞, the Sobolev spaces W 1,p(Ω) consist of all functions whose (weak) derivatives are in Lp(Ω), namely W 1,p(Ω) := {u ∈ Lp(Ω) : ∂iu ∈ Lp(Ω) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', n} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer to the excellent books [Eva98, Bre11] for the definition of weak derivatives and a detailed exposition on Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A few useful properties of Sobolev spaces are the following (see [Eva98]): (S1) The spaces W 1,p(Ω) are complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (S2) The inclusion W 1,p(Ω) ⊂ Lp(Ω) is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (S3) The space H1(Ω) := W 1,2(Ω) is a Hilbert space with the scalar product (u, v)H1(Ω) = � Ω uv + � Ω ∇u · ∇v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (S4) Any bounded sequence {uk} in the Hilbert space H1(Ω) contains a weakly convergent subsequence {ukj}, that is, there exists u ∈ H1(Ω) such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3) (ukj, v)H1(Ω) → (u, v)H1(Ω) for all v ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 1We refer to the Notation section (page 221) for the definition of C1 domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Preliminaries: Sobolev and H¨older spaces 5 In addition, such u will satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4) ∥u∥H1(Ω) ≤ lim inf j→∞ ∥ukj∥H1(Ω), and since H1(Ω) is compactly embedded in L2(Ω) one has (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5) ∥u∥L2(Ω) = lim j→∞ ∥ukj∥L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (S5) Let Ω be any bounded Lipschitz domain, and 1 ≤ p ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, there is a continuous (and compact for p > 1) trace operator from W 1,p(Ω) to Lp(∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For C0 functions, such trace operator is simply u �→ u|∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Because of this, for any function u ∈ H1(Ω) we will still denote by u|∂Ω its trace on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (S6) For 1 ≤ p < ∞, C∞(Ω) functions are dense in W 1,p(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, if Ω is bounded and Lipschitz, C∞(Ω) functions are dense in W 1,p(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (S7) For 1 ≤ p < ∞, we define the space W 1,p 0 (Ω) as the closure of C∞ c (Ω) in W 1,p(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Similarly, we denote H1 0(Ω) := W 1,2 0 (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' When Ω is bounded and Lipschitz, it is the space of functions u ∈ W 1,p(Ω) such that u|∂Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (S8) If u ∈ W 1,p(Ω), 1 ≤ p ≤ ∞, then for any subdomain K ⊂⊂ Ω we have ���� u(x + h) − u(x) |h| ���� Lp(K) ≤ C ∥∇u∥Lp(Ω) for all h ∈ Bδ, with δ > 0 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Conversely, if u ∈ Lp(Ω), 1 < p ≤ ∞, and ���� u(x + h) − u(x) |h| ���� Lp(K) ≤ C for every h ∈ Bδ, then u ∈ W 1,p(K) and ∥∇u∥Lp(Ω) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (However, this property fails when p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') (S9) Given any function u, define u+ = max{u, 0} and u− = max{−u, 0}, so that u = u+ − u−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, for any u ∈ W 1,p(Ω) we have u+, u− ∈ W 1,p(Ω), and ∇u = ∇u+ − ∇u− a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, the gradient of Sobolev functions vanishes almost everywhere on level sets, ∇u(x) = 0 for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' x ∈ {u = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' An important inequality in this context is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4 (Sobolev inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If p < n, then �� Rn |u|p∗dx �1/p∗ ≤ C �� Rn |∇u|pdx �1/p , 1 p∗ = 1 p − 1 n, for some constant C depending only on n and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, we have a continuous inclusion W 1,p(Rn) ⊂ Lp∗(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Overview and Preliminaries Notice that, as p ↑ n we have p∗ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In the limiting case p = n, however, it is not true that W 1,n functions are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This can be seen by taking, for example, u(x) = log log � 1 + 1 |x| � ∈ W 1,n(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Still, in case p > n, the following occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5 (Morrey inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If p > n, then sup x̸=y ��u(x) − u(y) �� |x − y|α ≤ C �� Rn |∇u|pdx �1/p , α = 1 − n p , for some constant C depending only on n and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, when p > n any function in W 1,p is continuous (after possibly being redefined on a set of measure 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, we will also use the following inequalities in bounded domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6 (Poincar´e inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ Rn be any bounded Lipschitz domain, and let p ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, for any u ∈ W 1,p(Ω) we have � Ω |u − uΩ|pdx ≤ CΩ,p � Ω |∇u|pdx, where uΩ := � Ω u, and � Ω |u|pdx ≤ C′ Ω,p �� Ω |∇u|pdx + � ∂Ω ��u|∂Ω ��pdσ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The constants CΩ,p and C′ Ω,p depend only on n, p, and Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' H¨older spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Given α ∈ (0, 1), the H¨older space C0,α(Ω) is the set of continuous functions u ∈ C(Ω) such that the H¨older semi-norm is finite, [u]C0,α(Ω) := sup x,y∈Ω x̸=y ��u(x) − u(y) �� |x − y|α < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The H¨older norm is ∥u∥C0,α(Ω) := ∥u∥L∞(Ω) + [u]C0,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' When α = 1, this is the usual space of Lipschitz continuous functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' More generally, given k ∈ N and α ∈ (0, 1), the space Ck,α(Ω) is the set of functions u ∈ Ck(Ω) such that the following norm is finite ∥u∥Ck,α(Ω) = ∥u∥Ck(Ω) + [Dku]C0,α(Ω), where ∥u∥Ck(Ω) := k � j=1 ∥Dju∥L∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Preliminaries: Sobolev and H¨older spaces 7 Notice that this yields the inclusions C0 ⊃ C0,α ⊃ Lip ⊃ C1 ⊃ C1,α ⊃ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' ⊃ C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will often write ∥u∥Ck,α(Ω) instead of ∥u∥Ck,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, it is sometimes convenient to use the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' When β > 0 is not an integer, we define Cβ(Ω) := Ck,α(Ω), where β = k + α, k ∈ N, α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' There are many properties or alternative definitions of H¨older spaces that will be used throughout the book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' They are valid for all α ∈ (0, 1), and are proved in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (H1) Assume oscBr(x)u ≤ C◦rα for all Br(x) ⊂ B1, where oscAu := supA u − infA u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ C0,α(B1) and [u]C0,α(B1) ≤ CC◦, with C depending only on n, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (H2) Let ux,r := � Br(x) u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume ∥u − ux,r∥L∞(Br(x)) ≤ C◦rα for all Br(x) ⊂ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ C0,α(B1) and [u]C0,α(B1) ≤ CC◦, with C depending only on n, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (H3) Let ux,r := � Br(x) u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume � � Br(x) |u − ux,r|2 �1/2 ≤ C◦rα for all Br(x) ⊂ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ C0,α(B1) and [u]C0,α(B1) ≤ CC◦, with C depending only on n, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (H4) Assume that for every x there is a constant Cx such that ∥u − Cx∥L∞(Br(x)) ≤ C◦rα for all Br(x) ⊂ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ C0,α(B1) and [u]C0,α(B1) ≤ CC◦, with C depending only on n, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that for every x there is a linear function ℓx(y) = ax + bx · (y − x) such that ∥u − ℓx∥L∞(Br(x)) ≤ C◦r1+α for all Br(x) ⊂ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ C1,α(B1) and [Du]C0,α(B1) ≤ CC◦, with C depending only on n, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Overview and Preliminaries Assume that for every x there is a quadratic polynomial Px(y) such that ∥u − Px∥L∞(Br(x)) ≤ C◦r2+α for all Br(x) ⊂ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ C2,α(B1) and [D2u]C0,α(B1) ≤ CC◦, with C depending only on n, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (H5) Let ρ◦ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that, for every x ∈ B1/2, there exists a sequence of quadratic polynomials, (Pk)k∈N, such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6) ∥u − Pk∥L∞(Bρk◦ (x)) ≤ C◦ρk(2+α) for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ C2,α(B1/2) and [D2u]C0,α(B1/2) ≤ CC◦, with C depending only on n, α, and ρ◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (H6) Assume that α ∈ (0, 1), ∥u∥L∞(B1) ≤ C◦, and sup x∈B1 x±h∈B1 ��u(x + h) + u(x − h) − 2u(x) �� |h|α ≤ C◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ C0,α(B1) and ∥u∥C0,α(B1) ≤ CC◦, with C depending only on n, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that α ∈ (0, 1), ∥u∥L∞(B1) ≤ C◦, and sup x∈B1 x±h∈B1 ��u(x + h) + u(x − h) − 2u(x) �� |h|1+α ≤ C◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ C1,α(B1) and ∥u∥C1,α(B1) ≤ CC◦, with C depending only on n, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' However, such property fails when α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (H7) Assume that α ∈ (0, 1], ∥u∥L∞(B1) ≤ C◦, and that for every h ∈ B1 we have (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7) ���� u(x + h) − u(x) |h|α ���� Cβ(B1−|h|) ≤ C◦, with C◦ independent of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume in addition that α + β is not an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ Cα+β(B1) and ∥u∥Cα+β(B1) ≤ CC◦, with C depending only on n, α, β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' However, such property fails when α + β is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (H8) Assume that ui → u0 uniformly in Ω ⊂ Rn, and that ∥ui∥Ck,α(Ω) ≤ C◦, with α ∈ (0, 1] and for some C◦ independent of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, we have that u0 ∈ Ck,α(Ω), and ∥u0∥Ck,α(Ω) ≤ C◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A review on the Laplace equation 9 Finally, an important result in this context is the following particular case of the Arzel`a–Ascoli theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7 (Arzel`a–Ascoli).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ Rn, α ∈ (0, 1), and let {fi}i∈N be any sequence of functions fi satisfying ∥fi∥C0,α(Ω) ≤ C◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, there exists a subsequence fij which converges uniformly to a func- tion f ∈ C0,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' More generally, this result — combined with (H8) — implies that if ∥ui∥Ck,α(Ω) ≤ C◦, with α ∈ (0, 1), then a subsequence uij will converge in the Ck(Ω) norm to a function u ∈ Ck,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Interpolation inequalities in H¨older spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A useful tool that will be used throughout the book is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For each 0 ≤ γ < α < β ≤ 1 and every ε > 0, we have (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8) ∥u∥C0,α(Ω) ≤ Cε∥u∥C0,γ(Ω) + ε∥u∥C0,β(Ω), where C is a constant depending only on n and ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (When γ = 0, C0,γ should be replaced by L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') This follows from the interpolation inequality ∥u∥C0,α(Ω) ≤ ∥u∥t C0,γ(Ω)∥u∥1−t C0,β(Ω) t = β − α β − γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' More generally, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8) holds for higher-order H¨older norms too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In par- ticular, we will use that for any ε > 0 and α ∈ (0, 1) ∥∇u∥L∞(Ω) ≤ Cε∥u∥L∞(Ω) + ε[∇u]C0,α(Ω), and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9) ∥u∥C2(Ω) = ∥u∥C1,1(Ω) ≤ Cε∥u∥L∞(Ω) + ε[D2u]C0,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer to [GT77, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='35] for a proof of such inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A review on the Laplace equation Elliptic equations are those that share some common properties with the Laplace equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (We will be more rigorous about this in the subsequent chapters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') Thus, we start with a quick review about the Laplace equation and harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The Dirichlet problem for this equation is the following: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10) � ∆u = 0 in Ω u = g on ∂Ω, — DRAFT — 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Overview and Preliminaries where the boundary condition g is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The domain Ω ⊂ Rn is bounded and smooth (or at least Lipschitz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The Dirichlet problem is solvable, and it has a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A useful way to think of the Laplacian ∆ is to notice that, up to a multiplicative constant, it is the only linear operator of second order which is translation invariant and rotation invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, it can be seen as an operator which measures (infinitesimally) the difference between u at x and the average of u around x, in the following sense: for any C2 function w we have ∆w(x) = lim r→0 cn r2 � � Br(x) w(y)dy − w(x) � = lim r→0 cn r2 � Br(x) � w(y) − w(x) � dy, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11) for some positive constant cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This can be shown, for example, by using the Taylor expansion of w(y) around x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, a similar formula holds with integrals in ∂Br(x) instead of Br(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' See, for example, [DV21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Actually, one can show by using the divergence theorem that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12) n r d dr � ∂Br(x) w dσ = � Br(x) ∆w, from which (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11) also follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Existence of solutions: energy methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The most classical way to construct solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10) is by “energy methods”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Namely, we consider the convex functional E(u) := 1 2 � Ω |∇u|2dx among functions satisfying u|∂Ω = g, and then look for the function u that minimizes the functional — see Theo- rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10 below for more details about the existence of a minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that such minimizer u will clearly satisfy the boundary condition u = g on ∂Ω, so we only have to check that it will satisfy in addition ∆u = 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If u is the minimizer, then E(u) ≤ E(u+εv) for every v ∈ C∞ c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since, for every fixed v, such function in ε has a minimum at ε = 0, we have d dε ���� ε=0 E(u + εv) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A review on the Laplace equation 11 Thus, 0 = d dε ���� ε=0 E(u + εv) = d dε ���� ε=0 1 2 � Ω |∇u + εv|2dx = d dε ���� ε=0 1 2 � Ω � |∇u|2 + 2ε∇u · ∇v + ε2|∇v|2� dx = � Ω ∇u · ∇v dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Hence, if u is the minimizer of the functional, then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13) � Ω ∇u · ∇v dx = 0 for all v ∈ C∞ c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If u is regular enough (say, u ∈ C2), then we can integrate by parts (Theo- rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3) to find that � Ω ∆u v dx = 0 for all v ∈ C∞ c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, using Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2 we deduce that ∆u = 0 in Ω, as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As mentioned above, one should prove regularity of u before integrating by parts — a priori the minimizer u will only satisfy u ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will prove this in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If no extra regularity of u is available, then the above argument shows that any minimizer u of E is a weak solution, in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We say that u is a weak solution of the Dirichlet problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10) whenever u ∈ H1(Ω), u|∂Ω = g, and � Ω ∇u · ∇v dx = 0 for all v ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Here, u|∂Ω is the trace of u on ∂Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' recall (S5) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' More generally, given f ∈ L2(Ω), we say that u satisfies −∆u = f in Ω in the weak sense whenever u ∈ H1(Ω) and � Ω ∇u · ∇v dx = � Ω fv for all v ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, we say that u is weakly superharmonic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' weakly subharmonic) in Ω, or satisfies ∆u ≤ 0 in Ω in the weak sense (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' ∆u ≥ 0 in the weak sense) if � Ω ∇u·∇v dx ≥ 0 � resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' � Ω ∇u · ∇v dx ≤ 0 � for all v ∈ H1 0(Ω), v ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Overview and Preliminaries Notice that, if H1(Ω) ∋ uk ⇀ u ∈ H1(Ω) weakly in H1, and L2(Ω) ∋ fk ⇀ f ∈ L2(Ω) weakly in L2 are such that ∆uk = fk in Ω in the weak sense, then ∆u = f in the weak sense as well (by taking the limits in the pre- vious definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Similarly, the weak limit of weakly (sub-)superharmonic functions is (sub-)superharmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We next show the following: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10 (Existence and uniqueness of weak solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that Ω ⊂ Rn is any bounded Lipschitz domain, and that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14) � w ∈ H1(Ω) : w|∂Ω = g � ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, there exists a unique weak solution to the Dirichlet problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let θ◦ := inf �1 2 � Ω |∇w|2dx : w ∈ H1(Ω), w|∂Ω = g � , that is, the infimum value of E(w) among all admissible functions w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us take a sequence of functions {uk} such that uk ∈ H1(Ω) uk|∂Ω = g E(uk) → θ◦ as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By the Poincar´e inequality (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6 with p = 2), the sequence {uk} is uniformly bounded in H1(Ω), and therefore a subsequence {ukj} will converge to a certain function u strongly in L2(Ω) and weakly in H1(Ω) (recall (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5) in (S4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, by compactness of the trace operator, we will have ukj|∂Ω → u|∂Ω in L2(∂Ω), so that u|∂Ω = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Furthermore, such function u will satisfy E(u) ≤ lim infj→∞ E(ukj) (by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5)), and therefore it will be a minimizer of the energy functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, we have constructed a minimizer u of the energy functional E(u) satisfying the boundary condition u|∂Ω = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By the argument above, for any minimizer u we have that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since C∞ c (Ω) is dense in H1 0(Ω), it follows that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13) holds for all v ∈ H1 0(Ω), and thus it is a weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A review on the Laplace equation 13 Uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If u is any weak solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10), then for every v ∈ H1 0(Ω) we have E(u + v) = 1 2 � Ω |∇u + ∇v|2dx = 1 2 � Ω |∇u|2dx + � Ω ∇u · ∇v dx + 1 2 � Ω |∇v|2dx = E(u) + 0 + 1 2 � Ω |∇v|2dx ≥ E(u), with strict inequality if v ̸≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, if u solves (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10), then it is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ In other words, we have shown that u is a weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10) if and only if it minimizes the functional E(u) and, moreover, the minimizer of such energy functional exists and it is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' An interesting question is to determine the set of possible boundary data g : ∂Ω → R such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Of course, when Ω is any bounded Lipschitz domain, and g is Lipschitz, then it is easy to show that g has a Lipschitz extension inside Ω, and in particular (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' However, if g is very irregular then it might happen that it is not the trace of any H1(Ω) function, so that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14) fails in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It turns out that the right condition on g is the following: Given any bounded Lipschitz domain Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14) holds if and only if � ∂Ω � ∂Ω |g(x) − g(y)|2 |x − y|n+1 dx dy < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer to [Eva98] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Poisson kernel and fundamental solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The unique weak solution to the Dirichlet problem in a ball is explicit: � ∆u = 0 in B1 u = g on ∂B1, =⇒ u(x) = cn � ∂B1 (1 − |x|2)g(σ) |x − σ|n dσ, where cn is a positive dimensional constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By an easy rescaling argument, a similar formula holds in any ball Br(x◦) ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, we deduce that for any harmonic function ∆u = 0 in Ω, with Br ⊂ Ω, we have (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15) u(x) = cn r � ∂Br (r2 − |x|2)u(y) |x − y|n dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By taking x = 0, this yields the mean value property u(0) = � ∂Br u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' More- over, an immediate consequence of the Poisson kernel representation is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Overview and Preliminaries Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ Rn be any open set, and u ∈ H1(Ω) be any function satisfying ∆u = 0 in Ω in the weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u is C∞ inside Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, if u is bounded and ∆u = 0 in B1 in the weak sense, then we have the estimates (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16) ∥u∥Ck(B1/2) ≤ Ck∥u∥L∞(B1), for all k ∈ N, and for some constant Ck depending only on k and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For any ball Br(x◦) ⊂ Ω, we will have (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thanks to such representation, it is immediate to see then that u ∈ C∞(Br/2(x◦)) and the estimates (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since this can be done for any ball Br(x◦) ⊂ Ω, we deduce that u is C∞ inside Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ On the other hand, we recall that the fundamental solution for the Lapla- cian is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17) Φ(x) := � � � � � κn |x|n−2 if n ≥ 3 κ2 log 1 |x| if n = 2, for some explicit positive dimensional constant κn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Such function satisfies ∆Φ = 0 in Rn \\ {0}, but it is singular at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In fact, it satisfies −∆Φ = δ0 in Rn, where δ0 is the Dirac delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, we have that w := Φ ∗ f solves −∆w = f in Rn, for any given f with appropriate decay at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Maximum principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The maximum principle states the following: If ∆u ≥ 0 in Ω, and u ∈ C(Ω), then max Ω u = max ∂Ω u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, we also deduce the comparison principle: if ∆u ≥ ∆v in Ω, and u ≤ v on ∂Ω, then u ≤ v in the whole domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Recall that a function is said to be subharmonic if −∆u ≤ 0, and super- harmonic if −∆u ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As shown next, the maximum principle actually holds for any weak solution u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ Rn be any bounded open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that u ∈ H1(Ω) satisfies, in the weak sense, � −∆u ≥ 0 in Ω u ≥ 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ≥ 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A review on the Laplace equation 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that −∆u ≥ 0 in Ω if and only if (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18) � Ω ∇u · ∇v dx ≥ 0 for all v ≥ 0, v ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us consider u− := max{−u, 0} and u+ := max{u, 0}, so that u = u+ − u−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By (S9) we have that u± ∈ H1(Ω) whenever u ∈ H1(Ω), and thus we can choose v = u− ≥ 0 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Namely, using that u+u− = 0 and ∇u = ∇u+ − ∇u−, we get 0 ≤ � Ω ∇u · ∇u− dx = − � Ω |∇u−|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since u−|∂Ω ≡ 0 this implies u− ≡ 0 in Ω, that is, u ≥ 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ A useful consequence of the maximum principle is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any weak solution of � ∆u = f in Ω u = g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, ∥u∥L∞(Ω) ≤ C � ∥f∥L∞(Ω) + ∥g∥L∞(∂Ω) � , for a constant C depending only on the diameter of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us consider the function ˜u(x) := u(x)/ � ∥f∥L∞(Ω) + ∥g∥L∞(∂Ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We want to prove that |˜u| ≤ C in Ω, for some constant C depending only on the diameter of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that such function ˜u solves � ∆˜u = ˜f in Ω u = ˜g on ∂Ω, with |˜g| ≤ 1 and | ˜f| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us choose R large enough so that BR ⊃ Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' after a translation, we can take R = 1 2diam(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In BR, let us consider the function w(x) = R2 − x2 1 2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Such function w satisfies� ∆w = −1 in Ω w ≥ 1 on ∂Ω, Therefore, by the comparison principle, we deduce that ˜u ≤ w in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Overview and Preliminaries Since w ≤ C (with C depending only on R), we deduce that ˜u ≤ C in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, repeating the same argument with −˜u instead of ˜u, we find that |˜u| ≤ C in Ω, and thus we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Finally, another important result which follows from the maximum prin- ciple is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Here, we say that Ω satisfies the interior ball condition whenever there exists ρ◦ > 0 such that every point on ∂Ω can be touched from inside with a ball of radius ρ◦ contained in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, for any x◦ ∈ ∂Ω there exists Bρ◦(y◦) ⊂ Ω with x◦ ∈ ∂Bρ◦(y◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It is not difficult to see that any C2 domain satisfies such condition, and also any domain which is the complement of a convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15 (Hopf Lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ Rn be any domain satisfying the interior ball condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u ∈ C(Ω) be any positive harmonic function in Ω ∩ B2, with u ≥ 0 on ∂Ω ∩ B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ≥ c◦d in Ω ∩ B1 for some c◦ > 0, where d(x) := dist(x, Ωc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since u is positive and continuous in Ω∩B2, we have that u ≥ c1 > 0 in {d ≥ ρ◦/2} ∩ B3/2 for some c1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us consider the solution of ∆w = 0 in Bρ◦ \\ Bρ◦/2, with w = 0 on ∂Bρ◦ and w = 1 on ∂Bρ◦/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Such function w is explicit — it is simply a truncated and rescaled version of the fundamental solution Φ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, it is immediate to check that w ≥ c2(ρ◦ − |x|) in Bρ◦ for some c2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By using the function c1w(x◦+x) as a subsolution in any ball Bρ◦(x◦) ⊂ Ω ∩ B3/2, we deduce that u(x) ≥ c1w(x◦ + x) ≥ c1c2(ρ◦ − |x − x◦|) ≥ c1c2d in Bρ◦(x◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Setting c◦ = c1c2 and using the previous inequality for every ball Bρ◦(x◦) ⊂ Ω ∩ B3/2, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Mean value property and Liouville theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If u is harmonic in Ω (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', ∆u = 0 in Ω), then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19) u(x) = � Br(x) u(y)dy for any ball Br(x) ⊂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is called the mean value property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Conversely, if u ∈ C2(Ω) satisfies the mean value property, then ∆u = 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This can be seen for example by using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In fact, the mean value property (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19) can be used to give yet another (weak) definition of harmonic functions that only requires u to be locally integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Similarly, it is not difficult to deduce the corresponding pro- perty arising from the definitions of weak super- and subharmonicity (see Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9): — DRAFT — 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A review on the Laplace equation 17 From (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12), if u is weakly superharmonic in Ω (∆u ≤ 0 in Ω in the weak sense) then for all x ∈ Ω (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20) r �→ � Br(x) u(y) dy is monotone non-increasing for r ∈ (0, dist(x, ∂Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (And it is monotone non-decreasing for weakly subharmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') Thus, we can define (weak) super- and subharmonicity for L1 loc functions: we say that u ∈ L1 loc(Ω) is superharmonic in Ω if (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20) holds for all x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Similarly, we say that u ∈ L1 loc(Ω) is subharmonic in Ω if the map in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20) is monotone non-decreasing for all x ∈ Ω and r ∈ (0, dist(x, ∂Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We now give two lemmas that will be used in Chapter 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The first lemma says that the pointwise limit of a sequence of superharmonic uni- formly bounded functions is superharmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ Rn, and let {wn}n∈N be a sequence of uniformly bounded functions wn : Ω → R satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20), converging pointwise to some w : Ω → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then w satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let w∞ := w and let us define for n ∈ N ∪ {∞}, ϕx,n(r) := � Br(x) wn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that ϕx,n(r) is non-increasing in r for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, given 0 < r1 < r2 < Rx, we have that ϕx,n(r1) ≥ ϕx,n(r2) for n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now we let n → ∞ and use that wn → w pointwise to deduce, by the dominated convergence theorem (notice that wn are uniformly bounded), that ϕx,∞(r1) ≥ ϕx,∞(r2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, w∞ = w satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ The second lemma shows that superharmonic functions are lower semi- continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us assume that w is bounded and satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20) in Ω ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, up to changing w in a set of measure 0, w is lower semi- continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The proof is standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If we define w0(x) := limr↓0 � Br(x) w (which is well defined, since the average is monotone non-increasing), then w0(x) = w(x) if x is a Lebesgue point, and thus w0 = w almost everywhere in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us now consider x◦ ∈ Ω, and let xk → x◦ as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, by the dominated convergence theorem we have that � Br(x◦) w = lim k→∞ � Br(xk) w ≤ lim inf k→∞ w0(xk) for 0 < r < 1 2dist(x◦, ∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, by letting r ↓ 0 on the left-hand side, we reach that w0(x◦) ≤ lim inf k→∞ w0(xk), — DRAFT — 18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Overview and Preliminaries that is, w0 is lower semi-continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ On the other hand, a well-known theorem that can be deduced from the mean value property is the classification of global bounded harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18 (Liouville’s theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Any bounded solution of ∆u = 0 in Rn is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any global bounded solution of ∆u = 0 in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since u is smooth (by Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12), each derivative ∂iu is well-defined and is harmonic too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, thanks to the mean-value property and the divergence theorem, for any x ∈ Rn and R ≥ 1 we have |∂iu(x)| = ����� cn Rn � BR(x) ∂iu ����� = ����� cn Rn � ∂BR(x) u(y) yi |y| dy ����� ≤ C Rn � ∂BR(x) |u|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, using that |u| ≤ M in Rn, we find |∂iu(x)| ≤ cn Rn |∂BR(x)|M = cn Rn |∂B1|Rn−1M = cnM R → 0, as R → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, ∂iu(x) = 0 for all x ∈ Rn, and u is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ More generally, one can even prove a classification result for functions with polynomial growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Here, for γ ∈ R, ⌊γ⌋ denotes the floor function, that is, the largest integer less or equal to γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19 (Liouville’s theorem with growth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that u is a solution of ∆u = 0 in Rn satisfying |u(x)| ≤ C(1+|x|γ) for all x ∈ Rn, with γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u is a polynomial of degree at most ⌊γ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us define uR(x) := u(Rx), and notice that ∆uR = 0 in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' From Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12 and the growth assumption Rk∥Dku∥L∞(BR/2) = ∥DkuR∥L∞(B1/2) ≤ Ck∥uR∥L∞(B1) = Ck∥u∥L∞(BR) ≤ CkRγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, if k = ⌊γ⌋ + 1, ∥Dku∥L∞(BR/2) ≤ CkRγ−k → 0 as R → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, Dku ≡ 0 in Rn, and u is a polynomial of degree k − 1 = ⌊γ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ — DRAFT — 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A review on the Laplace equation 19 u x◦ y◦ v w Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' v touches u from above at x◦, w touches u from above at y◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Existence of solutions: comparison principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We saw that one way to prove existence of solutions to the Dirichlet problem for the Laplacian is by using energy methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' With such approach, one proves in fact the existence of a weak solution u ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, we will see an alternative way to construct solutions: via the com- parison principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' With this method, one can show the existence of a vis- cosity solution u ∈ C(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For the Laplace equation, these solutions (weak or viscosity) can then be proved to be C∞(Ω), and thus they coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We start by giving the definition of sub- and superharmonicity in the viscosity sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It is important to remark that in such definition the function u is only required to be continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A function u ∈ C(Ω) is subharmonic (in the viscosity sense) if for every function v ∈ C2 such that v touches u from above at x◦ ∈ Ω (that is, v ≥ u in Ω and v(x◦) = u(x◦)), we have ∆v(x◦) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' See Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The definition of superharmonicity for u ∈ C(Ω) is analogous (touching from below and with ∆v(x◦) ≤ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A function u ∈ C(Ω) is harmonic if it is both sub- and superharmonic in the above viscosity sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This definition obviously coincides with the one we know in case u ∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' However, it allows non-C2 functions u, for example u(x) = |x| is subhar- monic and −|x| is superharmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A useful property of viscosity sub-/supersolutions is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Overview and Preliminaries u1 u2 max{u1, u2} Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The maximum of two functions u1 and u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The maximum of two subharmonic functions is also subharmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, if u1, u2 ∈ C(Ω) are subharmonic, then the function v := max{u1, u2} is subharmonic as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' See Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Similarly, the minimum of two superharmonic functions is superhar- monic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The proof follows easily from Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20 above, and it is left as an exercise to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, we also have the following: Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ Rn be a bounded domain, and assume that u ∈ C(Ω) satisfies, in the viscosity sense, � −∆u ≥ 0 in Ω u ≥ 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ≥ 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' After a rescaling, we may assume Ω ⊂ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume by contradiction that u has a negative minimum in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, since u ≥ 0 on ∂Ω, we have minΩ u = −δ, with δ > 0, and the minimum is achieved in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us now consider 0 < ε < δ, and v(x) := −κ + ε(|x|2 − 1), with κ > 0 (that is, a sufficiently flat paraboloid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, notice that u − v > 0 on ∂Ω, and that we can choose κ > 0 so that minΩ(u − v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, we can slide the paraboloid from below the solution u until we touch it, by assumption, at an interior point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, there exists x◦ ∈ Ω such that u(x◦) − v(x◦) = minΩ(u − v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, with such choice of κ, the function v touches u from below at x◦ ∈ Ω, and hence, by definition of viscosity solution, we must have ∆v(x◦) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' However, a direct computation gives ∆v ≡ 2nε > 0 in Ω, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Thanks to these two propositions, the existence of a (viscosity) solution to the Dirichlet problem can be shown as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Probabilistic interpretation of harmonic functions 21 Let Sg := � v ∈ C(Ω) : v is subharmonic, and v ≤ g on ∂Ω � , and define the pointwise supremum u(x) := sup v∈Sg v(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, it can be shown that, if Ω is regular and g is continuous, then u ∈ C(Ω), and ∆u = 0 in Ω, with u = g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is the so-called Perron method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer to [HL97] for a complete description of the method in case of the Laplace operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In Chapter 3 we will study the existence of viscosity solutions in the more general setting of fully nonlinear elliptic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Short summary on existence of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We have two completely dif- ferent ways to construct solutions: by energy methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' or by the maximum (or comparison) principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In the first case, the constructed solution belongs to H1(Ω), in the second case to C(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In any case, one can then prove that u ∈ C∞(Ω)∩C(Ω) — as long as Ω and g are regular enough — and therefore u solves the Dirichlet problem in the usual sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Probabilistic interpretation of harmonic functions To end this introductory chapter, we give a well-known probabilistic inter- pretation of harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The discussion will be mostly heuristic, just to give an intuition on the Laplace equation in terms of stochastic pro- cesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer to Appendix C for further probabilistic interpretations for fully nonlinear equations and for the obstacle problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Recall that the Brownian motion is a stochastic process Xt, t ≥ 0, satisfying the following properties: (1) X0 = 0 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (2) Xt has no memory (is independent of the past, or it has indepen- dent increments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (3) Xt has stationary increments: Xt+s − Xs is equal in distribution to Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (4) Xt has continuous paths (t �→ Xt is continuous) almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (5) Xt is isotropic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', it is rotationally symmetric in distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The previous properties actually determine the stochastic process Xt up to a multiplicative constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Another important property of Brownian motion is that it is scale invariant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', — DRAFT — 22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Overview and Preliminaries x z ∂Ω Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A stochastic process Xx t defined in Ω starting at x until it hits the first point on the boundary z ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (6) r−1Xr2t equals Xt in distribution, for any r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As we will see next, there is a strong connection between the Brownian motion and the Laplace operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Expected payoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Given a regular domain Ω ⊂ Rn, and a Brownian motion Xx t starting at x (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', Xx t := x+Xt), we play the following stochastic game: When the process Xx t hits the boundary ∂Ω for the first time we get a payoff g(z), depending on the hitting point z ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (See Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') We then ask ourselves: What is the expected payoff?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' To answer this question, we define τ := first hitting time of Xx t , u(x) := E [g(Xx τ )] (value function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The value of u(x) is, by definition, the answer to the question above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Namely, it is the expected value of g at the first point where Xx t hits the boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' To find u(x), we try to relate it with values of u(y) for y ̸= x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, we will see that this yields a PDE for u, and by solving it we can find u(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, let us consider a ball Br(x) ⊂ Ω, with r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For any such ball, we know that the process Xx t will hit (before reaching ∂Ω, by property (4)) — DRAFT — 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Probabilistic interpretation of harmonic functions 23 some point on ∂Br(x), and moreover any point on ∂Br(x) will be hit with the same probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is because the process is rotationally symmetric in distribution, (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since the process has no memory, (2), and stationary increments, (3), this means that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='21) u(x) = � ∂Br(x) u(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Heuristically, this is because when the process hits the boundary ∂Br(x) at a point y, it simply starts again the game from such point y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' But because all points y ∈ ∂Br(x) are reached for the first time with the same probability, then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='21) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, since this can be done for every x ∈ Ω and r > 0, we deduce that u(x) satisfies the mean value property, and therefore it is harmonic, ∆u = 0 in Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, since we also know that u = g on ∂Ω (since when we hit the boundary we get the payoff g surely), then u must be the unique solution of � ∆u = 0 in Ω u = g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer to [Law10] for a nice introduction to this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Expected hitting time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A similar stochastic problem is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Given a smooth domain Ω ⊂ Rn, and a Brownian motion Xx t , we ask: What is the expected first time at which Xx t will hit ∂Ω ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' To answer this question, we argue as before, using that the process must first hit the boundary of balls Br(x) ⊂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, we first denote by u(x) the expected hitting time that we are looking for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, for any such ball we have that the process Xx t will hit (before reaching ∂Ω) some point on ∂Br(x), and moreover any point y ∈ ∂Br(x) will be hit with the same probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, the total expected time u(x) will be the expected time it takes to hit ∂Br(x) for the first time, plus the expected time when we start from the corresponding point y ∈ ∂Br(x), which is u(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In other words, we have u(x) = T(r) + � ∂Br(x) u(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Here, T(r) is the expected first time at which Xx t hits ∂Br(x) — which clearly depends only on r and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, using the scale-invariance property of the Brownian motion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' r−1Xr2t ∼ Xt, we see that T(r) = T(1)r2 = c1r2 for some constant c1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Overview and Preliminaries Thus, we have u(x) = c1r2 + � ∂Br(x) u(y)dy, and by rearranging terms we find − 1 r2 � � Br(x) u(y)dy − u(x) � = c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, taking r → 0 and using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11), we deduce that −∆u = c2, for some constant c2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since we clearly have u = 0 on ∂Ω, the expected hitting time u(x) is the unique solution of the problem � −∆u = c2 in Ω u = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By considering a non-homogeneous medium (in which it takes more time to move in some regions than others), the same argument leads to the prob- lem with a right-hand side � −∆u = f(x) in Ω u = 0 on ∂Ω, with f ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — Chapter 2 Linear elliptic PDE In this chapter we will study linear elliptic PDEs of the type (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) tr � A(x)D2u(x) � = n � i,j=1 aij(x)∂iju = f(x) in Ω ⊂ Rn, as well as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2) div � A(x)∇u(x) � = n � i,j=1 ∂i � aij(x)∂ju(x) � = f(x) in Ω ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' These are elliptic PDEs in non-divergence and divergence form, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The coefficients (aij(x))ij and the right-hand side f(x) satisfy appropri- ate regularity assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In addition, we will assume that the coefficient matrix A(x) = (aij(x))ij satisfies the uniform ellipticity condition 0 < λ Id ≤ (aij(x))ij ≤ Λ Id, for some ellipticity constants 0 < λ ≤ Λ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (For two matrices A, B ∈ Mn, we say A ≥ B if the matrix A − B is positive semi-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') We will show that, under appropriate regularity assumptions on A(x), solutions u to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) “gain two derivatives” with respect to f and the coef- ficients A(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, for the divergence-form equation, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2), we expect solutions to “gain one derivative” with respect to the coefficients A(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In order to do that, we will use perturbative methods, by “freezing” the coefficients around a certain point and studying the constant coefficient equation first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' After a change of variables, one can transform the constant coefficient equation into the most ubiquitous and simple elliptic equation: 25 — DRAFT — 26 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Linear elliptic PDE Laplace’s equation, where (aij(x))ij is the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, we will begin the chapter by studying properties of Laplace’s equation such as Harnack’s inequality and the H¨older regularity with bounded right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' After that, we proceed by showing Schauder estimates for the Laplacian to con- tinue with the main theorems of the current chapter: Schauder estimates for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We finish the chapter by studying equations of the type (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2) with continuous coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In this case we do not gain two (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' one) derivatives, and instead we lose an arbitrarily small H¨older exponent of regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Equations in non-divergence and divergence form will become particu- larly useful in Chapters 3 and 4 in the context of nonlinear variational PDEs and fully nonlinear elliptic PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For both equations in non-divergence and divergence form, we establish a priori estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, rather than proving that the solution is regular, we show that if the solution is regular, then one can actually estimate the norm of respectively two and one derivative higher in terms of the H¨older norms of the coefficients (aij(x))ij and the right-hand side f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is enough for the application to nonlinear equations in Chapters 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' When the operator is the Laplacian, thanks to the a priori estimates, and by means of an approximation argument, we show that weak solutions are in fact smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For more general elliptic operators, a priori estimates together with the continuity method yield the existence of regular solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer the reader to [GT77] for such an approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Harnack’s inequality We start this chapter with one of the most basic estimates for harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It essentially gives a kind of “maximum principle in quantitative form”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will usually write that u ∈ H1 is harmonic, meaning in the weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Recall from the introduction, however, that as soon as a function is harmonic, it is immediately C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1 (Harnack’s inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume u ∈ H1(B1) is a non-negative, harmonic function in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then the infimum and the supremum of u are comparable in B1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, � ∆u = 0 in B1 u ≥ 0 in B1 ⇒ sup B1/2 u ≤ C inf B1/2 u, for some constant C depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Harnack’s inequality 27 ∂B1 ∂B1 ∂B1/2 ∂B1/2 O 1 C 1 u Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Graphic representation of Harnack’s inequality for a har- monic function u > 0 such that supB1 u = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This can be proved by the mean value property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Alternatively, we can also use the Poisson kernel representation, u(x) = cn � ∂B1 (1 − |x|2)u(z) |x − z|n dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that, for any x ∈ B1/2 and z ∈ ∂B1, we have 2−n ≤ |x−z|n ≤ (3/2)n and 3/4 ≤ 1 − |x|2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, since u ≥ 0 in B1, C−1 � ∂B1 u(z) dz ≤ u(x) ≤ C � ∂B1 u(z) dz, for all x ∈ B1/2, for some dimensional constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, for any x1, x2 ∈ B1/2 we have that u(x1) ≤ C2u(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Taking the infimum for x2 ∈ B1/2 and the supremum for x1 ∈ B1/2, we reach that supB1/2 u ≤ ˜C infB1/2 u, for some dimensional constant ˜C, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This inequality says that, if u ≥ 0 in B1, then not only u > 0 in B1/2 (strong maximum principle), but also we get quantitative information: u ≥ C−1 supB1/2 u in B1/2, for some constant C depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' See Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that there is nothing special about B1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We can obtain a similar inequality in Bρ with ρ < 1, but the constant C would depend on ρ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, repeating the previous argument, one gets that if ∆u = 0 and u ≥ 0 in B1, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3) sup Bρ u ≤ C (1 − ρ)n inf Bρ u, for some C depending only on n, and where ρ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Linear elliptic PDE From Harnack’s inequality, we deduce the oscillation decay for harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, the oscillation of a harmonic function is reduced (quan- titatively) in smaller domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The oscillation in a domain Ω is defined as osc Ω u := sup Ω u − inf Ω u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We remark that the following lemma is valid for all harmonic functions, not necessarily positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3 (Oscillation decay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u ∈ H1(B1) be a harmonic function in B1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' ∆u = 0 in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then osc B1/2 u ≤ (1 − θ) osc B1 u for some small θ > 0 depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let w(x) := u(x) − inf B1 u, which satisfies w ≥ 0 in B1 and oscB1/2 w = oscB1/2 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since ∆w = 0 in B1, we get by Harnack’s inequality sup B1/2 w ≤ C inf B1/2 w, so that osc B1/2 u = osc B1/2 w = sup B1/2 w − inf B1/2 w ≤ � 1 − 1 C � sup B1/2 w ≤ � 1 − 1 C � sup B1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now notice that supB1 w = oscB1 u, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4 (Alternative proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Alternatively, we can rewrite the previous proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3 by taking advantage of the in- variance of the estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, the function u−infB1 u is non-negative and harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since the estimate we want to prove is invariant under addition and multiplication by constants, we may assume that infB1 u = 0 and supB1 u = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let θ := 1 C+1, where C is the constant in Harnack’s inequality, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now we have two options: If supB1/2 u ≤ 1 − θ, we are done, If supB1/2 u ≥ 1 − θ we use Harnack’s inequality to get inf B1/2 u ≥ 1 C (1 − θ) ≥ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In any case, we get oscB1/2 u ≤ 1 − θ, so we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Harnack’s inequality 29 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We have proved that Harnack’s inequality implies the oscil- lation decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is always true, we did not use the fact that we are dealing with harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In general, we have � Harnack’s inequality � =⇒ � Oscillation decay � =⇒ � H¨older regularity � Harnack’s inequality and the oscillation decay are scale invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, the following corollary holds: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6 (Rescaled versions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u ∈ H1(Br) be such that ∆u = 0 in Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then (Harnack’s inequality) If u ≥ 0 in Br, then sup Br/2 u ≤ C inf Br/2 u, for some C depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (Oscillation decay) One has osc Br/2 u ≤ (1 − θ) osc Br u, for some small θ > 0 depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Define ˜u(x) := u(rx), which fulfills ∆˜u = 0 in B1 and therefore sup Br/2 u = sup B1/2 ˜u ≤ C inf B1/2 ˜u = C inf Br/2 u, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Similarly, osc Br/2 u = osc B1/2 ˜u ≤ (1 − θ) osc B1 ˜u = (1 − θ) osc Br u by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ A standard consequence of the quantitative oscillation decay proved above is the H¨older regularity of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7 (H¨older regularity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u ∈ H1(B1) ∩ L∞(B1) be such that ∆u = 0 in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then ∥u∥C0,α(B1/2) ≤ C∥u∥L∞(B1) for some constants α > 0 and C depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If we denote ˜u := (2∥u∥L∞(B1))−1u, then ˜u ∈ H1 ∩ L∞(B1) fulfills ∆˜u = 0 in B1 and ∥˜u∥L∞(B1) ≤ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If we show ∥˜u∥C0,α(B1/2) ≤ C, then the result will follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Linear elliptic PDE O 0 1 B1 B 1 2 B 1 4 B 1 8 � Oscillation decay � ⇓ � H¨older regularity � Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Graphical representation of the fact that oscillation decay- type lemmas imply H¨older regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, dividing u by a constant if necessary, we may assume that ∥u∥L∞(B1) ≤ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We need to prove that |u(x) − u(y)| ≤ C|x − y|α for all x, y ∈ B1/2, for some small α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We do it at y = 0 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let x ∈ B1/2 and let k ∈ N be such that x ∈ B2−k \\ B2−k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, |u(x) − u(0)| ≤ osc B2−k u ≤ (1 − θ)k osc B1 u ≤ (1 − θ)k = 2−αk, with α = − log2(1 − θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (Notice that we are using Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6 k-times, where the constant θ is independent from the radius of the oscillation decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') Now, since 2−k ≤ 2|x|, we find |u(x) − u(0)| ≤ (2|x|)α ≤ C|x|α, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' See Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2 for a graphical representation of this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Finally, another important consequence of Harnack’s inequality is the Liouville theorem for non-negative harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be a non-negative harmonic function, that is, u ≥ 0 and ∆u = 0 in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let v = u − inf Rn u, where infRn u is well-defined and finite since u ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, thanks to Har- nack’s inequality in arbitrary balls from Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6, we get that for any — DRAFT — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Harnack’s inequality 31 R > 0, sup BR v ≤ C inf BR v = C � inf BR u − inf Rn u � → 0, as R → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, supRn u = infRn u and therefore u is constant in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Of course, the previous result also holds if u ≥ −M in Rn, for some constant M, since then u + M is non-negative and harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Harnack’s inequality with a right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We can prove a Harnack inequality for equations with a right-hand side, that is, when the Laplacian is not necessarily zero, ∆u = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Again, we will be dealing with functions u ∈ H1, so that we have to understand the equation ∆u = f in the weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let f ∈ L∞(B1), and u ∈ H1(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, � ∆u = f in B1 u ≥ 0 in B1 ⇒ sup B1/2 u ≤ C � inf B1/2 u + ∥f∥L∞(B1) � , for some C depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We express u as u = v + w with � ∆v = 0 in B1 v = u on ∂B1 � ∆w = f in B1 w = 0 on ∂B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, we have sup B1/2 v ≤ C inf B1/2 v and sup B1 w ≤ C∥f∥L∞(B1) by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1 and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, sup B1/2 u ≤ sup B1/2 v + C∥f∥L∞(B1) ≤ C inf B1/2 v + C∥f∥L∞(B1) ≤ C � inf B1/2 u + ∥f∥L∞(B1) � , where we are taking a larger constant if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that we have also used here that v ≤ u + C∥f∥L∞(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Thus, as before, we also get an oscillation decay, but now involving an error term of size ∥f∥L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let f ∈ L∞(B1) and u ∈ H1(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If ∆u = f in B1 and f ∈ L∞(B1), then osc B1/2 u ≤ (1 − θ) osc B1 u + 2∥f∥L∞(B1), for some θ > 0 depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Linear elliptic PDE Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The proof is the same as in the case f ≡ 0, see the proof of Corol- lary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, with the right-hand side f = f(x), the equation ∆u = f and Harnack’s inequality are not invariant under rescalings in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In fact, as we zoom-in, the right-hand side gets smaller!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Namely, if ∆u = f in Br, then ˜u(x) := u(rx) satisfies ∆˜u(x) = r2f(rx) in B1 so that sup B1/2 ˜u ≤ C � inf B1/2 ˜u + 2r2∥f∥L∞(Br) � , and therefore sup Br/2 u ≤ C � inf Br/2 u + 2r2∥f∥L∞(Br) � for some constant C depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Even if the previous oscillation decay contains an error depending on f, it is enough to show H¨older regularity of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12 (H¨older regularity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let f ∈ L∞(B1) and u ∈ H1∩L∞(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If ∆u = f in B1, then ∥u∥C0,α(B1/2) ≤ C � ∥u∥L∞(B1) + ∥f∥L∞(B1) � , for some constants α > 0 and C depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If we denote ˜u := (2∥u∥L∞(B1)+2∥f∥L∞(B1))−1u, then ˜u ∈ H1(B1)∩ L∞(B1) fulfills ∆˜u = ˜f in B1 with ∥˜u∥L∞(B1) ≤ 1 2 and ∥ ˜f∥L∞(B1) ≤ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If we show that ∥˜u∥C0,α(B1/2) ≤ C, then the result will follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, after dividing u by a constant if necessary, we may assume that ∥u∥L∞(B1) ≤ 1 2 and ∥f∥L∞(B1) ≤ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7 we want to prove that |u(x◦)−u(0)| ≤ C|x◦|α for all x◦ ∈ B1/2 and for some constant C depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us show that it is enough to prove that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4) osc B2−k u ≤ C2−αk for all k ∈ N, k ≥ k◦, for some α > 0, C, and for some fixed k◦, all three depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, let k ∈ N be such that x ∈ B2−k \\ B2−k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If k < k◦, then |x| ≥ 2−k◦−1 and |u(x) − u(0)| ≤ osc B1 u ≤ 1 ≤ 2α(k◦+1)|x|α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, if k ≥ k◦, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4) |u(x) − u(0)| ≤ osc B2−k u ≤ C2−αk ≤ C(2|x|)α, — DRAFT — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Schauder estimates for the Laplacian 33 where in the last inequality we used that |x| ≥ 2−k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, it will be enough to show (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let k ∈ N and k ≥ k◦ for some k◦ to be chosen, and define ˜u(x) := u(rx), r = 2−k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then ˜u satisfies ∆˜u = r2f(rx) in B1 (in fact, in B2k−1), and thus, by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10 osc B1/2 ˜u ≤ (1 − θ) osc B1 ˜u + 2r2∥f∥L∞(Br).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since oscB1 ˜u = oscB2−k+1 u and ∥f∥L∞(Br) ≤ 1 2, we find osc B2−k u ≤ (1 − θ) osc B2−k+1 u + 4−k+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, take k◦ ∈ N large enough so that 4−k◦+1 ≤ θ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, osc B2−k u ≤ (1 − θ) osc B2−k+1 u + θ 24k◦−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It is immediate to check by induction that this yields osc B2−k+1 u ≤ 2α(k◦−k), for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, the induction step follows as osc B2−k u ≤ (1 − θ)2α(k◦−k) + θ 24k◦−k ≤ (1 − θ)2α(k◦−k) + θ 22α(k◦−k) = � 1 − θ 2 � 2α(k◦−k) = 2α(k◦−k−1) if 1 − θ 2 = 2−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4) holds with C = 2αk◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Summarizing, we have checked that Harnack’s inequality for harmonic functions yields the H¨older regularity of solutions, even with a right-hand side f ∈ L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is a general fact, and holds for other types of elliptic equations, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Schauder estimates for the Laplacian We now want to establish sharp results for the equation ∆u = f(x) in B1 (or in Ω ⊂ Rn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This will serve as an introduction for the more general case of equations in non-divergence and divergence form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The philosophy is that the sharp results should state that “u is two derivatives more regular than f(x)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The main known results in that directions are the following: — DRAFT — 34 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Linear elliptic PDE (a) Schauder estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If f ∈ C0,α then u ∈ C2,α, for α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (b) Calder´on–Zygmund estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If f ∈ Lp then u ∈ W 2,p, for p ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (c) When α is an integer, or when p ∈ {1, ∞}, the above results do not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For example, if f ∈ C0, it is not true in general that u ∈ C2, not even C1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (In that case, u ∈ C1,1−ε for all ε > 0, and u ∈ W 2,p for all p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') Two counterexamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us provide two counterexamples to show that Schauder and Calder´on–Zygmund estimates in general do not hold for the limiting values, α = 0 and p = 1 or p = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We start with an example of a function u whose Laplacian is bounded (∆u ∈ L∞), but whose second derivatives are not bounded (u /∈ W 2,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, we give a counterexample to Calder´on–Zygmund estimates for p = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u(x, y) = (x2 − y2) log(x2 + y2) in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, ∂xxu = 2 log(x2 + y2) + 8x2 x2 + y2 − 2 �x2 − y2 x2 + y2 �2 , ∂yyu = −2 log(x2 + y2) − 8y2 x2 + y2 + 2 �x2 − y2 x2 + y2 �2 , that is, both ∂xxu and ∂yyu are unbounded, and u /∈ W 2,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' However, ∆u = ∂xxu + ∂yyu = 8x2 − y2 x2 + y2 ∈ L∞(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' One can modify such construction in order to make ∆u continuous and u /∈ C1,1, thus giving a counterexample to Schauder estimates for α = 0, by taking u(x, y) = (x2 − y2) log | log(x2 + y2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (However, recall that Schauder estimates tell us that this is not possible if ∆u is H¨older continuous (C0,α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') Let us now provide a counterexample for Calder´on–Zygmund estimates when p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The fact that the estimate does not hold can be seen by taking smooth approximations of the Dirac delta (with constant integral) as right- hand side, so that the solution converges to the fundamental solution, which is not in W 2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us, however, give a specific example of a function u whose Laplacian is integrable (∆u ∈ L1) but whose second derivatives are not (u /∈ W 2,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u(x, y) = log log 1 x2 + y2 = log log r−2 in R2, — DRAFT — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Schauder estimates for the Laplacian 35 where we are using polar coordinates and denote r2 := x2 + y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since u = u(r) and ur = 1 r log r we have that ∆u = urr + 1 rur = − log r + 1 r2(log r)2 + 1 r2 log r = − 1 r2(log r)2 ∈ L1(B1/2), since � B1/2 ∆u = −2π � 1/2 0 dr r(log r)2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, a direct com- putation gives that ∂xxu (and ∂yyu) are not absolutely integrable around the origin, and thus u /∈ W 2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (Alternatively, since one has the embed- ding W 2,1(R2) ⊂ L∞(R2) [Bre11, Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13] and u /∈ L∞, we deduce u /∈ W 2,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A similar counterexample can be built in any dimension n ≥ 2, by taking as function u an appropriate primitive of r1−n log r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In this book we focus our attention on proving (a) Schauder estimates, but not (b) Calder´on–Zygmund estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Later in the book we will see applications of Schauder-type estimates to nonlinear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13 (Calder´on-Zygmund estimates for p = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In the case p = 2, one can prove a priori Calder´on-Zygmund estimates with a simple compu- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, let u, f ∈ C∞(B1), be such that ∆u = f in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5) ∥u∥W 2,2(B1/2) ≤ C � ∥u∥L2(B1) + ∥f∥L2(B1) � for some constant C depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, let w := ηu for some fixed test function η ∈ C∞ c (B1) such that η ≡ 1 in B1/2, η ≡ 0 in B1 \\ B3/4 and η ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, integrating by parts gives ∥D2u∥L2(B1/2) = n � i,j=1 � B1/2 |D2 iju|2 ≤ n � i,j=1 � B1 |D2 ijw|2 = − n � i,j=1 � B1 (Diijw)(Djw) = n � i,j=1 � B1/2 (Diiw)(Djjw) = � B1 (∆w)2 ≤ C � B1 � u2 + (∆u)2 + |∇η|2|∇u|2� , where in the last equality we can take C = C′ supB1 � η2 + |∆η|2� for some dimensional constant C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, again integrating by parts twice and using 2ab ≤ a2 + b2, we get � B1 |∇η|2|∇u|2 = − � B1 |∇η|2u∆u + � B1 1 2u2∆|∇η|2 ≤ ˜C � B1 � u2 + (∆u)2� , where ˜C = supB1 � |∇η|2 + ∆|∇η|2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 36 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Linear elliptic PDE This directly yields the result (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5) for smooth functions u and f such that ∆u = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Arguing as in the proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16 below, the same result also holds as long as u ∈ H1(B1) is a weak solution to ∆u = f in B1 for f ∈ L2(B1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proofs of Schauder estimates: some comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' There are various proofs of Schauder estimates, mainly using: (1) integral representation of solutions (fundamental solutions);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (2) energy considerations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (3) comparison principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The most flexible approaches are (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Here, we will see different proofs of type (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The common traits in proofs of type (2)-(3) are their “perturbative char- acter”, that is, that by zooming in around any point the equation gets closer and closer to ∆u = constant, and thus (after subtracting a paraboloid) close to ∆u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, the result can be proved by using the information that we have on harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us start by stating the results we want to prove in this section: Schauder estimates for the Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14 (Schauder estimates for the Laplacian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let α ∈ (0, 1), and let u ∈ C2,α(B1) satisfy ∆u = f in B1, with f ∈ C0,α(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6) ∥u∥C2,α(B1/2) ≤ C � ∥u∥L∞(B1) + ∥f∥C0,α(B1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The constant C depends only on α and the dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will, in general, state our estimates in balls B1/2 and B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By means of a covering argument explained below, this allows us to obtain interior regularity estimates in general domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15 (Covering argument).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us assume that we have an esti- mate, like the one in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6), but in a ball Br1 for some r1 ∈ (0, 1), which will be typically very close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Namely, we know that if ∆u = f in B1, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7) ∥u∥C2,α(Br1) ≤ C � ∥u∥L∞(B1) + ∥f∥C0,α(B1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us suppose that we are interested in finding an estimate for a bigger ball, Br2 with r1 < r2 ∈ (0, 1), where r2 will be typically close to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We do that via a “covering argument”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (See Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') That is, let us cover the ball Br2 with smaller balls Br(xi) such that xi ∈ Br2 and r = (1 − r2)r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We can do so with a finite number of balls, — DRAFT — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Schauder estimates for the Laplacian 37 B1/4 B1/2 xi Br(xi) B4r(xi) B1 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Graphical representation of the “covering argument” in the case r1 = 1 4, r2 = 1 2, and r = 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' so that i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' , N}, for some N depending on r1, r2, and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that Br/r1(xi) ⊂ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We apply our estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7) (translated and rescaled) at each of these balls Br/r1(xi) (we can do so, because ∆u = f in Br/r1(xi) ⊂ B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, we obtain a bound for ∥u∥C2,α(Br(xi)) ∥u∥C2,α(Br(xi)) ≤ C(r1, r2) � ∥u∥L∞(Br/r1(xi)) + ∥f∥C0,α(Br/r1(xi)) � ≤ C(r1, r2) � ∥u∥L∞(B1) + ∥f∥C0,α(B1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, since Br2 can be covered by a finite number of these balls, we obtain ∥u∥C2,α(Br2) ≤ n � i=1 ∥u∥C2,α(Br(xi)) ≤ NC(r1, r2) � ∥u∥L∞(B1) + ∥f∥C0,α(B1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is the type of bound we wanted, where the constant now also depends on r1 and r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As a consequence of the “a priori” estimate for the Laplacian we will show: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any bounded weak solution to ∆u = f in B1, — DRAFT — 38 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Linear elliptic PDE with f ∈ C0,α(B1) for some α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u is in C2,α inside B1, and the estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Furthermore, iterating the previous estimate we will establish the fol- lowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17 (Higher order regularity estimates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any bounded weak solution to ∆u = f in B1, with f ∈ Ck,α(B1) for some α ∈ (0, 1), and k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u is in Ck+2,α inside B1 and ∥u∥Ck+2,α(B1/2) ≤ C � ∥u∥L∞(B1) + ∥f∥Ck,α(B1) � , for some constant C that depends only on k, α, and the dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In case f ∈ L∞, we will prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to ∆u = f in B1, with f ∈ L∞(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u is in C1,1−ε inside B1, for any ε > 0, with the estimate ∥u∥C1,1−ε(B1/2) ≤ Cε � ∥u∥L∞(B1) + ∥f∥L∞(B1) � for some constant Cε depending only on ε and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will give two different proofs of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The first proof follows a method introduced by Wang in [Wan06] and shows the a priori estimate using a very much self-contained approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For the second proof we use an approach `a la Caffarelli from [Moo12, Caf89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Before doing so, let us observe the following: If ∆u = f ∈ L∞ then ˜u(x) := u(rx) solves ∆˜u = r2f(rx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In other words, if |∆u| ≤ C, then |∆˜u| ≤ Cr2 (and if r is small, the right-hand side becomes smaller and smaller).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If ∆u = f ∈ C0,α, ˜u(x) = u(rx) − f(0) 2n |x|2 solves ∆˜u = r2(f(rx) − f(0)), so that |∆˜u| ≤ Cr2+α in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This, by the comparison prin- ciple, means that ˜u is “very close” to a harmonic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us now show that Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16 holds assuming Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This follows by an approximation argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will deduce the result from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to ∆u = f in B1, with f ∈ C0,α(B1), and let η ∈ C∞ c (B1) be any smooth function with η ≥ 0 and � B1 η = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let ηε(x) := ε−nη �x ε � , — DRAFT — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Schauder estimates for the Laplacian 39 which satisfies � Bε ηε = 1, ηε ∈ C∞ c (Bε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Consider the convolution uε(x) := u ∗ ηε(x) = � Bε u(x − y)ηε(y) dy, which is C∞ and satisfies ∆uε = f ∗ ηε =: fε in B1−ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (Notice that for smooth functions, derivatives and convolutions commute;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' the same can be done for weak derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') Since uε ∈ C∞, we can use Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14 to get ∥uε∥C2,α(B1/2) ≤ C � ∥uε∥L∞(B1−ε) + ∥fε∥C0,α(B1−ε) � , where we are also using the covering argument in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15 to write it in a ball B1−ε in the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Observe now that for any x, y ∈ B1−ε |uε(x)| ≤ � Bε |u(x − z)|ηε(z) dy ≤ ∥u∥L∞(B1) � Bε ηε(z) dz = ∥u∥L∞(B1), and |fε(x) − fε(y)| ≤ � Bε |f(x − z) − f(y − z)|ηε(z) dz = [f]C0,α(B1)|x − y|α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' From here, we deduce ∥uε∥L∞(B1−ε) ≤ ∥u∥L∞(B1) and ∥fε∥C0,α(B1−ε) ≤ ∥f∥C0,α(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, the sequence uε is uniformly bounded in C2,α(B1/2), ∥uε∥C2,α(B1/2) ≤ C � ∥u∥L∞(B1) + ∥f∥C0,α(B1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, since u is continuous (see Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12), arguing as before we get ∥uε − u∥L∞(B1) → 0 as ε ↓ 0, so that uε → u uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We can use (H8) from Chapter 1 to deduce that u ∈ C2,α and ∥u∥C2,α(B1/2) ≤ C � ∥u∥L∞(B1) + ∥f∥C0,α(B1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By a covering argument (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15) we can get a similar estimate in any ball Bρ with ρ < 1, ∥u∥C2,α(Bρ) ≤ Cρ � ∥u∥L∞(B1) + ∥f∥C0,α(B1) � , where now the constant Cρ depends also on ρ, and in fact, blows up when ρ ↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In any case, we have that u ∈ C2,α(Bρ) for any ρ < 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', u is in C2,α inside B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ The previous proof is an example of a recurring phenomenon when prov- ing regularity estimates for PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If one can get estimates of the kind ∥u∥C2,α ≤ C (∥u∥L∞ + ∥f∥C0,α) , for all C∞ functions u, and with a constant C that depends only on α and n (but independent of u and f), then, in general, the estimate holds as well for all solutions u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, if one wants to prove the higher-order regularity — DRAFT — 40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Linear elliptic PDE estimates from Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17, it is enough to get a priori estimates in the spirit of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As a consequence, assuming that Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14 holds, we can prove Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As mentioned above, we just need to show that for any u ∈ C∞ such that ∆u = f, one has (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8) ∥u∥Ck+2,α(B1/2) ≤ C � ∥u∥L∞(B1) + ∥f∥Ck,α(B1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' for some constant C depending only on n, α, and k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' and then we are done by a covering argument (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We prove it by induction on k, and it follows applying the induction hypothesis to derivatives of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8) deals with balls B1/2 and B1, but after a rescaling and covering argument (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15), it could also be stated in balls B1/2 and B3/4 (we will use it in this setting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The base case, k = 0, already holds by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us now assume that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8) holds for k = m − 1, and we will show it for k = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In this case, let us differentiate ∆u = f to get ∆∂iu = ∂if, for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Applying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8) for k = m − 1 to ∂iu in balls B1/2 and B3/4, we get ∥∂iu∥Cm+1,α(B1/2) ≤ C � ∥∂iu∥L∞(B3/4) + ∥∂if∥Cm−1,α(B3/4) � ≤ C � ∥u∥C2,α(B3/4) + ∥f∥Cm,α(B3/4) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Using now Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14 in balls B3/4 and B1, ∥∂iu∥Cm+1,α(B1/2) ≤ C � ∥u∥L∞(B1) + ∥f∥Cα(B1) + ∥f∥Cm,α(B3/4) � This, together with the basic estimate from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14 for ∆u = f, and used for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' , n}, directly yields that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8) holds for k = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Similarly, if one wants to prove regularity estimates in other contexts, it is often enough to obtain the corresponding a priori estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For instance, using an estimate that we prove later in the chapter (in the more general context of non-divergence-form equations) we can immediately obtain also the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The proof is exactly the same as the proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16 but using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='31 from below instead of Theo- rem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (Alternatively, see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') □ Let us now provide the first proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The method used here was introduced by Wang in [Wan06].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Schauder estimates for the Laplacian 41 First proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will prove that |D2u(z) − D2u(y)| ≤ C|z − y|α � ∥u∥L∞(B1) + [f]C0,α(B1) � , for all y, z ∈ B1/32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' After a translation, we assume that y = 0, so that the proof can be centered around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This will prove our theorem with estimates in a ball B1/32, and the desired result in a ball of radius 1 2 follows by a covering argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, after dividing the solution u by ∥u∥L∞(B1) + [f]C0,α(B1) if necessary, we may assume that ∥u∥L∞(B1) ≤ 1 and [f]C0,α(B1) ≤ 1, and we just need to prove that for all z ∈ B1/16, |D2u(z) − D2u(0)| ≤ C|z|α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Throughout the proof, we will use the following basic estimates for har- monic functions: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9) ∆w = 0 in Br ⇒ ∥Dκw∥L∞(Br/2) ≤ Cr−κ∥w∥L∞(Br), where C depends only on n and κ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (In fact, we will only use κ ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') Such estimate follows by rescaling the estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16) — which corresponds to the case r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will also use the estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10) ∆w = λ in Br ⇒ ∥D2w∥L∞(Br/2) ≤ C � r−2∥w∥L∞(Br) + |λ| � , for some constant C depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This estimate follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9) after subtracting λ 2n|x|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' , let uk be the solution to � ∆uk = f(0) in B2−k uk = u on ∂B2−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, ∆(uk − u) = f(0) − f, and by the rescaled version of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11) ∥uk − u∥L∞(B2−k) ≤ C(2−k)2∥f(0) − f∥L∞(B2−k) ≤ C2−k(2+α), where we are using that [f]C0,α(B1) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Hence, the triangle inequality yields ∥uk+1 − uk∥L∞(B2−k−1) ≤ C2−k(2+α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since uk+1 − uk is harmonic, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12) ∥D2(uk+1 − uk)∥L∞(B2−k−2) ≤ C22(k+1)∥uk+1 − uk∥L∞(B2−k−1) ≤ C2−kα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, notice that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13) D2u(0) = lim k→∞ D2uk(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 42 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Linear elliptic PDE Indeed, let ˜u(x) := u(0) + x · ∇u(0) + 1 2x · D2u(0)x be the second order expansion of u at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, since u ∈ C2,α, we have ∥˜u−u∥L∞(Br) ≤ Cr2+α = o(r2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Using that ˜u − uk is harmonic together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9) we deduce |D2uk(0) − D2u(0)| ≤ ∥D2(uk − ˜u)∥L∞(B2−k−1) ≤ C22k∥uk − ˜u∥L∞(B2−k) = C22k∥u − ˜u∥L∞(∂B2−k) → 0 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, for any point z near the origin, we have |D2u(z) − D2u(0)| ≤ |D2uk(0) − D2u(0)| + |D2uk(0) − D2uk(z)| + |D2uk(z) − D2u(z)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For a given z ∈ B1/16, we choose k ∈ N such that 2−k−4 ≤ |z| ≤ 2−k−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thanks to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13), and by the triangle inequality, we get |D2uk(0) − D2u(0)| ≤ ∞ � j=k |D2uj(0) − D2uj+1(0)| ≤ C ∞ � j=k 2−jα = C2−kα, where we use that α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In order to estimate |D2u(z) − D2uk(z)|, the same argument can be repeated around z instead of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, take solutions of ∆vj = f(z) in B2−j(z) and vj = u on ∂B2−j(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, |D2uk(z) − D2u(z)| ≤ |D2uk(z) − D2vk(z)| + |D2vk(z) − D2u(z)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The second term above can be bounded by C2−kα arguing as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For the first term, we use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10) by noticing that ∆(uk − vk) = f(0) − f(z) in B2−k ∩ B2−k(z) ⊃ B2−k−1(z) (recall |z| ≤ 2−k−3), so that, in B2−2−k(z) we have |D2uk(z) − D2vk(z)| ≤ ∥D2(uk − vk)∥L∞(B2−2−k(z)) ≤ C22k∥uk − vk∥L∞(B2−k−1(z)) + C|f(z) − f(0)| ≤ C22k∥uk − vk∥L∞(B2−k−1(z)) + C2−kα, where we use, again, that |z| ≤ 2−k−3, and [f]C0,α(B1) ≤ 1 Finally, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11), we know that ∥uk − u∥L∞(B2−k−1(z)) ≤ ∥uk − u∥L∞(B2−k) ≤ C2−k(2+α), and ∥u − vk∥L∞(B2−k−1(z)) ≤ C2−k(2+α), — DRAFT — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Schauder estimates for the Laplacian 43 which gives ∥uk − vk∥L∞(B2−k−1(z)) ≤ C2−k(2+α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, we deduce that |D2uk(z) − D2u(z)| ≤ C2−kα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, to estimate |D2uk(z) − D2uk(0)|, we denote hj := uj − uj−1 for j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since hj are harmonic, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9) with κ = 3 and using that B2−k−3 ⊂ B2−j−1, we see that ���� D2hj(z) − D2hj(0) |z| ���� ≤ ∥D3hj∥L∞(B2−k−3) ≤ C23j∥hj∥L∞(B2−j ) ≤ C2j(1−α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Hence, |D2uk(0) − D2uk(z)| ≤ |D2u0(z) − D2u0(0)| + k � j=1 |D2hj(z) − D2hj(0)| ≤ C|z|∥u0∥L∞(B1) + C|z| k � j=1 2j(1−α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We have also used here that, if we define w := u0 − f(0) 2n |x|2 + f(0) 2n then w is harmonic, D3w = D3u0, and w = u0 on ∂B1, and |z|−1|D2u0(z) − D2u0(0)| ≤ ∥D3u0∥L∞(B1/2) = ∥D3w∥L∞(B1/2) ≤ C∥w∥L∞(B1) = C∥u0∥L∞(∂B1) ≤ C∥u0∥L∞(B1), by higher order regularity estimates for harmonic functions (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9)) and the maximum principle (Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Note that here the constant C de- pends only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Combined with the fact that, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11), ∥u0∥L∞(B1) ≤ C, (where we also use ∥u∥L∞(B1) ≤ 1) and |z| ≤ 2−k−3, we deduce that |D2uk(0) − D2uk(z)| ≤ C|z| + C|z|2k(1−α) ≤ C2−kα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We finish by noticing that |z| ≥ 2−k−4 and combining all the last in- equalities we reach |D2u(z) − D2u(0)| ≤ C2−kα ≤ C|z|α for all z ∈ B1/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, [D2u]C0,α(B1/16) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 44 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Linear elliptic PDE Now, thanks to the interpolation inequalities (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9) with ε = 1), ∥u∥C2,α(B1/16) = ∥u∥C2(B1/16) + [D2u]C0,α(B1/16) ≤ C∥u∥L∞(B1/16) + 2[D2u]C0,α(B1/16) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We finish by recalling that we divided the solution u by ∥u∥L∞(B1)+[f]C0,α(B1), and we use a covering argument to get the desired result (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ For the second proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14, we use the methods from [Moo12], originally from [Caf89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Second proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' After subtracting f(0) 2n |x|2 we may as- sume that f(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' After dividing u by ∥u∥L∞(B1) + ε−1∥f∥C0,α(B1) if necessary, we may also assume that ∥u∥L∞(B1) ≤ 1 and ∥f∥C0,α(B1) ≤ ε, where ε > 0 is a constant to be chosen depending only on n and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' After these simplifications, it is enough to show that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14) ∥u∥C2,α(B1/2) ≤ C for some constant C depending only on n and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will show that, for every x ∈ B1/2, there exist a sequence of quadratic polynomials, (Pk)k∈N, and a ρ◦ < 1 such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15) ∥u − Pk∥L∞(Bρk◦ (x)) ≤ C◦ρk(2+α) for all k ∈ N, for some constant C◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By property (H5) from Chapter 1, this yields that [D2u]C0,α(B1/2) ≤ CC◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' After using an interpolation inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9), we get (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15) for x = 0 (after a translation, it follows for all x ∈ B1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We are going to use that ∆u = f, ∥u∥L∞(B1) ≤ 1, f(0) = 0 and [f]C0,α(B1) ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that ∥∆u∥C0,α(B1) = ∥f∥C0,α(B1) ≤ 2ε, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', u is 2ε-close in H¨older norm to a harmonic function: let w be such that ∆w = 0 and w = u on ∂B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, ∆(u − w) = f in B1, and u − v = 0 on ∂B1, so that by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16) ∥u − w∥L∞(B1) ≤ C′∥f∥L∞(B1) ≤ Cε, for some C universal (we are only using ∥f∥L∞(B1) ≤ 2ε, and not using its Cα norm at this point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The function w is harmonic and |w| ≤ 1 (since |u| ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, it has a quadratic Taylor polynomial P1 at the origin, which satisfies ∆P1 ≡ 0 and |P1| ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, since w is harmonic (and in particular w ∈ C3), we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17) ∥w − P1∥L∞(Br) ≤ Cr3 for all r ≤ 1, for some C depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Schauder estimates for the Laplacian 45 Combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17) we obtain ∥u − P1∥L∞(Br) ≤ C(r3 + ε) for all r ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Choose now r◦ small enough such that Cr3 ≤ 1 2r2+α (notice α < 1), and ε small enough such that Cε < 1 2r2+α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (Notice that both r◦ and ε can be chosen depending only on n and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') Then, ∥u − P1∥L∞(Br◦) ≤ r2+α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us now define u2(x) := (u − P1)(r◦x) r2+α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that ∥u2∥L∞(B1) ≤ 1 and ∆u2(x) = r−α f(r◦x) =: f2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, f2(0) = 0 and [f2]C0,α(B1) ≤ [f]C0,α(B1) ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, the same hypotheses as before are fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Repeating the same procedure, there exists a polynomial P2 such that ∥u2 − P2∥L∞(Br◦) ≤ r2+α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, substituting back, ��u − P1 − r2+α P2(x/r◦) �� L∞(Br2◦) ≤ r2(2+α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Continuing iteratively, for every k ∈ N we can define uk+1(x) := (uk − Pk)(r◦x) r2+α , which satisfies ∥uk+1∥L∞(B1) ≤ 1, ∆uk+1(x) = r−α fk(r◦x) = r−kα f(rk x) =: fk+1(x), and there exists some Pk+1 such that ∥uk+1 − Pk+1∥L∞(Br◦) ≤ r2+α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Substituting back, ��u − P1 − r2+α P2(x/r◦) − · · · − rk(2+α) Pk+1(x/rk ) �� L∞(Brk+1 ) ≤ r(k+1)(2+α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, we have constructed a sequence of quadratic polynomials approxi- mating u in a decreasing sequence of balls around 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' which shows that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15) holds around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' After a translation, the same argument can be repeated around any point x ∈ B1/2, so that, by (H5) we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' When α = 0, the previous proof implies that if f ∈ L∞(B1) then, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15), ∇u is in the Zygmund space Λ1(B1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' see Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1 in the Appendix A for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, we also get a proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Linear elliptic PDE Notice that in the previous proof we have not directly used that u is C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In fact, the only properties of u (and the Laplacian) we have used are that the maximum principle holds and that ∆(u(rx)) = r2(∆u)(rx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, the second proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14 is not an a priori esti- mate, and rather it says that any weak solution to the Laplace equation with Cα right-hand side is C2,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, we have directly proved Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Schauder estimates for operators in non-divergence form After proving the Schauder estimates for the Laplacian, we will study now more general second order linear elliptic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We start with operators in non-divergence form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The type of equation we are interested in is tr � A(x)D2u(x) � = n � i,j=1 aij(x)∂iju(x) = f(x) in B1 where the matrix A(x) = (aij(x))ij is uniformly elliptic — in the sense that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18) below holds — and aij(x) ∈ C0,α(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will prove the following a priori estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20 (Schauder estimates in non-divergence form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let α ∈ (0, 1), and let u ∈ C2,α be any solution to n � i,j=1 aij(x)∂iju = f(x) in B1, with f ∈ C0,α(B1) and aij(x) ∈ C0,α(B1), and (aij(x))ij fulfilling the ellip- ticity condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18) 0 < λ Id ≤ (aij(x))ij ≤ Λ Id in B1, for some 0 < λ ≤ Λ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, ∥u∥C2,α(B1/2) ≤ C � ∥u∥L∞(B1) + ∥f∥C0,α(B1) � for some constant C depending only on α, n, λ, Λ, and ∥aij∥C0,α(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As for the Laplacian, we will provide two different proofs of the previous result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, as a consequence of the previous result, we also obtain higher order Schauder estimates in non-divergence form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='21 (Higher order Schauder estimates in non-divergence form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u ∈ Ck+2,α be a solution to n � i,j=1 aij(x)∂iju = f(x) in B1, — DRAFT — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Schauder estimates for operators in non-divergence form 47 with f ∈ Ck,α(B1) and aij(x) ∈ Ck,α(B1) for some α ∈ (0, 1), k ∈ N, and (aij(x))ij fulfilling the ellipticity conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18) for some 0 < λ ≤ Λ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, ∥u∥Ck+2,α(B1/2) ≤ C � ∥u∥L∞(B1) + ∥f∥Ck,α(B1) � for some constant C depending only on α, k, n, λ, Λ, and ∥aij∥Ck,α(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='22 (Ellipticity condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The uniform ellipticity condition in B1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18), is a quantification of the fact that the matrix A(x) := (aij(x))ij is uniformly positive definite and uniformly bounded as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that we can always assume that A(x) is symmetric (from ∂iju = ∂jiu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We recall that the inequality A1 ≤ A2 for symmetric matrices A1, A2 ∈ Mn has to be understood in the sense that A2−A1 is positive semi-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Alternatively, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18) will hold if 0 < λ|ξ|2 ≤ n � i,j=1 ξiξjaij(x) ≤ Λ|ξ|2 for all x ∈ B1 for all ξ ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='23 (Constant coefficients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us start by understanding the case of constant coefficients, n � i,j=1 aij∂iju(x) = 0 in B1, where aij are constants and satisfy the uniform ellipticity assumption, 0 < λId ≤ (aij)ij ≤ ΛId, for 0 < λ ≤ Λ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us denote A := (aij)ij ∈ Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, A is a symmetric positive definite matrix, and therefore has a unique positive definite square root A1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' After an affine change of variables z = A1/2x, the equation n � i,j=1 aij∂xixju = 0 becomes n � i=1 ∂ziziu = 0 or ∆zu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, n � i,j=1 aij∂xixju = tr(AD2 xu) = tr(A1/2D2 xuA1/2) = tr(D2 zu) = ∆zu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 48 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Linear elliptic PDE Therefore (and since 0 < λId ≤ A ≤ ΛId), the case of constant coefficients (uniformly elliptic) can be reduced to the case of harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thanks to the uniform ellipticity, the change of variables is not degen- erate, and thus the estimates on ∥u∥C2,α that we get depend only on α, n, λ, and Λ (but not on A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Similarly, after changing variables, there could be a shrinking of the domain, say that the C2,α norm of u is bounded in Bρ instead of B1/2, for some ρ < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Once again, since the change is non- degenerate, such ρ depends only on n, λ, and Λ, and one can complete the proof by a covering argument in B1/2 (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The maximum principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We state the maximum principle for equations in non-divergence form, which will be used in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='24 (Maximum Principle in non-divergence form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ Rn be any bounded open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Suppose that u ∈ C0(Ω) ∩ C2(Ω) satisfies n � i,j=1 aij(x)∂iju ≥ 0 in Ω, where (aij(x))ij satisfy 0 < λ Id ≤ (aij(x))ij in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, sup Ω u = sup ∂Ω u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us begin by showing the maximum principle in the case (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19) n � i,j=1 aij(x)∂iju > 0 in B1, that is, when we have a strict inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We show it by contradiction: suppose that there exists some x◦ ∈ Ω such that supΩ u = u(x◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since it is an interior maximum, we must have ∇u(x◦) = 0 and D2u(x◦) ≤ 0, that is, D2u(x◦) is a negative semi-definite symmetric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, all its eigenvalues are non-positive, and after a change of variables we have that P T D2u(x◦)P = diag(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' , λn) := Dx◦ for some orthogonal n × n matrix P, and with λi ≤ 0 for all 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let A(x) = (aij(x))ij, and let AP (x◦) := P T A(x◦)P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, since A(x◦) is positive definite, so is AP (x◦) = (aP ij(x◦))ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, aP ii(x◦) ≥ 0 for all 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, tr(A(x◦)D2u(x◦)) = tr(A(x◦)PDx◦P T ) = tr(P T A(x◦)PDx◦) — DRAFT — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Schauder estimates for operators in non-divergence form 49 and, therefore 0 < tr(A(x◦)D2u(x◦)) = tr(AP (x◦)Dx◦) = n � i=1 aP ii(x◦)λi ≤ 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Here, we used that aP ii(x◦) ≥ 0 and λi ≤ 0 for all 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This shows that the maximum principle holds when the strict inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us now remove this hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let R be large enough such that BR ⊃ Ω — after a translation, we can take R = 1 2diam(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Consider now the function uε(x) := u(x) + εex1 for x ∈ Ω, for ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that, n � i,j=1 aij(x)∂ijuε(x) ≥ λεex1 > 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, we can apply the result for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19) to obtain that sup Ω u ≤ sup Ω uε = sup ∂Ω uε ≤ sup ∂Ω u + εeR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By letting ε ↓ 0, we obtain the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ As a consequence, we find: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ Rn be a bounded open set, and let u ∈ C0(Ω)∩C2(Ω) be a function satisfying � �n i,j=1 aij(x)∂iju = f in Ω u = g on ∂Ω, where (aij)ij fulfill the ellipticity conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18) for some 0 < λ ≤ Λ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, ∥u∥L∞(Ω) ≤ C � ∥f∥L∞(Ω) + ∥g∥L∞(∂Ω) � , for a constant C depending only on the diameter of Ω, λ, and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This follows exactly as in the proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14 using Proposi- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Proof of Schauder estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us now proceed with the proof of Schauder estimates for equations in non-divergence form, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will first prove (in two ways) the following proposition, which is a weaker version of the estimate we want to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will later prove that, in fact, such estimate is enough to prove The- orem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Linear elliptic PDE Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u ∈ C2,α be a solution to n � i,j=1 aij(x)∂iju = f(x) in B1, with f ∈ C0,α(B1) and aij(x) ∈ C0,α(B1) for some α ∈ (0, 1), and (aij(x))ij fulfilling the ellipticity condition 0 < λId ≤ (aij(x))ij ≤ ΛId in B1, for some 0 < λ ≤ Λ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, for any δ > 0, [D2u]C0,α(B1/2) ≤ δ[D2u]C0,α(B1) + Cδ � ∥u∥L∞(B1) + ∥f∥C0,α(B1) � , for some constant Cδ depending only on δ, α, n, λ, Λ, and ∥aij∥C0,α(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that, the previous statement is almost what we want: if we could let δ ↓ 0 and Cδ remained bounded, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20 would be proved (after using interpolation inequalities (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, if the H¨older norm was in B1/2 instead of B1, choosing δ = 1 2 would also complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As we will see, although it is not so straightforward, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26 is just one step away from the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us provide two different proofs of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The first proof is a sketch that follows the same spirit as the first proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The second proof is through a blow-up argument (by contradiction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' First Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The proof is very similar to the case of the Laplacian, the first proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We define uk as the solution to � �n i,j=1 aij(0)∂ijuk = f(0) in B2−k uk = u on ∂B2−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (We freeze the coefficients at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') Then, vk := u − uk satisfies n � i,j=1 aij(0)∂ijvk = f(x) − f(0) + n � i,j=1 � aij(0) − aij(x) � ∂iju in B2−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By the maximum principle (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='25) we get ∥u − uk∥L∞(B2−k) ≤ C2−2k � 2−αk∥f∥C0,α(B2−k) + 2−αk∥D2u∥L∞(B2−k) n � i,j=1 ∥aij∥C0,α(B2−k) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Schauder estimates for operators in non-divergence form 51 Thus, ∥uk − uk+1∥L∞(2−k−1) ≤ C2−k(2+α) � ∥f∥C0,α(B2−k) + ∥D2u∥L∞(B2−k) � , where the constant C depends only on α, n, λ, Λ, and ∥aij∥C0,α(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Following the exact same proof as in the case of the Laplacian, ∆u = f(x), we now get [D2u]C0,α(B1/2) ≤ C � ∥u∥L∞(B1) + ∥f∥C0,α(B1) + ∥D2u∥L∞(B1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is almost exactly what we wanted to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' However, we have an extra term ∥D2u∥L∞(B1) on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This can be dealt with by means of interpolation inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We use that, for any ε > 0, there is Cε such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20) ∥D2u∥L∞(B1) ≤ ε[D2u]C0,α(B1) + Cε∥u∥L∞(B1) see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9) in Chapter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The idea is that, since the ∥D2u∥L∞ term is lower order, we can absorb it in the left-hand side by paying the price of adding more ∥u∥L∞ norm on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Namely, we have [D2u]C0,α(B1/2) ≤ C � ∥u∥L∞(B1) + ∥f∥C0,α(B1) + ∥D2u∥L∞(B1) � (by interpolation) ≤ C � Cε∥u∥L∞(B1) + ∥f∥C0,α(B1) + ε[D2u]C0,α(B1) � ≤ Cδ � ∥u∥L∞(B1) + ∥f∥C0,α(B1) � + δ[D2u]C0,α(B1), where we have used the interpolation inequality, and in the last step we have chosen ε = δ/C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The constant Cδ depends only on δ, α, n, λ, Λ, and ∥aij∥C0,α(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ For the second proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26 we use a robust blow-up method due to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Simon, [Sim97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For simplicity, we will first prove it for the Laplacian case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' After proving it for the Laplacian, we explain in detail how to adapt the method for the more general non-divergence operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Second Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume first that (aij(x))ij = Id, that is, ∆u = f in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We then explain the modifications needed to show the result in the general case, �n i,j=1 aij(x)∂iju(x) = f(x) in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thanks to interpolation inequalities we only need to prove the following estimate for any δ > 0 sufficiently small, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='21) [D2u]C0,α(B1/2) ≤ δ[D2u]C0,α(B1) + Cδ � ∥D2u∥L∞(B1) + [f]C0,α(B1) � — DRAFT — 52 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Linear elliptic PDE for all u ∈ C2,α(B1) with ∆u = f in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='21) holds then, by interpolation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20), with ε = δ/Cδ, [D2u]C0,α(B1/2) ≤ 2δ[D2u]C0,α(B1) + Cδ � ∥u∥L∞(B1) + [f]C0,α(B1) � for some new Cδ depending only on δ, n and α, which is the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will now show that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='21) holds by contradiction, for some Cδ de- pending only on δ, n, and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, suppose that it does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, there exist sequences uk ∈ C2,α(B1) and fk ∈ C0,α(B1) for k ∈ N such that ∆uk = fk in B1, and for a fixed small constant δ◦ > 0 we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='22) [D2uk]C0,α(B1/2) > δ◦[D2uk]C0,α(B1) + k � ∥D2uk∥L∞(B1) + [fk]C0,α(B1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We now have to reach a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Select xk, yk ∈ B1/2 such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='23) |D2uk(xk) − D2uk(yk)| |xk − yk|α ≥ 1 2[D2uk]C0,α(B1/2) and let ρk := |xk − yk|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Observe that we must necessarily have ρk → 0 as k → ∞, since 1 2[D2uk]C0,α(B1/2) ≤ |D2uk(xk) − D2uk(yk)| ρα k ≤ 2∥D2uk∥L∞(B1) ρα k ≤ 2[D2uk]C0,α(B1/2) kρα k , where we have used (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='22) in the last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, ρk ≤ Ck− 1 α → 0 as k → ∞ Now, we rescale and blow up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Define ˜uk(x) := uk(xk + ρkx) − pk(x) ρ2+α k [D2uk]C0,α(B1) , ˜fk(x) := fk(xk + ρkx) − fk(xk) ρα k[D2uk]C0,α(B1) , where the quadratic polynomial pk is chosen so that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='24) ˜uk(0) = |∇˜uk(0)| = |D2˜uk(0)| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Namely, pk(z) := uk(xk) + ρk n � i=1 ∂iuk(xk)zi + 1 2ρ2 k n � i,j=1 ∂ijuk(xk)zizj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It is now a simple computation to check that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='25) ∆˜uk = ˜fk in B1/(2ρk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Schauder estimates for operators in non-divergence form 53 Let us also denote ξk := yk − xk ρk ∈ Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26) [D2˜uk]C0,α� B1/(2ρk) � ≤ 1, and ��D2˜uk(ξk) �� > δ◦ 2 , where for the second inequality we use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='22)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since ˜uk are uniformly bounded in compact subsets, and bounded in the C2,α norm (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='24)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26)), we have by Arzel`a–Ascoli that the sequence ˜uk converges (up to a subsequence and in the C2 norm) to a C2,α function ˜u on compact subsets of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, again up to a subsequence, we have that ξk → ξ ∈ Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By the properties of ˜uk, we deduce that ˜u satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='27) ˜u(0) = |∇˜u(0)| = |D2˜u(0)| = 0, [D2˜u]C0,α(Rn) ≤ 1, |D2˜u(ξ)| > δ◦ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, for any R ≥ 1 we have ∥ ˜fk∥L∞(BR) = sup x∈BR |fk(xk + ρkx) − fk(xk)| ρα k[D2uk]C0,α(B1) ≤ (ρkR)α[fk]C0,α(B1) ρα k[D2uk]C0,α(B1) ≤ Rα[D2uk]C0,α(B1/2) k[D2uk]C0,α(B1) ≤ Rα k → 0, as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, ˜fk → 0 uniformly on compact sets of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Together with the fact that ˜uk → ˜u in the C2 norm in compact sets, we deduce (recall (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='25)) ∆˜u = 0 in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, ˜u is harmonic and, in particular, so is ∂ij ˜u for any i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us now use the three properties in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='27) to get a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' First notice that we have [D2˜u]C0,α(Rn) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, D2˜u has sub-linear growth at infinity, and by Liouville’s theorem (Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19) we find that D2˜u is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, ˜u is a quadratic polynomial, which also fulfills ˜u(0) = |∇˜u(0)| = |D2˜u(0)| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The only possibility is that ˜u ≡ 0 in Rn, which is a contradiction with |D2˜u(ξ)| > δ◦ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, the proposition is proved in the case of the Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We now treat the case of variable coefficients, n � i,j=1 aij(x)∂iju(x) = f(x) in B1, with aij(x) uniformly elliptic in B1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', 0 < λId ≤ (aij(x))ij ≤ ΛId for x ∈ B1) and with ∥aij∥C0,α(B1) ≤ M < ∞ for some M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The proof is — DRAFT — 54 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Linear elliptic PDE essentially the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As before, we proceed by contradiction, by assuming that there exist sequences uk, fk, and a(k) ij such that n � i,j=1 a(k) ij (x)∂ijuk(x) = fk(x) in B1, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='22) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The only difference with respect to the Laplacian case is the equation satisfied by ˜uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us define, ˜a(k) ij (x) := a(k) ij (xk + ρkx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that [˜a(k) ij ]C0,α(B1/(2ρk)) ≤ ρα k[a(k) ij ]C0,α(B1) → 0, as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, up to subsequences, ˜a(k) ij converges uniformly in compact sets to some ˜aij with [˜a(k) ij ]C0,α(Rn) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', ˜aij is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then ˜uk satisfies n � i,j=1 ˜a(k) ij ∂ij ˜uk = ˜fk(x) − n � i,j=1 � a(k) ij (xk + ρkx) − a(k) ij (xk) � ∂ijuk(xk) ρα k[D2uk]C0,α(B1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, ������ n � i,j=1 ˜a(k) ij ∂ij ˜uk − ˜fk(x) ������ ≤ n � i,j=1 |x|αρα k[a(k) ij ]C0,α(B1)∥∂ijuk∥L∞(B1) ρα k[D2uk]C0,α(B1) ≤ C|x|α ∥D2uk∥L∞(B1) [D2uk]C0,α(B1) ≤ C|x|α ∥D2uk∥L∞(B1) [D2uk]C0,α(B1/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='22) we deduce that, for any x ∈ Bσ for some fixed σ ∈ (0, ∞), and for k large enough, ������ n � i,j=1 ˜a(k) ij ∂ij ˜uk − ˜fk(x) ������ ≤ C(σ) ∥D2uk∥L∞(B1) [D2uk]C0,α(B1/2) ≤ C(σ) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Taking the limit k → ∞ (and recalling that ˜fk → 0 uniformly in compact sets) we get n � i,j=1 ˜aij∂ij ˜u = 0 in Rn, an equation with constant coefficients, which is equivalent to ∆˜u = 0 in Rn (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='23), and we reach a contradiction as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ — DRAFT — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Schauder estimates for operators in non-divergence form 55 We can now proceed with the proof of the Schauder estimates in non- divergence form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Namely, we will show how to go from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26 to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As with the previous results, we will do it in two different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In this case, however, both ways reduce to the same idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' First Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Define the semi-norm [D2u]∗ α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='B1 := sup Bρ(x◦)⊂B1 ρ2+α[D2u]C0,α(Bρ/2(x◦)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that this norm measures in a precise way how the C2,α norm of U blows up as we approach ∂B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' From the fact that H¨older semi-norms are sub-additive with respect to unions of convex sets, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='28) [D2u]∗ α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='B1 ≤ C sup Bρ(x◦)⊂B1 ρ2+α[D2u]C0,α(Bρ/4(x◦)) (and, in fact, they are comparable) for some constant C depending only on α and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, for any fixed ball Bρ(x◦) ⊂ B1, we cover Bρ/2(x◦) with N smaller balls (Bρ/8(zj))1≤j≤N, which, since Bρ/2(zj) ⊂ B1, gives �ρ 2 �2+α [D2u]C0,α(Bρ/8(zj)) ≤ sup Bρ(x◦)⊂B1 ρ2+α[D2u]C0,α(Bρ/4(x◦)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, ρ2+α[D2u]C0,α(Bρ/2(x◦)) ≤ ρ2+α N � j=1 [D2u]C0,α(Bρ/8(zj)) ≤ 22+αN sup Bρ(x◦)⊂B1 ρ2+α[D2u]C0,α(Bρ/4(x◦)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Taking the supremum on the left-hand side gives (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Applying the inequality [D2u]C0,α(B1/2) ≤ δ[D2u]C0,α(B1) + Cδ � ∥u∥L∞(B1) + ∥f∥C0,α(B1) � from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26 to any ball Bρ/2(x◦) ⊂ Bρ(x◦) ⊂ B1 we get ρ2+α[D2u]C0,α(Bρ/4) ≤ δρ2+α[D2u]C0,α(Bρ/2) + Cδ � ∥u∥L∞(B1) + ∥f∥C0,α(B1) � ≤ δ[D2u]∗ α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='B1 + Cδ � ∥u∥L∞(B1) + ∥f∥C0,α(B1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Taking the supremum and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='28) we get 1 C [D2u]∗ α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='B1 ≤ sup Bρ(x◦)⊂B1 ρ2+α[D2u]C0,α(Bρ/4(x◦)) ≤ δ[D2u]∗ α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='B1 + C � ∥u∥L∞(B1) + ∥f∥C0,α(B1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 56 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Linear elliptic PDE Now, if we fix a small enough δ > 0, we can absorb the [D2u]∗ α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='B1 term on the left-hand side to get [D2u]C0,α(B1/2) ≤ [D2u]∗ α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='B1 ≤ Cδ � ∥u∥L∞(B1) + ∥f∥C0,α(B1) � , which, after interpolation (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9)) gives the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ We also give an alternative proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20 by directly using the following abstract lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Such lemma constitutes a generalization of the previous proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let k ∈ R and γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let S be a non-negative function on the class of open convex subsets of B1, and suppose that S is sub-additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, if A, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' , AN are open convex subsets of B1 with A ⊂ �N j=1 Aj, then S(A) ≤ �N j=1 S(Aj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, there is δ > 0 small (depending only on n and k) such that, if ρkS(Bρ/2(x◦)) ≤ δρkS(Bρ(x◦)) + γ for all Bρ(x◦) ⊂ B1, then S(B1/2) ≤ Cγ, for some C depending only on n and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Q := sup Bρ(x◦)⊂B1 ρkS(Bρ/2(x◦)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thanks to the assumption in the Lemma, we get �ρ 2 �k S(Bρ/4(x◦)) ≤ δ �ρ 2 �k S(Bρ/2(x◦))+γ ≤ δQ+γ, for all Bρ(x◦) ⊂ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Taking now the supremum for all Bρ(x◦) ⊂ B1 we get ˜Q := sup Bρ(x◦)⊂B1 �ρ 2 �k S(Bρ/4(x◦)) ≤ δQ + γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We now claim that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='29) Q ≤ C ˜Q, for some C depending only on n and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This will yield 1 C Q ≤ ˜Q ≤ δQ + γ ⇒ Q ≤ ˜Cγ if δ > 0 is small enough depending only on n and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, we have to show (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Take any Bρ(x◦) ⊂ B1, and cover Bρ/2(x◦) with a finite collection of smaller balls Bρ/8(zj) (j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' , N), with zj ∈ Bρ/2(x◦) and N ≤ C — DRAFT — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Schauder estimates for operators in non-divergence form 57 (universally bounded depending only on the dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since Bρ/2(zj) ⊂ B1 we then have �ρ 4 �k S(Bρ/8(zj)) ≤ ˜Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Adding up over all indices j, and using the sub-additivity of S, we obtain ρkS(Bρ/2(x◦)) ≤ N � j=1 ρkS(Bρ/8(zj)) ≤ N4k ˜Q = C ˜Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Taking the supremum, we reach (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Second Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='27, with k = α and S(A) := [D2u]C0,α(A), which is sub-additive on open convex subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' From the estimate in Propo- sition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26, fixing δ > 0 from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='27 (which depends only on α and n) we know [D2u]C0,α(B1/2) ≤ C � ∥u∥L∞(B1) + ∥f∥C0,α(B1) � + δ[D2u]C0,α(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Rescaling1 to Bρ(x◦) with ρ ≤ 1 we obtain ρ2+α[D2u]C0,α(Bρ/2(x◦)) ≤ ≤ δρ2+α[D2u]C0,α(Bρ(x◦)) + C � ∥u∥L∞(Bρ(x◦)) + ρ2∥f∥L∞(Bρ(x◦)) + ρ2+α[f]C0,α(Bρ(x◦)) � ≤ δρ2+α[D2u]C0,α(Bρ) + C � ∥u∥L∞(B1) + ∥f∥C0,α(B1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is exactly ρkS(Bρ/2(x◦)) ≤ δρkS(Bρ(x◦)) + γ, with γ = Cδ � ∥u∥L∞(B1) + ∥f∥C0,α(B1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, thanks to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='27, we immediately deduce S(B1/2) ≤ Cγ, that is, [D2u]C0,α(B1/2) ≤ C � ∥u∥L∞(B1) + ∥f∥C0,α(B1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, after using interpolation inequalities (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9)) we get ∥u∥C2,α(B1/2) ≤ C � ∥u∥L∞(B1) + ∥f∥C0,α(B1) � as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ 1The rescaling is done by considering the estimate on uρ(x) = u(x◦ + ρx), which ful- fills � a(ρ) ij (x)∂ijuρ(x) = ρ2f(x◦ + ρx) =: fρ(x) in B1, with a(ρ) ij (x) = aij(x◦ + ρx) (notice that ∥a(ρ) ij ∥C0,α(B1) ≤ ∥aij∥C0,α(B1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, [D2uρ]C0,α(B1/2) = ρ2+α[D2u]C0,α(Bρ/2(x◦)) and [fρ]C0,α(B1) = ρ2+α[f]C0,α(Bρ(x◦)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 58 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Linear elliptic PDE We finish this section by proving Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We follow the proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will show by induction on k that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='30) ∥u∥Ck+2,α(B1/2) ≤ C � ∥u∥L∞(B1) + ∥f∥Ck,α(B1) � for some constant C depending only on n, α, k, λ, Λ, and ∥aij∥Ck,α(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We apply the induction hypothesis to derivatives of the equation in non- divergence form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As in the proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='30) deals with balls B1/2 and B1, but after a rescaling and covering argument (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15), it could also be stated in balls B1/2 and B3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The base case, k = 0, already holds by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us now assume that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='30) holds for k = m − 1, and we will show it for k = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We differentiate the non-divergence-form equation with respect to ∂e to get n � i,j=1 aij(x)∂ij∂eu(x) = ∂ef(x) − n � i,j=1 ∂eaij(x)∂iju(x) in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, we apply the estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='30) with k = m−1 to ∂eu in the previous expression, in balls B1/2 and B3/4, to get ∥∂eu∥Cm+1,α(B1/2) ≤ C � ∥∂eu∥L∞(B3/4) + ∥∂ef∥Cm−1,α(B3/4) + n � i,j=1 ∥∂eaij∂iju∥Cm−1,α(B3/4) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that ∥∂eaij∂iju∥Cm−1,α(B3/4) ≤ ∥∂eaij∥Cm−1,α(B3/4)∥∂iju∥Cm−1,α(B3/4) = C∥∂iju∥Cm−1,α(B3/4) ≤ C � ∥u∥L∞(B1) + ∥f∥Cm−1,α(B1) � , where in the last inequality we have used the induction hypothesis in balls B3/4 and B1 (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Using that ∥∂eu∥L∞(B3/4) ≤ ∥u∥C2,α(B3/4) we can use the base case (with balls B3/4 and B1) of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='30) to bound this term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In all, we obtain that ∥∂eu∥Cm+1,α(B1/2) ≤ C � ∥u∥L∞(B1) + ∥f∥Cm−1,α(B1) � , which, combined with the base case, and for every e ∈ Sn−1, yields the desired estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ — DRAFT — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Schauder estimates for operators in divergence form 59 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Schauder estimates for operators in divergence form We will next prove Schauder estimates for operators in divergence form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, we will study the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='31) div � A(x)∇u(x) � = n � i,j=1 ∂i � aij(x)∂ju(x) � = f(x) in B1, where A(x) := (aij(x))ij is uniformly elliptic, and aij(x) ∈ C0,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that, a priori, the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='31) does not make sense even for C∞ functions u: we are taking derivatives of aij(x), which is only C0,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is why we need to define a weak notion of solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, we will say that u ∈ H1(B1) solves (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='31) weakly if � B1 ∇φ(y) · A(y)∇u(y) dy = − � B1 φ(y)f(y) dy for all φ ∈ C∞ c (B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will prove the following: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='28 (Schauder estimates in divergence form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u ∈ C1,α be a weak solution to n � i,j=1 ∂i � aij(x)∂ju(x) � = f(x) in B1, with f ∈ Lq(B1) for q ≥ n 1−α, and aij(x) ∈ C0,α(B1) for some α ∈ (0, 1), such that (aij(x))ij fulfills the ellipticity condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='32) 0 < λ Id ≤ (aij(x))ij ≤ Λ Id in B1, for some 0 < λ ≤ Λ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, ∥u∥C1,α(B1/2) ≤ C � ∥u∥L∞(B1) + ∥f∥Lq(B1) � for some constant C depending only on α, n, λ, Λ, and ∥aij∥C0,α(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' And as a consequence, we also get higher order Schauder estimates for operators in divergence form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='29 (Higher order Schauder estimates in divergence form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u ∈ Ck+1,α be a weak solution to n � i,j=1 ∂i � aij(x)∂ju(x) � = f(x) in B1, with f ∈ Ck−1+α(B1) and aij(x) ∈ Ck,α(B1) for some α ∈ (0, 1), k ∈ N, such that (aij(x))ij fulfills the ellipticity condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='32) for some 0 < λ ≤ Λ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, ∥u∥Ck+1,α(B1/2) ≤ C � ∥u∥L∞(B1) + ∥f∥Ck−1,α(B1) � — DRAFT — 60 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Linear elliptic PDE for some constant C depending only on α, k, n, λ, Λ, and ∥aij∥Ck,α(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The maximum principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As in the case of operators in non-divergence form, we also have a maximum principle for equations in divergence form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='30 (Maximum Principle in divergence form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ Rn be a bounded open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Suppose that u ∈ H1(Ω) satisfies, in the weak sense, n � i,j=1 ∂i � aij(x)∂ju(x) � ≥ 0 in Ω, where (aij(x))ij ∈ L∞(Ω) fulfill the pointwise ellipticity condition, 0 < (aij(x))ij in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, sup Ω u = sup ∂Ω u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We know that, denoting A(x) = (aij(x))ij, � Ω ∇φ · A(x)∇u dx ≤ 0 for all φ ∈ C∞ c (Ω), φ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, by approximation (see (S6) in Chapter 1), the previous ex- pression holds for all φ ∈ H1 0(Ω) such that φ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We take, as test function, φ(x) := (u − sup∂Ω u)+ ∈ H1 0(Ω), where f+ := max{f, 0} denotes the posi- tive part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then,� Ω ∇φ · A(x)∇φ dx = � Ω ∇φ · A(x)∇u dx ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since A(x) > 0, this implies that ∇φ ≡ 0, and φ is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since φ ∈ H1 0(Ω), this implies that φ ≡ 0, that is, u ≤ sup∂Ω u in Ω, as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Proof of Schauder estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We proceed with the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will do so via a blow-up argument, in the spirit of the second proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As in the (second) proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26, we will show that, for any δ > 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='33) [∇u]C0,α(B1/2) ≤ δ[∇u]C0,α(B1) + Cδ � ∥∇u∥L∞(B1) + ∥f∥Lq(B1) � for all u ∈ C1,α(B1) such that div(A(x)∇u(x)) = n � i,j=1 ∂i (aij(x)∂ju(x)) = f(x), weakly in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This yields ∥u∥C1,α(B1/2) ≤ δ[∇u]C0,α(B1) + Cδ � ∥u∥L∞(B1) + ∥f∥Lq(B1) � — DRAFT — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Schauder estimates for operators in divergence form 61 and so, proceeding as in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20 by using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='27, (or, alternatively, adapting the first proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20), we get the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us focus, therefore, on the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='33): Suppose that it does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, there exist sequences uk ∈ C1,α(B1) and fk ∈ Lq(B1) for k ∈ N such that div � Ak(x)uk(x) � = fk(x) weakly in B1, and for a fixed small constant δ◦ > 0 we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='34) [∇uk]C0,α(B1/2) > δ◦[∇uk]C0,α(B1) + k � ∥∇uk∥L∞(B1) + ∥fk∥Lq(B1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We now have to reach a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Select xk, yk ∈ B1/2 such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='35) |∇uk(xk) − ∇uk(yk)| |xk − yk|α ≥ 1 2[∇uk]C0,α(B1/2) and let ρk := |xk − yk| 2 , and zk := xk + yk 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, as in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26, ρk ≤ Ck− 1 α → 0 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Define ˜uk(x) := uk(zk + ρkx) + uk(zk − ρkx) − 2uk(zk) ρ1+α k [∇uk]C0,α(B1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='36) ˜uk(0) = |∇˜uk(0)| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We remark that here, instead of defining ˜uk as in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', subtracting a quadratic polynomial), we have used second order incremental quotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us also denote ξk := yk − xk 2ρk ∈ Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='37) [∇˜uk]C0,α(B1/(2ρk)) ≤ 2, and |∇˜uk(ξk)| > δ◦ 2 , where for the second inequality we use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='34) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since ˜uk are uniformly bounded in compact subsets, and bounded in the C1,α norm (due to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='36) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='37)), it follows by Arzel`a–Ascoli that the sequence ˜uk converges (in the C1 norm) to a C1,α function ˜u on compact subsets of Rn (up to a subsequence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, again up to a subsequence, we have that ξk → ξ ∈ Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By the properties of ˜uk, we deduce that ˜u satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='38) ˜u(0) = |∇˜u(0)| = 0, [∇˜u]C0,α(Rn) ≤ 2, |∇˜u(ξ)| > δ◦ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 62 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Linear elliptic PDE Let us check which equation does ˜uk satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let ˜a(k) ij (x) := a(k) ij (zk + ρkx), so that, as in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26, ˜a(k) ij converges uniformly in compact sets to some ˜aij constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For any φ ∈ C∞ c (B1), we know that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='39) � B1 ∇φ · Ak(x)∇uk = − � B1 fkφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let ˜Ak(x) := Ak(zk + ρkx) = (˜a(k) ij (x))ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let φ ∈ C∞ c (Rn), and let k be large enough so that supp φ ⊂ B1/(2ρk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let � ∇φ · ˜Ak(x)∇˜uk = I − II, where I = 1 ρα k[∇uk]C0,α(B1) � ∇φ(x) · Ak(zk + ρkx)∇uk(zk + ρkx) dx = 1 ρα k[∇uk]C0,α(B1) � ∇y � φ � ρ−1 k (y − zk) �� Ak(y)∇uk(y)ρ−n+1 k dy = −ρ1−α k [∇uk]C0,α(B1) � φ(x)fk(zk + ρkx) dx, thanks to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='39), and II = 1 ρα k[∇uk]C0,α(B1) � ∇φ(x) · Ak(zk + ρkx)∇uk(zk − ρkx) dx = IIi + IIii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Here, we have denoted by IIi and IIii the following quantities: IIi = 1 ρα k[∇uk]C0,α(B1) � ∇φ(x)·(Ak(zk+ρkx)−Ak(zk−ρkx))∇uk(zk−ρkx) dx and IIii = 1 ρα k[∇uk]C0,α(B1) � ∇φ(x) · Ak(zk − ρkx)∇uk(zk − ρkx) dx = ρ1−α k [∇uk]C0,α(B1) � φ(x)fk(zk − ρkx) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us now show that ���� � ∇φ · ˜Ak∇˜uk ���� → 0, as k → ∞ for all φ ∈ C∞ c (Rn), by bounding each term separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Schauder estimates for operators in divergence form 63 Notice that, for 1 q + 1 q′ = 1, ���� � φ(x)fk(zk + ρkx) dx ���� ≤ �� |φ|q′� 1 q′ �� |fk(zk + ρkx)|q dx � 1 q ≤ C(φ)∥fk∥Lq(B1)ρ − n q k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, |I| ≤ C(φ)ρ 1−α− n q k ∥fk∥Lq(B1) [∇uk]C0,α(B1) ≤ C(φ)ρ 1−α− n q k k−1 → 0, as k → ∞, as long as 1 − α − n q ≥ 0, that is, q ≥ n 1−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In the last step we have used (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Similarly, |IIii| → 0, as k → ∞, since q ≥ n 1−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, |IIi| ≤ [Ak]C0,α(B1) [∇uk]C0,α(B1) � |∇φ||x|α∥∇u∥L∞(B1) dx ≤ C(φ) ∥∇u∥L∞(B1) [∇uk]C0,α(B1) ≤ C(φ) k → 0, as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Here, we used again (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, |IIi| → 0 uniformly in compact sets of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then we conclude that, for any φ ∈ C∞ c (Rn), ���� � ∇φ · ˜Ak(x)∇˜uk ���� → 0, as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By taking limits, up to a subsequence we will have that ˜Ak → ˜A uniformly in compact sets, where ˜A is a constant coefficient matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, we deduce that � ∇φ · ˜A∇˜u = 0 for all φ ∈ C∞ c (Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This means that, after a change of variables, ˜u is harmonic (recall Re- mark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By Liouville’s theorem (Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19) we obtain that ∇˜u must be constant (since it is harmonic, and [∇˜u]C0,α(Rn) ≤ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' However, ∇˜u(0) = 0 and ∇˜u(ξ) ̸= 0 (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='38)), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We proceed by induction on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The case k = 0 is due to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, let us assume that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='40) ∥u∥Ck+1,α(B1/2) ≤ C � ∥u∥L∞(B1) + ∥f∥Ck−1,α(B1) � holds for all k ≤ m − 1, and let us show it for k = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 64 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Linear elliptic PDE To do so, notice that, since aij(x) ∈ Cm,α, and m ≥ 1, we can compute the derivatives in the divergence-form equation, to get n � i,j=1 aij(x)∂iju = f(x) − n � i,j=1 ∂iaij(x)∂ju in B1, that is, a non-divergence-form equation, where the right-hand side is in Cm−1,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Applying the higher order Schauder estimates for equations in non-divergence form, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='21 (in balls B1/2 and B3/4), we get that ∥u∥Cm+1,α(B1/2) ≤ C � ∥u∥L∞(B3/4) + ∥f∥Cm−1,α(B3/4)+ + n � i,j=1 ∥∂i(aij)∥Cm−1,α(B3/4)∥∂ju∥Cm−1,α(B3/4) � , that is, ∥u∥Cm+1,α(B1/2) ≤ C � ∥u∥L∞(B3/4) + ∥f∥Cm−1,α(B3/4) + ∥u∥Cm,α(B3/4) � , where the constant C depends only on n, α, λ, Λ, and ∥aij∥Cm,α(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Using now the hypothesis induction, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='40) for k = m − 1, in balls B3/4 and B1, completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The case of continuous coefficients Let us finish this chapter by studying equations in divergence and non- divergence form with continuous coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In this section we establish a priori Schauder estimates for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2) whenever aij ∈ C0(B1) (and the right-hand side is bounded or in Ln respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This kind of estimates will be useful in the next chapters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In this limiting case (when α ↓ 0), one could extrapolate from the pre- vious results that the solution has respectively bounded C2 and C1 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' However, this is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will show, instead, that we gain almost two derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Namely, for any ε > 0, the solution has bounded C2−ε and C1−ε norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' More precisely, we prove below the following results: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u ∈ C2 be any solution to n � i,j=1 aij(x)∂iju = f(x) in B1, with f ∈ L∞(B1) and aij ∈ C0(B1) for some (aij(x))ij satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18) for some 0 < λ ≤ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, for any ε > 0, ∥u∥C1,1−ε(B1/2) ≤ Cε � ∥u∥L∞(B1) + ∥f∥L∞(B1) � — DRAFT — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The case of continuous coefficients 65 for some constant Cε depending only on ε, n, λ, Λ, and (aij)ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, we are not gaining two full derivatives, but instead we are losing an arbitrarily small factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This loss is paired with the fact that the constant Cε diverges when ε ↓ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' see [JMV09, EM17] for counterexamples in the case ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is also consistent with what occurs with the Laplacian (see the counterexample at the beginning of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We remark that the dependence of C on (aij)i,j in the previous propo- sition is a dependence on the modulus of continuity of (aij)i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, if ω : [0, ∞) → [0, ∞) is a continuous monotone function with ω(0) = 0 and such that |aij(x) − aij(y)| ≤ ω(|x − y|), for all x, y ∈ B1, then the constant in the previous proposition depends on ω rather than on (aij)i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For divergence-form equations we have the following: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u ∈ C1 be a weak solution to n � i,j=1 ∂i � aij(x)∂ju � = f(x) in B1, with f ∈ Ln(B1) and aij(x) ∈ C0(B1) satisfying the ellipticity conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='32) for some 0 < λ ≤ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, for any ε > 0, ∥u∥C1−ε(B1/2) ≤ Cε � ∥u∥L∞(B1) + ∥f∥L∞(B1) � for some constant Cε depending only on ε, n, λ, Λ, and (aij)ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The proofs of the previous two propositions are analogous to those of the Schauder estimates for operators in non-divergence and divergence form respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We give short sketches of the proofs of Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='31 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='32 that contain all the essential information regarding the steps to take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Sketch of the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We give a short sketch of the proof in the case (aij(x))ij = Id, and leave the details to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The proof sketched follows the same steps and arguments as the second proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proceeding analogously, and after using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='27, (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' first or second proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20), we just need to show that for any δ > 0 [∇u]C1−ε(B1/2) ≤ δ[∇u]C1−ε(B1) + Cδ(∥∇u∥L∞(B1) + ∥f∥L∞(B1)), for some Cδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 66 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Linear elliptic PDE By contradiction, suppose that we have a sequence fk ∈ L∞(B1), uk ∈ C2(B1), and coefficients (a(k) ij )ij with a common modulus of continuity, such that �n i,j=1 a(k) ij ∂ijuk = fk and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='41) [∇uk]C1−ε(B1/2) > δ◦[∇uk]C1−ε(B1) + k(∥∇uk∥L∞(B1) + ∥fk∥L∞(B1)), for some δ◦ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Select xk, yk ∈ B1/2 such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='42) |∇uk(xk) − ∇uk(yk)| |xk − yk|1−ε ≥ 1 2[∇uk]C1−ε(B1/2) and let ρk := |xk − yk|, so that as in the second proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26 ρk ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Define ˜uk(x) := uk(xk + ρkx) − uk(xk) − ρk∇uk(xk) · x ρ2−ε k [∇uk]C1−ε(B1) and ˜fk(x) := ρε k fk(xk + ρkx) − fk(xk) [∇uk]C1−ε(B1) , so that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='43) ˜uk(0) = |∇˜uk(0)| = 0, n � i,j=1 ˜a(k) ij ∂ij ˜uk = ˜fk in B1/(2ρk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' where ˜a(k) ij (x) := a(k) ij (zk + ρkx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Denoting ξk := yk−xk ρk ∈ Sn−1, we have [∇˜uk]C1−ε� B 1 2ρk � ≤ 1, and ��∇˜uk(ξk) �� > δ◦ 2 , by means of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='41) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26, ˜uk converges (up to a subsequence and in the C1 norm) to a C1,1−ε function ˜u on compact subsets of Rn, and ξk → ξ ∈ Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Furthermore, ˜u(0) = |∇˜u(0)| = 0, [∇˜u]C1−ε(Rn) ≤ 1, |∇˜u(ξ)| > δ◦ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, for any R ≥ 1 we have ∥ ˜fk∥L∞(BR) ≤ ρε k k → 0, as k → ∞, and that, from the uniform modulus of continuity of a(k) ij , ˜a(k) ij (x) → ˜aij locally uniformly in Rn, where the limiting coefficients ˜aij are constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' At this point, in the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='43) the coefficients converge locally uniformly to constant coefficients, and the solutions ˜uk converge simply in C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The — DRAFT — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The case of continuous coefficients 67 passage to the limit is now more involved than before: in order to do it, we need the notion of viscosity solutions (see Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20 and Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3) and the fact that they are stable under uniform limits (see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In all, we can show that the limiting ˜u satisfies n � i,j=1 ˜aij∂ij ˜u = 0 in Rn (in the viscosity sense).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Hence, the limiting solution ˜u is harmonic (after changing variables) and we reach a contradiction as in the second proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ In order to prove the convergence of the sequence in the proof of Propo- sition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='32 we will need the following lemma: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u ∈ H1(B1) satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='44) div(A(x)∇u(x)) = f(x) in B1, in the weak sense, for some f ∈ L2(B1) and A(x) = (aij(x))ij uniformly elliptic with ellipticity constants λ and Λ (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then ∥∇u∥L2(B1/2) ≤ C(∥u∥L2(B1) + ∥f∥L2(B1)) for some C depending only on λ, Λ, and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us prove the lemma in the case A(x) is symmetric for all x ∈ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let η ∈ C∞ c (B1) be arbitrary with η ≡ 1 in B1/2, and observe that � B1/2 |∇u|2 ≤ C � B1 ∇(uη) · A(x)∇(uη) dx by ellipticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, since A(x) is symmetric for all x ∈ B1 we can use that ∇(uη) · A∇(uη) = u2∇η · A∇η + ∇(uη2) · A∇u and the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='44) to get � B1/2 |∇u|2 ≤ C � B1 u2|∇η|2 + C � B1 |fu|η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By H¨older’s inequality, we get the desired estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer to the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8 for more details on the proof and on the non-symmetric case in a very similar situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Let us now give the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The proof is by contradiction and proceeds as the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='28, with the analogous modifications introduced in the Sketch of the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='31 with respect to the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 68 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Linear elliptic PDE Observe that, in this case, we should define ˜uk(x) as first order incre- mental quotients: ˜uk(x) = uk(xk + ρkx) − uk(xk) ρ1−ε k [uk]C1−ε(B1) so that we directly have (differently from the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='28) that ˜uk satisfies: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='45) div( ˜Ak(x)∇˜uk(x)) = ˜fk(x) in B1/(2ρk), ˜fk(x) = ρ1+ε k fk(xk + ρkx) [uk]C1−ε(B1) , in the weak sense, where ˜Ak(x) := Ak(xk + ρkx) and ∥ ˜fk∥Ln(B1/(2ρk)) ↓ 0 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, ˜Ak(x) converges to some constant matrix ˜A∞ locally uniformly by uniform continuity of Ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, observe that each ˜uk is in H1 (since they are C1 by assumption), and they are locally uniformly in L2 (since they are uniformly locally bounded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Hence, we can apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='33 to get that ˜uk are locally uniformly bounded in H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, by (S4) from Chapter 1 (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3)) ∇˜uk converges weakly to ∇˜u∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus: � Rn ∇φ · ˜Ak∇˜uk → � Rn ∇φ · ˜A∞∇˜u∞ as k → ∞, for all φ ∈ C∞ c (Rn), and from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='45) we have that u∞ is harmonic (after changing variables) in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The contradiction is now reached, again, by the Liouville theorem, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The blow-up technique is a common tool in analysis that has great versatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, the technique presented in this section is due to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Simon, [Sim97], and can be applied in a similar fashion to many different situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We have seen the technique applied in interior a priori estimates for linear second-order equations, both in divergence and non-divergence form, and blow-up arguments like the one presented above can be adapted also to boundary estimates, parabolic equations, nonlinear equations, and even integro-differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Boundary regularity We finish the chapter by stating the corresponding results to Corollaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17 for the global (up to the boundary) estimates, for a sufficiently smooth domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For the sake of readability we state the result for the Laplacian, but there exists an analogous result for uniformly elliptic equations in non-divergence form (with the corresponding regularity on the coefficients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Boundary regularity 69 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='35 (Boundary regularity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let α ∈ (0, 1) and k ∈ N with k ≥ 2, and let Ω be a bounded Ck,α domain of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u ∈ H1(Ω) be a weak solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='46) � ∆u = f in Ω u = g on ∂Ω, for some f ∈ Ck−2,α(Ω), g ∈ Ck,α(∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ Ck,α(Ω) and ∥u∥Ck,α(Ω) ≤ C � ∥f∥Ck−2,α(Ω) + ∥g∥Ck,α(∂Ω) � , for some constant C depending only on α, n, k, and Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that in this case we do not need a term ∥u∥L∞(Ω) on the right-hand side because, thanks to the maximum principle (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='25), max Ω u ≤ C � max ∂Ω g + ∥f∥L∞(Ω) � for some C depending only on Ω, λ, Λ, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='35 can be proved using similar techniques (correspondingly adapted) to the ones in the previous sections: after a blow-up, points near the boundary behave like in a local problem in the half-space (that is, the blow-up flattens ∂Ω), and we can reach a contradiction with Liouville’s theorem in the half-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' One might wonder what happens under lower regularity assumptions on the domain (we refer to [Ken94, Kry96] for further reading in this direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In such case, similar regularity results hold in C1,α (and even C1) domains, but when Ω is merely Lipschitz, almost all regularity is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Namely, assume that u solves (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='46), with f and g smooth enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, If Ω is a C1,α domain, then solutions are C1,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If Ω is a C1 domain, then solutions are C1−ε(Ω) for all ε > 0, but not C0,1(Ω) in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If Ω is a Lipschitz domain, then solutions are Cγ(Ω) for some small γ > 0 that depends on the Lipschitz norm of the domain, and this is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We see that, if Ω is a Lipschitz domain, then essentially all regularity is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If one thinks on the blow-up and compactness method, it is clear that Lipschitz domains are quite different from C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, Lipschitz domains do not get flatter by doing a blow-up (they remain Lipschitz, with the same Lipschitz norm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, one cannot improve regularity by blowing up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Solutions turn out to be Cγ for some small γ > 0 and, in general, not better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — — DRAFT — Chapter 3 Nonlinear variational PDE & Hilbert’s XIXth problem Eine der begrifflich merkw¨urdigsten Thatsachen in den Elementen der Theorie der analytischen Functionen erblicke ich darin, daß es partielle Differentialgleichungen giebt, deren Integrale s¨amtlich notwendig analytische Funktionen der unabh¨angigen Variabeln sind, die also, kurz gesagt, nur analytischer L¨osungen f¨ahig sind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — David Hilbert (1900).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Up until this point, we have studied linear elliptic PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In this chapter we start the study of nonlinear elliptic PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' More precisely, we study variational nonlinear PDEs, that is, those that appear in the Calculus of Variations (minimizing an energy functional).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, our main goal is to introduce and solve Hilbert’s XIXth problem1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Hilbert’s XIXth problem (1900): Consider any local min- imizer of energy functionals of the form E(w) := � Ω L(∇w) dx, where L : Rn → R is smooth and uniformly convex, and Ω ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Is is true that all local minimizers to this type of problems are smooth?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 1The original statement by Hilbert says that “there exist partial differential equations whose integrals are all of necessity analytic functions of the independent variables, that is, in short, equations susceptible of none but analytic solutions”, and refers to solutions to what he calls “regular variational problems”, involving convex (in ∇w) and analytic operators of the form L(∇w, w, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We deal here with L(∇w) for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 71 — DRAFT — 72 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Nonlinear variational PDE & Hilbert’s XIXth problem Notice that, given a boundary condition u = g on ∂Ω, one can show that there is a unique minimizer to this problem, u ∈ H1(Ω), with u|∂Ω = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, there exists a unique u ∈ H1(Ω) such that u minimizes the functional E(w) := � Ω L(∇w) dx, among all functions w ∈ H1(Ω) such that w|∂Ω = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will be more precise about this in the first two sections of this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The question in Hilbert’s XIXth problem is that of regularity: Is such minimizer u smooth?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1 (On the convexity assumption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The uniform convexity of the function is what gives us existence and uniqueness of a minimizer (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, from the point of view of regularity, if L is not convex and reaches its minimum at two different points, then even in dimension n = 1 there exist counterexamples to regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If n = 1 and L has a minimum at two points p1 < p2, then we can construct Lipschitz only minimizers zigzagging with slopes p1 and p2 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', if p1 = −1 and p2 = 1, then u(x) = |x| would be a minimizer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, the convexity assumption is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Overview Hilbert’s XIXth problem as posed above is a generalization of the minimiza- tion of the Dirichlet integral, � Ω |∇w|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Local minimizers of the Dirichlet integral verify the corresponding Euler– Lagrange equation, which in this case is the Laplace equation ∆w = 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Solutions to this PDE, as seen in Chapter 2, are known to be C∞ in the interior of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, the Dirichlet integral case L(p) = |p|2 is extremely simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Sur- prisingly, the general case is far more difficult, and its resolution took more than 50 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' First, let us be more precise about the problem: by a local minimizer of E(w) = � Ω L(∇w) dx, we mean a function u ∈ H1(Ω) such that E(u) ≤ E(u + φ) for all φ ∈ C∞ c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The uniform convexity of the functional is equivalent to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) 0 < λId ≤ D2L(p) ≤ ΛId for all p ∈ Rn, — DRAFT — 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Overview 73 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', uniform convexity of L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice the analogy with the uniform ellipticity from the previous chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, what is the PDE satisfied by minimizers of E(u)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (Namely, the Euler–Lagrange equation of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') If u ∈ H1(Ω) is a local minimizer, then E(u) ≤ E(u + εφ) for all φ ∈ C∞ c (Ω), and all ε ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Hence, � Ω L(∇u) dx ≤ � Ω L(∇u + ε∇φ) dx for all φ ∈ C∞ c (Ω), and all ε ∈ R, and thus, as a function of ε, it has a minimum at ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Taking derivatives in ε we reach 0 = d dε ���� ε=0 � Ω L(∇u + ε∇φ) dx = � Ω DL(∇u)∇φ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The weak formulation of the Euler–Lagrange equation is then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2) � Ω DL(∇u)∇φ dx = 0 for all φ ∈ C∞ c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, u solves in the weak sense the PDE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3) div (DL(∇u)) = 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (This derivation will be properly justified in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') If u is C2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3) is equivalent to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4) n � i,j=1 (∂ijL)(∇u)∂iju = 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By uniform convexity of L, this is a (nonlinear) uniformly elliptic PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' What can we say about the regularity of u?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Regularity of local minimizers: First approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us assume that u is smooth enough so that it solves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We can regard (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4) as a linear equation with variable coefficients, by denoting aij(x) := (∂ijL)(∇u(x)), and we notice that, by uniform convexity of L, we have 0 < λ Id ≤ (aij(x))ij ≤ Λ Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, if ∇u ∈ C0,α, then aij ∈ C0,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, using Schauder estimates (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5) u ∈ C1,α ⇒ aij ∈ C0,α ⇒ u ∈ C2,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 74 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Nonlinear variational PDE & Hilbert’s XIXth problem We can then bootstrap the regularity and get C∞: u ∈ C2,α ⇒ ∇u ∈ C1,α ⇒ aij ∈ C1,α ⇒ u ∈ C3,α ⇒ · · · ⇒ u ∈ C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In fact, using the linear estimates for continuous coefficients, one can actually get u ∈ C1 ⇒ aij ∈ C0 ⇒ u ∈ C1,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We remark that while the previous implications are true at a formal level, we did not properly argue the use of Schauder estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, our results for Schauder estimates in both non-divergence form (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20) and divergence form (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='28) are a priori, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', they already assume regularity on u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We show how to use them in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5 below to prove the results we want and expect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Equations with bounded measurable coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We have argued that using perturbative results for linear equations (Schauder estimates), one expects to prove that u ∈ C1 =⇒ u ∈ C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' However, this approach does not allow us to prove any regularity if we do not know a priori that u ∈ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The main open question in Hilbert’s XIXth problem was then is it true that u ∈ H1 ⇒ u ∈ C1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This problem was open for many years, and it was finally solved (inde- pendently and almost at the same time) by De Giorgi [DeG57] and Nash [Nas57, Nas58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2 (De Giorgi–Nash).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be a local minimizer of E(w) = � Ω L(∇w) dx, with L uniformly convex and smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ C1,α for some α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This theorem solved Hilbert’s XIXth problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In order to show regularity of local minimizers u of E(w) = � Ω L(∇w) dx, with w ∈ H1(Ω), we first notice that they solve (in the weak sense) the nonlinear elliptic equation div (DL(∇u)) = 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The first idea in the proof is to consider derivatives of u, v = ∂eu, and to show that they solve an elliptic PDE as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If we differentiate the equation div (DL(∇u)) = 0 with respect to e ∈ Sn−1, we get div � D2L(∇u)∇∂eu � = 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Existence and basic estimates 75 Denoting (as before) v := ∂eu, aij(x) := ∂ijL(∇u(x)) and A(x) := (aij(x))ij, we can write this equation as div (A(x)∇v) = 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is a linear, uniformly elliptic equation in divergence form, but we do not have any regularity of A(x) in the x-variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We only know that the equation is uniformly elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is called a (uniformly elliptic) equation in divergence form with bounded measurable coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (Recall that the uniform convexity of L yields 0 < λId ≤ A(x) ≤ ΛId.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') De Giorgi and Nash established a new regularity result for such type of equations, see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The aim of this Chapter is to provide a complete and detailed proof of the solution to Hilbert’s XIXth problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will follow De Giorgi’s approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Existence and basic estimates We start by showing the existence and uniqueness of minimizers of E among the class of H1(Ω) functions with prescribed boundary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, we want a statement analogous to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10, but with the functional in- volving L instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We recall that we denote by u|∂Ω the trace of u on ∂Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' see (S5) in Chapter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3 (Existence and uniqueness of minimizers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that Ω ⊂ Rn is any bounded Lipschitz domain, and that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6) � w ∈ H1(Ω) : w|∂Ω = g � ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let L : Rn → R be smooth and uniformly convex, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let E(w) := � Ω L(∇w) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, there exists a unique minimizer u ∈ H1(Ω) with u|∂Ω = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, u solves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3) in the weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In order to prove the existence and uniqueness theorem for minimizers, we need first to show the following result on the lower semi-continuity of the energy in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We provide two different proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4 (Lower semi-continuity of the functional).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ Rn be a bounded domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let L : Rn → R be smooth and uniformly convex, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' and let E(w) := � Ω L(∇w) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 76 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Nonlinear variational PDE & Hilbert’s XIXth problem Then, E is weakly lower semi-continuous in H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, if H1(Ω) ∋ wk ⇀ w ∈ H1(Ω) weakly in H1(Ω), then E(w) ≤ lim inf k→∞ E(wk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' First proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us define the set A(t) := � v ∈ H1(Ω) : E(v) ≤ t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that, by convexity of E, A(t) is convex as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us show that it is closed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', if A(t) ∋ wk → w strongly in H1(Ω), then w ∈ A(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This simply follows by noticing that, up to a subsequence, ∇wk → ∇w almost everywhere, so that, by Fatou’s lemma, E(w) = � Ω L(∇w) ≤ lim inf k→∞ � Ω L(∇wk) ≤ t, that is, w ∈ A(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, A(t) is closed (with respect to the H1(Ω) convergence), and it is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By a standard result in functional analysis (closed and convex sets are weakly closed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' see, for example, [Bre11, The- orem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7]), A(t) is also closed under weak convergence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' namely, if A(t) ∋ wk ⇀ w weakly in H1(Ω) then w ∈ A(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us now consider a sequence weakly converging in H1(Ω), wk ⇀ w, and let us denote t∗ := lim infk→∞ E(wk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For any ε > 0, there exists some subsequence kj,ε such that wkj,ε ⇀ w weakly in H1(Ω) and E(wkj,ε) ≤ t∗ +ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, wkj,ε ∈ A(t∗ + ε), and therefore, since A(t) is weakly closed (in H1(Ω)) for all t, we have w ∈ A(t∗ +ε) and E(w) ≤ t∗ +ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since this can be done for any ε > 0, we reach that E(w) ≤ t∗, and therefore, we have shown the weak lower semi-continuity of E in H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Second proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us prove the lower semi-continuity of the functional by means of a different proof, from [Mag11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will actually show that if uk, u ∈ W 1,1(Ω) and uk → u in L1 loc(Ω), then � Ω L(∇u) ≤ lim inf k→∞ � Ω L(∇uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, since Ω is bounded, we can apply this result to the sequences in H1(Ω) converging weakly in H1(Ω) (by (S2) from Chapter 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let η ∈ C∞ c (B1) be a smooth function with η ≥ 0 and � B1 η = 1, and let ηε(x) = ε−nη(x/ε), so that we can consider the mollifications (uk)ε(x) := (uk ∗ ηε)(x) = � Bε u(x − y)ηε(y) dy, uε(x) := (u ∗ ηε)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω′ ⊂ Ω be such that for all x ∈ Ω′, Bε(x) ⊂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, since uk → u in L1 loc(Ω), we have ∇(uk)ε(x) → ∇uε(x) for every x ∈ Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' From — DRAFT — 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Existence and basic estimates 77 the smoothness of L we also have that L(∇(uk)ε(x)) → L(∇uε(x)) and by Fatou’s lemma (recall that we may assume L ≥ 0) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7) � Ω′ L(∇uε) ≤ lim inf k→∞ � Ω′ L(∇(uk)ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Noticing now that ∇(uk)ε = (∇uk)ε and using Jensen’s inequality (since L is convex and � ηε = 1) we have L(∇(uk)ε) = L �� Bε(x) ηε(x − y)∇uk(y) dy � ≤ � Bε(x) ηε(x−y)L(∇uk(y)) dy which leads to � Ω′ L(∇(uk)ε) ≤ � Ω′ �� Bε(x) ηε(x − y)L(∇uk(y)) dy � dx ≤ � Iε(Ω′) L(∇uk(y)) � Bε(y)∩Ω′ ηε(x − y) dx dy ≤ � Ω L(∇uk), where Iε(Ω′) ⊂ Ω denotes an ε-neighborhood of Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Combined with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7), this yields � Ω′ L(∇uε) ≤ lim inf k→∞ � Ω L(∇uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, since u ∈ W 1,1(Ω), we have ∇uε → ∇u as ε ↓ 0 almost everywhere in Ω′ and so, again by Fatou’s Lemma, we can let ε ↓ 0 to deduce � Ω′ L(∇u) ≤ lim inf k→∞ � Ω L(∇uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By taking an increasing sequence of sets Ω′ whose union is Ω we reach the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ We can now prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We divide the proof into three different parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If u is a local minimizer, then it solves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3) in the weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This follows from the fact that � Ω L(∇u) dx ≤ � Ω L(∇u + ε∇φ) dx for all ε, for all φ ∈ C∞ c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, notice that the integrals are bounded (L being uniformly convex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', at most quadratic at infinity, and ∇u ∈ L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since L is smooth, we can take a Taylor expansion L(∇u + ε∇φ) ≤ L(∇u) + εDL(∇u)∇φ + ε2 2 |∇φ|2 sup p∈Rn ��D2L(p) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 78 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Nonlinear variational PDE & Hilbert’s XIXth problem Recalling from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) that ��D2L �� is bounded by Λ, and plugging it back into the integral we obtain −Λε 2|∇φ|2 ≤ � Ω DL(∇u)∇φ dx for all ε > 0, for all φ ∈ C∞ c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Letting ε go to zero, we reach that � Ω DL(∇u)∇φ dx ≥ 0 for all φ ∈ C∞ c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, taking −φ instead of φ, we reach the equality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2), as we wanted to see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us now show the existence of a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since L is uniformly convex (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1)) it has a unique minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, there exists pL ∈ Rn such that L(p) ≥ L(pL) for all p ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, since L is smooth, ∇L(pL) = 0 and thus, from the uniform convexity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) we have that 0 < λ|p|2 ≤ L(p − pL) − L(pL) ≤ Λ|p|2, for all p ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Without loss of generality, by taking ˜L(p) = L(p − pL) − L(pL) if neces- sary, we may assume that L(0) = 0 and ∇L(0) = 0, so that we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8) 0 < λ|p|2 ≤ L(p) ≤ Λ|p|2, for all p ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (Notice that we may assume that because if u is a minimizer for L, then u + ⟨pL, x⟩ is a minimizer for ˜L, since the domain is bounded and therefore the integral of L(pL) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') Let E◦ = inf �� Ω L(∇w) dx : w ∈ H1(Ω), w|∂Ω = g � , that is, the infimum value of E(w) among all admissible functions w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that, by assumption (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6), such infimum exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, if w ∈ H1(Ω), by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8) we have that E(w) = � Ω L(∇w) ≤ Λ � Ω |∇w|2 = Λ∥∇w∥2 L2(Ω) < ∞ that is, the energy functional is bounded for functions in H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us take a minimizing sequence of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, we take {uk} such that uk ∈ H1(Ω), uk|∂Ω = g, and E(uk) → E◦ as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We begin by showing that E(uk) are bounded, and that uk is a sequence bounded in H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8), λ∥∇uk∥2 L2(Ω) ≤ λ � Ω |∇uk|2 ≤ � Ω L(∇uk) ≤ E(uk) < ∞, — DRAFT — 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Existence and basic estimates 79 That is, since E(uk) is uniformly bounded (being a convergent sequence with non-infinite elements), we reach that ∥∇uk∥2 L2(Ω) is uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, by the Poincar´e inequality (see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6) the sequence uk is uni- formly bounded in H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, there exists a subsequence ukj converging strongly in L2(Ω) and weakly in H1(Ω) to some u ∈ H1(Ω), uk ⇀ u weakly in H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By the weak lower semi-continuity (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4) we reach that E(u) ≤ lim inf k→∞ E(uk) = E◦, so that E(u) = E◦ (by minimality) and therefore u is a minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We finish the proof by showing the uniqueness of such minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This follows from the uniform convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, since L is uniformly convex, if p ̸= q, then L(p) + L(q) 2 > L �p + q 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u, v ∈ H1(Ω) be two distinct minimizers with the same boundary data (E(u) = E(v) = E◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, ∇u ̸≡ ∇v in Ω, so that U := {x ∈ Ω : ∇u ̸= ∇v} ⊂ Ω has positive measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, L(∇u) + L(∇v) 2 > L �∇u + ∇v 2 � in U, so that, since |U| > 0, 1 2 � U � L(∇u) + L(∇v) � > � U L �∇u + ∇v 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since the integrals are equal in Ω \\ U, we reach E◦ = 1 2 � Ω � L(∇u) + L(∇v) � > � Ω L �∇u + ∇v 2 � ≥ E◦, where the last inequality comes from the minimality of E◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We have reached a contradiction, and thus, the minimizer is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ We next give a complete and rigorous proof of the formal argumentation from the previous section, where we explained that C1 solutions are C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u ∈ H1(Ω) be a local minimizer of E(w) = � Ω L(∇w) dx, with L uniformly convex and smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that u ∈ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then u ∈ C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 80 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Nonlinear variational PDE & Hilbert’s XIXth problem Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We know that if u ∈ C1 and u is a minimizer of E(w), then � Ω DL(∇u(x))∇φ(x) dx = 0 for all φ ∈ C∞ c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let h ∈ Rn, and assume that |h| is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We have, in particular, that � Ω � DL(∇u(x + h)) − DL(∇u(x)) � ∇φ(x) dx = 0 for all φ ∈ C∞ c (Ωh), where Ωh := {x ∈ Ω : dist(x, ∂Ω) > |h|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that, by the fundamental theorem of calculus for line integrals, we can write DL(∇u(x + h)) − DL(∇u(x)) = = � 1 0 D2L � t∇u(x + h) + (1 − t)∇u(x) �� ∇u(x + h) − ∇u(x) � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If we define ˜A(x) := � 1 0 D2L � t∇u(x + h) + (1 − t)∇u(x) � dt, then ˜A(x) is uniformly elliptic (since L is uniformly convex), and continuous (since L is smooth and ∇u is continuous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, by the previous argumen- tation, � Ω ∇ �u(x + h) − u(x) |h| � ˜A(x)∇φ(x) dx = 0 for all φ ∈ C∞ c (Ωh), that is, u(·+h)−u |h| solves weakly div � ˜A(x)∇ �u(x + h) − u(x) |h| �� = 0 for x ∈ Ωh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, notice that u(·+h)−u |h| is C1 for all h ̸= 0, since u is C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, by the Schauder-type estimates for operators in divergence form and continuous coefficients(Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='32), ���� u(· + h) − u |h| ���� Cβ(Bρ/2(x◦)) ≤ C(ρ) ���� u(· + h) − u |h| ���� L∞(Bρ(x◦)) ≤ C, for all Bρ(x◦) ⊂ Ωh and β ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In the last inequality we used that ∇u is continuous (and thus, bounded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that the constant C(ρ) is independent of h (but might depend on β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, from (H7) in Chapter 1, namely (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7) with α = 1, we obtain that u ∈ C1,β(Ωh) for all h ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Letting |h| ↓ 0 we get that u ∈ C1,β inside Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We want to repeat the previous reasoning, noticing now that ˜A(x) is C0,β (since ∇u ∈ C0,β and L is smooth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, u(·+h)−u |h| ∈ C1,β(Ω) and — DRAFT — 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' De Giorgi’s proof 81 fulfills div � ˜A(x)∇ �u(x + h) − u(x) |h| �� = 0 for x ∈ Ωh, in the weak sense, with ˜A ∈ Cβ and uniformly elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='28, ���� u(· + h) − u |h| ���� C1,β(Bρ/2) ≤ C(ρ) ���� u(· + h) − u |h| ���� L∞(Bρ) ≤ C, for all Bρ ⊂ Ωh, and again, thanks to (H7), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7), we obtain that u ∈ C2,β(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We can now proceed iteratively using the higher order interior Schauder estimates in divergence form (Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='29) to obtain that u ∈ Ck(Ω) for all k ∈ N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e, u ∈ C∞ inside Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that in the formal proof (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5) we were using Schauder estimates in non-divergence form, since we were already assuming that the solution u was C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Here, in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5, we need to differentiate the equation (in incremental quotients) and then we obtain an equation in divergence form whose coefficients have the right regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, in the actual proof we are using Schauder estimates for equations in divergence form instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' De Giorgi’s proof The result of De Giorgi and Nash regarding the regularity of solutions to equations with bounded measurable coefficients is the following (see the discussion in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7 (De Giorgi–Nash).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let v ∈ H1(Ω) be any weak solution to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9) div (A(x)∇v) = 0 in Ω, with 0 < λId ≤ A(x) ≤ ΛId.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, there exists some α > 0 such that v ∈ C0,α(˜Ω) for any ˜Ω ⊂⊂ Ω, with ∥v∥C0,α(˜Ω) ≤ C∥v∥L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The constant C depends only on n, λ, Λ, Ω, and ˜Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The constant α > 0 depends only on n, λ, and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This theorem yields Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2, and combined with previous discus- sions, solved Hilbert’s XIXth problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, if u ∈ H1(Ω) is any local minimizer of E(w) = � Ω L(∇w) dx, then any derivative of u, v = ∂eu, solves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thanks to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7 we will have u ∈ H1(Ω) ⇒ v ∈ L2(Ω) ======⇒ DeGiorgi −Nash v ∈ C0,α(˜Ω) ⇒ u ∈ C1,α ======⇒ Schauder u ∈ C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 82 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Nonlinear variational PDE & Hilbert’s XIXth problem This will be proved in detail in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7 is significantly different in spirit than all the results on elliptic regularity which existed before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Most of the previous results can be seen as perturbation of the Laplace equation (they are perturbative results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In Schauder-type estimates, we always use that, when zooming in a solution at a point, the operator gets closer and closer to the Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In De Giorgi’s theorem, this is not true anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The uniform ellipticity is preserved by scaling, but the equation is not better, nor closer to the Laplace equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' General ideas of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will follow the approach of De Giorgi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' From now on, we denote L any operator of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10) Lv := −div(A(x)∇v), where A(x) is uniformly elliptic with ellipticity constants 0 < λ ≤ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By a standard covering argument (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15), we only need to prove the estimate for Ω = B1 and ˜Ω = B1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Throughout the proof, we will use that, if v solves Lv = 0, then ˜v(x) := Cv(x◦+rx) solves an equation of the same kind, ˜L˜v = 0, for some operator ˜L with the same ellipticity constants as L — given by ˜L˜v = div � A(x◦+rx)∇˜v � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' De Giorgi’s proof is split into two steps: First step: Show that ∥v∥L∞ ≤ C∥v∥L2 Second step: Show that ∥v∥C0,α ≤ C∥v∥L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In the first step, we work on the family of balls (see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) ˜Bk := � x : |x| ≤ 1 2 + 2−k−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Note that ˜B0 = B1, and ˜Bk converges to B1/2 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We assume ∥v∥L2(B1) ≤ δ ≪ 1 and then consider the truncated functions vk := (v − Ck)+ with Ck := 1 − 2−k, and the numbers Vk ≈ � ˜Bk |vk|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, the main point is to derive an estimate of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11) Vk ≤ CkV β k−1 for some β > 1, for some constant C depending only on n, λ, and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This previous inequality implies that Vk → 0 as k → ∞ if V0 is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, v∞ = (v − 1)+ is equal to zero in B1/2, and so v ≤ 1 in B1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' De Giorgi’s proof 83 B1/2 B1 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Representation of the family of balls ˜Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that our equation Lv = 0 in B1 is linear, while the bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11) is nonlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The “game” consists in using the Sobolev inequality (which gives control of Lp norms of vk in terms of L2 norms of ∇vk), combined with an energy inequality, which gives a “reversed” Poincar´e inequality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', a control of ∥∇vk∥L2 in terms of ∥vk∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Once we have the first step v ∈ L2 ⇒ v ∈ L∞, the second step consists of showing an oscillation-decay lemma Lv = 0 in B1 =⇒ osc B1/2 v ≤ (1 − θ) osc B1 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This implies the C0,α regularity of v (as we saw in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In the next proofs we follow [CV10, Vas16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' De Giorgi’s first step: from L2 to L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The two main ingredients are the Sobolev inequality ∥v∥Lp(Rn) ≤ C∥∇v∥L2(Rn), p = 2n n − 2, (see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4) and the following energy inequality (the Caccioppoli in- equality): — DRAFT — 84 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Nonlinear variational PDE & Hilbert’s XIXth problem Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8 (Energy inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let v ∈ H1(B1) with v ≥ 0 such that Lv ≤ 0 in B1, for some L of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, for any ϕ ∈ C∞ c (B1) we have � B1 |∇(ϕv)|2 dx ≤ C∥∇ϕ∥2 L∞(B1) � B1∩supp ϕ v2 dx, where C depends only on n, λ, and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that the weak formulation of −div(A(x)∇v) ≤ 0 in B1 is � B1 ∇η · A(x)∇v dx ≤ 0 for all η ∈ H1 0(B1), η ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Take η = ϕ2v, to get � B1 ∇(ϕ2v) · A(x)∇v dx ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, we want to “bring one of the ϕ from the first gradient to the second gradient”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, using ∇(ϕ2v) = ϕ∇(ϕv) + (ϕv)∇ϕ, ∇(ϕv) = ϕ∇v + v∇ϕ, we get 0 ≥ � B1 ∇(ϕ2v) · A(x)∇v dx = � B1 ϕ∇(ϕv) · A(x)∇v dx + � B1 ϕv ∇ϕ · A(x)∇v dx = � B1 ∇(ϕv) · A(x)∇(ϕv) dx − � B1 v∇(ϕv) · A(x)∇ϕ dx + � B1 ϕv ∇ϕ · A(x)∇v dx = � B1 ∇(ϕv) · A(x)∇(ϕv) dx − � B1 v∇(ϕv) · (A(x) − AT (x))∇ϕ dx − � B1 v2∇ϕ · A(x)∇ϕ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' De Giorgi’s proof 85 Let us first bound the term involving (A − AT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By H¨older’s inequality, using the uniform ellipticity of A and that (A − AT )2 ≤ 4Λ2Id, we get � B1 v∇(ϕv) · (A(x) − AT (x))∇ϕ dx ≤ �� B1 |v (A(x) − AT (x))∇ϕ|2 dx � 1 2 �� B1 |∇(ϕv)|2 dx � 1 2 ≤ 2 Λ λ 1 2 �� B1 |v∇ϕ|2 dx � 1 2 �� B1 ∇(ϕv)A(x)∇(ϕv) dx � 1 2 ≤ 1 2 � B1 ∇(ϕv)A(x)∇(ϕv) dx + 2Λ2 λ � B1 |v∇ϕ|2 dx, where in the last inequality we are using that 2ab ≤ a2 + b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Combining the previous inequalities, we obtain that 2Λ2 λ � B1 |v∇ϕ|2 dx ≥ 1 2 � B1 ∇(ϕv) · A(x)∇(ϕv) dx − � B1 v2∇ϕ · A(x)∇ϕ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, we deduce λ � B1 |∇(ϕv)|2 dx ≤ � B1 ∇(ϕv) · A(x)∇(ϕv) dx ≤ 2 � B1 v2 ∇ϕ · A(x)∇ϕ dx + 4Λ2 λ � B1 |v∇ϕ|2 dx ≤ � 2Λ + 4Λ2 λ � ∥∇ϕ∥2 L∞(B1) � B1∩supp ϕ v2 dx, and the lemma is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ We will use the energy inequality (from the previous lemma) applied to the function v+ := max{v, 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Before doing so, let us show that if Lv ≤ 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', v is a subsolution), then Lv+ ≤ 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', v+ is a subsolution as well).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (More generally, the maximum of two subsolutions is always a subsolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let L be of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10), let v ∈ H1(B1) be such that Lv ≤ 0 in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, Lv+ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We proceed by approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let F ∈ C∞(R) be a smooth, non- decreasing, convex function, with globally bounded first derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We start by showing that L(F(v)) ≤ 0 in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that if v ∈ W 1,2(B1), then F(v) ∈ W 1,2(B1) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 86 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Nonlinear variational PDE & Hilbert’s XIXth problem We know that L(v) ≤ 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', � B1 ∇η · A∇v dx ≤ 0 for all η ∈ H1 0(B1), η ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us now compute, for any fixed η ∈ H1 0(Ω) satisfying η ≥ 0, L(F(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that the weak formulation still makes sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' � B1 ∇η · A∇F(v) dx = � B1 F ′(v)∇η · A∇v dx = � B1 ∇(F ′(v)η) · A∇v dx − � B1 ηF ′′(v)∇v · A∇v dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The first term is non-positive, since F ′(v)η ∈ H1 0(B1) and F ′(v) ≥ 0 (F is non-decreasing), so that F ′(v)η is an admissible test function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The second term is also non-positive, since ηF ′′(v) ≥ 0 and ∇v · A∇v ≥ 0 by ellipticity (and the integral is well defined, since ηF ′′(v) can be assumed to be bounded by approximation, and � B1 ∇v · A∇v ≤ Λ∥∇v∥2 L2(B1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, � B1 ∇η · A∇F(v) dx ≤ 0, and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We finish by taking smooth approximations of the positive part function, Fε, converging uniformly in compact sets to F(x) = max{x, 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that this can be done in such a way that ∥Fε(v)∥W 1,2(B1) ≤ C, for some C independent of ε > 0, which gives the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ We want to prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10 (from L2 to L∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let L be of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10), and let v ∈ H1(B1) be a solution to Lv ≤ 0 in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then ∥v+∥L∞(B1/2) ≤ C∥v+∥L2(B1), for some constant C depending only on n, λ, and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will prove, in fact, the following (which is actually equivalent): Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11 (from L2 to L∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let L be of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' There exists a constant δ > 0 depending only on n, λ, and Λ, such that if v ∈ H1(B1) solves Lv ≤ 0 in B1 and � B1 v2 + ≤ δ, then v ≤ 1 in B1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' De Giorgi’s proof 87 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Define, as introduced in the general ideas of the proof, for k ≥ 0, ˜Bk := � |x| ≤ 1 2 + 2−k−1 � , vk := (v − Ck)+ with Ck = 1 − 2−k, and let ϕk be a family of shrinking cut-off functions 0 ≤ ϕk ≤ 1 that fulfill ϕk ∈ C∞ c (B1), ϕk = � 1 in ˜Bk 0 in ˜Bc k−1 , and |∇ϕk| ≤ C2k in ˜Bk−1\\ ˜Bk, where C here depends only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Vk := � B1 ϕ2 kv2 k dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, the Sobolev inequality, and the energy inequality (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8) give �� B1 |ϕk+1vk+1|p dx � 2 p ≤ C �� B1 |∇(ϕk+1vk+1)|2 dx � ≤ C22k � ˜Bk |vk+1|2 dx ≤ C22k � B1 (ϕkvk)2 dx = C22kVk, for p = 2n n−2 if n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If n = 1 or n = 2, we can take p = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, by H¨older’s inequality, Vk+1 = � B1 ϕ2 k+1v2 k+1 dx ≤ �� B1 (ϕk+1vk+1)p dx � 2 p ��{ϕk+1vk+1 > 0} ��γ, where γ := 2 n (if n = 1 or n = 2, γ = 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Here, we are using that � A f ≤ ( � A |f|p/2)2/p|A|γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, from Chebyshev’s inequality and the definition of vk and ϕk, ��{ϕk+1vk+1 > 0} �� ≤ ��{ϕkvk > 2−k−1} �� = ��{ϕ2 kv2 k > 2−2k−2} �� ≤ 22(k+1) � B1 ϕ2 kv2 k dx = 22(k+1)Vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Apart from Chebyshev’s inequality, we are using here that if vk+1 > 0 and ϕk+1 > 0, then vk > 2−k−1 and ϕk = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, combining the previous — DRAFT — 88 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Nonlinear variational PDE & Hilbert’s XIXth problem inequalities, we get Vk+1 ≤ �� B1 (ϕk+1vk+1)p dx � 2 p ��{ϕk+1vk+1 > 0} ��γ ≤ C22kVk � 22(k+1)Vk �γ ≤ Ck+1V 1+γ k , where we recall γ = 2 n if n ≥ 3, and γ = 1 2 otherwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' and C depends only on n, λ, and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, we claim that, if δ > 0 is small enough, then � 0 ≤ Vk+1 ≤ Ck+1V 1+γ k 0 ≤ V0 ≤ δ =⇒ Vk → 0 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, in order to see this it is enough to check by induction that if V0 ≤ C−1/γ−1/γ2 then V γ k ≤ C−k−1 (2C) 1 γ , which is a simple computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Alternatively, one could check by induction that Vk ≤ C (1+γ)k��k i=1 i (1+γ)i � V (1+γ)k 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Hence, we have proved that Vk = � B1 (ϕkvk)2 dx → 0 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Passing to the limit, we get� B1/2 (v − 1)2 + dx = 0, and thus, v ≤ 1 in B1/2, as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' To deduce the Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10 from Propo- sition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11, just use ˜v := √ δ v/∥v+∥L2(B1) (which solves the same equa- tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ This proves the first part of the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12) Lv ≤ 0 in B1 =⇒ ∥v+∥L∞(B1/2) ≤ C∥v+∥L2(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that, as a direct consequence, we have the L2 to L∞ estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, if Lv = 0 then Lv+ ≤ 0 (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9) but also Lv− ≤ 0, where v− := max{0, −v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, ∥v−∥L∞(B1/2) ≤ C∥v−∥L2(B1), and since ∥v∥L2(B1) = ∥v+∥L2(B1) + ∥v−∥L2(B1), combining the estimate for v+ and v− we get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13) Lv = 0 in B1 =⇒ ∥v∥L∞(B1/2) ≤ C∥v∥L2(B1), as we wanted to see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' De Giorgi’s proof 89 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12 (Moser’s proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12) here presented is the original proof of De Giorgi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The first ingredient in the proof was to use ϕ2v+ as a test function in the weak formulation of our PDE to get the energy inequality from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8, � B1 |∇(ϕv)2| dx ≤ C � B1 v2|∇ϕ|2 dx for all ϕ ∈ C∞ c (B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Roughly speaking, this inequality said that v cannot jump too quickly (the gradient is controlled by v itself).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moser did something similar, but taking η = ϕ2(v+)β instead, for some β ≥ 1, to get the inequality � B1 ����∇ � v β+1 2 ϕ �2���� dx ≤ C � B1 vβ+1|∇ϕ|2 dx for all ϕ ∈ C∞ c (B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Combining this with Sobolev’s inequality, one gets �� Br1 vqγ dx � 1 qγ ≤ � C |r2 − r1|2 � Br2 vq dx � 1 q , where γ = 2∗ 2 > 1 and q = β + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Taking a sequence of rk ↓ 1 2 as in De Giorgi’s proof, one obtains ∥v∥L2γk(Brk) ≤ C∥v∥L2(B1), and taking k → ∞ we obtain the L∞ bound in B1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer the interested reader to [HL97, Chapter 4] for a full proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' De Giorgi’s second step: L∞ to C0,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We next prove the second step of De Giorgi’s estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We want to prove: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13 (Oscillation decay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let L be of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let v ∈ H1(B2) be a solution to Lv = 0 in B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, osc B1/2 v ≤ (1 − θ) osc B2 v for some θ > 0 small depending only on n, λ, and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As we saw in Chapter 2 (see Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7), this proposition immediately implies C0,α regularity of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As shown next, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13 follows from the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 90 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Nonlinear variational PDE & Hilbert’s XIXth problem B2 1 v ≤ 1 B1 v ≤ 1 − γ in B1/2 |{v ≤ 0} ∩ B1| ≥ µ > 0 1 − γ B1/2 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Graphical representation of v with Lv ≤ 0 from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let L be of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10), and let v ∈ H1(B2) be such that v ≤ 1 in B2, and Lv ≤ 0 in B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that ��{v ≤ 0} ∩ B1 �� ≥ µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, sup B1/2 v ≤ 1 − γ, for some small γ > 0 depending only on n, λ, Λ, and µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In other words, if v ≤ 1, and it is “far from 1” in a set of non-zero measure, then v cannot be close to 1 in B1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (See Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') Let us show how this lemma yields the oscillation decay: Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Consider the function w(x) := 2 oscB2 v � v(x) − supB2 v + infB2 v 2 � and notice that −1 ≤ w ≤ 1 in B2, (in fact, oscB2 w = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us assume that ��{w ≤ 0}∩B1 �� ≥ 1 2|B1| (otherwise, we can take −w instead).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14, we get w ≤ 1 − γ in B1/2, — DRAFT — 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' De Giorgi’s proof 91 and thus osc B1/2 w ≤ 2 − γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This yields osc B1/2 v ≤ � 1 − γ 2 � osc B2 v, and thus the proposition is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ To prove Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14, we will need the following De Giorgi isoperimetric inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It is a kind of a quantitative version of the fact that an H1 function cannot have a jump discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let w ∈ H1(B1) be such that � B1 |∇w|2 dx ≤ C◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let A := {w ≤ 0} ∩ B1, D := � w ≥ 1 2 � ∩ B1, E := � 0 < w < 1 2 � ∩ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, we have C◦|E| ≥ c|A|2 · |D|2 for some constant c depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We define ¯w in B1 as ¯w = w in E, ¯w ≡ 0 in A and ¯w = 1 2 in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In this way, ∇ ¯w ≡ 0 in B1 \\ E and � B1 |∇ ¯w|2 ≤ C◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us denote the average of ¯w in B1 by ¯wB1 := � B1 ¯w(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, |A| · |D| ≤ 2 � A � D | ¯w(x) − ¯w(y)| dx dy ≤ 2 � B1 � B1 ��� ¯w(x) − ¯wB1 �� + �� ¯w(y) − ¯wB1 ��� dx dy = 4|B1| � B1 �� ¯w(x) − ¯wB1 �� dx ≤ C � E |∇ ¯w(x)| dx, where in the last step we have used the Poincar´e inequality (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6 with p = 1) and the fact that ∇ ¯w ≡ 0 in B1\\E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, by H¨older’s inequality we reach |A| · |D| ≤ C � E |∇ ¯w| ≤ C �� E |∇ ¯w|2 �1/2 |E|1/2 ≤ CC1/2 |E|1/2, as we wanted to see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Finally, we prove Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14: — DRAFT — 92 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Nonlinear variational PDE & Hilbert’s XIXth problem Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Consider the sequence wk := 2k � v − (1 − 2−k) � + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that wk ≤ 1 in B2 since v ≤ 1 in B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, Lwk ≤ 0 in B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Using the energy inequality (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8), we easily get that � B1 |∇wk|2 ≤ C � B2 w2 k ≤ C◦ (notice 0 ≤ wk ≤ 1 in B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We also have ��{wk ≤ 0} ∩ B1 �� ≥ µ > 0 (by the assumption on v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We now apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15 recursively to wk, as long as � B1 w2 k+1 ≥ δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We get ���� � wk ≥ 1 2 � ∩ B1 ���� ≥ ��{wk+1 > 0} ∩ B1 �� ≥ � B1 w2 k+1 ≥ δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15, ���� � 0 < wk < 1 2 � ∩ B1 ���� ≥ c C◦ δ4µ2 = β > 0, where β > 0 is independent of k, and depends only on n, δ, and µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' But notice that the sets � 0 < wk < 1 2 � are disjoint for all k ∈ N, therefore we cannot have the previous inequality for every k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This means that, for some k◦ ∈ N (depending only on n and β) we have � B1 w2 k◦ < δ2 and, hence, by the L2 to L∞ estimate from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10 ∥w+∥L∞(B1/2) ≤ C∥w+∥L2(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We get ∥wk◦∥L∞(B1/2) ≤ Cδ ≤ 1 2, provided that δ > 0 is small enough, depending only on n, λ, and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This means that wk◦ ≤ 1 2 in B1/2, and thus v ≤ 1 22−k◦ + � 1 − 2−k◦� ≤ 1 − 2−k◦−1 = 1 − γ as desired, where k◦ (and therefore, γ) depends only on n, λ, Λ, and µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ — DRAFT — 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' De Giorgi’s proof 93 Summarizing, we have now proved Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14 (by using the L2 to L∞ estimate and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14 implies the oscillation decay, and the oscillation decay implies the H¨older regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let L be of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10), and let v ∈ H1(B1) solve Lv = 0 in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, ∥v∥C0,α(B1/2) ≤ C∥v∥L∞(B1) for some α > 0 and C depending only on n, λ, and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The theorem follows from the oscillation decay, in much the same way as Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Combining this last result with the L2 to L∞ estimate, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10, we finally obtain the theorem of De Giorgi–Nash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let v ∈ H1(B1) be a weak solution to div (A(x)∇v) = 0 in B1, with 0 < λ Id ≤ A(x) ≤ Λ Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, there exists some α > 0 such that v ∈ C0,α(B1/2) and ∥v∥C0,α(B1/2) ≤ C∥v∥L2(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The constants C and α > 0 depend only on n, λ, and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The result follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16 combined with Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10 (by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ As a consequence of the previous result, we have: Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It follows by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17 by a covering argu- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ In particular, as shown below, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17 solved Hilbert’s XIXth problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is one of the main results for which De Giorgi got the Wolf Prize in 1990, and Nash got the Abel Prize in 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It has been speculated that if only one of them had solved Hilbert’s XIXth problem, he would also have received the Fields Medal for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18 (Harnack’s inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Even though it is not needed to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17, it is interesting to notice that with some more work one can also prove Harnack’s inequality for operators of the form div (A(x)∇v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' see [LZ17, Mos61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 94 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Nonlinear variational PDE & Hilbert’s XIXth problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Solution to Hilbert’s XIXth problem In this chapter, we have proved the interior regularity result for v ∈ H1(B1) div � A(x)∇v � = 0 in B1 =⇒ ∥v∥C0,α(B1/2) ≤ C∥v∥L2(B1), for some small α > 0 depending only on n, λ, and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For a general domain Ω ⊂ Rn, this gives the estimate for v ∈ H1(Ω) div � A(x)∇v � = 0 in Ω =⇒ ∥v∥C0,α(˜Ω) ≤ C∥v∥L2(Ω), for any ˜Ω ⊂⊂ Ω (with a constant C that depends only on n, λ, Λ, Ω, and ˜Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thanks to this, one can in fact solve Hilbert’s XIXth problem: Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u ∈ H1(Ω) be any local minimizer of E(w) := � Ω L(∇w) dx, where L is smooth and uniformly convex, and Ω ⊂ Rn is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u is C∞ in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Any local minimizer u satisfies � Ω DL(∇u)∇φ dx = 0, for all φ ∈ C∞ c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (This is the weak formulation of div(DL(∇u)) = 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') As in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5, if we define for any h ∈ Rn, ˜A(x) := � 1 0 D2L � t∇u(x + h) + (1 − t)∇u(x) � dt, then ˜A(x) is uniformly elliptic (since L is uniformly convex, 0 < λId ≤ D2L(p) ≤ ΛId).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We have that u(·+h)−u |h| ∈ H1(Ωh) fulfills (again, see Theo- rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5) � Ω ∇ �u(x + h) − u(x) |h| � ˜A(x)∇φ(x) dx = 0 for all φ ∈ C∞ c (Ωh), that is, u(·+h)−u |h| solves weakly div � ˜A∇ �u(· + h) − u |h| �� = 0, in Ωh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (We recall Ωh := {x ∈ Ω : dist(x, ∂Ω) > |h|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') — DRAFT — 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Further results and open problems 95 By the estimate of De Giorgi and Nash (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7), we find that ���� u(· + h) − u |h| ���� C0,α(˜Ω) ≤ C ���� u(· + h) − u |h| ���� L2(Ωh) ≤ C∥∇u∥L2(Ω) for any ˜Ω ⊂⊂ Ωh (see (S8) in Chapter 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By (H7), since the constant C is independent of h, this yields ∥u∥C1,α(˜Ω) ≤ C∥u∥H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, once u is C1,α, we are done by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Further results and open problems Let us finish this chapter by mentioning some state-of-the art results and open problems regarding the minimization of convex energy functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As we have explained, the minimization of a convex functional is a clas- sical problem in the Calculus of Variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Namely, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14) min w∈W � Ω L(∇w) dx, with L : Rn → R convex, Ω ⊂ Rn, and some appropriate class of functions W, say, with prescribed trace on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Hilbert’s XIXth problem deals with the case in which L is uniformly convex and smooth, to obtain nice reg- ularity results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1 we discuss that lack of convexity can yield non-uniqueness of minimizers, but it is not that clear what occurs if we sim- ply remove the condition on the uniform convexity, but maintain the strict convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In fact, functionals involving functions L that only involve strict convexity (that is, D2L could have 0 and ∞ as eigenvalues in some sets) appear naturally in some applications: anisotropic surface tensions, traffic flow, and statistical mechanics (see [Moo20] and the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Minimizers of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14) are known to be Lipschitz (under enough smooth- ness of the domain and boundary data) by the comparison principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, the following natural question is to whether first derivatives of minimizers are continuous: If L is strictly convex, are minimizers to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14) C1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The answer to that question has been investigated in the last years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The problem was first addressed by De Silva and Savin in [DS10], where they studied the case of dimension 2: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19 ([DS10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be a Lipschitz minimizer to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14) in R2, and suppose that L is strictly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that the set of points where D2L has some eigenvalue equal to 0 or ∞ is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 96 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Nonlinear variational PDE & Hilbert’s XIXth problem Later, Mooney in [Moo20] studied the problem in higher dimensions and showed that the question has a negative answer, in general, in dimen- sions n ≥ 4: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20 ([Moo20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In R4 there exists a Lipschitz minimizer to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14), with L strictly convex, that is not C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In the example by Mooney, the minimizer is analytic outside the origin (having a singularity there), and the corresponding functional has a Hessian with an eigenvalue going to ∞ in {x2 1 +x2 2 = x2 3 +x2 4}∩ √ 2S3, but otherwise, the eigenvalues are uniformly bounded from below away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It is currently an open question what happens in dimension n = 3, as well as what happens for general strictly convex functionals in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — Chapter 4 Fully nonlinear elliptic PDE Second order nonlinear elliptic PDEs in their most general form can be written as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) F(D2u, ∇u, u, x) = 0 in Ω ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Understanding the regularity of solutions to these equations has been a major research direction since the mid-20th century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' These are called fully nonlinear elliptic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Besides their own interest in PDE theory, they arise in Probability Theory (stochastic control, differential games;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' see Appendix C for a probabilistic interpretation), and in Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thanks to Schauder-type estimates, under natural assumptions on the dependence on ∇u, u, and x, the regularity for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) can be reduced to understanding solutions to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2) F(D2u) = 0 in Ω ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, some of the “perturbative” methods that we used in Chapter 2 to prove Schauder estimates for linear equations � aij(x)∂iju = f(x) in Ω ⊂ Rn work in such fully nonlinear setting, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For simplicity, we will focus here on the study of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In the next sections we will discuss the following: – What is ellipticity for solutions to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' – Existence and uniqueness of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' – Regularity of solutions to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 97 — DRAFT — 98 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Fully nonlinear elliptic PDE We will not prove all the main known results of this Chapter, but only give an overview of what is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer to the books [CC95] and [NTV14] for more details about this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' What is ellipticity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' There are (at least) two possible ways to define ellipticity: – Linearizing the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' – “Imposing” that the comparison principle holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will see that they are essentially the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let F : Rn×n → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We say that F is elliptic if for any two symmetric matrices A, B ∈ Rn×n such that A ≥ B (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', A − B is positive semi-definite) we have F(A) ≥ F(B), with strict inequality if A > B (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', A − B positive definite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The Laplace equation ∆u = 0 corresponds to the case F(M) = tr M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For a linear equation (with constant coefficients) n � i,j=1 aij∂iju = 0, F is given by F(M) = tr (AM), where A = (aij)i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This equation is elliptic if and only if the coefficient matrix A is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, it coincides with our notion of ellipticity for linear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2 (Comparison Principle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If a C2 function v touches u ∈ C2 from below at a point x◦ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' u ≥ v everywhere, and u(x◦) = v(x◦);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1), then it follows that ∇u(x◦) = ∇v(x◦), D2u(x◦) ≥ D2v(x◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, for these functions we would have F(D2u(x◦)) ≥ F(D2v(x◦)) if F is elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is essential when proving the comparison principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3 (Comparison Principle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that F is elliptic, and Ω ⊂ Rn is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u, v ∈ C2(Ω) ∩ C0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, � u ≥ v on ∂Ω F(D2u) ≤ F(D2v) in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' =⇒ u ≥ v in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We separate into two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume first that F(D2u) < F(D2v) in Ω (with strict inequal- ity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If the conclusion is false, then the function u − v would have an interior minimum inside Ω, say at x◦ ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, we would have D2(u − — DRAFT — 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' What is ellipticity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 99 v x◦ u Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The function v touches u from below at x◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' v)(x◦) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, D2u(x◦) ≥ D2v(x◦) and by ellipticity of F, this yields F(D2u(x◦)) ≥ F(D2v(x◦)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is a contradiction with F(D2u) < F(D2v) in Ω, and hence u ≥ v in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume now F(D2u) ≤ F(D2v) in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, we can define ¯u(x) := u(x) + ε � cΩ − |x|2� , where cΩ > 0 is a constant such that cΩ − |x|2 > 0 in Ω (recall that Ω is bounded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, we have ¯u ≥ u on ∂Ω, and D2¯u = D2u−2εId.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, by ellipticity, F(D2¯u) = F(D2u − 2εId) < F(D2u) ≤ F(D2v) in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By Case 1, � ¯u ≥ v on ∂Ω, F(D2¯u) < F(D2v) in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' =⇒ ¯u ≥ v in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This gives u(x) + ε � cΩ − |x|2� ≥ v(x) in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Letting ε ↓ 0 we deduce that u ≥ v in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Thus, we see that ellipticity is exactly what we need in order to prove the comparison principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will see that uniform ellipticity (analogously to the case of linear equations) implies, in fact, the regularity of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let F : Rn×n → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then F is uniformly elliptic if there are 0 < λ ≤ Λ (the ellipticity constants), such that for every symmetric matrices M, N with N ≥ 0 (that is, positive semi-definite), we have λ∥N∥ ≤ F(M + N) − F(M) ≤ Λ∥N∥, where ∥N∥ := tr � (NT N)1/2� = tr(N) is the sum of the (absolute value of the) eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 100 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Fully nonlinear elliptic PDE We remark that our choice of matrix norm in the previous definition is not standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In Rn, all norms are equivalent and thus we could have chosen any other norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This definition of norm, however, avoids dealing with constants in future computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Of course, uniform ellipticity implies ellipticity, in a quantitative way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For linear equations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' F(M) = tr (AM), uniform ellipticity is equiv- alent to 0 < λId ≤ A ≤ ΛId, as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The alternative way to see ellipticity is by linearizing the equation: Assume F ∈ C1 (which is not always the case!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We consider the func- tions Fij(M) := ∂F ∂Mij (M), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', the first derivative of F(M) with respect to the component Mij of the matrix M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, it is immediate to see that F is uniformly elliptic ⇐⇒ 0 < λ Id ≤ (Fij(M))i,j ≤ Λ Id, ∀M ⇐⇒ the linearized equation is uniformly elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, at least when F is C1, uniform ellipticity can be seen as uniform ellipticity of the linearized equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In general, though, the uniform ellipticity condition implies that F is Lipschitz, but not always C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' There are, in fact, important examples of equations F(D2u) = 0 in which the corresponding F is Lipschitz but not C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In this case, the previous characterization of ellipticity through the deriva- tives of F still holds, understanding now that they are defined almost every- where.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5 (Convex (or concave) equations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' An important subclass of equations F(D2u) = 0 are those for which F is convex (or concave).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Namely, F(M) as a function F : Rn×n → R is convex (or concave).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In this case, the equation can be written as a Bellman equation (see (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3)), as F(D2u) = max α∈A{Lαu} = 0, where {Lα}α∈A is a family of linear operators of the form Lαu := n � i,j=1 aα ij∂iju + cα, for a family of coefficients {aα ij}α∈A uniformly elliptic, with ellipticity con- stants λ and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' What is ellipticity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 101 Notice that if u solves F(D2u) = 0, with F convex, then v = −u solves G(D2v) = 0, with G(M) = −F(−M), and therefore, G is concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Pucci operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Within the class of fully nonlinear uniformly elliptic operators with ellipticity constants λ and Λ, the extremal or Pucci operators, denoted by M+ and M−, are those that attain the extreme values (from above and below, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Alternatively, every other elliptic operator with the same ellipticity constants is ordered with respect to them in the sense of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We define M± as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Given 0 < λ ≤ Λ, the extremal or Pucci operators with ellipticity constants λ and Λ, M± : Rn×n → R, are defined as M−(M) := inf λId≤(aij)i,j≤ΛId � n � i,j=1 aijMij � = inf λId≤A≤ΛId {tr (AM)} M+(M) := sup λId≤(aij)i,j≤ΛId � n � i,j=1 aijMij � = sup λId≤A≤ΛId {tr (AM)} , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3) for any symmetric matrix M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' They are uniformly elliptic operators, with ellipticity constants λ and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, from the definition we have M±(αM) = αM±(M), for all α ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that M± = M± n,λ,Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In general, however, the dependence on the ellipticity constants and the dimension will be clear in the corresponding context, and thus we will drop it in the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Sometimes, it is easier to define the Pucci operators through the eigenval- ues of the corresponding matrix, appropriately weighted with the ellipticity constants, in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The Pucci operators as defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3) can be equivalently defined as M−(M) = λ � µi>0 µi + Λ � µi<0 µi = λ∥M+∥ − Λ∥M−∥, M+(M) = Λ � µi>0 µi + λ � µi<0 µi = Λ∥M+∥ − λ∥M−∥, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4) where µi = µi(M) denote the eigenvalues of the symmetric matrix M, the matrices M+ and M− are such that M± ≥ 0, M = M+ − M−, and ∥A∥ = tr � (AT A)1/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 102 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Fully nonlinear elliptic PDE Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The proof follows directly using the following rearrangement-type inequalities involving the eigenvalues and the product of two symmetric matrices A and B: n � i=1 λi(A)λn−i(B) ≤ tr(AB) ≤ n � i=1 λi(A)λi(B), where λ1(A) ≤ · · · ≤ λn(A) denote the ordered eigenvalues of A, and λ1(B) ≤ · · · ≤ λn(B) denote the ordered eigenvalues of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ From the definition of uniform ellipticity of F (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4) it follows that, given two symmetric matrices M, N, λ∥N+∥ − Λ∥N−∥ ≤ F(M + N) − F(M) ≤ Λ∥N+∥ − λ∥N−∥, where N = N+ − N−, and N± ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5) M−(N) ≤ F(M + N) − F(M) ≤ M+(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If we take M = 0, we see that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6) M−(N) ≤ F(N) − F(0) ≤ M+(N), so these operators are like the “worse case” from above and below — up to a constant, F(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (Recall that M± are fully nonlinear uniformly elliptic operators with ellipticity constants λ, Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') If we further assume that F(0) = 0, we see that if u solves any equation of the form F(D2u) = 0 then in particular (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7) M−(D2u) ≤ 0 ≤ M+(D2u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7) is called equation in non-divergence form with bounded measurable coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, notice that given some uniformly elliptic coefficients (aij(x))i,j with no regularity assumption on x, if u ∈ C2 fulfills � i,j aij(x)∂iju then in particular (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, if (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7) holds for some u ∈ C2, one can recover some uniformly elliptic coefficients (aij(x))i,j such that � i,j aij(x)∂iju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Equations in two variables Before going into the general theory of existence and regularity for fully non- linear equations in Rn, let us study a simpler case: fully nonlinear equations in two variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The main regularity estimate in this context is due to Nirenberg [Nir53], and reads as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let F : R2×2 → R be uniformly elliptic with ellipticity constants λ and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u ∈ C2(B1) solve F(D2u) = 0 in B1 ⊂ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Equations in two variables 103 Then, ∥u∥C2,α(B1/2) ≤ C∥u∥L∞(B1), for some constants α > 0 and C depending only on λ and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The idea of the proof is the following: define v := ∂eu, and differentiate the equation F(D2u) = 0 in the e direction, to get (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8) 2 � i,j=1 aij(x)∂ijv(x) = 0 in B1 ⊂ R2, where aij(x) := Fij(D2u(x)) for i, j ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since F is uniformly elliptic, we have a22(x) ≥ λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, we can divide (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8) by a22(x) to obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9) a(x)∂11v(x) + b(x)∂12v(x) + ∂22v(x) = 0, for some coefficients a(x) = a11(x) a22(x) and b(x) = a12(x) + a21(x) a22(x) = 2a12(x) a22(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If we write w := ∂1v and differentiate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9) with respect to x1, we get ∂1 � a(x)∂1w(x) + b(x)∂2w(x) � + ∂22w(x) = div(A(x)∇w) = 0, where A(x) := �a(x) b(x) 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, w solves an equation in divergence form, and A is uniformly elliptic, with ellipticity constants depending on λ and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, by the De Giorgi– Nash result (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7) one has ∂1v = w ∈ C0,α(B1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since the roles of x1 and x2 can be changed, and since v = ∂eu (with e ∈ Sn−1 arbitrary), we deduce that u ∈ C2,α(B1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us now formally prove it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The idea is the one presented in the lines above, where we used that u ∈ C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In reality we can only use that u ∈ C2, so we proceed by means of incremental quotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us define v(x) = u(x + h) − u(x) |h| ∈ C2(B1−|h|), with |h| < 1 4, and proceed similarly to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since F is translation invariant, we have F(D2u(x)) = 0, F(D2u(x + h)) = 0 in B1−|h|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, by the fundamental theorem of calculus for line integrals, 0 = F(D2u(x + h)) − F(D2u(x)) = 2 � i,j=1 aij(x)∂ij � u(x + h) − u(x) � , — DRAFT — 104 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Fully nonlinear elliptic PDE where aij(x) = � 1 0 Fij � tD2u(x + h) + (1 − t)D2u(x) � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since F is uniformly elliptic, (aij)i,j is uniformly elliptic (with the same ellipticity constants).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, v ∈ C2(B1−|h|) solves an equation in non- divergence form a11(x)∂11v(x) + 2a12(x)∂12v(x) + a22(x)∂22v(x) = 0 in B1−|h|, where a12 = a21 and ∂12v = ∂21v because v ∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' From the ellipticity conditions, we have λ ≤ a22(x) ≤ Λ, and we can divide by a22(x) to get a(x)∂11v(x) + b(x)∂12v(x) + ∂22v(x) = 0 in B1−|h|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let A(x) := �a(x) b(x) 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It is straightforward to check that A is uniformly elliptic, with ellipticity constants λ/Λ and Λ/λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let η ∈ C2 c (B1−|h|) and notice that, by integration by parts, � B1−|h| ∂2η ∂12v = � B1−|h| ∂1η ∂22v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, � B1−|h| ∇η · A(x)∇∂1v = � B1−|h| ∇η(x) · �a(x)∂11v(x) + b(x)∂12v(x) ∂12v(x) � dx = � B1−|h| � ∂1η � a(x)∂11v + b(x)∂12v � + ∂2η ∂12v � dx = � B1−|h| ∂1η � a(x)∂11v + b(x)∂12v + ∂22v � dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, ∂1v solves an equation with bounded measurable coefficients A(x) in divergence form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, by the De Giorgi–Nash theorem (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16), we know that ∂1v ∈ Cα and ∥∂1v∥C0,α(B1/2) ≤ C∥∂1v∥L∞(B1−|h|) ≤ C∥∂1u∥C0,1(B1), (notice that we can go from B1 to B1−|h| in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16 by a covering argument for |h| small), for some constant C depending only on λ and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By letting |h| → 0, thanks to (H7), we obtain that ∥∇∂1u∥C0,α(B1/2) ≤ C∥∂1v∥L∞(B1−|h|) ≤ C∥∂1u∥C0,1(B1), for some constant C depending only on λ and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By symmetry, the same inequality is true for ∂2v (and ∂2u), so that ∥u∥C2,α(B1/2) ≤ C∥u∥C1,1(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Existence of solutions 105 Notice that, by interpolation inequalities (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9)), for each ε > 0, there exists some Cε > 0 such that ∥u∥C1,1(B1/2) ≤ ε∥u∥C2,α(B1) + Cε∥u∥L∞(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, the proof can be concluded by means of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='27 analogously to what has been done in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Thus, as we can see, in the two-dimensional case it is rather easy to show a priori C2,α estimates for solutions to the fully nonlinear equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thanks to these estimates, by means of the continuity method (see [GT77] or [HL97]) one can actually show the existence of C2,α solutions for the Dirichlet problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Nonetheless, as we will see, it turns out that in higher dimensions such an a priori estimate is no longer available, and one needs to prove existence of solutions in a different way, by introducing a new notion of weak solution (viscosity solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is what we do in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Existence of solutions We now turn our attention to fully nonlinear elliptic equations in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The first question to understand is the existence of solutions: given a nice domain Ω ⊂ Rn, and a nice boundary data g : ∂Ω → R, can we always solve the following Dirichlet problem?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' � F(D2u) = 0 in Ω u = g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that here we cannot construct the solution by minimizing a func- tional, since these fully nonlinear equations do not come, in general, from any energy functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' To construct the solution, we only have two options: – Prove “a priori estimates” and then use the continuity method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' – Use the comparison principle and Perron’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The continuity method is reasonably easy to use, but we need C2,α estimates for solutions up to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is a very difficult problem, and in fact, in general we do not have C2,α estimates for these equations in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, we need to construct some kind of generalized notion of so- lution: viscosity solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The right concept of solution must be so that we have Existence of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 106 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Fully nonlinear elliptic PDE Comparison principle (and in particular, uniqueness of solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Stability (so that limits of solutions are solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that if we consider only C2 solutions, then we have the comparison principle (and it is easy to prove), but we may not be able to prove existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, if we relax the notion of solution, then we may be able to easily prove the existence of a solution, but then it will be more difficult to prove the uniqueness/comparison principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The right notion of generalized solution is the one given in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10 below, known as viscosity solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For subsolutions in the viscosity sense, this notion only requires that the function is upper semi-continuous (USC), while for supersolutions in the viscosity sense, this notion can be checked on lower semi-continuous (LSC) functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is important in the proof of existence of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We recall that a function f is said to be upper semi-continuous at x◦ if lim sup x→x◦ f(x) ≤ f(x◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Similarly, it is lower semi-continuous at x◦ if lim inf x→x◦ f(x) ≥ f(x◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer to [Sil15] for a nice introduction to viscosity solutions to elliptic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10 (Viscosity solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let F : Rn×n → R be uniformly elliptic, and consider the PDE F(D2u) = 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We say that u ∈ USC(Ω) is a subsolution (in the viscosity sense), and we write F(D2u) ≥ 0, if for any φ ∈ C2(Ω) such that φ ≥ u in Ω and φ(x◦) = u(x◦), x◦ ∈ Ω, we have F(D2φ(x◦)) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We say that u ∈ LSC(Ω) is a supersolution (in the viscosity sense), and we write F(D2u) ≤ 0, if for any φ ∈ C2(Ω) such that φ ≤ u in Ω and φ(x◦) = u(x◦), x◦ ∈ Ω, we have F(D2φ(x◦)) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We say that u ∈ C(Ω) solves F(D2u) = 0 in Ω in the viscosity sense if it is both a subsolution and a supersolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that there may be points x◦ ∈ Ω at which no function φ ∈ C2 touches u at x◦ (from above and/or from below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is allowed by the previous definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11 (Some history).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The concept of viscosity solution was in- troduced in 1983 by Crandall and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lions in the study of first-order — DRAFT — 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Existence of solutions 107 equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' During a few years, the work on viscosity solutions focused on first-order equations, because it was not known whether second-order uni- formly elliptic PDEs would have a unique viscosity solution (or if the com- parison principle would hold for these solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In 1988 the comparison principle for viscosity solutions was finally proved by Jensen [Jen88], and in subsequent years the concept has become prevalent in the analysis of elliptic PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In 1994, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lions received the Fields Medal for his contributions to nonlinear PDEs, one of his major contributions being his work on viscosity solutions [ICM94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A key result in the theory of viscosity solutions is the following (see [Jen88, CC95]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12 (Comparison principle for viscosity solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ Rn be any bounded domain, and F : Rn×n → R be uniformly elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that u ∈ LSC(Ω) and v ∈ USC(Ω) satisfy (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10) u ≥ v on ∂Ω, and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11) F(D2u) ≤ 0 ≤ F(D2v) in Ω in the viscosity sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ≥ v in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We already proved this for C2 functions u in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3, and the proof was very simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For viscosity solutions the proof is more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The main step in the proof of the comparison principle is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ Rn be any bounded domain, and F : Rn×n → R be uniformly elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that u ∈ LSC(Ω) and v ∈ USC(Ω) are bounded functions that satisfy (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, M−(D2(u − v)) ≤ 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer the reader to [CC95, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3] for a proof of such result, where it is proved assuming that u, v ∈ C(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The same proof works under the hypotheses here presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The comparison principle follows using Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13 and the next lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ Rn be any bounded domain, and assume that w ∈ LSC(Ω) satisfies w ≥ 0 on ∂Ω, — DRAFT — 108 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Fully nonlinear elliptic PDE v w x◦ Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We slide v from below until it touches w at a point x◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' and M−(D2w) ≤ 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, w ≥ 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The proof is similar to that of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, first notice that after a rescaling we may assume Ω ⊂ B1, and assume by contradiction that w has a negative minimum in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, since w ≥ 0 on ∂Ω, we have minΩ w = −δ, with δ > 0, and the minimum is achieved in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us now consider 0 < ε < δ, and v(x) := −κ + ε(|x|2 − 1), with κ > 0 (that is, a sufficiently flat paraboloid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, notice that v < 0 on ∂Ω, and we can choose κ > 0 so that v touches w from below at a point inside Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In other words, there is κ > 0 such that w ≥ v in Ω, and w(x◦) = v(x◦) for some x◦ ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (See Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') Then, by definition of viscosity supersolution, we have M−(D2v)(x◦) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' However, a direct computation gives M−(D2v) = M−(2εId) ≡ 2λnε > 0 in Ω, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Once we have the comparison principle for viscosity solutions, we can use Perron’s method to prove existence of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We next do this, following [Sil15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' First let us notice that, for any bounded function u in Ω ⊂ Rn, we may define its upper semi-continuous envelope as u∗(x) := sup{lim sup k u(xk) : xk → x}, where the supremum is taken among all sequences Ω ∋ xk → x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that u∗ is the smallest function satisfying u∗ ∈ USC(Ω) and u∗ ≥ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Similarly, we define the lower semi-continuous envelope of u as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12) u∗(x) := inf{lim inf k u(xk) : xk → x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will need the following lemma, which is a generalization of the fact that the maximum of subsolutions is also a subsolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Existence of solutions 109 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let F : Rn×n → R be uniformly elliptic, and let Ω ⊂ Rn be any bounded domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let (ua)a∈A be a family of subsolutions: ua ∈ USC(Ω), and F(D2ua) ≥ 0 in Ω, for all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u(x) := sup a∈A ua, and let u∗(x) := sup � lim sup k→∞ u(xk) : xk → x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u∗ ∈ USC(Ω) is a subsolution: F(D2u∗) ≥ 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We divide the proof into two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In the first part, we show that if u∗ has a strict local maximum at x◦, then one can extract sequences of indices ak ∈ A for k ∈ N, and of points xk ∈ Ω, such that xk → x◦, uak has a local maximum at xk, and uak(xk) → u∗(x◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By definition of u∗(x◦), we can extract a sequence of indices (aj)j∈N, aj ∈ A, and of points yj → x◦, such that uaj(yj) → u∗(x◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now let us prove that we can extract a further subsequence ak := ajk such that our desired conclusion holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, let r > 0 be such that u∗(y) < u∗(x◦) for y ∈ Br(x◦) \\ {x◦}, and let ρ > 0 be so small that, if Kρ := Br(x◦) \\ Bρ(x◦), then max Kρ u∗ ≤ u∗(x◦) − δ, for some δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now notice that, for j large enough, uaj ≤ u∗(x◦)−δ/2 in Kρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Otherwise, there would be jm → ∞ and zm such that uajm(zm) > u∗(x◦) − δ/2 ≥ maxKρ u∗ + δ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since Kρ is compact, up to a subsequence, zm → z∞ for some z∞ in Kρ such that u∗(z∞) ≥ lim sup m→∞ uajm(zm) > max Kρ u∗ + δ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, uaj ≤ u∗(x◦) − δ/2 in Kρ for j large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let now xj ∈ Br(x◦) be the point where the maximum of uaj in Br(x◦) is attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, uaj(xj) ≥ uaj(yj) → u∗(x◦), that is, uaj(xj) ≥ u∗(x◦) − δ/4 for j large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since uaj ≤ u∗(x◦) − δ/2 in Kρ (again, for j large enough), this implies that xj ∈ Bρ(x◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, ukj attains its maximum in Br(x◦), inside Bρ(x◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By repeating this argument choosing smaller ρ > 0, we can extract a subsequence ak := ajk to get the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that xj → x◦, and that by construction, uaj(xj) ≥ uaj(yj) → u∗(x◦), so that uaj(xj) → u∗(x◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This completes the first part of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that so far we have not used that ua are subsolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 110 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Fully nonlinear elliptic PDE Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us now proceed with the second part of the proof, which proves the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let φ ∈ C2 be such that φ(x◦) = u∗(x◦) and u ≤ φ around x◦ (that is, u−φ attains its local maximum at x◦), with x◦ ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By considering ¯φ(x) = φ(x) + |x − x◦|4, we have that u − ¯φ attains a strict local maximum at x◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We apply now the first part of the proof with va := ua − ¯φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, there exist sequences of indices (ak)k∈N, and points xk → x◦ such that uak − ¯φ attains its local maximum at xk and uak(xk) → u∗(x◦) (since ¯φ is continuous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, since uak are subsolutions in the viscosity sense, we have F � D2 ¯φ(xk) � ≥ 0 =⇒ F � D2 ¯φ(x◦) � = F � D2φ(x◦) � ≥ 0, by continuity of F and D2φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, u is a viscosity subsolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ We can now prove the existence of viscosity solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' To do so, we assume that we are given a bounded domain Ω ⊂ Rn such that for every x◦ ∈ ∂Ω, there exists some ψ+ ∈ C2(Ω) such that ψ+(x◦) = 0, ψ+|∂Ω\\{x◦} > 0, and M+(D2ψ+) ≤ 0 in Ω, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13) where we recall that M+ is the Pucci operator defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3) with ellipticity constants λ and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that, if (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13) holds, then we also have that for every x◦ ∈ ∂Ω, there exists some ψ− ∈ C2(Ω) such that ψ−(x◦) = 0, ψ−|∂Ω\\{x◦} < 0, and M−(D2ψ−) ≥ 0 in Ω, where ψ− is simply given by ψ− = −ψ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will later show that any bounded C2 domain satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13), for any constants 0 < λ ≤ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In the following results, we will often assume that F(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Otherwise, if F(0) ̸= 0, we can consider the uniformly elliptic operator ˜Ft(D2u) := F � D2(u + t|x|2/2) � = F(D2u + tId) instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, ˜Ft(0) = F(tId), and we can choose t ∈ R such that F(tId) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, if F(0) > 0, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6) ˜Ft(0) = F(tId) ≤ M+(tId) + F(0) = tnλ + F(0) < 0 for t < 0 negative enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since ˜F0(0) = F(0) > 0, by continuity of ˜Ft in t, we are done for some t ∈ � − F(0) nλ , 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The case F(0) < 0 follows analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17 (Existence and uniqueness of viscosity solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let F : Rn×n → R be uniformly elliptic with ellipticity constants λ and Λ, let Ω ⊂ Rn be any bounded domain such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13) holds, and let g ∈ C(∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, there exists a (unique) viscosity solution to the Dirichlet problem � F(D2u) = 0 in Ω u = g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Existence of solutions 111 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The uniqueness follows directly from the comparison principle, The- orem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thanks to Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16, we will assume F(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The proof of existence follows by means of Perron’s method, as shown next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us define the set of all subsolutions as A := � v ∈ USC(Ω) : F(D2v) ≥ 0 in Ω, v ≤ g on ∂Ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, we can define the pointwise supremum of all subsolutions in A, u(x) := sup v∈A v(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that since the constant function −∥g∥L∞(∂Ω) belongs to A, such set is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice also that all elements of A must be below the constant ∥g∥L∞(∂Ω) by the comparison principle, and thus u is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We define the upper semi-continuous envelope u∗(x) = sup � lim sup k→∞ u(xk) : xk → x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15, we have F(D2u∗) ≥ 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The strategy of the proof is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We first prove that u∗ = g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This implies that u∗ ∈ A, and therefore u∗ = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, once this is done, we will define u∗ as the lower semi-continuous envelope of u, and show that u∗ is a supersolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By the comparison principle, this will imply that u∗ ≥ u, and thus u∗ = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This means that u is continuous, and that it is both a subsolution and a supersolution, as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us start by showing that u∗ = g on ∂Ω, and that u∗ is continuous on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Namely, we show that for every x◦ ∈ ∂Ω, and every xk → x◦ with xk ∈ Ω, then lim infk→∞ u∗(xk) = lim supk→∞ u∗(xk) = g(x◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let ε > 0, and let us define w− ε := g(x◦) − ε + kεψ− = g(x◦) − ε − kεψ+, where kε > 0 is chosen large enough (depending on ε but also on g and Ω) such that w− ε ≤ g on ∂Ω, and ψ− = −ψ+ is the function given by property (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13) at x◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us also define w+ ε := g(x◦) + ε + kεψ+, where kε > 0 is such that w+ ε ≥ g on ∂Ω (without loss of generality, by taking it larger if necessary, we can assume it is the same as before).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By the properties of the extremal operators (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3), we have M−(D2w− ε ) = kεM−(D2ψ−) ≥ 0 and M+(D2w+ ε ) = kεM+(D2ψ+) ≤ 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particu- lar, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6) (recall F(0) = 0), F(D2w− ε ) ≥ 0 and F(D2w+ ε ) ≤ 0 in Ω, — DRAFT — 112 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Fully nonlinear elliptic PDE and w− ε ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that, by continuity of ψ−, for each ε > 0 there exists some δ > 0 such that w− ε ≥ g(x◦) − 2ε in Bδ(x◦) ∩ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This yields, u∗ ≥ w− ε ≥ g(x◦) − 2ε in Bδ(x◦) ∩ Ω, so that if xk → x◦, then lim inf k→∞ u(xk) ≥ g(x◦) − 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, by the comparison principle, all elements in A are below w+ ε for any ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Again, by continuity of ψ+, for each ε > 0 there exists some δ > 0 such that w+ ε ≤ g(x◦) + 2ε in Bδ(x◦) ∩ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This yields, u∗ ≤ w+ ε ≤ g(x◦) + 2ε in Bδ(x◦) ∩ Ω, so that if xk → x◦, then lim sup k→∞ u(xk) ≤ g(x◦) + 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since ε > 0 is arbitrary, we have that if xk → x◦, then lim k→∞ u∗(xk) = g(x◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, u∗ = g on ∂Ω and u is continuous on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, we have u∗ ∈ A and (since u∗ ≥ u) u∗ ≡ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This means that u ∈ USC(Ω) and F(D2u) ≥ 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, we show that u is a supersolution as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' To do so, we consider its lower semi-continuous envelope u∗, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12), and prove that F(D2u∗) ≤ 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We start by noticing that, since u is continuous on the boundary (by Step 1), then u∗ = g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume by contradiction that u∗ is not a supersolution, that is, there exists some x◦ ∈ Ω such that for some φ ∈ C2 we have φ(x◦) = u∗(x◦), φ ≤ u∗, but F(D2φ(x◦)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By taking ¯φ = φ − |x − x◦|4 if necessary, we may assume that φ < u∗ if x ̸= x◦, and we still have F(D2φ(x◦)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that, by continuity of F and D2φ, we have F(D2φ) > 0 in Bρ(x◦) for some small ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, consider φ+δ for δ > 0, and define uδ := max{u, φ+ δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since φ(x) < u∗(x) ≤ u(x) for x ̸= x◦, we have for δ > 0 small enough that φδ < u outside Bρ(x◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, notice that uδ is a subsolution, since it coincides with u outside Bρ(x◦) and it is the maximum of two subsolutions in Bρ(x◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This means that uδ ∈ A, and thus uδ ≤ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' However, this means that φ + δ ≤ u everywhere in Ω, and thus φ + δ ≤ u∗, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, u∗ had to be a supersolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' But then, again by the comparison principle, since u is a subsolution and u = u∗ = g on ∂Ω, we get that u∗ ≥ u in Ω, which means that u = u∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, u is continuous, both a subsolution and a supersolution, and u = g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ — DRAFT — 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Existence of solutions 113 x◦ zx◦ Bρ(zx◦) Ω Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Representation of the construction from the proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As a consequence, we find the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω be any bounded C2 domain, and F : Rn×n → R be uniformly elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, for any continuous g ∈ C(∂Ω), the Dirichlet problem � F(D2u) = 0 in Ω u = g on ∂Ω, has a unique viscosity solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The result follows from the previous theorem, we just need to check that any C2 domain fulfils (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' To do so, we need to construct an appro- priate barrier at every boundary point z ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice, that in the very simple case that Ω is strictly convex, such barrier ψ+ can simply be a hyperplane with zero level set tangent to Ω at a given boundary point, such that it is positive in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In general, since Ω is a bounded C2 domain, it satisfies the exterior ball condition for some uniform radius ρ > 0: that is, for each point x◦ ∈ ∂Ω there exist some point zx◦ = z(x◦) ∈ Ωc and a ball Bρ(zx◦) such that Bρ(zx◦) ⊂ Ωc and Bρ(zx◦) ∩ ∂Ω = {x◦}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' See Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us construct the barrier ψ+ from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13) for C2 domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We consider the function ψ in Rn \\ Bρ, for ρ > 0 given by the exterior ball condition, ψ(x) = e−αρ2 − e−α|x|2, for some α > 0 also to be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 114 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Fully nonlinear elliptic PDE Notice that eα|x|2D2ψ(x) = −4α2 � � � � � x2 1 x1x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' x1xn x2x1 x2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' x2xn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' xnx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' x2 n � � � � � + 2αId = 2αId − 4α2xxT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, for |x| ≥ ρ we have eα|x|2M+(D2ψ) ≤ 2αM+(Id) − 4α2M−(xxT ) = 2αnΛ − 4α2λ|x|2 ≤ 2α(nΛ − 2αλρ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, if we choose α ≥ nΛ 2λρ2 , we have M+(D2ψ) ≤ 0 in Bc ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, translations of ψ are good candidates for the function ψ+ from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let now x◦ ∈ ∂Ω be any point on the boundary, and take ψ+(x) := ψ(x − zx◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It is clear that ψ+(x◦) = 0, and that ψ+(x) > 0 for any x ∈ Ω \\ {x◦}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, from the discussion above we know that M+(D2ψ+) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, Ω fulfills (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19 (Lipschitz domains).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It is actually possible to show that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13) holds for any bounded Lipschitz domain, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, this yields the existence of viscosity solutions in such class of domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, we also have the following: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20 (Stability of viscosity solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Fk be a sequence of uniformly elliptic operators (with ellipticity constants λ and Λ), and let uk ∈ C(Ω) be such that Fk(D2uk) = 0 in Ω in the viscosity sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that Fk converges to F uniformly in compact sets, and uk → u uniformly in compact sets of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, F(D2u) = 0 in Ω in the viscosity sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The proof uses the same ideas as the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let x◦ ∈ Ω and φ ∈ C2 be such that φ(x◦) = u(x◦) and φ ≤ u in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By taking ¯φ(x) = φ(x) + |x − x◦|4 we have that u − ¯φ attains a strict local maximum at x◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let now vk := uk − ¯φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Up to a subsequence, by Step 1 in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15, there exists a sequence xk → x◦ such that uk − ¯φ attains a local maximum at xk, and from the uniform convergence of uk to u, we also have uk(xk) → u(x◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since uk are, in particular, subsolutions in the — DRAFT — 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Regularity of solutions: an overview 115 viscosity sense for the operator Fk, we have that Fk(D2 ¯φ(xk)) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, since xk → x◦, and Fk converges uniformly to F, we get that, letting k → ∞, F(D2 ¯φ(x◦)) = F(D2φ(x◦)) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, u is a viscosity subsolution for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Doing the same for −u, we reach that u is a viscosity solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We have seen that for fully nonlinear equations F(D2u) = 0 we have existence, uniqueness, and stability of viscosity solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The same can be done for more general equations like F(D2u, x) = f(x), with continuous coefficients in x, see [CC95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' However, when we want to study linear equations in non-divergence form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14) � aij(x)∂iju(x) = 0 with bounded measurable coefficients, it turns out that viscosity solutions do not behave so well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' see the counterexample in [Nad97] (see also [CCKS96]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is the reason why, instead of defining viscosity solutions for a specific equation of the type (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14), what we do is to say that u solves an equation with bounded measurable coefficients (in non-divergence form) whenever it satisfies M−(D2u) ≤ 0 ≤ M+(D2u) in viscosity sense, where M± are the Pucci extremal operators (recall Def- inition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As explained in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8, for C2 functions u these two in- equalities are equivalent to saying that u solves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14) for some coefficients aij(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Summarizing: For viscosity solutions we now have all we need in order to study regularity issues: – Existence of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' – Comparison principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' – Stability under uniform limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Regularity of solutions: an overview In the last section we saw that for any (smooth) domain Ω ⊂ Rn and any (continuous) boundary data g, one can find a unique viscosity solution u ∈ C(Ω) to the Dirichlet problem � F(D2u) = 0 in Ω u = g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, the main question is that of regularity: If u ∈ C(B1) solves F(D2u) = 0 in B1, what can we say about the regularity of u?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 116 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Fully nonlinear elliptic PDE Is the following implication true?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15) � � � � � � � F ∈ C∞ and uniformly elliptic & F(D2u) = 0 in B1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' =====⇒ u ∈ C∞(B1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is in some sense a question analogous to Hilbert’s XIXth problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Regularity for fully nonlinear equations: first results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that u has some initial regularity, and that F is C∞ and uniformly elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, F(D2u) = 0 −→ ∂e n � i,j=1 Fij(D2u)∂ij(∂eu) = 0, where Fij := ∂F ∂Mij is the derivative of F(M) with respect to Mij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, if we denote aij(x) := Fij(D2u(x)), we will then have aij(x) is uniformly elliptic, 0 < λId ≤ (aij(x))i,j ≤ ΛId, thanks to the uniform ellipticity of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Denoting v = ∂eu, we have F(D2u) = 0 =⇒ v = ∂eu solves n � i,j=1 aij(x)∂ij(∂eu) = 0, where aij(x) = Fij(D2u(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, if u ∈ C2 (or C2,α), then the coefficients aij(x) are continuous (or C0,α), and therefore we get, by Schauder-type estimates, u ∈ C2 ⇒ aij ∈ C0 ⇒ v ∈ C1,α ⇒ u ∈ C2,α ⇒ · · · ⇒ u ∈ C∞, where we use the bootstrap argument u ∈ C2,α ⇒ aij ∈ C0,α ⇒ u ∈ C3,α ⇒ aij ∈ C1,α ⇒ · · · ⇒ u ∈ C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In other words, this suggests that the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let F be uniformly elliptic and C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution of F(D2u) = 0 in B1, and assume that u ∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Regularity of solutions: an overview 117 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The idea is the one presented in the lines above, but we can only use that u ∈ C2 (in the previous argumentation, we used that u is C3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' To do so, we make use of incremental quotients, as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u ∈ C2(B1), and let h ∈ Rn with |h| small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that F is translation invariant, so F(D2u(x)) = 0, F(D2u(x + h)) = 0 in B1−|h|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, 0 = F(D2u(x + h)) − F(D2u(x)) = n � i,j=1 aij(x)∂ij � u(x + h) − u(x) � , where aij(x) = � 1 0 Fij � tD2u(x + h) + (1 − t)D2u(x) � dt (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5 or Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is just the fundamental theorem of calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, since F is uniformly elliptic, (aij)i,j is uniformly elliptic (with the same ellipticity constants).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since u ∈ C2 and F is smooth, aij are continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, u(·+h)−u |h| solves the equation in non-divergence form n � i,j=1 aij(x)∂ij �u(x + h) − u(x) |h| � = 0 in B1−|h|, for some continuous and uniformly elliptic coefficients aij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By the a priori estimates for equations with continuous coefficients (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='31), we know that for any α ∈ (0, 1) we have ���� u(· + h) − u |h| ���� C1,α(B1/2) ≤ C ���� u(· + h) − u |h| ���� L∞(B3/4) ≤ C∥u∥C0,1(B3/4), for some constant C that is independent of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By (H7), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7), from Chap- ter 1, we reach that u ∈ C2,α(B1/2), and by a covering argument u ∈ C2,α inside B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, we proceed iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since u ∈ C2,α inside B1, we have that u(·+h)−u |h| ∈ C2,α inside B1−|h| for all h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Together with F being smooth, this implies that aij ∈ C0,α inside B1−|h|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, now u(·+h)−u |h| solves a non-divergence-form equation with H¨older continuous coefficients, and from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20 we get uniform bounds in the C2,α norm for u(·+h)−u |h| , thus yielding that u ∈ C3,α inside B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We can repeat this argument iteratively, using the higher order estimates from Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='21, to reach the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ — DRAFT — 118 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Fully nonlinear elliptic PDE This is similar to what happened in Hilbert’s XIXth problem: in that case we proved C1 ⇒ C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice, however, that for fully nonlinear equations, the “gap to be filled” (from C0 to C2) is “bigger” than in Hilbert’s XIXth problem (from H1 to C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, the central question to be answered is: Is it true that solutions are always C2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, we wonder whether viscosity solutions are always classical solutions or not, and thus, whether the Dirichlet problem always admits a classical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Regularity for fully nonlinear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' An important observation in the previous argument was the following: � u solves F(D2u) = 0 =⇒ � v = ∂eu solves �n i,j=1 aij(x)∂ijv = 0, with aij(x) = Fij(D2u(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This means that, at least formally, the derivatives of any solution to any fully nonlinear equation solve an equation with bounded measurable coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This can be argued properly by looking at incremental quotients: Recall from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5) the equivalence F is uniformly elliptic �� M−(D2(u − v)) ≤ F(D2u) − F(D2v) ≤ M+(D2(u − v)), where M± are the Pucci operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, M−� D2(u(x + h) − u(x)) � ≤ F(D2u(x + h)) − F(D2u(x)) ≤ ≤ M+� D2(u(x + h) − u(x)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Using F(D2u) = 0 and denoting vh(x) = u(x + h) − u(x) |h| , we then reach � M+(D2vh) ≥ 0 M−(D2vh) ≤ 0 � equation with bounded measurable coefficients � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The question is now: in case of divergence-form equations we proved � equation with bounded meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' coeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' div(A(x)∇v) = 0 =⇒ v ∈ C0,α (De Giorgi-Nash).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Regularity of solutions: an overview 119 Is there a similar result for equations in non-divergence form?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The answer is Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='23 (Krylov–Safonov, 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let 0 < λ ≤ Λ be the ellipticity constants, and v ∈ C(B1) be any solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16) � M+(D2v) ≥ 0 in B1 M−(D2v) ≤ 0 in B1, in the viscosity sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, ∥v∥C0,α(B1/2) ≤ C∥v∥L∞(B1) for some small α > 0 and C depending only on n, λ, and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This result was proved in [KS79] (for classical solutions);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' see also [Moo19] for a more recent and simplified proof, and [DS21] for an extension of the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Recall that (see the end of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1), for C2 functions, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16) is actu- ally equivalent to v solving an equation of the type � i,j aij(x)∂ijv for some uniformly elliptic coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is why (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16) is called an equation in non-divergence form with bounded measurable coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As a consequence of this result, we find the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We assume for simplicity F(0) = 0, otherwise see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='24 (Krylov–Safonov, 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let F be uniformly elliptic, F(0) = 0, and u ∈ C(B1) be any viscosity solution to F(D2u) = 0 in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, ∥u∥C1,α(B1/2) ≤ C∥u∥L∞(B1) for some small α > 0 and C depending only on n, λ, and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13 (with v ≡ 0), the function u ∈ C(B1) solves itself an equation with bounded measurable coefficients � M+(D2u) ≥ 0 in B1 M−(D2u) ≤ 0 in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='23, u ∈ C0,α inside B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, for β ∈ (0, 1] take vh(x) := u(x + h) − u(x) |h|β , which (again by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13) also solves an equation with bounded measurable coefficients, � M+(D2vh) ≥ 0 in B1−|h| M−(D2vh) ≤ 0 in B1−|h|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 120 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Fully nonlinear elliptic PDE Then, again by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='23, we have ∥vh∥C0,α(B1/2) ≤ C∥vh∥L∞(B1−|h|) ≤ C∥u∥Cβ(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By (H7), we deduce that ∥u∥Cα+β(B1/2) ≤ C∥u∥Cβ(B1), provided that α + β is not an integer and β ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Using this estimate with β = α, 2α, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', kα, one gets C1,α regularity in a finite number of steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Observe that: The C0,α estimate for bounded measurable coefficients, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='23, is the best one can get in dimensions n ≥ 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' see [Saf87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In a sense, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='23 is the analogue of the result of De Giorgi– Nash for divergence-form equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' However, it is not enough to get C2 regularity for solutions to fully nonlinear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Summary: We have F(D2u) = 0 ⇒ u ∈ C1,α (for some small α > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, u ∈ C2 ⇒ u ∈ C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' However, we have no idea (yet) if u ∈ C1,α ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='=⇒ u ∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In the two-dimensional case, as we have seen in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9 (as an a priori estimate), it turns out that one can do something better, and all solu- tions are C2,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is because, in R2, solutions to equations with bounded measurable coefficients are not only C0,α, but C1,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As a consequence, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let F : R2×2 → R be uniformly elliptic and smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u ∈ C(B1) be any viscosity solution to F(D2u) = 0 in B1 ⊂ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then u ∈ C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This completely answers question (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15) in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In higher dimensions, a famous result established (independently) by Evans [Eva82] and Krylov [Kry82] gives the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='27 (Evans–Krylov, 1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let F be any convex (or concave) uniformly elliptic operator, with F(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u ∈ C(B1) be any viscosity solution to F(D2u) = 0 in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, ∥u∥C2,α(B1/2) ≤ C∥u∥L∞(B1), — DRAFT — 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Further results and open problems 121 for some α > 0 and C depending only on n, λ, and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, if F is smooth then u ∈ C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer to [CS10] for a shorter proof of such result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, for any solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2), with F uniformly elliptic and smooth, we have: If u ∈ C2, then u ∈ C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' u ∈ C1,α always (Krylov–Safonov, 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In two dimensions, u ∈ C∞ (Nirenberg, 1952).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If F is convex, then u ∈ C∞ (Evans–Krylov, 1982) Question: What happens in general?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For decades it was an open problem to decide whether all solutions are C2 or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The question was finally answered by Nadirashvili and Vladuts in the 2000s [NV07, NV08, NV13]: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='28 (Nadirashvili–Vladuts, 2007-2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' There are solutions to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2) that are not C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' These counterexamples exist in dimensions n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, for every τ > 0, there exists a dimension n and ellipticity constants λ and Λ, such that there are solutions u to F(D2u) = 0 with u /∈ C1,τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer to the monograph [NTV14] for more references and details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It is not known what happens in R3 and R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is one of the most remarkable open problems in elliptic PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Further results and open problems As explained above, one of the main open questions regarding the problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17) F(D2u) = 0 in B1 ⊂ Rn is the following: Let u be any solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17) in R3 or R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Is it true that u ∈ C2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We have seen that it is in general not true that solutions to fully nonlinear equations (in dimension n ≥ 5) are C2 under the assumption that F is simply uniformly elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Convexity, on the other hand, is a strong condition under which C2 regularity is achieved, which, unfortunately, does not hold in some important applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Even with this, it is still unclear what the optimal regularity of solutions is when F is convex and uniformly elliptic (not necessarily smooth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='27 only gives, a priori, C2,α regularity for some small α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 122 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Fully nonlinear elliptic PDE These observations motivate, on the one hand, a more refined study for the regularity (and size of singularity) of solutions to general fully nonlinear elliptic equations, and on the other hand, the study of the optimal regularity under the convexity assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Partial regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Recall that the ellipticity requirement for F implies that F is Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Under the slightly more restrictive requirement that F is also C1, the following partial regularity result was proved by Armstrong, Silvestre, and Smart in [ASS12]: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='29 ([ASS12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let F be uniformly elliptic, and assume in addition that F ∈ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u ∈ C0(B1) be any viscosity solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, there exist some ε > 0 depending only on n, λ, Λ, and a closed subset Σ ⊂ B1 with dimH Σ ≤ n − ε, such that u ∈ C2(B1 \\ Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Here, dimH denotes the Hausdorff dimension of a set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' see [Mat95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that if dimH Σ ≤ n − ε then in particular Σ has zero measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This result is the best known partial regularity result for solutions of (non-convex) fully nonlinear equations in dimensions n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that the size of the singular set is not known to be optimal (it could be much smaller!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, it is an important open problem to decide whether the same statement holds without the regularity assumption F ∈ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Optimal regularity when F is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' When F is convex and uniformly elliptic, solutions to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17) are known to be C2,α for some small α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If F ∈ C∞, a bootstrap argument then yields higher regularity for u, but the higher regularity of F is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' What happens if we just require F to be convex and uniformly elliptic?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since F is convex, the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17) can be reformulated as a supre- mum of linear uniformly elliptic operators as sup a∈A Lau = 0 in B1 ⊂ Rn, also known as Bellman equation (see (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3) in the Appendix C), where each of the operators La is a linear uniformly elliptic operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The question that remains open here is: What is the optimal regularity of solutions to Bellman equations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In the simpler model of just two different operators, the previous equa- tion is (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18) max{L1u, L2u} = 0 in B1 ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Further results and open problems 123 The best known result in this direction was proved by Caffarelli, De Silva, and Savin in 2018, and establishes the optimal regularity of solutions to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18) in two dimensions: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='30 ([CDS18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any viscosity solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18) in B1 ⊂ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then ∥u∥C2,1(B1/2) ≤ C∥u∥L∞(B1), for some constant C depending only on λ and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The approach used in [CDS18] to show this result does not work in higher dimensions n ≥ 3, and thus the following question remains open: Let u be any solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18), with n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Is is true that u ∈ C2,1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — — DRAFT — Chapter 5 The obstacle problem In this last chapter we focus our attention on a third type of nonlinear elliptic PDE: a free boundary problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In this kind of problems we are no longer only interested in the regularity of a solution u, but also in the study of an a priori unknown interphase Γ (the free boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As explained later, there is a wide variety of problems in physics, indus- try, biology, finance, and other areas which can be described by PDEs that exhibit free boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Many of such problems can be written as variational inequalities, for which the solution is obtained by minimizing a constrained energy functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' And one of the most important and canonical examples is the obstacle problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1 Given a smooth function ϕ, the obstacle problem is the following: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) minimize 1 2 � Ω |∇v|2dx among all functions v ≥ ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Here, the minimization is subject to boundary conditions v|∂Ω = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The interpretation of such problem is clear: One looks for the least ener- gy function v, but the set of admissible functions consists only of functions that are above a certain “obstacle” ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In the two-dimensional case, one can think of the solution v as a “mem- brane” which is elastic and is constrained to be above ϕ (see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 1Other examples of important free boundary problems include the one-phase or Bernoulli problem, the thin or fractional obstacle problem, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer the interested reader to [CS05, PSU12, Vel23, Fer22] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 125 — DRAFT — 126 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem v ϕ −∆v ≥ 0 everywhere v ≥ ϕ everywhere ∆v = 0 in {v > ϕ} Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The function v minimizes the Dirichlet energy among all functions with the same boundary values situated above the obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The Euler–Lagrange equation of the minimization problem is the follow- ing: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2) � � � v ≥ ϕ in Ω ∆v ≤ 0 in Ω ∆v = 0 in the set {v > ϕ}, together with the boundary conditions v|∂Ω = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, notice that if we denote E(v) = 1 2 � Ω |∇v|2dx, then we will have E(v + εη) ≥ E(v) for every ε ≥ 0 and η ≥ 0, η ∈ C∞ c (Ω), which yields ∆v ≤ 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, we can perturb v with nonnegative functions (εη) and we always get admissible functions (v + εη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' However, due to the constraint v ≥ ϕ, we cannot perturb v with negative functions in all of Ω, but only in the set {v > ϕ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is why we get ∆v ≤ 0 everywhere in Ω, but ∆v = 0 only in {v > ϕ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (We will show later that any minimizer v of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) is continuous, so that {v > ϕ} is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') Alternatively, we may consider u := v−ϕ, and the problem is equivalent to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3) � � � u ≥ 0 in Ω ∆u ≤ f in Ω ∆u = f in the set {u > 0}, where f := −∆ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Such solution u can be obtained as follows: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4) minimize � Ω �1 2|∇u|2 + fu � dx among all functions u ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem 127 In other words, we can make the obstacle just zero, by adding a right- hand side f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Here, the minimization is subject to the boundary conditions u|∂Ω = ˜g, with ˜g := g − ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the Euler–Lagrange equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As said above, the Euler–Lagrange equations of the minimization problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) are: (i) v ≥ ϕ in Ω (v is above the obstacle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (ii) ∆v ≤ 0 in Ω (v is a supersolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (iii) ∆v = 0 in {v > ϕ} (v is harmonic where it does not touch the obstacle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' These are inequalities, rather than a single PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Alternatively, one can write also the Euler–Lagrange equations in the following way: min{−∆v, v − ϕ} = 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (Notice that this resembles a fully nonlinear equation min{L1u, L2u} = 0, but in the present situation one of the two operators is of order zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') Of course, the same can be done for the equivalent problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In that case, moreover, the minimization problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4) is equivalent to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5) minimize � Ω �1 2|∇u|2 + fu+ � dx, where u+ = max{u, 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In this way, we can see the problem not as a constrained minimization but as a minimization problem with a non-smooth term u+ in the functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The Euler–Lagrange equation for this functional is then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6) ∆u = fχ{u>0} in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (Here, χA denotes the characteristic function of a set A ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') We will show this in detail later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The free boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us take a closer look at the obstacle problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' One of the most important features of such problem is that it has two unknowns: the solution u, and the contact set {u = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In other words, there are two regions in Ω: one in which u = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' and one in which ∆u = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' These regions are characterized by the minimization problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' More- over, if we denote Γ := ∂{u > 0} ∩ Ω, then this is called the free boundary, see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem is a free boundary problem, as it involves an un- known interface Γ as part of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 128 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem {u = 0} {u > 0} ∆u = f Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The free boundary could, a priori, be very irregular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, it is not difficult to see that the fact that u is a nonnegative supersolution must imply ∇u = 0 on Γ, that is, we will have that u ≥ 0 solves � � � ∆u = f in {u > 0} u = 0 on Γ ∇u = 0 on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is an alternative way to write the Euler–Lagrange equation of the prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In this way, the interface Γ appears clearly, and we see that we have both Dirichlet and Neumann conditions on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This would usually be an over-determined problem (too many boundary conditions on Γ), but since Γ is also free, it turns out that the problem has a unique solution (where Γ is part of the solution, of course).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Some motivations and applications Let us briefly comment on some of the main motivations and applications in the study of the obstacle problem, which are further developed in Appen- dix D (see also Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer to the books [DL76, KS80, Rod87, Fri88, PSU12], for more details and further applications of obstacle-type problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Fluid filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The so-called Dam problem aims to describe the filtration of water inside a porous dam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' One considers a dam separating two reservoirs of water at different heights, made of a porous medium (permeable to water).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then there is some transfer of water across the dam, and the interior of the dam has a wet part, where water flows, and a dry part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In this setting, an integral of the pressure (with respect to the height of the column of water at each point) solves the obstacle problem, and the free boundary corresponds precisely to the interphase separating the wet and dry parts of the dam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The Stefan problem, dating back to the 19th century, is one of the most classical and important free boundary problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It describes — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Some motivations and applications 129 the temperature of a homogeneous medium undergoing a phase change, typically a body of ice at zero degrees submerged in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In this context, it turns out that the integral of the temperature θ(x, t), namely u(x, t) := � t 0 θ, solves the parabolic version of the obstacle problem, � � � ut − ∆u = χ{u>0} in Ω × (0, T) ⊂ R3 × R, ∂tu ≥ 0, u ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The moving interphase separating the solid and liquid is exactly the free boundary ∂{u > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Hele-Shaw flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This two-dimensional model, dating back to 1898, de- scribes a fluid flow between two flat parallel plates separated by a very thin gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Various problems in fluid mechanics can be approximated to Hele-Shaw flows, and that is why understanding these flows is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A Hele-Shaw cell is an experimental device in which a viscous fluid is sandwiched in a narrow gap between two parallel plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In certain regions, the gap is filled with fluid while in others the gap is filled with air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' When liquid is injected inside the device through some sinks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' through a small hole on the top plate) the region filled with liquid grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In this context, an integral of the pressure solves, for each fixed time t, the obstacle problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In a similar way to the Dam problem, the free boundary corresponds to the interface between the fluid and the air regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Optimal stopping, finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In probability and finance, the obstacle prob- lem appears when considering optimal stopping problems for stochastic pro- cesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, consider a random walk (Brownian motion) inside a domain Ω ⊂ Rn, and a payoff function ϕ defined on the same domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We can stop the random walk at any moment, and we get the payoff at that position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We want to maximize the expected payoff (by choosing appropriately the stopping strategy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, it turns out that the highest expected payoff v(x) starting at a given position x satisfies the obstacle problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2), where the contact set {v = ϕ} is the region where we should immediately stop the random walk and get the payoff, while {v > ϕ} is the region where we should wait (see Appendix C for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Interacting particle systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Large systems of interacting particles arise in physical, biological, or material sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In some models, the particles attract each other when they are far, and experience a repulsive force when they are close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In other related models in statistical mechanics, the particles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' electrons) repel with a Coulomb — DRAFT — 130 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem force and one wants to understand their behavior in presence of some exter- nal field that confines them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In this kind of models, a natural and interesting question is to deter- mine the “equilibrium configurations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For instance, in Coulomb systems the charges accumulate in some region with a well defined boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Inter- estingly, these problems are equivalent to the obstacle problem — namely, the electric potential u = u(x) generated by the charges solves such prob- lem — and the contact set {u = 0} corresponds to the region in which the particles concentrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Quasi-Steady Electrochemical Shaping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Consider a metal inside an electrolyte under the action of an electric potential, in such a way that the metal shrinks with time due to a chemical reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, the integral (in time) of the potential satisfies, for each fixed time, the obstacle problem, whose free boundary corresponds to the shape of the metal at that moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Heat control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Trying to automatically control the temperature of a room using only heating devices, under suitable conditions, also yields the obsta- cle problem (in this case, for the temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Here, the free boundary separates the region where the heating devices are active and where they are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, in elasticity theory we probably find the most visual representation of the obstacle problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Given a thin membrane that is affected only by tension forces (thus tries to minimize area), it approximately satisfies the obstacle problem, where the contact region is the area where the membrane touches the obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Basic properties of solutions I We proceed now to study the basic properties of solutions to the obstacle problem: existence of solutions, optimal regularity, and nondegeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will first study all these properties for minimizers v ≥ ϕ of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1), and then in the next section we will study independently minimizers u ≥ 0 of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4) or (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is not only for completeness and clarity of presentation, but also to have both points of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For instance, the proof of the optimal regularity of solutions can be done in two completely different ways, one for each of the settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Existence of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Existence and uniqueness of solutions follows eas- ily from the fact that the functional � Ω |∇v|2dx is convex, and that we want to minimize it in the closed convex set {v ∈ H1(Ω) : v ≥ ϕ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Basic properties of solutions I 131 Recall that w|∂Ω denotes the trace of w on ∂Ω whenever it is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1 (Existence and uniqueness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ Rn be any bounded Lipschitz domain, and let g : ∂Ω → R and ϕ ∈ H1(Ω) be such that C = � w ∈ H1(Ω) : w ≥ ϕ in Ω, w|∂Ω = g � ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, there exists a unique minimizer of � Ω |∇v|2dx among all functions v ∈ H1(Ω) satisfying v ≥ ϕ in Ω and v|∂Ω = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The proof is quite similar to that of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, let θ◦ := inf �1 2 � Ω |∇w|2dx : w ∈ H1(Ω), w|∂Ω = g, w ≥ ϕ in Ω � , that is, the infimum value of E(w) = 1 2 � Ω |∇w|2dx among all admissible functions w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us take a sequence of functions {vk} such that vk ∈ H1(Ω) vk|∂Ω = g and vk ≥ ϕ in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' E(vk) → θ◦ as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By the Poincar´e inequality (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6), the sequence {vk} is uniformly bounded in H1(Ω), and therefore a subsequence {vkj} will converge to a certain function v strongly in L2(Ω) and weakly in H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, by compactness of the trace operator (see (S5) in Chapter 1), we will have vkj|∂Ω → v|∂Ω in L2(∂Ω), so that v|∂Ω = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Furthermore, such function v will satisfy E(v) ≤ lim infj→∞ E(vkj) (by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5) from (S4) in Chapter 1), and therefore it will be a minimizer of the energy functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since vkj ≥ ϕ in Ω and vkj → v in L2(Ω), we have v ≥ ϕ in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, we have proved the existence of a minimizer v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The uniqueness of the minimizer follows from the strict convexity of the functional E(v), exactly as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ As in the case of harmonic functions, it is easy to show that if a function v satisfies � � � v ≥ ϕ in Ω ∆v ≤ 0 in Ω ∆v = 0 in the set {v > ϕ}, then it must actually be the minimizer of the functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' There are two alternative ways to construct the solution to the obstacle problem: as the “least supersolution above the obstacle”, or with a “penal- ized problem”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us briefly describe them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 132 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem t βε(t) = e−t/ε Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The function βε → β0 as ε ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Least supersolution: This is related to the existence of viscosity solutions described in Chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, we consider v(x) := inf � w(x) : w ∈ C(Ω), −∆w ≥ 0 in Ω, w ≥ ϕ in Ω, w|∂Ω ≥ g � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Here, the inequality −∆w ≥ 0 in Ω has to be understood in the viscosity sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, as in Perron’s method (recall Chapters 1 and 4), it turns out that v is itself a continuous supersolution, it satisfies ∆v = 0 in {v > ϕ}, and thus it solves the obstacle problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, � least supersolution � ←→ � minimizer of the functional � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Penalized problem: We consider βε : R → R smooth and convex, converg- ing to β0(t) := � 0 if t ≥ 0 ∞ if t < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We may take for example βε(t) := e−t/ε, see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, we minimize the functional Jε(v) := 1 2 � Ω |∇v|2dx + � Ω βε(v − ϕ)dx, subject to the appropriate boundary conditions on ∂Ω, and get a solution vε ∈ C∞ of ∆vε = β′ ε(vε − ϕ) in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since β′ ε ≤ 0 everywhere, and β′ ε(t) = 0 for t ≥ 0, we have � −∆vε ≥ 0 everywhere in Ω ∆vε = 0 in {vε > ϕ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Basic properties of solutions I 133 As ε → 0, we have vε → v, where v is the solution to the obstacle problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer to [PSU12] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Basic properties of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us next prove that any minimizer v of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) is actually continuous and solves (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' From now on we will “forget” about the regularity of the obstacle, and assume that it is as smooth as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is why we will always be dealing with obstacles ϕ ∈ C∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' One gets analogous results under much weaker regularity assumptions on ϕ, which depend on the type of result to be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The role of the regularity of the obstacle is beyond the scope of this book, and thus we will always assume ϕ to be smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We start with the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ Rn be any bounded Lipschitz domain, ϕ ∈ C∞(Ω), and v ∈ H1(Ω) be any minimizer of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) subject to the boundary conditions v|∂Ω = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, −∆v ≥ 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let E(v) = 1 2 � Ω |∇v|2dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, since v minimizes E among all functions above the obstacle ϕ (and with fixed boundary conditions on ∂Ω), we have that E(v + εη) ≥ E(v) for every ε ≥ 0 and η ≥ 0, η ∈ C∞ c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This yields ε � Ω ∇v · ∇η + ε2 2 � Ω |∇η|2dx ≥ 0 for every ε ≥ 0 and η ≥ 0, η ∈ C∞ c (Ω), and thus � Ω ∇v · ∇η ≥ 0 for every η ≥ 0, η ∈ C∞ c (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This means that −∆v ≥ 0 in Ω in the weak sense, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ From here, by showing first that {v > ϕ} is open, we obtain the Euler– Lagrange equations for the functional: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ Rn be any bounded Lipschitz domain, ϕ ∈ C∞(Ω), and v ∈ H1(Ω) be any minimizer of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) subject to the bound- ary conditions v|∂Ω = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, v ∈ C(Ω) and it satisfies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7) � � � v ≥ ϕ in Ω ∆v ≤ 0 in Ω ∆v = 0 in {v > ϕ} ∩ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 134 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By construction, we already know that v ≥ ϕ in Ω and, thanks to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2, −∆v ≥ 0 in Ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e, v is (weakly) superharmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Up to replacing v in a set of measure zero, we may also assume that v is lower semi-continuous (by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, we only need to prove that ∆v = 0 in {v > ϕ} ∩ Ω and that v is, in fact, continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In order to do that, let us show first that {v > ϕ} ∩ Ω is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let x◦ ∈ {v > ϕ} ∩ Ω be such that v(x◦) − ϕ(x◦) > ε◦ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since v is lower semi-continuous and ϕ is continuous, there exists some δ > 0 such that v(x) − ϕ(x) > ε◦/2 for all x ∈ Bδ(x◦), and hence Bδ(x◦) ⊂ {v > ϕ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since x◦ was arbitrary, this means that {v > ϕ} is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This implies, also, that ∆v = 0 weakly in {v > ϕ} ∩ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, for any x◦ ∈ {v > ϕ} and η ∈ C∞ c (Bδ(x◦)) with |η| ≤ 1, we have v ± εη ≥ ϕ in Ω for all |ε| < ε◦/2, and therefore it is an admissible competitor to the minimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, we have E(v + εη) ≥ E(v) for all |ε| < ε◦, and differentiating in ε we deduce that v is harmonic in {v > ϕ} ∩ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, let us show that v is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We already know, by the regularity of harmonic functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12), that v is continuous in {v > ϕ} ∩ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us now show that v is continuous in {v = ϕ} ∩ Ω as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let y◦ ∈ {v = ϕ} ∩ Ω, and let us argue by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, since v is lower semi-continuous, let us assume that there is a sequence yk → y◦ such that v(yk) → v(y◦) + ε◦ = ϕ(y◦) + ε◦ for some ε◦ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since ϕ is continuous, we may assume also that yk ∈ {v > ϕ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us denote by zk the projection of yk towards {v = ϕ}, so that δk := |zk − y◦| ≤ 2|yk − y◦| ↓ 0 and v(zk) → v(y◦) = ϕ(y◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, since v is superharmonic by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20), v(zk) ≥ � B2δk(yk) v = (1 − 2−n) � B2δk(yk)\\Bδk(yk) v + 2−n � Bδk(yk) v = I1 + I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Observe that, for the first term, since v is lower semi-continuous and δk ↓ 0, we can assume that, for k large enough, v ≥ ϕ(y◦) − 2−nε◦ in B2δk, so that I1 ≥ (1 − 2−n)[ϕ(y◦) − 2−nε◦].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, since v is harmonic in Bδk(yk), we have by the mean-value property that I2 = 2−nv(yk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Combin- ing everything, we get v(zk) ≥ (1 − 2−n)[ϕ(y◦) − 2−nε◦] + 2−nv(yk) → ϕ(y◦) + 2−2nε◦ which contradicts the fact that we had v(zk) → v(y◦) = ϕ(y◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Hence, v is continuous in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ We next prove the following result, which says that v can be character- ized as the least supersolution above the obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Basic properties of solutions I 135 v ϕ ∆v = 0 ∆v = ∆ϕ Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Second derivatives are in general discontinuous across the free boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4 (Least supersolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ Rn be any bounded Lips- chitz domain, ϕ ∈ H1(Ω), and v ∈ H1(Ω) be any minimizer of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) subject to the boundary conditions v|∂Ω = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, for any function w satisfying −∆w ≥ 0 in Ω, w ≥ ϕ in Ω, and w|∂Ω ≥ v|∂Ω, we have w ≥ v in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In other words, if w is any supersolution above the obstacle ϕ, then w ≥ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If w is any function satisfying −∆w ≥ 0 in Ω, w ≥ ϕ in Ω, and w|∂Ω ≥ v|∂Ω, it simply follows from the maximum principle (Propo- sition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13) that w ≥ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, we have −∆w ≥ −∆v in Ω ∩ {v > ϕ}, and on the boundary of such set we have w|∂Ω ≥ v|∂Ω and w ≥ ϕ = v on {v = ϕ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Optimal regularity of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thanks to Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3, we know that any minimizer of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) is continuous and solves (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' From now on, we will actually localize the problem and study it in a ball: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8) � � � v ≥ ϕ in B1 ∆v ≤ 0 in B1 ∆v = 0 in {v > ϕ} ∩ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Our next goal is to answer the following question: Question: What is the optimal regularity of solutions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' First, a few important considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that in the set {v > ϕ} we have ∆v = 0, while in the interior of {v = ϕ} we have ∆v = ∆ϕ (since v = ϕ there);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, since ∆ϕ is in general not zero, ∆v is discontinuous across the free boundary ∂{v > ϕ} in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, v /∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 136 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem We will now prove that any minimizer of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) is actually C1,1, which gives the: Answer: v ∈ C1,1 (second derivatives are bounded but not continuous) The precise statement and proof are given next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5 (Optimal regularity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let ϕ ∈ C∞(B1), and v be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, v is C1,1 in B1/2, with the estimate ∥v∥C1,1(B1/2) ≤ C � ∥v∥L∞(B1) + ∥ϕ∥C1,1(B1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The constant C depends only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' To prove this, the main step is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let ϕ ∈ C∞(B1), and v be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let x◦ ∈ B1/2 be any point on {v = ϕ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, for any r ∈ (0, 1 4) we have 0 ≤ sup Br(x◦) (v − ϕ) ≤ Cr2, with C depending only on n and ∥ϕ∥C1,1(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' After dividing v by a constant if necessary, we may assume that ∥ϕ∥C1,1(B1) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let ℓ(x) := ϕ(x◦) + ∇ϕ(x◦) · (x − x◦) be the linear part of ϕ at x◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let r ∈ (0, 1 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, by C1,1 regularity of ϕ, in Br(x◦) we have ℓ(x) − r2 ≤ ϕ(x) ≤ v(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We want to show that, in the ball Br(x◦) (see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5), we have v(x) ≤ ℓ(x) + Cr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For this, consider w(x) := v(x) − � ℓ(x) − r2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This function w satisfies w ≥ 0 in Br(x◦), and −∆w = −∆v ≥ 0 in Br(x◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us split w into w = w1 + w2, with � ∆w1 = 0 in Br(x◦) w1 = w on ∂Br(x◦) and � −∆w2 ≥ 0 in Br(x◦) w2 = 0 on ∂Br(x◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that 0 ≤ w1 ≤ w and 0 ≤ w2 ≤ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Basic properties of solutions I 137 x◦ {v = ϕ} ∂Br(x◦) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The solution v and a free boundary point x◦ We have that w1(x◦) ≤ w(x◦) = v(x◦) − � ℓ(x◦) − r2� = r2, and thus by the Harnack inequality ∥w1∥L∞(Br/2(x◦)) ≤ Cr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For w2, notice that ∆w2 = ∆v, and in particular ∆w2 = 0 in {v > ϕ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This means that w2 attains its maximum on {v = ϕ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' But in the set {v = ϕ} we have w2 ≤ w = ϕ − � ℓ − r2� ≤ Cr2, and therefore we deduce that ∥w2∥L∞(Br(x◦)) ≤ Cr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Combining the bounds for w1 and w2, we get ∥w∥L∞(Br(x◦)) ≤ Cr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Translating this into v, and using that ∥ϕ∥C1,1(B1) ≤ 1, we find v − ϕ ≤ Cr2 in Br/2(x◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Therefore, we have proved that: At every free boundary point x◦, v separates from ϕ at most quadratically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As shown next, this easily implies the C1,1 regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Dividing v by a constant if necessary, we may assume that ∥v∥L∞(B1) + ∥ϕ∥C1,1(B1) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We already know that v ∈ C∞ in the set {v > ϕ} (since v is harmonic), and also in the interior of the set {v = ϕ} (since ϕ ∈ C∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, on the interface Γ = ∂{v > ϕ} we have proved the quadratic growth supBr(x◦)(v − ϕ) ≤ Cr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us prove that this yields the C1,1 bound we want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 138 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem {v = ϕ} x◦ x1 ρ Bρ(x1) {v > ϕ} ∆v = 0 Γ Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A solution v satisfying ∆v = 0 in Bρ(x1) ⊂ {v > ϕ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let x1 ∈ {v > ϕ} ∩ B1/2, and let x◦ ∈ Γ be the closest free boundary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Denote ρ = |x1 − x◦|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, we have ∆v = 0 in Bρ(x1) (see the setting in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6), and thus we have also ∆(v − ℓ) = 0 in Bρ(x1), where ℓ is the linear part of ϕ at x◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By estimates for harmonic functions, we find ∥D2v∥L∞(Bρ/2(x1)) = ∥D2(v − ℓ)∥L∞(Bρ/2(x1)) ≤ C ρ2 ∥v − ℓ∥L∞(Bρ(x1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' But by the growth proved in the previous Lemma, we have ∥v−ℓ∥L∞(Bρ(x1)) ≤ Cρ2, which yields ∥D2v∥L∞(Bρ/2(x1)) ≤ C ρ2 ρ2 = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, |D2v(x1)| ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We can do this for all x1 ∈ {v > ϕ} ∩ B1/2, and on ∂{v > ϕ} we have quadratic growth by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6, hence it follows that ∥v∥C1,1(B1/2) ≤ C, as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ The overall strategy of the proof of optimal regularity is summarized in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Nondegeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We now want to prove that, at all free boundary points, v separates from ϕ at least quadratically (we already know at most quadrat- ically).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, we want (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9) 0 < cr2 ≤ sup Br(x◦) (v − ϕ) ≤ Cr2 for all free boundary points x◦ ∈ ∂{v > ϕ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This property is essential in order to study the free boundary later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Basic properties of solutions I 139 {u = 0} {u = 0} {u = 0} ∂{u = 0} ∂{u = 0} ∂{u = 0} quadratic growth by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6 u ∈ C1,1 by interior estimates u u u Cr2 Cr2 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Strategy of the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since −∆v ≥ 0 everywhere, it is clear that if x◦ ∈ ∂{v > ϕ} is a free boundary point, then necessarily −∆ϕ(x◦) ≥ 0 (otherwise we would have −∆ϕ(x◦) < 0, and since u touches ϕ from above at x◦, also −∆v(x◦) < 0, a contradiction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover it can be proved that, in fact, if ∆ϕ and ∇∆ϕ do not vanish simultaneously, then −∆ϕ > 0 near all free boundary points [Caf98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This motivates the following: Assumption: The obstacle ϕ satisfies −∆ϕ ≥ c◦ > 0 in the ball B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, by Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7, if ∆ϕ and ∇∆ϕ do not vanish simultane- ously, then we have −∆ϕ > 0 near any free boundary point, and thus by zooming in if necessary, we will always have that the assumption is satisfied in B1, for some small c◦ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, the only real assumption here is that ∆ϕ and ∇∆ϕ do not vanish simultaneously, which is a very mild assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, this is in a sense a necessary assumption: without this, the nondegeneracy (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9) does not hold, and no regularity result can be proved for the free boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (Without the assumption, one can actually construct counterexamples in which the free boundary is a fractal set with infinite perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8 (Nondegeneracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let ϕ ∈ C∞(B1), and v be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that ϕ satisfies −∆ϕ ≥ c◦ > 0 in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, for every free boundary point x◦ ∈ ∂{v > ϕ} ∩ B1/2, we have 0 < cr2 ≤ sup Br(x◦) (v − ϕ) ≤ Cr2 for all r ∈ (0, 1 4), with a constant c > 0 depending only on n and c◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let x1 ∈ {v > ϕ} be any point close to x◦ (we will then let x1 → x◦ at the end of the proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 140 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem Consider the function w(x) := v(x) − ϕ(x) − c◦ 2n|x − x1|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, in {v > ϕ} we have ∆w = ∆v − ∆ϕ − c◦ = −∆ϕ − c◦ ≥ 0 and hence −∆w ≤ 0 in {v > ϕ} ∩ Br(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, w(x1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By the maximum principle, w attains a positive maximum on ∂ � {v > ϕ} ∩ Br(x1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' But on the free boundary ∂{v > ϕ} we clearly have w < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, there is a point on ∂Br(x1) at which w > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In other words, 0 < sup ∂Br(x1) w = sup ∂Br(x1) (v − ϕ) − c◦ 2n r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Letting now x1 → x◦, we find sup∂Br(x◦)(v − ϕ) ≥ cr2 > 0, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Summary of basic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let v be any solution to the obstacle problem � � � v ≥ ϕ in B1 ∆v ≤ 0 in B1 ∆v = 0 in {v > ϕ} ∩ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, we have: Optimal regularity: ∥v∥C1,1(B1/2) ≤ C � ∥v∥L∞(B1) + ∥ϕ∥C1,1(B1) � Nondegeneracy: If −∆ϕ ≥ c◦ > 0, then 0 < cr2 ≤ sup Br(x◦) (v − ϕ) ≤ Cr2 for all r ∈ (0, 1 2) at all free boundary points x◦ ∈ ∂{v > ϕ} ∩ B1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Equivalence with zero obstacle: The problem is equivalent to � � � u ≥ 0 in B1 ∆u ≤ f in B1 ∆u = f in {u > 0} ∩ B1, where f = −∆ϕ ≥ c◦ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will next provide an alternative approach to the optimal regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Basic properties of solutions II 141 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Basic properties of solutions II We proceed now to study the basic properties of solutions u ≥ 0 to the obstacle problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4) or (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As explained before, the main point here is that we prove optimal regularity independently from the previous Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Throughout this section we will always assume f ≥ 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Existence of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4) is equivalent to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1), ex- istence and uniqueness of solutions follow easily from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1, as shown next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9 (Existence and uniqueness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ Rn be any bounded Lipschitz domain, and let g : ∂Ω → R be such that C = � u ∈ H1(Ω) : u ≥ 0 in Ω, u|∂Ω = g � ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, for any f ∈ L2(Ω) there exists a unique minimizer of 1 2 � Ω |∇u|2dx + � Ω fu among all functions u ∈ H1(Ω) satisfying u ≥ 0 in Ω and u|∂Ω = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We follow the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let θ◦ := inf �1 2 � Ω |∇w|2dx + � Ω fw : w ∈ H1(Ω), w|∂Ω = g, w ≥ 0 in Ω � , that is, the infimum value of E(w) = 1 2 � Ω |∇w|2dx+ � Ω fw among all admis- sible functions w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that, by H¨older’s inequality, E(w) < +∞ if w ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We take again a sequence of functions {vk} such that vk ∈ H1(Ω), vk|∂Ω = g, vk ≥ 0 in Ω, and E(vk) → θ◦ as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By the Poincar´e inequal- ity (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6), H¨older’s inequality, and the fact that E(vk) ≤ θ◦ + 1, for k large enough ∥vk∥2 H1(Ω) ≤ C �� Ω |∇vk|2 + � ∂Ω g2 � ≤ C � θ◦ + 1 + � Ω |fvk| + 1 2 � ∂Ω g2 � ≤ C � θ◦ + 1 + ∥f∥L2(Ω)∥vk∥H1(Ω) + 1 2 � ∂Ω g2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, ∥vk∥H1(Ω) ≤ C for some constant C depending only on n, Ω, g, f, and θ◦ (recall that g ∈ L2(∂Ω) by the trace theorem, (S5) in Chapter 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Hence, a subsequence {vkj} converges to a certain function v strongly in L2(Ω) and weakly in H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By compactness of the trace operator vkj|∂Ω → v|∂Ω = g in L2(∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Furthermore, v satisfies E(v) ≤ lim infj→∞ E(vkj) (by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5) from (S4) and weak convergence), and therefore it will be a — DRAFT — 142 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem minimizer of the energy functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since vkj ≥ 0 in Ω and vkj → v in L2(Ω), we have v ≥ 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, there is a minimizer v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The uniqueness of the minimizer follows from the strict convexity of the functional E(v), exactly as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Alternatively, we could have denoted v := u+ϕ with ϕ such that −∆ϕ = f in Ω, and use Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Furthermore, we have the following equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (Recall that we denote u+ = max{u, 0}, and u− = max{−u, 0}, so that u = u+ − u−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ Rn be any bounded Lipschitz domain, and let g : ∂Ω → R be such that C = � u ∈ H1(Ω) : u ≥ 0 in Ω, u|∂Ω = g � ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, the following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (i) u minimizes 1 2 � Ω |∇u|2+ � Ω fu among all functions satisfying u ≥ 0 in Ω and u|∂Ω = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (ii) u minimizes 1 2 � Ω |∇u|2 + � Ω fu+ among all functions satisfying u|∂Ω = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The two functionals coincide whenever u ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, the only key point is to prove that the minimizer in (ii) must be nonnegative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', u = u+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (Notice that C ̸= ∅ implies that g ≥ 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') To show this, recall that the positive part of any H1 function is still in H1, and moreover |∇u|2 = |∇u+|2 + |∇u−|2 (see (S9) in Chapter 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, we have that (recall that f ≥ 0 in Ω) 1 2 � Ω |∇u+|2 + � Ω fu+ ≤ 1 2 � Ω |∇u|2 + � Ω fu+, with strict inequality unless u = u+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This means that any minimizer u of the functional in (ii) must be nonnegative, and thus we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Basic properties of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us next prove that any minimizer of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4) is actually a solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We recall that we are always assuming that obstacles are as smooth as necessary, ϕ ∈ C∞(Ω), and therefore we assume here that f ∈ C∞(Ω) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ Rn be any bounded Lipschitz domain, f ∈ C∞(Ω), and u ∈ H1(Ω) be any minimizer of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4) subject to the boundary conditions u|∂Ω = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u solves (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10) ∆u = fχ{u>0} in Ω — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Basic properties of solutions II 143 in the weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, u is C1,α inside Ω, for every α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that, by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11, u is actually a minimizer of E(u) = 1 2 � Ω |∇u|2 + � Ω fu+ subject to the boundary conditions u|∂Ω = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, for any η ∈ H1 0(Ω) and ε > 0 we have E(u + εη) ≥ E(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, we obtain 0 ≤ lim ε↓0 E(u + εη) − E(u) ε = � Ω ∇u · ∇η + lim ε↓0 � Ω f (u + εη)+ − u+ ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that lim ε↓0 (u + εη)+ − u+ ε = � η in {u > 0} η+ in {u = 0}, so that we have � Ω ∇u · ∇η + � Ω fηχ{u>0} + � Ω fη+χ{u=0} ≥ 0 for all η ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume first that η ≥ 0, so that � Ω ∇u · ∇η + � Ω fη ≥ 0 for all η ∈ H1 0(Ω), η ≥ 0, which implies that ∆u ≤ f in the weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, if η ≤ 0, then � Ω ∇u · ∇η + � Ω fηχ{u>0} ≥ 0 for all η ∈ H1 0(Ω), η ≤ 0, which implies that ∆u ≥ fχ{u>0} in the weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In all (recall that f ≥ 0), fχ{u>0} ≤ ∆u ≤ f in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (In particular, notice that ∆u = f in {u > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') Now, since f is smooth, this implies that ∆u ∈ L∞ loc(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18 we deduce that u ∈ C1,1−ε for every ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, since ∆u ∈ L∞ loc(Ω) we have ∆u ∈ L2 loc(Ω) and by Calder´on-Zygmund estimates (see, for example, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13) we have u ∈ W 2,2 loc (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, ∆u = 0 almost everywhere in the level set {u = 0} (see (S9) in Chapter 1) and we have ∆u = fχ{u>0} a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' From here we deduce that ∆u = fχ{u>0} in Ω in the weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ — DRAFT — 144 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem Notice that in the previous Section, when dealing with minimizers v of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1), it was not easy to prove that v is continuous (see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Here, instead, thanks to Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12 we simply used Schauder-type estimates for the Laplacian to directly deduce that u is C1,1−ε, which is the almost-optimal regularity of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Optimal regularity of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thanks to the previous results, we know that any minimizer of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4) is continuous and solves (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' From now on, we will localize the problem and study it in a ball: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11) � u ≥ 0 in B1 ∆u = fχ{u>0} in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Our next goal is to answer the following question: Question: What is the optimal regularity of solutions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' First, a few important considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that in the set {u > 0} we have ∆u = f, while in the interior of {u = 0} we have ∆u = 0 (since u ≡ 0 there).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, since f is in general not zero, ∆u is discontinuous across the free boundary ∂{u > 0} in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, u /∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will now prove that any minimizer of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4) is actually C1,1, which gives the: Answer: u ∈ C1,1 (second derivatives are bounded but not continuous) The precise statement and proof are given next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13 (Optimal regularity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let f ∈ C∞(B1), and let u be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u is C1,1 inside B1/2, with the estimate ∥u∥C1,1(B1/2) ≤ C � ∥u∥L∞(B1) + ∥f∥Lip(B1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The constant C depends only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' To prove this, the main step is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let x◦ ∈ B1/2 be any point on {u = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, for any r ∈ (0, 1 4) we have 0 ≤ sup Br(x◦) u ≤ Cr2, with C depending only on n and ∥f∥L∞(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Basic properties of solutions II 145 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We have that ∆u = fχ{u>0} in B1, with fχ{u>0} ∈ L∞(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, since u ≥ 0, we can use the Harnack inequality (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9) for the equa- tion ∆u = fχ{u>0} in B2r(x◦), to find sup Br(x◦) u ≤ C � inf Br(x◦) u + r2∥fχ{u>0}∥L∞(B2r(x◦)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since u ≥ 0 and u(x◦) = 0, this yields supBr(x◦) u ≤ C∥f∥L∞(B1)r2, as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Notice that this proof is significantly shorter than the one given in the previous Section (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is an advantage of using the formula- tion (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We have proved the following: At every free boundary point x◦, u grows (at most) quadratically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As shown next, this easily implies the C1,1 regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Dividing u by a constant if necessary, we may assume that ∥u∥L∞(B1) + ∥f∥Lip(B1) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We already know that u ∈ C∞ in the set {u > 0} (since ∆u = f ∈ C∞ there), and also inside the set {u = 0} (since u = 0 there).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, on the interface Γ = ∂{u > 0} we have proved the quadratic growth supBr(x◦) u ≤ Cr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us prove that this yields the C1,1 bound we want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let x1 ∈ {u > 0} ∩ B1/2, and let x◦ ∈ Γ be the closest free boundary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Denote ρ = |x1 − x◦|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, we have ∆u = f in Bρ(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By Schauder estimates, we find ∥D2u∥L∞(Bρ/2(x1)) ≤ C � 1 ρ2 ∥u∥L∞(Bρ(x1)) + ∥f∥Lip(B1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' But by the growth proved in the previous Lemma, we have ∥u∥L∞(Bρ(x1)) ≤ Cρ2, which yields ∥D2u∥L∞(Bρ/2(x1)) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, |D2u(x1)| ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We can do this for each x1 ∈ {u > 0}∩B1/2, and therefore ∥u∥C1,1(B1/2) ≤ C, as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Also, notice that as a consequence of the previous results, we have that as soon as the solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11) has non-empty contact set, then its C1,1 norm is universally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 146 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11), and let us assume that u(0) = 0 and ∥f∥Lip(B1) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, ∥u∥C1,1(B1/2) ≤ C for some C depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It is an immediate consequence of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13 combined with Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Nondegeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For completeness, we now state the nondegeneracy in this setting (analogously to Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, at all free boundary points, u grows at least quadratically (we already know at most quadrati- cally).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We want: 0 < cr2 ≤ sup Br(x◦) u ≤ Cr2 for all free boundary points x◦ ∈ ∂{u > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This property is essential in order to study the free boundary later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As before, for this we need the following natural assumption: Assumption: The right-hand side f satisfies f ≥ c◦ > 0 in the ball B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16 (Nondegeneracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As- sume that f ≥ c◦ > 0 in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, for every free boundary point x◦ ∈ ∂{u > 0} ∩ B1/2, we have 0 < cr2 ≤ sup Br(x◦) u ≤ Cr2 for all r ∈ (0, 1 2), with a constant c > 0 depending only on n and c◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The proof is the one from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Summary of basic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to the obstacle problem � u ≥ 0 in B1, ∆u = fχ{u>0} in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, we have: Optimal regularity: ∥u∥C1,1(B1/2) ≤ C � ∥u∥L∞(B1) + ∥f∥Lip(B1) � Nondegeneracy: If f ≥ c◦ > 0, then 0 < cr2 ≤ sup Br(x◦) u ≤ Cr2 for all r ∈ (0, 1 2) — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Regularity of free boundaries: an overview 147 {u = 0} ∂B1 ∆u = f Γ u ∈ C1,1 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A solution to the obstacle problem in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' at all free boundary points x◦ ∈ ∂{u > 0} ∩ B1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Using these properties, we can now start the study of the free boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Regularity of free boundaries: an overview From now on, we consider any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12) � � � � � u ∈ C1,1(B1), u ≥ 0 in B1, ∆u = f in {u > 0}, (see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8) with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13) f ≥ c◦ > 0 and f ∈ C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that on the interface Γ = ∂{u > 0} ∩ B1 we have that u = 0 on Γ, ∇u = 0 on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The central mathematical challenge in the obstacle problem is to understand the geometry/regularity of the free boundary Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that, even if we already know the optimal regularity of u (it is C1,1), we know nothing about the free boundary Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A priori Γ could be a very irregular object, even a fractal set with infinite perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As we will see, under the natural assumption f ≥ c◦ > 0, it turns out that free boundaries are always smooth, possibly outside a certain set of — DRAFT — 148 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem {u = 0} {u = 0} ∆u = f in {u > 0} all regular points one singular point (the contact set has zero density) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Singular points are those where the contact set has zero density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In fact, in our proofs we will assume for simplicity that f ≡ 1 (or constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We do that in order to avoid x-dependence and the technicalities associated to it, which gives cleaner proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In this way, the main ideas behind the regularity of free boundaries are exposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Regularity of free boundaries: main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume from now on that u solves (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12)-(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, the main known results on the free bound- ary Γ = ∂{u > 0} can be summarized as follows: At every free boundary point x◦ ∈ Γ, we have 0 < cr2 ≤ sup Br(x◦) u ≤ Cr2 ∀r ∈ (0, r◦) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The free boundary Γ splits into regular points and singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The set of regular points is an open subset of the free boundary, and Γ is C∞ near these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Singular points are those at which the contact set {u = 0} has zero density, and these points (if any) are contained in an (n−1)-dimensional C1 manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Summarizing, the free boundary is smooth, possibly outside a certain set of singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' See Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' So far, we have not even proved that Γ has finite perimeter, or anything at all about Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Our goal will be to prove that Γ is C∞ near regular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is the main and most important result in the obstacle problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It was proved by Caffarelli in 1977, and it is one of the major results for which he received the Wolf Prize in 2012 and the Shaw Prize in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Overview of the strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' To prove these regularity results for the free boundary, one considers blow-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Namely, given any free boundary point x◦ — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Regularity of free boundaries: an overview 149 u0(x) = 1 2(x · e)2 + u0(x) = 1 2x2 1 e Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Possible blow-ups of the solution to the obstacle problem at free boundary points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' for a solution u of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12)-(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13), one takes the rescalings ur(x) := u(x◦ + rx) r2 , with r > 0 small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is like “zooming in” at a free boundary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The factor r−2 is chosen so that ∥ur∥L∞(B1) ≈ 1 as r → 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' recall that 0 < cr2 ≤ supBr(x◦) u ≤ Cr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, by C1,1 estimates, we will prove that a subsequence of ur converges to a function u0 locally uniformly in Rn as r → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Such function u0 is called a blow-up of u at x◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Any blow-up u0 is a global solution to the obstacle problem, with f ≡ 1 (or with f ≡ constant > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, the main issue is to classify blow-ups: that is, to show that either u0(x) = 1 2(x · e)2 + (this happens at regular points) or u0(x) = 1 2xT Ax (this happens at singular points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Here, e ∈ Sn−1 is a unit vector, and A ≥ 0 is a positive semi-definite matrix satisfying trA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that the contact set {u0 = 0} becomes a half- space in case of regular points, while it has zero measure in case of singular points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Once this is done, one has to “transfer” the information from the blow- up u0 to the original solution u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Namely, one shows that, in fact, the free boundary is C1,α near regular points (for some small α > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, once we know that the free boundary is C1,α, we will “boot- strap” the regularity to C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is in a somewhat similar spirit as in Hilbert’s XIXth problem (Chapter 3), where the really difficult point was to — DRAFT — 150 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem prove that minimizers are always C1,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Once this was done, by Schauder estimates (Chapter 2) and a bootstrap argument we saw that solutions are actually C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Classifying blow-ups is not easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Generally speaking, classifying blow- ups is of similar difficulty to proving regularity estimates — recall the blow- up arguments in Chapter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, how can we classify blow-ups?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Do we get any extra information on u0 that we did not have for u?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (Otherwise it seems hopeless!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') The answer is yes: Convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will prove that all blow-ups are always convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is a huge improvement, since this yields that the contact set {u0 = 0} is also convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Prior to that, we will also show that blow-ups are also homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' So, before the blow-up we had no information on the set {u = 0}, but after the blow-up we get that {u0 = 0} is a convex cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thanks to this we will be able to classify blow-ups, and thus to prove the regularity of the free boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The main steps in the proof of the regularity of the free boundary will be the following: (1) 0 < cr2 ≤ supBr(x◦) u ≤ Cr2 (2) Blow-ups u0 are homogeneous and convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (3) If the contact set has positive density at x◦, then u0(x) = 1 2(x·e)2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (4) Deduce that the free boundary is C1,α near x◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (5) Deduce that the free boundary is C∞ near x◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The proof we will present here for the convexity of blow-ups is new, based on the fact that they are homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer to [Caf98], [PSU12], [Wei99], and [KN77], for different proofs of the classification of blow-ups and/or of the regularity of free boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Classification of blow-ups The aim of this Section is to classify all possible blow-ups u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For this, we will first prove that blow-ups are homogeneous, then we will prove that they are convex, and finally we will establish their complete classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Homogeneity of blow-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We start by proving that blow-ups are ho- mogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is not essential in the proof of the regularity of the free boundary (see [Caf98]), but it actually simplifies it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Classification of blow-ups 151 0 {u = 0} B1 ∆u = 1 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A solution u to the obstacle problem with f ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Recall that, for simplicity, from now on we will assume that f ≡ 1 in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is only to avoid x-dependence in the equation, it simplifies some proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, from now on we consider a solution u satisfying (see Fig- ure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='11): (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14) u ∈ C1,1(B1) u ≥ 0 in B1 ∆u = 1 in {u > 0} 0 is a free boundary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will prove all the results around the origin (without loss of generality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will show that, for the original solution u in B1, the closer we look at a free boundary point x◦, the closer is the solution to being homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17 (Homogeneity of blow-ups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, any blow-up of u at 0 is homogeneous of degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It is important to remark that not all global solutions to the obstacle problem in Rn are homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' There exist global solutions u0 that are convex, C1,1, and whose contact set {u0 = 0} is an ellipsoid, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' However, thanks to the previous result, we find that such non-homogeneous solutions cannot appear as blow-ups, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', that all blow-ups must be homo- geneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We provide two different proofs of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The first one uses a monotonicity formula as introduced by Weiss;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' while the second one does not require any monotonicity formula and is due to Spruck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Homogeneity of blow-ups `a la Weiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For the first proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17, we need the following monotonicity formula due to Weiss [Wei99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 152 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18 (Weiss’ monotonicity formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, the quantity (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15) Wu(r) := 1 rn+2 � Br � 1 2|∇u|2 + u � − 1 rn+3 � ∂Br u2 is monotone in r, that is, d drWu(r) = 1 rn+4 � ∂Br (x · ∇u − 2u)2dx ≥ 0 for r ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let ur(x) = r−2u(rx), and observe that Wu(r) = � B1 � 1 2|∇ur|2 + ur � − � ∂B1 u2 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Using this, together with d dr(∇ur) = ∇ d drur, we find d drWu(r) = � B1 � ∇ur · ∇ d drur + d drur � − 2 � ∂B1 ur d drur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, integrating by parts we get � B1 ∇ur · ∇ d drur = − � B1 ∆ur d drur + � ∂B1 ∂ν(ur) d drur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since ∆ur = 1 in {ur > 0} and d drur = 0 in {ur = 0}, we have � B1 ∇ur · ∇ d drur = − � B1 d drur + � ∂B1 ∂ν(ur) d drur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, we deduce d drWu(r) = � ∂B1 ∂ν(ur) d drur − 2 � ∂B1 ur d drur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Using that on ∂B1 we have ∂ν = x · ∇, combined with d drur = 1 r {x · ∇ur − 2ur} yields d drWu(r) = 1 r � ∂B1 (x · ∇ur − 2ur)2 , which gives the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ We now give the: — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Classification of blow-ups 153 First proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let ur(x) = r−2u(rx), and notice that we have the scaling property Wur(ρ) = Wu(ρr), for any r, ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If u0 is any blow-up of u at 0 then there is a sequence rj → 0 satisfying urj → u0 in C1 loc(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, for any ρ > 0 we have Wu0(ρ) = lim rj→0 Wurj (ρ) = lim rj→0 Wu(ρrj) = Wu(0+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that the limit Wu(0+) := limr→0 Wu(r) exists by monotonicity of W and since u ∈ C1,1 implies Wu(r) ≥ −C for all r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Hence, the function Wu0(ρ) is constant in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' However, by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18 this yields that x · ∇u0 − 2u0 ≡ 0 in Rn, and therefore u0 is homogeneous of degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Here, we used that a C1 function u0 is 2-homogeneous (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' u0(λx) = λ2u0(x) for all λ ∈ R+) if and only if x · ∇u0 ≡ 2u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is because ∂λ|λ=1 � λ−2u0(λx) � = x · ∇u0 − 2u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Homogeneity of blow-ups `a la Spruck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We present an alternative (and quite different) proof of the homogeneity of blow-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Such proof is due to Spruck [Spr83] and is not based on any monotonicity formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Second proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u0 be a blow-up given by the limit along a sequence rk ↓ 0, u0(x) := lim k→∞ r−2 k u(rkx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By taking polar coordinates (ϱ, θ) ∈ [0, +∞)×Sn−1 with x = ϱθ, and by denoting ˜u0(ϱ, θ) = u0(ϱθ) = u0(x), we will prove that u0(x) = ϱ2˜u0(1, θ) = |x|2u0(x/|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us define τ := − log ϱ, ˜u(ϱ, θ) = u(x), and ψ = ψ(τ, θ) as ψ(τ, θ) := ϱ−2˜u(ϱ, θ) = e2τu(e−τθ) for τ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We observe that, since ∥u∥L∞(Br) ≤ Cr2, ψ is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, ψ ∈ C1((0, ∞) × Sn−1) ∩ C2({ψ > 0}) from the regularity of u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' and ∂τψ and ∇θψ are not only continuous, but also uniformly bounded in [0, ∞) × Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, ��∇θψ(τ, θ) �� ≤ eτ��∇u(e−τθ) �� ≤ C, since ∥∇u∥L∞(Br) ≤ Cr by C1,1 regularity and the fact that ∇u(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For the same reason we also obtain ��∂τψ(τ, θ) �� ≤ 2ψ(τ, θ) + eτ��∇u(e−τθ) �� ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 154 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem Observe that, by assumption, if we denote τk := − log rk, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16) ψ(τk, θ) → ˜u0(1, θ) uniformly on Sn−1, as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us now write an equation for ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In order to do that, since we know that ∆u = χ{u>0} and χ{u>0} = χ{ψ>0}, we have ∆ � ϱ2ψ(− log ϱ, θ) � = χ{ψ>0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By expanding the Laplacian in polar coordinates, ∆ = ∂ϱϱ + n−1 ϱ ∂ϱ + ϱ−2∆Sn−1 (where ∆Sn−1 denotes the spherical Laplacian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' the Laplace– Beltrami operator on Sn−1) we obtain (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17) 2nψ − (n + 2)∂τψ + ∂ττψ + ∆Sn−1ψ = χ{ψ>0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We multiply the previous equality by ∂τψ, and integrate in [0, τ]×Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We can consider the terms separately, integrating in τ first, 2n � Sn−1 � τ 0 ψ∂τψ = n � Sn−1 � ψ2(τ, θ) − ψ2(0, θ) � dθ and � Sn−1 � τ 0 ∂ττψ∂τψ = 1 2 � Sn−1 � (∂τψ)2(τ, θ) − (∂τψ)2(0, θ) � dθ, and then integrating by parts in θ first, to integrate in τ afterwards: � τ 0 � Sn−1 ∆Sn−1ψ∂τψ = −1 2 � τ 0 � Sn−1 ∂τ|∇θψ|2 = 1 2 � Sn−1 � |∇θψ|2(0, θ) − |∇θψ|2(τ, θ) � dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, since ∂τψ = 0 whenever ψ = 0, we have χ{ψ>0}∂τψ = ∂τψ and � Sn−1 � τ 0 χ{ψ>0}∂τψ = � Sn−1 � ψ(τ, θ) − ψ(0, θ) � dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In all, plugging back in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17) the previous expressions, and using that ∂τψ and ∇θψ are uniformly bounded in [0, ∞) × Sn−1, we deduce that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18) � ∞ 0 � Sn−1(∂τψ)2 = � ∞ 0 ∥∂τψ∥2 L2(Sn−1) ≤ C < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' To finish, now observe that for any |s| ≤ C∗ fixed and for a sufficiently large k (such that τk + s ≥ 0), ∥ψ(τk + s, ·) − ˜u0(1, ·)∥L2(Sn−1) ≤ ∥ψ(τk + s, ·) − ψ(τk, ·)∥L2(Sn−1) + ∥ψ(τk, ·) − ˜u0(1, ·)∥L2(Sn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Classification of blow-ups 155 The last term goes to zero, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, for the first term and by H¨older’s inequality ∥ψ(τk + s, ·) − ψ(τk, ·)∥2 L2(Sn−1) ≤ ���� � s 0 ∂τψ(τk + τ, ·) dτ ���� 2 L2(Sn−1) ≤ C∗ ���� � τk+s τk ∥∂τψ∥2 L2(Sn−1) ���� → 0, as k → ∞, where we are using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Hence, ψ(τk + s, ·) → ˜u0(1, ·) in L2(Sn−1) as k → ∞, for any fixed s ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, ψ(τk + s, θ) = e2sr−2 k u(e−2rkθ) → e2su0(e−sθ) = e2s˜u0(e−s, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, for any ρ = e−s > 0, ˜u0(1, ·) = ρ−2˜u0(ρ, θ), as we wanted to see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Convexity of blow-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By taking advantage of the fact that we know that blow-ups are 2-homogeneous, we can now give a short (and new) proof of the fact that they are also convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' More precisely, we will prove that 2-homogeneous global solutions to the obstacle problem are convex (and in particular, by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17, blow-ups are convex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u0 ∈ C1,1 be any 2-homogeneous global solution to � � � � � u0 ≥ 0 in Rn ∆u0 = 1 in {u0 > 0} 0 is a free boundary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u0 is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The heuristic idea behind the proof of the previous result is the following: second derivatives D2u0 are harmonic in {u0 > 0} and satisfy that D2u0 ≥ 0 on ∂{u0 > 0} (since u0 ≥ 0, it is “convex at the free boundary”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since D2u0 is also 0-homogeneous, we can apply the maximum principle and conclude that D2u0 ≥ 0 everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, u0 is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us formalize the previous heuristic idea into an actual proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We state a short lemma before providing the proof, which says that if w ≥ 0 is superharmonic in {w > 0}, then it is superharmonic everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For the sake of generality, we state the lemma for general H1 functions, but we will use it only for functions that are also continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Λ ⊂ B1 be closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let w ∈ H1(B1) be such that w ≥ 0 on Λ and such that w is superharmonic in the weak sense in B1 \\ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then min{w, 0} is superharmonic in the weak sense in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 156 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us start by assuming that w is, furthermore, continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In this case, we define wε = min{w, −ε} ∈ H1(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then notice that (by continuity) in a neighborhood of {w = −ε}, w is superharmonic (∆w ≤ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9 (we apply the lemma with v = −w − ε) we have that ∆wε ≤ 0 in the weak sense, namely, wε is superharmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, they are uniformly in H1, so up to subsequences they converge weakly to min{w, 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since the weak limit of weakly superharmonic functions is superharmonic, we deduce the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, to remove the continuity assumption on w ∈ H1(B1), we repeat the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The only thing we need to check is that F ′(v)η ∈ H1 0(B1 \\Λ), which follows from the fact that such function is in H1(B1) and vanishes in Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' see for example [AH96, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ We now give the: Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let e ∈ Sn−1 and consider the second derivatives ∂eeu0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We define w0 := min{∂eeu0, 0} and we claim that w0 is superharmonic in Rn, in the sense (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, let δ2 t u0(x) for t > 0 be defined by δ2 t u0(x) := u0(x + te) + u0(x − te) − 2u0(x) t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, since ∆u0 = χ{u0>0}, we have that ∆δ2 t u0 = 1 t2 � χ{u0( · +te)} + χ{u0( · −te)} − 2 � ≤ 0 in {u0 > 0} in the weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, δ2 t u0 ≥ 0 in {u0 = 0} and δ2 t u0 ∈ C1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='21, wt := min{δ2 t u0, 0} is weakly superharmonic, and hence it satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Also notice that δ2 t u0(x) is uniformly bounded independently of t, since u0 ∈ C1,1, and therefore wt is uniformly bounded in t and converges pointwise to w0 as t ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16 we have that w0 is superharmonic in the sense of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20), as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Up to changing it in a set of measure 0, w0 is lower semi-continuous by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, since w0 is 0-homogeneous, it must attain its minimum at a point y◦ ∈ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' But since � Br(y◦) w0 is non-increasing for r > 0, we must have that w0 is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since it vanishes on the free boundary, we have w0 ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, for any e ∈ Sn−1 we have that ∂eeu0 ≥ 0 and therefore u0 is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='22 (Convexity of blow-ups `a la Caffarelli).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The original proof by Caffarelli on the convexity of blow-ups, [Caf77, Caf98], is more involved than the previous one, but obtains a quantitative estimate on the convexity — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Classification of blow-ups 157 without using the homogeneity assumption (in particular, it is valid for any global solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' More precisely, for any solution u to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14) in B1 ∂eeu(x) ≥ − C �� log |x| ��ε for all e ∈ Sn−1, x ∈ B1/2, for some ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that C �� log |x| ��−ε → 0 as x → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, u becomes closer and closer to being convex as we approach to the free boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Rescaling this result to BR, and letting R → ∞, this implies that any global solution is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, we refer to [PSU12, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1] for yet another different proof of the convexity of blow-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Classification of blow-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We next want to classify all possible blow- ups for solutions to the obstacle problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' First, we will prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14), and let ur(x) := u(rx) r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, for any sequence rk → 0 there is a subsequence rkj → 0 such that urkj −→ u0 in C1 loc(Rn) as kj → ∞, for some function u0 satisfying � � � � � � � � � � � � � � � � � � � u0 ∈ C1,1 loc(Rn) u0 ≥ 0 in B1 ∆u0 = 1 in {u0 > 0} 0 is a free boundary point u0 is convex u0 is homogeneous of degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By C1,1 regularity of u, and by nondegeneracy, we have that 1 C ≤ sup B1 ur ≤ C for some C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, again by C1,1 regularity of u, we have ∥D2ur∥L∞(B1/(2r)) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since the sequence {urk}, for rk → 0, is uniformly bounded in C1,1(K) for each compact set K ⊂ Rn, there is a subsequence rkj → 0 such that urkj −→ u0 in C1 loc(Rn) — DRAFT — 158 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem for some u0 ∈ C1,1(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, such function u0 satisfies ∥D2u0∥L∞(K) ≤ C, with C independent of K, and clearly u0 ≥ 0 in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The fact that ∆u0 = 1 in {u0 > 0} ∩ K can be checked as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For any smooth function η ∈ C∞ c ({u0 > 0} ∩ K) we will have that, for kj large enough, urkj > 0 in the support of η, and thus � Rn ∇urkj · ∇η dx = − � Rn η dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since urkj → u0 in C1(K), we can take the limit kj → ∞ to get � Rn ∇u0 · ∇η dx = − � Rn η dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since this can be done for any η ∈ C∞ c ({u > 0}∩K), and for every K ⊂ Rn, it follows that ∆u0 = 1 in {u0 > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The fact that 0 is a free boundary point for u0 follows simply by taking limits to urkj (0) = 0 and ∥urkj ∥L∞(Bρ) ≈ ρ2 for all ρ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, the homogeneity and convexity of u0 follow from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17 and Theo- rem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Our next goal is to prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='24 (Classification of blow-ups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14), and let u0 be any blow-up of u at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, (a) either u0(x) = 1 2(x · e)2 + for some e ∈ Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (b) or u0(x) = 1 2xT Ax for some matrix A ≥ 0 with tr A = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It is important to remark here that, a priori, different subsequences could lead to different blow-ups u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In order to establish Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='24, we will need the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Σ ⊂ Rn be any closed convex cone with nonempty inte- rior, and with vertex at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let w ∈ C(Rn) be a function satisfying ∆w = 0 in Σc, w > 0 in Σc, and w = 0 in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume in addition that w is homogeneous of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, Σ must be a half-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Classification of blow-ups 159 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By convexity of Σ, there exists a half-space H = {x · e > 0}, with e ∈ Sn−1, such that H ⊂ Σc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let v(x) = (x · e)+, which is harmonic and positive in H, and vanishes in Hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By the Hopf Lemma (see Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15), we have that w ≥ c◦dΣ in Σc ∩ B1, where dΣ(x) = dist(x, Σ) and c◦ is a small positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, since both w and dΣ are homogeneous of degree 1, we deduce that w ≥ c◦dΣ in all of Σc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that, in order to apply the Hopf Lemma, we used that — by convexity of Σ — the domain Σc satisfies the interior ball condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, since dΣ ≥ dHc = v, we deduce that w ≥ c◦v, for some c◦ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The idea is now to consider the functions w and cv, and let c > 0 increase until the two functions touch at one point, which will give us a contradiction (recall that two harmonic functions cannot touch at an interior point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' To do this rigorously, define c∗ := sup{c > 0 : w ≥ cv in Σc}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that c∗ ≥ c◦ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, we consider the function w−c∗v ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that w − c∗v is not identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Such function is harmonic in H and hence, by the strict maximum principle, w − c∗v > 0 in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, using the Hopf Lemma in H (see Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15) we deduce that w−c∗v ≥ c◦dHc = c◦v, since v is exactly the distance to Hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' But then we get that w−(c∗+c◦)v ≥ 0, a contradiction with the definition of c∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, it must be w − c∗v ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This means that w is a multiple of v, and therefore Σ = Hc, a half-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26 (Alternative proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' An alternative way to argue in the pre- vious lemma could be the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Any function w which is harmonic in a cone Σc and homogeneous of degree α can be written as a function on the sphere, satisfying ∆Sn−1w = µw on Sn−1 ∩ Σc with µ = α(n + α − 2) — in our case α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (Here, ∆Sn−1 denotes the spherical Laplacian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' the Laplace–Beltrami operator on Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') In other words, homogeneous har- monic functions solve an eigenvalue problem on the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Using this, we notice that w > 0 in Σc and w = 0 in Σ imply that w is the first eigenfunction of Sn−1∩Σc, and that the first eigenvalue is µ = n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' But, on the other hand, the same happens for the domain H = {x · e > 0}, since v(x) = (x · e)+ is a positive harmonic function in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This means that both domains Sn−1 ∩Σc and Sn−1 ∩H have the same first eigenvalue µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' But then, by strict monotonicity of the first eigenvalue with respect to domain inclusions, we deduce that H ⊂ Σc implies H = Σc, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will also need the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 160 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that ∆u = 1 in Rn \\∂H, where ∂H is a hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If u ∈ C1(Rn), then ∆u = 1 in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume ∂H = {x1 = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For any ball BR ⊂ Rn, we consider the solution to ∆w = 1 in BR, w = u on ∂BR, and define v = u − w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, we have ∆v = 0 in BR \\ ∂H, and v = 0 on ∂BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We want to show that u coincides with w, that is, v ≡ 0 in BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For this, notice that since v is bounded, for κ > 0 large enough we have v(x) ≤ κ(2R − |x1|) in BR, where 2R − |x1| is positive in BR and harmonic in BR \\ {x1 = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, we may consider κ∗ := inf{κ ≥ 0 : v(x) ≤ κ(2R − |x1|) in BR}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume κ∗ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since v and 2R − |x1| are continuous in BR, and v = 0 on ∂BR, we must have a point p ∈ BR at which v(p) = κ∗(2R − |p1|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, since v is C1, and the function 2R − |x1| has a wedge on ∂H = {x1 = 0}, we must have p ∈ BR \\ ∂H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' However, this is not possible, as two harmonic functions cannot touch tangentially at an interior point p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This means that κ∗ = 0, and hence v ≤ 0 in BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Repeating the same argument with −v instead of v, we deduce that v ≡ 0 in BR, and thus the lemma is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Finally, we will use the following basic property of convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u : Rn → R be a convex function such that the set {u = 0} contains the straight line {te′ : t ∈ R}, e′ ∈ Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u(x + te′) = u(x) for all x ∈ Rn and all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' After a rotation, we may assume e′ = en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, writing x = (x′, xn) ∈ Rn−1 × R, we have that u(0, xn) = 0 for all xn ∈ R, and we want to prove that u(x′, xn) = u(x′, 0) for all x′ ∈ Rn−1 and all xn ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, by convexity, given x′ and xn, for every ε > 0 and M ∈ R we have (1 − ε)u(x′, xn) + εu(0, xn + M) ≥ u((1 − ε)x′, xn + εM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since u(0, xn + M) = 0, choosing M = λ/ε and letting ε → 0 we deduce that u(x′, xn) ≥ u(x′, xn + λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since this can be done for any λ ∈ R and xn ∈ R, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ We finally establish the classification of blow-ups at regular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u0 be any blow-up of u at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We already proved that u0 is convex and homogeneous of degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We divide the proof into two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that {u0 = 0} has nonempty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, we have {u0 = 0} = Σ, a closed convex cone with nonempty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Regularity of the free boundary 161 For any direction τ ∈ Sn−1 such that −τ ∈ ˚Σ, we claim that ∂τu0 ≥ 0 in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, for every x ∈ Rn we have that u0(x + τt) is zero for t ≪ −1, and therefore by convexity of u0 we get that ∂tu0(x + τt) is monotone non- decreasing in t, and zero for t ≪ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This means that ∂tu0 ≥ 0, and thus ∂τu0 ≥ 0 in Rn, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, for any such τ, we define w := ∂τu0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that, at least for some τ ∈ Sn−1 with −τ ∈ ˚Σ, the function w is not identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, since it is harmonic in Σc — recall that ∆u0 = 1 in Σc — then w > 0 in Σc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' But then, since w is homogeneous of degree 1, we can apply Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='25 to deduce that we must necessarily have that Σ is a half-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By convexity of u0 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='28, this means that u0 is a one- dimensional function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', u0(x) = U(x · e) for some U : R → R and some e ∈ Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, we have that U ∈ C1,1 solves U ′′(t) = 1 for t > 0, with U(t) = 0 for t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We deduce that U(t) = 1 2t2 +, and therefore u0(x) = 1 2(x · e)2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume now that {u0 = 0} has empty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, by convexity, {u0 = 0} is contained in a hyperplane ∂H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Hence, ∆u0 = 1 in Rn\\∂H, with ∂H being a hyperplane, and u0 ∈ C1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='27 that ∆u0 = 1 in all of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' But then all second derivatives of u0 are harmonic and globally bounded in Rn, so they must be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Hence, u0 is a quadratic polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, since u0(0) = 0, ∇u0(0) = 0, and u0 ≥ 0, we deduce that u0(x) = 1 2xT Ax for some A ≥ 0, and since ∆u0 = 1, we have tr A = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Regularity of the free boundary The aim of this Section is to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='38 below, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', that if u is any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14) satisfying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19) lim sup r→0 ��{u = 0} ∩ Br �� |Br| > 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', the contact set has positive density at the origin), then the free bound- ary ∂{u > 0} is C∞ in a neighborhood of the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For this, we will use the classification of blow-ups established in the previous Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' C1,α regularity of the free boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The first step here is to transfer the local information on u given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19) into a blow-up u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' More precisely, — DRAFT — 162 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem we next show that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19) =⇒ The contact set of a blow-up u0 has nonempty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14), and assume that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, there is at least one blow-up u0 of u at 0 such that the contact set {u0 = 0} has nonempty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let rk → 0 be a sequence along which lim rk→0 ��{u = 0} ∩ Brk �� |Brk| ≥ θ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Such sequence exists (with θ > 0 small enough) by assumption (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Recall that, thanks to Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='23, there exists a subsequence rkj ↓ 0 along which urkj → u0 uniformly on compact sets of Rn, where ur(x) = r−2u(rx) and u0 is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume by contradiction that {u0 = 0} has empty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, by convexity, we have that {u0 = 0} is contained in a hyperplane, say {u0 = 0} ⊂ {x1 = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since u0 > 0 in {x1 ̸= 0} and u0 is continuous, we have that for each δ > 0 u0 ≥ ε > 0 in {|x1| > δ} ∩ B1 for some ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, by uniform convergence of urkj to u0 in B1, there is rkj > 0 small enough such that urkj ≥ ε 2 > 0 in {|x1| > δ} ∩ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, the contact set of urkj is contained in {|x1| ≤ δ} ∩ B1, so ��{urkj = 0} ∩ B1 �� |B1| ≤ ��{|x1| ≤ δ} ∩ B1 �� |B1| ≤ Cδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Rescaling back to u, we find ��{u = 0} ∩ Brkj �� |Brkj | = ��{urkj = 0} ∩ B1 �� |B1| < Cδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since we can do this for every δ > 0, we find that limrkj →0 |{u=0}∩Brkj | |Brkj | = 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, the lemma is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Combining the previous lemma with the classification of blow-ups from the previous Section, we deduce: — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Regularity of the free boundary 163 τ e {x · e > 0} ∂τu0 > 0 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Derivatives ∂τu0 are nonnegative if τ · e ≥ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14), and assume that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, there is at least one blow-up of u at 0 of the form u0(x) = 1 2(x · e)2 +, e ∈ Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The result follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='29 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ We now want to use this information to show that the free boundary must be smooth in a neighborhood of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For this, we start with the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14), and assume that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Fix any ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, there exist e ∈ Sn−1 and r◦ > 0 such that ��ur◦(x) − 1 2(x · e)2 + �� ≤ ε in B1, and ��∂τur◦(x) − (x · e)+(τ · e) �� ≤ ε in B1 for all τ ∈ Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='30 and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='23, we know that there is a subsequence rj → 0 for which urj → 1 2(x · e)2 + in C1 loc(Rn), for some e ∈ Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, for every τ ∈ Sn−1 we have urj → 1 2(x · e)2 + and ∂τurj → ∂τ � 1 2(x · e)2 + � uniformly in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This means that, given ε > 0, there exists j◦ such that ��urj◦(x) − 1 2(x · e)2 + �� ≤ ε in B1, and ��∂τurj◦(x) − ∂τ � 1 2(x · e)2 + ��� ≤ ε in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since ∂τ � 1 2(x · e)2 + � = (x · e)+(τ · e), the proposition is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Now, notice that if (τ · e) > 0, then the derivatives ∂τu0 = (x · e)+(τ · e) are nonnegative, and strictly positive in {x · e > 0} (see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 164 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem We want to transfer this information to ur◦, and prove that ∂τur◦ ≥ 0 in B1 for all τ ∈ Sn−1 satisfying τ · e ≥ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For this, we need a lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14), and consider ur◦(x) = r−2 u(r◦x) and Ω = {ur◦ > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that a function w ∈ C(B1) satisfies: (a) w is bounded and harmonic in Ω ∩ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (b) w = 0 on ∂Ω ∩ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (c) Denoting Nδ := {x ∈ B1 : dist(x, ∂Ω) < δ}, we have w ≥ −c1 in Nδ and w ≥ C2 > 0 in Ω \\ Nδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If c1/C2 is small enough, and δ > 0 is small enough, then w ≥ 0 in B1/2∩Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that in Ω \\ Nδ we already know that w > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let y◦ ∈ Nδ ∩ Ω ∩ B1/2, and assume by contradiction that w(y0) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Consider, in B1/4(y◦), the function v(x) = w(x) − γ � ur◦(x) − 1 2n|x − y◦|2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, ∆v = 0 in B1/4(y◦) ∩ Ω, and v(y◦) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, v must have a negative minimum in ∂ � B1/4(y◦) ∩ Ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' However, if c1/C2 and δ are small enough, then we reach a contradiction as follows: On ∂Ω we have v ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On ∂B1/4(y◦) ∩ Nδ we have v ≥ −c1 − C◦γδ2 + γ 2n �1 4 �2 ≥ 0 on ∂B1/4(y◦) ∩ Nδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On ∂B1/4(y◦) ∩ � Ω \\ Nδ � we have v ≥ C2 − C◦γ ≥ 0 on ∂B1/4(y◦) ∩ � Ω \\ Nδ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Here, we used that ∥ur◦∥C1,1(B1) ≤ C◦, and chose C◦c1 ≤ γ ≤ C2/C◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Using the previous lemma, we can now show that there is a cone of directions τ in which the solution is monotone near the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14), and assume that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let ur(x) = r−2u(rx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, there exist r◦ > 0 and e ∈ Sn−1 such that ∂τur◦ ≥ 0 in B1/2 for every τ ∈ Sn−1 satisfying τ · e ≥ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Regularity of the free boundary 165 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='31, for any ε > 0 there exist e ∈ Sn−1 and r◦ > 0 such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20) ��ur◦(x) − 1 2(x · e)2 + �� ≤ ε in B1 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='21) ��∂τur◦(x) − (x · e)+(τ · e) �� ≤ ε in B1 for all τ ∈ Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We now want to use Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='32 to deduce that ∂τur◦ ≥ 0 if τ · e ≥ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' First, we claim that ur◦ > 0 in {x · e > C◦ √ε}, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='22) ur◦ = 0 in {x · e < −C◦ √ε}, and therefore the free boundary ∂Ω = ∂{ur◦ > 0} is contained in the strip {|x · e| ≤ C◦ √ε}, for some C◦ depending only on n (see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' To prove this, notice that if x · e > C◦ √ε then ur◦ > 1 2(C◦ √ε)2 − ε > 0, while if there was a free boundary point x◦ in {x · e < −C◦ε} then by nondegeneracy we would get sup BC◦ √ε(x◦) ur◦ ≥ c(C◦ √ε)2 > 2ε, a contradiction with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, we have ∂Ω ⊂ {|x · e| ≤ C◦ √ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, for each τ ∈ Sn−1 satisfying τ · e ≥ 1 2 we define w := ∂τur◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In order to use Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='32, we notice: (a) w is bounded and harmonic in Ω ∩ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (b) w = 0 on ∂Ω ∩ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (c) Thanks to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='21), if δ ≫ √ε then w satisfies w ≥ −ε in Nδ and w ≥ δ/4 > 0 in (Ω \\ Nδ) ∩ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 166 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem Ω 0 2C◦ √ε Nδ ∩ Ω ∂Ω Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The setting in which we use Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (We recall Nδ := {x ∈ B1 : dist(x, ∂Ω) < δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') Indeed, to check the last inequality we use that, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='22), we have {x · e < δ − C◦ √ε} ∩ Ω ⊂ Nδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='21), we get that for all x ∈ (Ω \\ Nδ) ∩ B1 w ≥ 1 2(x · e)+ − ε ≥ 1 2δ − 1 2C◦ √ε − ε ≥ 1 4δ, provided that δ ≫ √ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Using (a)-(b)-(c), we deduce from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='32 that w ≥ 0 in B1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since we can do this for every τ ∈ Sn−1 with τ · e ≥ 1 2, the proposition is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ As a consequence of the previous proposition, we find: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14), and assume that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, there exists r◦ > 0 such that the free boundary ∂{ur◦ > 0} is Lipschitz in B1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, the free boundary of u, ∂{u > 0}, is Lipschitz in Br◦/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This follows from the fact that ∂τur◦ ≥ 0 in B1/2 for all τ ∈ Sn−1 with τ · e ≥ 1 2 (by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='33), as explained next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Regularity of the free boundary 167 Σ1 Σ2 e x◦ τ Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Representation of Σ1 and Σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let x◦ ∈ B1/2 ∩ ∂{ur◦ > 0} be any free boundary point in B1/2, and let Θ := � τ ∈ Sn−1 : τ · e > 1 2 � , Σ1 := � x ∈ B1/2 : x = x◦ − tτ, with τ ∈ Θ, t > 0 � , and Σ2 := � x ∈ B1/2 : x = x◦ + tτ, with τ ∈ Θ, t > 0 � , see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We claim that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='23) � ur◦ = 0 in Σ1, ur◦ > 0 in Σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, since ur◦(x◦) = 0, it follows from the monotonicity property ∂τur◦ ≥ 0 — and the nonnegativity of ur◦ — that ur◦(x◦ − tτ) = 0 for all t > 0 and τ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, there cannot be any free boundary point in Σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, by the same argument, if ur◦(x1) = 0 for some x1 ∈ Σ2 then we would have ur◦ = 0 in � x ∈ B1/2 : x = x1 − tτ, with τ ∈ Θ, t > 0 � ∋ x◦, and in particular x◦ would not be a free boundary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, ur◦(x1) > 0 for all x1 ∈ Σ2, and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='23) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, notice that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='23) yields that the free boundary ∂{ur◦ > 0} ∩ B1/2 satisfies both the interior and exterior cone condition, and thus it is Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Once we know that the free boundary is Lipschitz, we may assume with- out loss of generality that e = en and that ∂{ur◦ > 0} ∩ B1/2 = {xn = g(x′)} ∩ B1/2 — DRAFT — 168 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem wi > 0 ∆wi = 0 wi = 0 B1 Ω Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Setting of the boundary Harnack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' for a Lipschitz function g : Rn−1 → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Here, x = (x′, xn), with x′ ∈ Rn−1 and xn ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, we want to prove that Lipschitz free boundaries are C1,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A key ingredient for this will be the following basic property of harmonic functions (see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15 for a representation of the setting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='35 (Boundary Harnack).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let w1 and w2 be positive harmonic functions in B1 ∩ Ω, where Ω ⊂ Rn is any Lipschitz domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that w1 and w2 vanish on ∂Ω ∩ B1, and C−1 ≤ ∥wi∥L∞(B1/2) ≤ C◦ for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, 1 C w2 ≤ w1 ≤ Cw2 in Ω ∩ B1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, ���� w1 w2 ���� C0,α(Ω∩B1/2) ≤ C for some small α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The constants α and C depend only on n, C◦, and Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For completeness, we provide in Appendix B a proof of this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer to [DS20] for the boundary Harnack for more general operators and to [AS19, RT21] for the boundary Harnack for equations with a right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The main point in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='35 is that Ω is allowed to be Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If Ω is smooth (say, C2 or even C1,α) then it follows from a simple barrier argument that both w1 and w2 would be comparable to the distance to ∂Ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', they vanish at a linear rate from ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' However, in Lipschitz — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Regularity of the free boundary 169 domains the result cannot be proved with a simple barrier argument, and it is much more delicate to establish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The boundary Harnack is a crucial tool in the study of free boundary problems, and in particular in the obstacle problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Here, we use it to prove that the free boundary is C1,α for some small α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14), and assume that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, there exists r◦ > 0 such that the free boundary ∂{ur◦ > 0} is C1,α in B1/4, for some small α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, the free boundary of u, ∂{u > 0}, is C1,α in Br◦/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω = {ur◦ > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='34, if r◦ > 0 is small enough then (possibly after a rotation) we have Ω ∩ B1/2 = {xn ≥ g(x′)} ∩ B1/2 and the free boundary is given by ∂Ω ∩ B1/2 = {xn = g(x′)} ∩ B1/2, where g is Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let w2 := ∂enur◦ and w1 := ∂eiur◦ + ∂enur◦, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since ∂τur◦ ≥ 0 in B1/2 for all τ ∈ Sn−1 with τ ·en ≥ 1 2, we have that w2 ≥ 0 in B1/2 and w1 ≥ 0 in B1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is because ∂ei + ∂en = ∂ei+en = √ 2∂τ, with τ · en = 1/ √ 2 > 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that we add the term ∂enur◦ in w1 in order to get a nonnegative function w2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now since w1 and w2 are positive harmonic functions in Ω ∩ B1/2, and vanish on ∂Ω ∩ B1/2, we can use the boundary Harnack, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='35 (or Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2), to get ���� w1 w2 ���� C0,α(Ω∩B1/4) ≤ C for some small α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, since w1/w2 = 1 + ∂eiur◦/∂enur◦, we deduce (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='24) ���� ∂eiur◦ ∂enur◦ ���� C0,α(Ω∩B1/4) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, we claim that this implies that the free boundary is C1,α in B1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, if ur◦(x) = t then the normal vector to the level set {ur◦ = t} is — DRAFT — 170 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem given by νi(x) = ∂eiur◦ |∇ur◦| = ∂eiur◦/∂enur◦ � 1 + �n−1 j=1 � ∂ejur◦/∂enur◦ �2 , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is a C0,α function by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='24), and therefore we can take t → 0 to find that the free boundary is C1,α (since the normal vector to the free boundary is given by a C0,α function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ So far we have proved that � {u = 0} has positive density at the origin � =⇒ � any blow-up is u0 = 1 2(x · e)2 + � =⇒ � free boundary is C1,α near 0 � As a last step in this section, we will now prove that C1,α free boundaries are actually C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Higher regularity of the free boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We want to finally prove the smoothness of free boundaries near regular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='38 (Smoothness of the free boundary near regular points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14), and assume that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, the free boundary ∂{u > 0} is C∞ in a neighborhood of the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For this, we need the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='39 (Higher order boundary Harnack).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ Rn be any Ck,α domain, with k ≥ 1 and α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let w1, w2 be two solutions of ∆wi = 0 in B1 ∩ Ω, wi = 0 on ∂Ω ∩ B1, with w2 > 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that C−1 ≤ ∥wi∥L∞(B1/2) ≤ C◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, ���� w1 w2 ���� Ck,α(Ω∩B1/2) ≤ C, where C depends only on n, k, α, C◦, and Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Contrary to Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='35, the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='39 is a perturba- tive argument, in the spirit of (but much more delicate than) the Schauder estimates from Chapter 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We will not prove the higher order boundary Harnack here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' we refer to [DS16] for the proof of such result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Using Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='39, we can finally prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='38: Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let ur◦(x) = r−2 u(r◦x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='37, we know that if r◦ > 0 is small enough then the free boundary ∂{ur◦ > 0} is C1,α in B1, and (possibly after a rotation) ∂enur◦ > 0 in {ur◦ > 0} ∩ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Singular points 171 Thus, using the higher order boundary Harnack (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='39) with w1 = ∂eiur◦ and w2 = ∂enur◦, we find that ���� ∂eiur◦ ∂enur◦ ���� C1,α(Ω∩B1/2) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Actually, by a simple covering argument we find that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='25) ���� ∂eiur◦ ∂enur◦ ���� C1,α(Ω∩B1−δ) ≤ Cδ for any δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, as in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='37, we notice that if ur◦(x) = t then the normal vector to the level set {ur◦ = t} is given by νi(x) = ∂eiur◦ |∇ur◦| = ∂eiur◦/∂enur◦ � 1 + �n j=1 � ∂ejur◦/∂enur◦ �2 , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='25), this is a C1,α function in B1−δ for any δ > 0, and therefore we can take t → 0 to find that the normal vector to the free boundary is C1,α inside B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' But this means that the free boundary is actually C2,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Repeating now the same argument, and using that the free boundary is C2,α in B1−δ for any δ > 0, we find that ���� ∂eiur◦ ∂enur◦ ���� C2,α(Ω∩B1−δ′) ≤ Cδ′, which yields that the normal vector is C2,α and thus the free boundary is C3,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Iterating this argument, we find that the free boundary ∂{ur◦ > 0} is C∞ inside B1, and hence ∂{u > 0} is C∞ in a neighborhood of the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ This completes the study of regular free boundary points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It remains to understand what happens at points where the contact set has density zero (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is the content of the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Singular points We finally study the behavior of the free boundary at singular points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', when (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26) lim r→0 ��{u = 0} ∩ Br �� |Br| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For this, we first notice that, as a consequence of the results of the previous Section, we get the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, we have the following dichotomy: — DRAFT — 172 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem (a) Either (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19) holds and all blow-ups of u at 0 are of the form u0(x) = 1 2(x · e)2 +, for some e ∈ Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (b) Or (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26) holds and all blow-ups of u at 0 are of the form u0(x) = 1 2xT Ax, for some matrix A ≥ 0 with tr A = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Points of type (a) were studied in the previous Section;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' they are called regular points and the free boundary is C∞ around them (in particular, the blow-up is unique).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Points of type (b) are those at which the contact set has zero density, and are called singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' To prove the result, we need the following: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14), and assume that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, every blow-up of u at 0 satisfies |{u0 = 0}| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u0 be a blow-up of u at 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', urk → u0 in C1 loc(Rn) along a sequence rk → 0, where ur(x) = r−2u(rx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that the functions ur solve ∆ur = χ{ur>0} in B1, in the sense that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='27) � B1 ∇ur · ∇η dx = � B1 χ{ur>0}η dx for all η ∈ C∞ c (B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, by assumption (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26), we have ��{ur = 0} ∩ B1 �� −→ 0, and thus taking limits rk → 0 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='27) we deduce that ∆u0 = 1 in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since we know that u0 is convex, nonnegative, and homogeneous, this implies that |{u0 = 0}| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ We can now give the: Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By the classification of blow-ups (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='24), the possible blow-ups can only have one of the two forms presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='19) holds for at least one blow-up, thanks to the smoothness of the free boundary (by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='37), it holds for all blow-ups, and thus, by Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='30, u0(x) = 1 2(x · e)2 + (and in fact, the smoothness of the free boundary yields uniqueness of the blow-up in this case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='26) holds, then by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='41 the blow-up u0 must satisfy ��{u0 = 0} �� = 0, and thus we are in case (b) (see the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Singular points 173 In the previous Section we proved that the free boundary is C∞ in a neighborhood of any regular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A natural question then is to understand better the solution u near singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' One of the main results in this direction is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='42 (Uniqueness of blow-ups at singular points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14), and assume that 0 is a singular free boundary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, there exists a homogeneous quadratic polynomial p2(x) = 1 2xT Ax, with A ≥ 0 and ∆p2 = 1, such that ur −→ p2 in C1 loc(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, the blow-up of u at 0 is unique, and u(x) = p2(x) + o(|x|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' To prove this, we need the following monotonicity formula due to Mon- neau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='43 (Monneau’s monotonicity formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14), and assume that 0 is a singular free boundary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let q be any homogeneous quadratic polynomial with q ≥ 0, q(0) = 0, and ∆q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, the quantity Mu,q(r) := 1 rn+3 � ∂Br (u − q)2 is monotone in r, that is, d drMu,q(r) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We sketch the argument here, and refer to [PSU12, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We first notice that Mu,q(r) = � ∂B1 (u − q)2(rx) r4 , and hence a direct computation yields d drMu,q(r) = 2 rn+4 � ∂Br (u − q) {x · ∇(u − q) − 2(u − q)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, it turns out that 1 rn+3 � ∂Br (u − q) {x · ∇(u − q) − 2(u − q)} = Wu(r) − Wu(0+)+ + 1 rn+2 � Br (u − q)∆(u − q), where Wu(r) (as defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='15)) is monotone increasing in r > 0 thanks to Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, we have d drMu,q(r) ≥ 2 rn+3 � Br (u − q)∆(u − q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 174 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem But since ∆u = ∆q = 1 in {u > 0}, and (u−q)∆(u−q) = q ≥ 0 in {u = 0}, we have d drMu,q(r) ≥ 2 rn+3 � Br∩{u=0} q ≥ 0, as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ We can now give the: Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='40 (and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='23), we know that at any singular point we have a subsequence rj → 0 along which urj → p in C1 loc(Rn), where p is a 2-homogeneous quadratic polynomial satisfying p(0) = 0, p ≥ 0, and ∆p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, we can use Monneau’s monotonicity formula with such polynomial p to find that Mu,p(r) := 1 rn+3 � ∂Br (u − p)2 is monotone increasing in r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, the limit limr→0 Mu,p(r) := Mu,p(0+) exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, recall that we have a sequence rj → 0 along which urj → p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, r−2 j {u(rjx) − p(rjx)} −→ 0 locally uniformly in Rn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', 1 r2 j ∥u − p∥L∞(Brj ) −→ 0 as rj → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This yields that Mu,p(rj) ≤ 1 rn+3 j � ∂Brj ∥u − p∥2 L∞(Brj ) −→ 0 along the subsequence rj → 0, and therefore Mu,p(0+) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us show that this implies the uniqueness of blow-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, if there was another subsequence rℓ → 0 along which urℓ → q in C1 loc(Rn), for a 2- homogeneous quadratic polynomial q, then we would repeat the argument above to find that Mu,q(0+) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' But then this yields, by homogeneity of p and q, � ∂B1 (p − q)2 = 1 rn+3 � ∂Br (p − q)2 ≤ 2Mu,p(r) + 2Mu,q(r) −→ 0, and hence � ∂B1 (p − q)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This means that p = q, and thus the blow-up of u at 0 is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us finally show that u(x) = p(x)+o(|x|2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', r−2∥u−p∥L∞(Br) → 0 as r → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, assume by contradiction that there is a subsequence — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the size of the singular set 175 rk → 0 along which r−2 k ∥u − p∥L∞(Brk) ≥ c1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, there would be a subsequence of rki along which urki → u0 in C1 loc(Rn), for a certain blow-up u0 satisfying ∥u0 − p∥L∞(B1) ≥ c1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' However, by uniqueness of blow-ups it must be u0 = p, and hence we reach a contradic- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ We refer to [SY23, Bon01] for an alternative approach to the unique- ness of blow-ups at singular points, not based on monotonicity formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Summarizing, we have proved the following result: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, we have the following dichotomy: (a) Either all blow-ups of u at 0 are of the form u0(x) = 1 2(x · e)2 + for some e ∈ Sn−1, and the free boundary is C∞ in a neighborhood of the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (b) Or there is a homogeneous quadratic polynomial p, with p(0) = 0, p ≥ 0, and ∆p = 1, such that ∥u − p∥L∞(Br) = o(r2) as r → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, when this happens we have lim r→0 ��{u = 0} ∩ Br �� |Br| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The last question that remains to be answered is: How large can the set of singular points be?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is the topic of the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the size of the singular set We finish this chapter with a discussion of more recent results (as well as some open problems) about the set of singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Recall that a free boundary point x◦ ∈ ∂{u > 0} is singular whenever lim r→0 ��{u = 0} ∩ Br(x◦) �� |Br(x◦)| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The main known result on the size of the singular set reads as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='45 ([Caf98]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Σ ⊂ B1 be the set of singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, Σ∩B1/2 is locally contained in a C1 manifold of dimension n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 176 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem This result is sharp, in the sense that it is not difficult to construct examples in which the singular set is (n − 1)-dimensional;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' see [Sch77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As explained below, such result essentially follows from the uniqueness of blow-ups at singular points, established in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, given any singular point x◦, let px◦ be the blow-up of u at x◦ (recall that px◦ is a nonnegative 2-homogeneous polynomial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let k be the dimension of the set {px◦ = 0} — notice that this is a proper linear subspace of Rn, so that k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', n − 1} — and define (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='28) Σk := � x◦ ∈ Σ : dim({px◦ = 0}) = k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Clearly, Σ = �n−1 k=0 Σk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The following result gives a more precise description of the singular set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='46 ([Caf98]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u be any solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Σk ⊂ B1 be defined by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='28), k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, Σk is locally contained in a C1 manifold of dimension k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The rough heuristic idea of the proof of this result is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume for simplicity that n = 2, so that Σ = Σ1 ∪ Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us take a point x◦ ∈ Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='44, we have the expansion (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='29) u(x) = px◦(x − x◦) + o � |x − x◦|2� where px◦ is the blow-up of u at x◦ (recall that this came from the uniqueness of blow-ups at x◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By definition of Σ0, the polynomial px◦ must be positive outside the origin, and thus by homogeneity satisfies px◦(x−x◦) ≥ c|x−x◦|2, with c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This, combined with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='29), yields then that u must be positive in a neighborhood of x◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, all points in Σ0 are isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, let us now take a point x◦ ∈ Σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, by definition of Σ1 the blow-up must necessarily be of the form px◦(x) = 1 2(x · ex◦)2, for some ex◦ ∈ Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Again by the expansion (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='29), we find that u is positive in a region of the form � x ∈ Bρ(x◦) : ��(x − x◦) · ex◦ �� > ω(|x − x◦|) � , where ω is a certain modulus of continuity, and ρ > 0 is small (see Fig- ure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is roughly saying that the set Σ1 “has a tangent plane” at x◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Repeating the same at any other point ˜x◦ ∈ Σ1 we find that the same happens at every point in Σ1 and, moreover, if ˜x◦ is close to x◦ then e˜x◦ must be close to ex◦ — otherwise the expansions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='29) at ˜x◦ and x◦ would not match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, since the modulus ω can be made independent of the — DRAFT — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the size of the singular set 177 x◦ u > 0 u > 0 Bρ(x◦) ex◦ Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' u is positive in {x ∈ Bρ(x◦) : |(x − x◦) · ex◦| > ω(|x − x◦|)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' x◦ ˜x◦ ex◦ e˜x◦ Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Singular points x◦, ˜x◦ ∈ Σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' point (by a compactness argument), it turns out that the set Σ1 is contained in a C1 curve (see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' What we discussed here is just an heuristic argument;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' the actual proof uses Whitney’s extension theorem and can be found for example in [PSU12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, we refer to [CSV18], [FS19], and [FZ21] (and the expository paper [Fig18b]) for some recent finer results about the set of singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Generic regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In PDE problems in which singularities may appear, it is very natural and important to understand whether these singularities appear “often”, or if instead “most” solutions have no singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In the context of the obstacle problem, the key question is to understand the generic regularity of free boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Explicit examples show that sin- gular points in the obstacle problem can form a very large set, of dimension — DRAFT — 178 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem n − 1 (as large as the regular set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Still, singular points are expected to be rare (see [Sch74]): Conjecture (Schaeffer, 1974): Generically, the weak solution of the obsta- cle problem is also a strong solution, in the sense that the free boundary is a C∞ manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In other words, the conjecture states that, generically, the free boundary has no singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The first result in this direction was established by Monneau in 2003, who proved the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='47 ([Mon03]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Schaeffer’s conjecture holds in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' More precisely, Monneau considers a 1-parameter family of solutions uλ, with λ ∈ (0, 1), such that � ∆uλ = χ{uλ>0} in Ω uλ = gλ on ∂Ω, with gλ = g + λ and g ≥ 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, the first step is to notice that not only each of the singular sets Σλ ⊂ Ω is contained in a C1 manifold of dimension (n − 1), but actually the union � λ∈(0,1) Σλ ⊂ Ω is still contained in an (n − 1)-dimensional manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' After that, we look at the free boundary as a set in Ω×(0, 1) ∋ (x, λ), and notice that it can be written as a graph {λ = h(x)}, for some function h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A second key step in the proof is to show that h is Lipschitz and, furthermore, it has zero gradient at any singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This, combined with the coarea formula, yields that in R2 the set of singular points is empty for almost every λ ∈ (0, 1), which implies Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, the best known result in this direction was established very recently by Figalli, Serra, and the second author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='48 ([FRS20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Schaeffer’s conjecture holds in R3 and R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The proof of this result is based on a new and very fine understanding of singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For this, [FRS20] combines Geometric Measure The- ory tools, PDE estimates, several dimension reduction arguments, and even several new monotonicity formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It remains an open problem to decide whether or not Schaeffer’s conjec- ture holds in dimensions n ≥ 5 or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — Appendix A Some properties of H¨older spaces In this appendix, we prove the properties (H1)-(H8) stated in Chapter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Recall that, given α ∈ (0, 1], the H¨older space C0,α(Ω) is the set of functions u ∈ C(Ω) such that [u]C0,α(Ω) := sup x,y∈Ω x̸=y ��u(x) − u(y) �� |x − y|α < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The H¨older norm is ∥u∥C0,α(Ω) := ∥u∥L∞(Ω) + [u]C0,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' When α = 1, this is the usual space of Lipschitz functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' More generally, given k ∈ N and α ∈ (0, 1], the space Ck,α(Ω) is the set of functions u ∈ Ck(Ω) such that the following norm is finite ∥u∥Ck,α(Ω) := k � j=1 ∥Dju∥L∞(Ω) + sup x,y∈Ω x̸=y ��Dku(x) − Dku(y) �� |x − y|α = ∥u∥Ck(Ω) + [Dku]C0,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, when β > 0 is not an integer, we denote Cβ(Ω) := Ck,α(Ω), where β = k + α, with k ∈ N, α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Next, we give the proofs of the properties of H¨older spaces that we have used throughout the book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Unless stated otherwise, in the following statements we assume α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 179 — DRAFT — 180 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Some properties of H¨older spaces (H1) Assume oscBr(x)u ≤ C◦rα for all Br(x) ⊂ B1, where oscAu := supA u − infA u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ C0,α(B1) and [u]C0,α(B1) ≤ CC◦, with C depending only on n, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of (H1) We want to prove that |u(z) − u(x)| ≤ CC◦|z − x|α for all z, x ∈ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Given z, x ∈ B1, let r = |z − x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For this, we may assume r < 1/10 and distinguish two cases: (i) If Br(x) ⊂ B1, then we simply use the assumption to get |u(z) − u(x)| ≤ oscBr(x)u ≤ C◦rα = C◦|z − x|α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (ii) Otherwise, we take ¯x and ¯z on the segments 0x and 0z, respectively, such that |x − ¯x| = r and |z − ¯z| = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, by assumption we have |u(x) − u(¯x)| ≤ C◦rα, |u(z) − u(¯z)| ≤ C◦rα, and |u(¯x) − u(¯z)| ≤ C◦rα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The last inequality holds because |¯x − ¯z| < r, which can be easily checked by construction of ¯x and ¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Combining the last three inequalities, we deduce that |u(x) − u(z)| ≤ 3C◦rα, as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ We also state and prove the following slight modification of (H1), which will be useful in later proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that the difference with respect to the previous statement is that now, given any ball in B1, we control the oscillation in the ball with half the radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (H1’) Assume oscBr(x)u ≤ C◦rα for all B2r(x) ⊂ B1, where oscAu := supA u − infA u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ C0,α(B1) and [u]C0,α(B1) ≤ CC◦, with C depending only on n, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of (H1’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We proceed analogously to the proof of (H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let z, x ∈ B1, and let r = |z − x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We may assume that r < 1/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If B2r(x) ⊂ B1, the result follows by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Otherwise, let us take ¯x and ¯z on the segments 0x and 0z, respectively, such that |x− ¯x| = 2r and |z − ¯z| = 2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us define xk = (1−2−k)x+2−k¯x and zk = (1 − 2−k)z + 2−k¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that |xk+1 − xk| = |zk+1 − zk| = 2−kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Also, |xk| = |x| − 2−k+1r, so that B2|xk+1−xk|(xk) ⊂ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, we can use our assumption on xk and xk+1 to get that |u(xk) − u(xk+1)| ≤ C◦|xk − xk+1|α = C◦2−kαrα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Some properties of H¨older spaces 181 (An analogous result holds for zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') On the other hand, by choice of ¯x and ¯z, they can also be compared in the oscillation of u as |u(¯x) − u(¯z)| ≤ C◦|¯x − ¯z|α ≤ C◦rα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Putting everything together, we reach that |u(x) − u(z)| ≤ � k≥0 |u(xk+1) − u(xk)| + |u(¯x) − u(¯z)| + � k≥0 |u(zk+1) − u(zk)| ≤ 2 � k≥0 C◦2−kαrα + C◦rα ≤ CC◦rα, for some constant C depending only on α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ (H2) Let ux,r := � Br(x) u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume ∥u − ux,r∥L∞(Br(x)) ≤ C◦rα for all Br(x) ⊂ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ C0,α(B1) and [u]C0,α(B1) ≤ CC◦, with C depending only on n, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of (H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By the triangle inequality we have oscBr(x)u ≤ 2∥u − ux,r∥L∞(Br(x)) ≤ 2C◦rα, and thus the result follows from (H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ (H3) Let ux,r := � Br(x) u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume � � Br(x) |u − ux,r|2 �1/2 ≤ C◦rα for all Br(x) ⊂ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ C0,α(B1) and [u]C0,α(B1) ≤ CC◦, with C depending only on n, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of (H3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that, for every z ∈ B1, ��ux,r − ux, r 2 ��2 ≤ 2|u(z) − ux,r|2 + 2 ��u(z) − ux, r 2 ��2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, integrating in Br/2(x) and using the assumption we deduce |ux,r − ux, r 2 |2 ≤ 2 � Br/2(x) |u − ux,r|2 + 2 � Br/2(x) ��u − ux, r 2 ��2 ≤ 2n+1 � Br(x) |u − ux,r|2 + 2 � Br/2(x) ��u − ux, r 2 ��2 ≤ CC2 r2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This means that ��ux,r − ux, r 2 �� ≤ CC◦rα, and summing a geometric series we get |ux,r − u(x)| ≤ � k≥0 ��ux, r 2k − ux, r 2k+1 �� ≤ � k≥0 CC◦ � r 2k �α = 2CC◦rα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 182 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Some properties of H¨older spaces Here we used that, up to redefining u on a set of measure zero, by Lebesgue differentiation theorem (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) we have that ux,r → u(x) as r → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let now x, y ∈ B1, r = 2|x − y|, and assume that Br(x) ⊂ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, we have |ux,r − uy,r|2 ≤ � Br/2(x) |u − ux,r|2 + � Br/2(x) ��u − uy,r ��2 ≤ 2n � Br(x) |u − ux,r|2 + 2n � Br(y) ��u − uy,r ��2 ≤ CC2 r2α, and thus ��ux,r − uy,r �� ≤ CC◦rα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Combining the previous estimates, we deduce that for every x, y ∈ B1 such that B2|x−y|(x) ⊂ B1, we have |u(x) − u(y)| ≤ |u(x) − ux,r| + |ux,r − uy,r| + |uy,r − u(y)| ≤ 3CC◦rα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Once we have this, by (H1’) we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ (H4) Assume that for every x there is a constant Cx such that ∥u − Cx∥L∞(Br(x)) ≤ C◦rα for all Br(x) ⊂ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ C0,α(B1) and [u]C0,α(B1) ≤ CC◦, with C depending only on n, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that for every x there is a linear function ℓx(y) = ax+bx·(y−x) such that ∥u − ℓx∥L∞(Br(x)) ≤ C◦r1+α for all Br(x) ⊂ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ C1,α(B1) and [Du]C0,α(B1) ≤ CC◦, with C depending only on n, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that for every x there is a quadratic polynomial Px(y) such that ∥u − Px∥L∞(Br(x)) ≤ C◦r2+α for all Br(x) ⊂ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ C2,α(B1) and [D2u]C0,α(B1) ≤ CC◦, with C depending only on n, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of (H4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (i) The first statement — with the C0,α norm — follows from (H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (ii) Let us sketch the proof of the second statement — with the C1,α norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let x, y ∈ B1 with y ∈ Br(x) ⊂ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that, dividing by r and taking r → 0 in the assumption, it follows that u is differentiable at x and that ℓx must be given by ℓx(y) = u(x) + ∇u(x) · (y − x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, by assumption, we have u(y) = u(x) + ∇u(x) · (y − x) + O(r1+α) — DRAFT — A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Some properties of H¨older spaces 183 and, for every z ∈ Br(x) such that |z − y| ≈ |z − x| ≈ |y − x| ≈ r, u(z) = u(x) + ∇u(x) · (z − x) + O(r1+α) = u(y) + ∇u(y) · (z − y) + O(r1+α) = u(x) + ∇u(x) · (y − x) + ∇u(y) · (z − y) + O(r1+α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' From this, we deduce that ∇u(x) · (z − y) = ∇u(y) · (z − y) + O(r1+α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Taking z such that z − y is parallel to ∇u(y) − ∇u(x), we get ∇u(x) = ∇u(y) + O(rα), as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (iii) Let us prove the third statement concerning the C2,α norm — the following proof is more general and works also in case (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let x, y ∈ Br(x◦) with |x − y| = r and suppose B2r(x◦) ⊂ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us rescale u around x◦, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', ur(z) := u(x◦ + rz), so that |¯x − ¯y| = 1, where x◦ + r¯x = x and x◦ + r¯y = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us define also Px,r(z) := Px(x◦ + rz) and Py,r := Py(x◦ + rz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, ∥ur − Px,r∥L∞(B1(¯x)) = ∥u − Px∥L∞(Br(x)) ≤ C◦r2+α, ∥ur − Py,r∥L∞(B1(¯y)) = ∥u − Py∥L∞(Br(y)) ≤ C◦r2+α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Hence, if we denote ¯w = ¯x+¯y 2 then B1/2( ¯w) ⊂ B1(¯x) ∩ B1(¯y) and ∥Px,r − Py,r∥L∞(B1/2( ¯w)) ≤ ∥ur − Px,r∥L∞(B1(¯x)) + ∥ur − Py,r∥L∞(B1(¯y)) ≤ CC◦r2+α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This means that all the coefficients of the polynomial Px,r − Py,r are con- trolled by ˜CC◦r2+α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, notice that if we denote Px(z) = ax+bx·(z−x)+(z−x)T Mx(z−x) then Px,r(z) = ax + rbx · (z − ¯x) + r2(z − ¯x)T Mx(z − ¯x), and an analogous expression holds for Py,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Hence, we can write Px,r(z) − Py,r(z) = � ax − ay + rbx · (¯y − ¯x) + r2(¯y − ¯x)T Mx(¯y − ¯x) � + r � bx − by + 2r(¯y − ¯x)T Mx � (z − ¯y) + r2(z − ¯y)T (Mx − My)(z − ¯y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, by looking at the quadratic and linear coefficients of such polynomial, we have proved that |Mx − My| ≤ ˜CC◦rα and ��bx − by + 2r(¯y − ¯x)T Mx �� ≤ CC◦r1+α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 184 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Some properties of H¨older spaces Since r(¯y − ¯x) = y − x, this is equivalent to ��by − bx − 2(y − x)T Mx �� ≤ CC◦r1+α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice, also, that ∥u − ax − bx · (· − x)∥L∞(Br(x)) ≤ C◦r2+α + Cxr2 ≤ 2Cxr2 if r small enough, so that, in particular, arguing as in (i), u is differentiable at x and ax = u(x), bx = ∇u(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, using that r = 2|x − y|, we have ��∇u(y) − ∇u(x) − 2(y − x)T Mx �� ≤ CC◦|y − x|1+α, and letting y → x we deduce that ∇u is differentiable at x, with D2u(x) = 2Mx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' An analogous result holds for My, so that we have shown that, for any x, y ∈ Br(x◦) with |x − y| = r and B2r(x◦) ⊂ B1, ��D2u(x) − D2u(y) �� ≤ ˜CC◦rα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The result now follows by (H1’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that the converse statement to (H4) also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For ex- ample, when k = 1, if u ∈ C1,α(B1) then we have ∥u − ℓx∥L∞(Br(x)) ≤ C◦r1+α for all Br(x) ⊂ B1, where ℓx(y) = u(x) + ∇u(x) · (y − x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, to show this, we use that u(y) = u(x) + � 1 0 ∇u(ty + (1 − t)x) · (y − x)dt, combined with ��∇u � ty + (1 − t)x � − ∇u(x) �� ≤ C◦|ty + (1 − t)x − x|α ≤ C◦|y − x|α, to get ��u(y) − u(x) − ∇u(x) · (y − x) �� ≤ � 1 0 C◦|y − x|α|y − x|dt = C◦|y − x|1+α, as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (H5) Let ρ◦ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that, for every x ∈ B1/2, there exists a se- quence of quadratic polynomials, (Pk)k∈N such that ∥u − Pk∥L∞(Bρk◦ (x)) ≤ C◦ρk(2+α) for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ C2,α(B1/2) and [D2u]C0,α(B1/2) ≤ CC◦, with C depending only on n, α, and ρ◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Some properties of H¨older spaces 185 Proof of (H5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us take x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By hypothesis, we have ∥Pk−1 − Pk∥L∞(Bρk◦ ) ≤ ∥u − Pk−1∥L∞(Bρk◦ ) + ∥u − Pk∥L∞(Bρk◦ ) ≤ CC◦ρk(2+α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, we use the following: Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that P is a quadratic polynomial satisfying ∥P∥L∞(Br) ≤ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If we denote P(z) = a + b · z + zT Mz, then we have that |a| ≤ Cγ, |b| ≤ Cγ r , |M| ≤ Cγ r2 , where C is a constant depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' To prove the claim, notice that, by rescaling, we have Pr(z) := P(rz) = ar + br · z + zT Mrz, where ar = a, br = rb, Mr = r2M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By assumption, we have that ∥Pr∥L∞(B1) ≤ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since the coefficients of polynomials on B1 are controlled by the L∞ norm, we get that |ar| ≤ Cγ, |br| ≤ Cγ, and |Mr| ≤ Cγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Using the previous claim and the bound on Pk−1 − Pk, we deduce that |ak−1 − ak| ≤ CC◦ρk(2+α) , |bk−1 − bk| ≤ CC◦ρk(1+α) , and |Mk−1 − Mk| ≤ CC◦ρkα , where Pk(z) = ak + bk · z + zT Mkz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It follows that Pk converge uniformly to a polynomial P(z) = a + b · z + zT Mz, and that ∥u − P∥L∞(Bρk◦ ) ≤ ∥u − Pk∥L∞(Bρk◦ ) + |ak − a| + ρk |bk − b| + ρ2k |Mk − M| ≤ CC◦ρk(2+α) for all k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' From this, it follows that for every r ∈ (0, 1) we have ∥u − P∥L∞(Br) ≤ CC◦r2+α (simply use that for any r we have ρk+1 ≤ r ≤ ρk for some k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, since we can do this for every x ∈ B1/2, it follows from (H4) that [D2u]C0,α(B1/2) ≤ CC◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ We refer to Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1 below for a generalization of property (H5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (H6) Assume that α ∈ (0, 1), ∥u∥L∞(B1) ≤ C◦, and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) sup h∈B1 x∈B1−|h| ��u(x + h) + u(x − h) − 2u(x) �� |h|α ≤ C◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ C0,α(B1) and ∥u∥C0,α(B1) ≤ CC◦, with C depending only on n, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 186 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Some properties of H¨older spaces Assume that α ∈ (0, 1), ∥u∥L∞(B1) ≤ C◦, and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2) sup h∈B1 x∈B1−|h| ��u(x + h) + u(x − h) − 2u(x) �� |h|1+α ≤ C◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ C1,α(B1) and ∥u∥C1,α(B1) ≤ CC◦, with C depending only on n, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' However, such property fails when α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of (H6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (i) Let us do the case (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2) first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Given h ∈ B1 and x ∈ B1−|h|, let w(h) := u(x + h) − u(x) |h| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, by assumption we have ��w(h) − w(h/2) �� = |u(x + h) + u(x) − 2u(x + h/2)| |h| ≤ C◦|h|α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, for every k ≥ 0, ��w(h/2k) − w(h/2k+1) �� ≤ C◦|h|α2−kα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This implies the existence of the limit limt→0 w(th), and by summing a geometric series we get ��w(h) − lim t→0 w(th) �� ≤ CC◦|h|α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since lim t→0 w(th) = lim t→0 u(x + th) − u(x) t|h| = h |h| · ∇u(x), this leads to ��u(x + h) − u(x) − h · ∇u(x) �� ≤ CC◦|h|1+α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Using (H4), we see that the last inequality implies that [Du]C0,α(B1) ≤ CC◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, using that ∥u∥L∞(B1) ≤ C◦, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (ii) Let us do now the case (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As before, let us define w(h) := u(x+h)−u(x) |h| and notice that ��w(h) − w(h/2) �� = |u(x + h) + u(x) − 2u(x + h/2)| |h| ≤ C◦|h|α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, for every k ≥ 0 we have ��w(2kh) − w(2k+1h) �� ≤ C◦|h|α−12−k(1−α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Some properties of H¨older spaces 187 Take k◦ ≥ 0 such that 2k◦|h| ≈ 1 (and so that1 still x + 2k◦h ∈ B1), and add the previous inequality for all 0 ≤ k < k◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, by summing a geometric series, we deduce that ��w(2k◦h) − w(h) �� ≤ CC◦|h|α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since ��w(2k◦h) �� ≤ C∥u∥L∞(B1) ≤ CC◦ ≤ CC◦|h|α−1, we finally get ��w(h) �� ≤ ��w(2k◦h) �� + CC◦|h|α−1 ≤ CC◦|h|α−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Translating back to u, this gives the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (iii) Finally, let us prove that the function u(x) = x log |x|, x ∈ (−1, 1), satisfies (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2) with α = 0, but it is not in C0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, let us show that ��(x + h) log |x + h| + (x − h) log |x − h| − 2x log |x| �� |h| ≤ C◦ for all x, h ∈ (−1, 1) and for some C◦ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For this, notice that (x + h) log |x + h| + (x − h) log |x − h| − 2x log |x| h = = � 1 + h x � log ��1 + h x �� + � 1 − h x � log ��1 − h x �� h x = (1 + t) log |1 + t| + (1 − t) log |1 − t| t , with t = h/x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Such function of t is smooth in R\\{0} and has finite limits at t = 0 and at t = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, it is globally bounded in R by some constant C◦ (actually, C◦ < 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer to [And97, Section 2] for higher order versions of the characterization (H6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (H7) Assume that α ∈ (0, 1], ∥u∥L∞(B1) ≤ C◦, and that for every h ∈ B1 we have ���� u(x + h) − u(x) |h|α ���� Cβ(B1−|h|) ≤ C◦, with C◦ independent of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume in addition that α + β is not an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ Cα+β(B1) and ∥u∥Cα+β(B1) ≤ CC◦, with C depending only on n, α, β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 1Note that this is always possible if x, y ∈ B9/10, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If x, y are close to the boundary ∂B1, then this is possible for example when (x − y) · x |x| > 1 2 |y − x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It is easy to see that we can always reduce to this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 188 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Some properties of H¨older spaces However, such property fails when α + β is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of (H7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We prove it in case β ∈ (0, 1], the proof for β > 1 is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us define vh(x) = u(x + h) − u(x) |h|α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, by assumption we have sup x,y∈B1−|h| |vh(x) − vh(y)| |x − y|β ≤ C◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is equivalent to sup x,y∈B1−|h| |u(x + h) − u(x) − u(y + h) + u(y)| |h|α+β ≤ C◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Taking y = x − h, this yields sup x∈B1−2|h| |u(x + h) + u(x + h) − 2u(x)| |h|α+β ≤ C◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By (H6), we deduce that ∥u∥Cα+β(B1) ≤ CC◦ — as long as α + β ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ (H8) Assume that ui → u uniformly in Ω ⊂ Rn, and that ∥ui∥Ck,α(Ω) ≤ C◦, with α ∈ (0, 1] and for some C◦ independent of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ∈ Ck,α(Ω), and ∥u∥Ck,α(Ω) ≤ C◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of (H8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume first k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, we have that for every x, y ∈ Ω, x ̸= y, ∥ui∥L∞(Ω) + |ui(x) − ui(y)| |x − y|α ≤ C◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Taking limits ui → u, we deduce that the same inequality holds for u, and thus ∥u∥C0,α(Ω) ≤ C◦, as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume now that k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, it follows from Arzel`a–Ascoli that Dmui → Dmu uniformly in Ω for m ≤ k and thus, as before, taking limits in the inequality ∥ui∥Ck(Ω) + |Dkui(x) − Dkui(y)| |x − y|α ≤ C◦, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In relation with property (H5), one can define L ∞,β as the set of functions u : B1 → R satisfying that, for each x ∈ R and each r ∈ (0, 1 − |x|), there exists some polynomial Px,r of degree ⌊β⌋ such that ∥u − Px,r∥L∞(Br(x)) ≤ Crβ — DRAFT — A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Some properties of H¨older spaces 189 for some C universal, and where ⌊β⌋ denotes the integer part of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' More generally, one can define2 L p,β for p ∈ [1, ∞] as the set of functions u satisfying r− n p ∥u − Px,r∥Lp(Br) ≤ Crβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, it turns out that, for any β > 0 and p ≥ 1, L p,β = L ∞,β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' see [JTW83, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, similarly to what we did in (H5), one can prove that if β = k + α, then L p,k+α = L ∞,k+α = Ck,α, if α ∈ (0, 1) and k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, when β is an integer these spaces do not coincide with H¨older spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, for β = 1 we have L p,1 = L ∞,1 = Λ1, (see [JW84, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6]), and for β > 1, u ∈ L p,β ⇐⇒ ∇u ∈ L p,β−1, (see [JTW83, Theorem 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') Here, Λ1 denotes the Zygmund space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' the set of functions u : B1 → R such that sup h∈B1 x∈B1−|h| ��u(x + h) + u(x − h) − 2u(x) �� |h| ≤ C, for some universal C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, when β = 0 we have L p,0 = L 1,0 = BMO, if p ∈ [1, ∞), where BMO denotes the space of bounded mean oscillation functions, see [JN61, JW84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice also that ∇u ∈ BMO implies u ∈ Λ1, but the opposite implication does not hold, see [Str80, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 2These spaces are called Morrey-Campanato spaces when p < ∞ and β < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — — DRAFT — Appendix B Proof of the boundary Harnack inequality The goal of this appendix is to prove the boundary Harnack inequality for Lipschitz domains, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The proof we present here is due to De Silva and Savin [DS20], and is different to the one given in the book [CS05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For simplicity, we consider domains Ω such that (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) Ω ∩ B1 is given by a Lipschitz graph in the en direction, with Lipschitz norm ≤ 1, and with 0 ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In other words, we consider (x′, xn) ∈ Rn−1 × R, and let (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2) g : Rn−1 → R, [g]C0,1(Rn−1) ≤ 1, g(0) = 0, Ω := {x ∈ Rn : xn > g(x′)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The boundary Harnack inequality in Lipschitz domains is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (See Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1 for a depiction of the setting in the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1 (Boundary Harnack).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let w1 and w2 be positive harmonic functions in B1 ∩ Ω, where Ω ⊂ Rn is a Lipschitz domain as in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1)-(B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that w1 and w2 vanish continuously on ∂Ω ∩ B1, and C−1 ≤ ∥wi∥L∞(B1/2) ≤ C◦ for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, C−1w2 ≤ w1 ≤ Cw2 in Ω ∩ B1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The constant C depends only on n and C◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, an appropriate iteration of the previous result gives the fol- lowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 191 — DRAFT — 192 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of the boundary Harnack inequality wi > 0 ∆wi = 0 wi = 0 B1 Ω xn x′ g(x′) Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Depiction of the setting in Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1 and Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let w1 and w2 be as in Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, ���� w1 w2 ���� C0,α(Ω∩B1/2) ≤ C for some small α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The constants α and C depend only on n and C◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that, for simplicity, we deal with Lipschitz domains with Lipschitz constant bounded by 1 and, as a consequence, none of the constants appearing in Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1 depend on the domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The same proof presented here can be adapted to the case of general Lipschitz domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The reasons we consider domains with Lipschitz constant bounded by 1 are to avoid introducing more notation and so that the domain Ω in B1 has a single connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Note, moreover, that when we apply the boundary Harnack in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='37, we are doing so to a Lipschitz domain with Lipschitz constant smaller than 1 (therefore, we can directly apply Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The following two (well-known) lemmas for sub- and superharmonic functions will be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that these are interior regularity properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4 (Weak Harnack Inequality for supersolutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u ∈ C(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, � −∆u ≥ 0 in B1 u ≥ 0 in B1 =⇒ inf B1/2 u ≥ c ∥u∥L1(B1/2), — DRAFT — B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of the boundary Harnack inequality 193 for some c > 0 depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By the mean value property of the Laplace equation, for any x◦ ∈ B1/3 we have u(x◦) ≥ 1 |B2/3| � B2/3(x◦) u = c∥u∥L1(B2/3)(x◦) ≥ c∥u∥L1(B1/3), with c a dimensional constant, so that we have proved the property in a ball of radius 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Take now any ¯x◦ ∈ ∂B1/3 and consider the ball B1/6(¯x◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that we can repeat the previous steps to derive inf B1/6(¯x◦) u ≥ c∥u∥L1(B1/6)(¯x◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, if we denote B := B1/3 ∩ B1/6(¯x◦), then ∥u∥L1(B1/6)(¯x◦) ≥ � B u ≥ |B| inf B u ≥ c inf B1/3 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' From the first result in this proof, we can conclude inf B1/2 u ≥ c1 inf B1/3 u ≥ c2∥u∥L1(B1/3) ≥ c3∥u∥L1(B1/2) for some dimensional constant c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In the last step we have used the mono- tonicity of averages with respect to the radius for superharmonic functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' see for example (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ The second lemma reads as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5 (L∞ bound for subsolutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u ∈ C(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, −∆u ≤ 0 in B1 ⇒ sup B1/2 u ≤ Cε∥u∥Lε(B1), for any ε > 0, and for some Cε depending only on n and ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Again, by the mean value property we have that, for any r > 0, ∥u∥L∞(Br/2) ≤ C � Br u ≤ C∥u∥1−ε L∞(Br) � Br |u|ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We now want to use an interpolation inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that, for any δ > 0, there exists some Cδ (depending only on δ and ε) such that ξ1−ε ≤ δξ + Cδ for all ξ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Taking ξ = A B with A = ∥u∥L∞(Br), B = � C � Br |u|ε � 1 ε we deduce that, for any δ > 0, there exists some Cδ such that ∥u∥L∞(Br/2) ≤ C∥u∥1−ε L∞(Br) � Br |u|ε ≤ δ ∥u∥L∞(Br) + Cδ � C � Br |u|ε � 1 ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 194 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of the boundary Harnack inequality Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A chain of balls to apply the Harnack inequality sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, ∥u∥L∞(Br/2) ≤ δ∥u∥L∞(Br) + Cδr− n ε ∥u∥Lε(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We are now in position to apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='27 with S(A) = ∥u∥L∞(A), k = n ε and γ = Cδ∥u∥Lε(B1), to deduce that ∥u∥L∞(B1/2) ≤ C∥u∥Lε(B1), for some constant C depending only on n and ε, as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ As a consequence of the previous lemmas we obtain the following two useful results, which are partial steps towards the proof of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The first one gives an L∞ bound for u in terms of the value of the function at an interior point in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u ∈ C(B1) be a positive harmonic function in B1∩Ω with u = 0 on B1 \\ Ω, where Ω ⊂ Rn is a Lipschitz domain as in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1)-(B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume, moreover, that u( 1 2en) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, ∥u∥L∞(B1/2) ≤ C, for some constant C depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that since u ≥ 0 is harmonic whenever u > 0, and it is continuous, we have ∆u ≥ 0 in B1 in the viscosity sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, since g in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) has Lipschitz constant bounded by 1, we have Bϱ( 1 2en) ⊂ {∆u = 0}, with ϱ = 1 2 √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, by Harnack’s inequality (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3)) we have that u ≤ Cn in B1/4( 1 2en).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, u(0, xn) ≤ Cn for xn ∈ � 1 4, 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Repeating iteratively, we get u(0, xn) ≤ Ck n for xn ∈ � 2−k−1, 2−k� (see Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2 for a sketch of this chain of inequalities), so that u(0, t) ≤ t−K for t ∈ � 0, 1 2 � , for some large dimensional constant K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We — DRAFT — B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of the boundary Harnack inequality 195 can repeat the same procedure at all points in B1/2 by iterating successive Harnack inequalities, to deduce that u ≤ d−K in B1/2, where d(x) := dist(x, Ωc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, for ε > 0 small enough we have � B1/2 |u|ε ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5, we deduce that ∥u∥L∞(B1/4) ≤ C, and the result in B1/2 follows from a simple covering argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ The second lemma reads as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let δ > 0 be small, let Ω ⊂ Rn be a Lipschitz domain as in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1)-(B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2), and let Ωδ := {x ∈ Ω : dist(x, Ωc) ≥ δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u ∈ C(B1) satisfy � ∆u = 0 in Ω ∩ B1 u = 0 on ∂Ω ∩ B1 and � u ≥ 1 in B1 ∩ Ωδ u ≥ −δ in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, for all k ∈ N such that kδ ≤ 3 4, we have u ≥ −δ(1 − c◦)k in B1−kδ for some constant c◦ depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let u− = min{u, 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that u− is superharmonic (in the vis- cosity sense) since ∆u− = 0 when u− < 0, and u− ≤ 0, so we have ∆u− ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let w = u− + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By assumption, w ≥ 0 and ∆w ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let x◦ ∈ ∂Ω ∩ B1−2δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us apply Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4 to a ball of radius 2δ around x◦, so that (after scaling) we deduce inf Bδ(x◦) w ≥ cδ−n∥w∥L1(Bδ(x◦)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice, now, that since the domain is Lipschitz and w ≥ δ in Ωc, we can bound ∥w∥L1(Bδ(x◦)) ≥ δ|{w ≥ δ} ∩ Bδ(x◦)| ≥ cδn+1 for some c (see Fig- ure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3) depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, inf Bδ(x◦) w ≥ c◦δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, since w ≥ δ in B1 ∩ Ωδ we have w ≥ c◦δ in B1−δ and therefore u ≥ −δ(1 − c◦) in B1−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Applying iteratively this inequality for balls of radius 1 − 2δ, 1 − 3δ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', we obtain the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ We can now show the following result, which is a key step in the proof of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 196 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of the boundary Harnack inequality ∂Ω u = 0 u ≥ −δ w ≥ 0 w = δ x◦ Bδ(x◦) ∥w∥L1(Bδ(x◦)) ≥ cδn+1 Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The fact that the domain is Lipschitz allows us to bound the L1 norm of w in Bδ(x◦) from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' There exists δ > 0, depending only on n, such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ Rn be a Lipschitz domain as in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1)-(B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2), and let Ωδ := {x ∈ Ω : dist(x, Ωc) ≥ δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Assume that u ∈ C(B1) satisfies � ∆u = 0 in Ω ∩ B1 u = 0 on ∂Ω ∩ B1 and � u ≥ 1 in B1 ∩ Ωδ u ≥ −δ in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, u ≥ 0 in B1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It is enough to show that, for some a > 0, we have (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3) � u ≥ a in B1/2 ∩ Ωδ/2 u ≥ −δa in B1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, iterating (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3) at all scales, and at all points z ∈ ∂Ω ∩ B1/2, we obtain � u ≥ ak in B2−k(z) ∩ Ω2−kδ u ≥ −δak in B2−k(z) for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, the first inequality yields that u(z + ten) ≥ 0 for z ∈ ∂Ω ∩ B1/2 and t > 0, and therefore u ≥ 0 in B1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us show (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We start with the first inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let x◦ ∈ B1/2 ∩ Ωδ/2, and let us suppose that δ 2 ≤ dist(x◦, Ωc) < δ (otherwise, we are done by assumption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Consider the function w = u + δ, which satisfies w ≥ 0 in Ω by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that we can connect the points x◦ and x◦ + 1 2δen with a sequence of (three) overlapping balls in Ω, so that we can apply Harnack’s inequality to w to deduce w(x◦) ≥ 1 C w � x◦ + 1 2δen � ≥ 1 C , — DRAFT — B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of the boundary Harnack inequality 197 for some dimensional constant C, where in the last step we are using that w � x◦ + 1 2δen � ≥ 1+δ by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, by taking δ > 0 smaller than 1 2C , we get u(x◦) ≥ 1 C − δ ≥ 1 2C for all x◦ ∈ B1/2 ∩ Ωδ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7 we know that u ≥ −δ(1 − c◦)k in B1−kδ as long as kδ ≤ 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If we take k = 1 2δ, we deduce u ≥ −δ(1 − c◦) 1 2δ in B1/2, and taking δ small enough such that (1 − c◦) 1 2δ ≤ 1 2C we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='9 (Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8 for small Lipschitz constants).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The proofs of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7 and Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8 can be simplified a lot in the case of a domain with small Lipschitz constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, let us assume that the hypotheses of Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8 hold, where the domain Ω satisfies (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1)-(B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2) but with Lipschitz constant L < 1 n−1, and let us consider the harmonic function ϕ(x) = x2 n − 1 n − 1 � x2 1 + x2 2 + · · · + x2 n−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, for δ small enough, ϕ ≤ u on ∂B1/2 ∩ Ω, and by assumption on the Lipschitz constant of the domain we have that ϕ ≤ 0 on ∂Ω∩B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In all, the maximum principle gives ϕ ≤ u in B1/2 ∩ Ω, which implies that u(ten) ≥ 0 for t ∈ � 0, 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By repeating the same argument at all boundary points in ∂Ω ∩ B1/2 we reach that u ≥ 0 in B1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We can now give the proof of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thanks to Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6, up to a constant depend- ing on C◦, we may assume w1( 1 2en) = w2( 1 2en) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, let us define v = Mw1 − εw2 for some constants M (large) and ε (small) to be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let δ > 0 be given by Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, since w2 is bounded, v ≥ −εw2 ≥ −δ in B1/2 for ε > 0 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, by the interior Harnack inequal- ity, we can take M large enough so that Mw1 ≥ 1 + δ in B1/2 ∩ Ωδ, where we recall that Ωδ = {x ∈ Ω : dist(x, Ωc) ≥ δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, v = Mw1 − εw2 ≥ 1 in B1/2 ∩ Ωδ, for M large enough depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, the hypotheses of Proposi- tion B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='8 are satisfied, and therefore we deduce that v ≥ 0 in B1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 198 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of the boundary Harnack inequality This means that, w2 ≤ Cw1 in B1/4 for some constant C depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The inequality in B1/2 follows by a covering argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, reversing the roles of w1 and w2, we obtain the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Finally, we give the: Proof of Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us denote W := w1 w2 , so that we have to prove H¨older regularity for W in Ω ∩ B1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that, by Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1, we know that 1 C ≤ W ≤ C in B1/2 ∩ Ω, for some C depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We start by claiming that, for some θ > 0 and all k ∈ N, we have (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4) osc B2−k−1 W ≤ (1 − θ) osc B2−k W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, let ak := sup B2−k W and bk := inf B2−k W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If we denote pk = 1 2k+1 en, then either W(pk) ≥ 1 2(ak + bk) or W(pk) ≤ 1 2(ak + bk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Suppose first that W(pk) ≥ 1 2(ak + bk), and let us define v := w1 − bkw2 ak − bk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that, by assumption, 1 2w2(pk) ≤ v(pk) ≤ w2(pk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, we can apply Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1 to the pair of functions v and w2 in the ball B2−k, to deduce that v ≥ 1 C w2 in B2−k−1, that is, w1 − bkw2 ak − bk ≥ 1 C w2 in B2−k−1 ⇐⇒ inf B2−k−1 W ≥ 1 C (ak − bk) + bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since supB2−k−1 W ≤ supB2−k W ≤ ak, we deduce that osc B2−k−1 W ≤ ak − 1 C (ak − bk) − bk = � 1 − 1 C � (ak − bk) = (1 − θ) osc B2−k W, with θ = 1 C , as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If we assume instead that W(pk) ≤ 1 2(ak + bk), then the argument is similar taking v := (akw2 − w1)/(ak − bk) instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In all, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of the boundary Harnack inequality 199 In particular, we have shown that, for some small α depending only on n, we have (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5) osc Br(x◦) W ≤ Crα for all r ∈ (0, 1 4) and x◦ ∈ ∂Ω ∩ B1/2, (compare with the proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We now need to combine (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5) with interior estimates for harmonic functions to deduce our desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, letting x, y ∈ Ω ∩ B1/2, we want to show that (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6) |W(x) − W(y)| ≤ C|x − y|α, for some constant C depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let 2r = dist(x, ∂Ω) = |x − x∗|, with x∗ ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We consider two cases: If |x − y| ≥ r 2, then we apply (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5) in a ball Bρ(x∗) with radius ρ = 2r + |x − y| to deduce that |W(x) − W(y)| ≤ osc Bρ(x∗) W ≤ C(2r + |x − y|)α ≤ C|x − y|α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If |x−y| ≤ r 2, then by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='5) we know that oscBr(x) W ≤ Crα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, if we denote c∗ := W(x), then ∥w1 − c∗w2∥L∞(Br(x)) = ∥w2 (W − c∗) ∥L∞(Br(x)) ≤ Crα∥w2∥L∞(Br(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' On the other hand, since w1 − c∗w2 is harmonic in Br(x), by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='7 (rescaled) we know that [w1 − c∗w2]C0,α(Br/2(x)) ≤ C rα ∥w1 − c∗w2∥L∞(Br(x)) ≤ C∥w2∥L∞(Br(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Hence, |W(y) − W(x)| = ���� w1(y) − c∗w2(y) w2(y) ���� ≤ C|x − y|α ∥w2∥L∞(Br(x)) w2(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We finish by noticing that, by Harnack’s inequality applied to w2 in B2r(x), we have ∥w2∥L∞(Br(x)) ≤ Cw2(y) for some C depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' With these two cases, we have shown (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' □ Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As said above, the proofs in this Appendix have been car- ried out in case that Ω is a Lipschitz domain as in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1), with Lipschitz constant bounded by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This slightly simplifies the notation, and we have that Ω ∩ B1 has only one connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In case of general Lipschitz domains (with Lipschitz constant bounded by L), the same proofs can be carried out, provided that one is slightly more careful with the underlying geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A simple way to do this is to prove all the results with B1/2 replaced by Bρ, with ρ > 0 small depending on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 200 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Proof of the boundary Harnack inequality An alternative way to do this is to work with cylinders, rather than balls, as in [DS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — Appendix C Probabilistic interpretation of fully nonlinear equations In this appendix, we heuristically describe the probabilistic interpretation of fully nonlinear elliptic PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This extends the discussion from Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3 in the context of the Laplace operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We start by recalling the following probabilistic interpretation of har- monic functions from Chapter 1: We have a Brownian motion Xx t , starting at x ∈ Ω, and a payoff function g : ∂Ω → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' When we hit the boundary ∂Ω (for the first time) at a point z ∈ ∂Ω, we get a paid g(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The question is then: What is the expected payoff?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It turns out that u(x) := � expected payoff � = E � g (Xx τ ) � satisfies � ∆u = 0 in Ω u = g on ∂Ω, where τ is the first time at which Xx t hits ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We already saw this in Chapter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, we will see more general “prob- abilistic games” that lead to more general elliptic PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Stochastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A stochastic process Xt is a collection of random variables indexed by a parameter, that for us is going to be t ≥ 0, taking values in a state space, that for us is going to be Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' One can think of them as simply a “particle” moving randomly in Rn, with t ≥ 0 being the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' 201 — DRAFT — 202 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Probabilistic interpretation of fully nonlinear equations The most famous and important stochastic process is the Brownian mo- tion, that we already introduced in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We recall that it is charac- terized by the following properties: (1) X0 = 0 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (2) Xt has no memory (is independent of the past, or it has indepen- dent increments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (3) Xt has stationary increments: Xt+s − Xs is equal in distribution to Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (4) Xt has continuous paths (t �→ Xt is continuous) almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (5) Xt is isotropic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', it is rotationally symmetric in distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A more general class of stochastic processes is obtained by removing the assumption (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Infinitesimal generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The infinitesimal generator of a stochastic pro- cess Xt is an operator L defined to act on functions u : Rn → R by (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1) Lu(x) := lim t↓0 E [u (x + Xt)] − u(x) t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It takes C2 functions u, and gives Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For the Brownian motion, we have that L is the Laplacian ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' More generally, under the assumptions (1)-(2)-(3)-(4), the infinitesimal generator L will be a second order elliptic operator of the form Lu = n � i,j=1 aij∂iju + n � i=1 bi∂iu + cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Why is this infinitesimal generator useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The infinitesimal generator of a stochastic process encodes all the infor- mation of such process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, it is a classical fact that the definition of L leads to the formula (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2) E [u(x + Xt)] = u(x) + E �� t 0 Lu(x + Xs) ds � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (This is analogous to the fundamental theorem of Calculus!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') We can come back to the “expected payoff” problem: Let Ω ⊂ Rn be a fixed domain, and consider a stochastic process (x+Xt) starting at x ∈ Ω, satisfying (2)-(3)-(4) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Given a payoff function g : ∂Ω → R, we have the following: when Xx t hits the boundary ∂Ω for the first time at z ∈ ∂Ω, we get a payoff g(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (See Figure C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') What is the expected payoff?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Probabilistic interpretation of fully nonlinear equations 203 x z ∂Ω Figure C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A stochastic process Xx t defined in Ω starting at x until it hits the first point on the boundary z ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Of course, the expected payoff will depend on x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For the Brown- ian motion, we defined u(x) to be the expected payoff when starting at x, E [g(Xx τ )], where τ is the first time we hit ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, we observed that, since the Brownian motion is isotropic, u must satisfy the mean value property, and thus u is harmonic: ∆u = 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, for more general stochastic processes, we must use (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Indeed, we define u as before (expected payoff), and notice that if t > 0 is small enough, then x+Xt will still be inside Ω, and therefore, the expected payoff is simply equal to E [u(x + Xt)] (up to a small error), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e, u(x) = E [u(x + Xt)] + o(t) (for t > 0 small enough).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' where the term o(t) is due to the fact that x+Xt could potentially lie outside of Ω, even for arbitrarily small times t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, using the definition of infinitesimal generator, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1), we obtain that Lu(x) = lim t↓0 E [u(x + Xt)] − u(x) t = lim t↓0 o(t) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, for every x ∈ Ω, we get Lu(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We clearly have u = g on ∂Ω, thus, u must be the solution of � Lu = 0 in Ω u = g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 204 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Probabilistic interpretation of fully nonlinear equations Summarizing: � Expected payoff for Xt � ←→ � Dirichlet problem for L (infinitesimal generator) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Something similar can be done to solve other probabilistic problems related to Xt: – What is the expected time it will take to exit Ω if we start at x?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' � −Lu = 1 in Ω u = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' – What is the probability density p(x, t) of Xt in Rn?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' � ∂tp − Lp = 0 in Rn × (0, ∞) p(·, 0) = δ{x=0} on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We next see what happens when we have a control, or a two-player game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In that case, we get nonlinear PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Optimal stopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We start with the optimal stopping problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This kind of problem appears very often in Mathematical Finance, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Given a process Xt in Rn, we can decide at each instant of time whether to stop it or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' When we stop, we get a payoff ϕ (which depends on the point we stopped at).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The goal is to discover what is the optimal strategy so that we maximize the payoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us consider the process x + Xt (starting at x ∈ Rn), and a payoff ϕ ∈ C∞ c (Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For any stopping time θ, we get a payoff E [ϕ(x + Xθ)], and therefore we want to maximize u(x) := max θ E [ϕ(x + Xθ)] among all possible stopping times θ (notice that a stopping time θ is actually a random variable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' see [Eva13] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Can we find a PDE for u(x)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Roughly speaking, the only important thing to decide here is: If we are at x, is it better to stop and get ϕ(x), or to continue and hope for a better payoff later?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us find the PDE for u: – First, since we can always stop (take θ = 0), we have u(x) ≥ ϕ(x) for every x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' – Second, since we can always continue for some time (take θ ≥ t◦ > 0), we have that u(x) ≥ E [u(x + Xt)] for t ≤ t◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This, combined with (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2) (or with the definition (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1)), gives Lu(x) ≤ 0 for every x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Probabilistic interpretation of fully nonlinear equations 205 u ϕ −Lu ≥ 0 everywhere u ≥ ϕ everywhere Lu = 0 in {u > ϕ} Figure C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The obstacle problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' – Third, at those points where we have u(x) > ϕ(x), we are clearly not stopping there, so we have u(x) = E [u(x + Xt)] + o(t) for t very small, and thus Lu(x) = 0 whenever u(x) > ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The PDE for u is � � � u ≥ ϕ in Rn, −Lu ≥ 0 in Rn, Lu = 0 in {u > ϕ}, ←→ min{−Lu, u − ϕ} = 0 in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is the obstacle problem in Rn from Chapter 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (See Figure C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') Notice that once we know u, we know the sets {u = ϕ} and {u > ϕ}, so we have the optimal strategy!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Controlled diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us now take a different problem, that nonethe- less is quite similar to the optimal stopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Consider two stochastic processes, X(1) t and X(2) t , with infinitesimal gen- erators L1 and L2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ Rn be a domain, and let g : ∂Ω → R be a payoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We have the same “game” as before (we get a payoff when we hit the boundary), but now we have a control: for every x ∈ Ω, we can choose to move according to X(1) t or X(2) t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The question is then: What is the optimal strategy if we want to maximize the payoff?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 206 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Probabilistic interpretation of fully nonlinear equations Notice that now the strategy consists of choosing between X(1) t and X(2) t for every x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As before, we define u(x) := max all possible choices of a : Ω → {1, 2} E [g(Xa τ )] (where τ is the time we hit the boundary ∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that for every a : Ω → {1, 2} we have Xa t , a process which could change from point to point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Is there any PDE for u?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The optimality conditions are: – First, when we are at x we can simply decide to continue with X(1) t , and therefore, u(x) ≥ E � u(x + X(1) t ) � for every x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This yields L1u(x) ≤ 0 for every x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' – Similarly, we can do the same for X(2) t , and get L2u(x) ≤ 0 for every x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' – Finally, it turns out that either u(x) = lim t↓0 E � u(x + X(1) t ) � or u(x) = lim t↓0 E � u(x + X(2) t ) � , since close to x we are taking either X(1) t or X(2) t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This means that either L1u(x) = 0 or L2u(x) = 0, for every x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Therefore, u satisfies � � � −L1u ≥ 0 in Ω, −L2u ≥ 0 in Ω, either L1u = 0 or L2u = 0 in Ω ←→ max{L1u, L2u} = 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' More generally, if we have a family of processes Xα t , with α ∈ A, then the PDE for u becomes (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3) max α∈A {Lαu} = 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Even more generally, we could have two players, one that wants to max- imize the payoff and the other one that wants to minimize the payoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' They have two parameters, Xαβ t , α ∈ A, β ∈ B, and each player controls one parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, the optimal payoff solves the PDE (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4) min β∈B max α∈A {Lαβu} = 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3) above is called the Bellman equation (stochastic control).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='4) above is called the Isaacs equation (differential games).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' These two equations are fully nonlinear elliptic equations!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Probabilistic interpretation of fully nonlinear equations 207 Indeed, assume that we have (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3), and that the infinitesimal generators Lαu are of the form Lαu = n � i,j=1 a(α) ij ∂iju, (α ∈ A) with a(α) ij uniformly elliptic: 0 < λId ≤ (a(α) ij )ij ≤ ΛId.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, the equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3) is max α∈A � � � n � i,j=1 a(α) ij ∂iju � � � = 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is a nonlinear function of the Hessian D2u: F(D2u) = 0 in Ω, with F(M) := max α∈A � � � n � i,j=1 a(α) ij Mij � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The function F : Rn×n → R is the maximum of linear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In partic- ular, F is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Moreover, F is uniformly elliptic: 0 < λ∥N∥ ≤ min α∈A � � � n � i,j=1 a(α) ij Nij � � � ≤ F(M + N) − F(M) ≤ ≤ max α∈A � � � n � i,j=1 a(α) ij Nij � � � ≤ Λ∥N∥ for any symmetric matrix N ≥ 0 (we are using here that max f + min g ≤ max(f + g) ≤ max f + max g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Furthermore, any convex function can be written as the maximum of linear functions (see Figure C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3), and thus: Remark C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Any F : Rn×n → R which is uniformly elliptic and convex can be written as F(M) = max α∈A � tr (A(α)M) + cα � (where cα are constants).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' (If F is homogeneous of degree 1, then we do not need the cα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=') In particular, every fully nonlinear uniformly elliptic equation F(D2u) = 0 in Ω, with F being convex, can be written as a Bellman equation max α∈A {Lαu} = 0 in Ω with Lαu = �n i,j=1 a(α) ij ∂iju + cα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Finally, for non-convex F it turns out that: — DRAFT — 208 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Probabilistic interpretation of fully nonlinear equations Figure C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Convex function as the maximum of linear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Any F : Rn×n → R which is uniformly elliptic (not neces- sarily convex), can be written as F(M) = min β∈B max α∈A � � � n � i,j=1 a(αβ) ij Mij + cαβ � � � = min β∈B max α∈A � tr � A(α,β)M � + cαβ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is because any Lipschitz function F can be written as the minimum of convex functions, and convex functions can be written as the maximum of linear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In particular, every fully nonlinear uniformly elliptic equation F(D2u) = 0 in Ω can be written as an Isaacs equation min β∈B max α∈A {Lαβu} = 0 in Ω with Lαβu = �n i,j=1 a(α,β) ij ∂iju + cαβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Summary: Every fully nonlinear elliptic PDE has an interpretation in terms of a probabilistic game!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Probabilistic interpretation of fully nonlinear equations 209 Probabilistic interpretation of PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Expected payoff ←→ Dirichlet problem � Lu = 0 in Ω u = g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Expected exit time (or running costs/ non-homogeneous environments) ←→ Dirichlet problem � −Lu = f in Ω u = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Distribution of the process ←→ Heat equation ∂tu − Lu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Optimal stopping ←→ Obstacle problem min{−Lu, u − ϕ} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Controlled diffusion ←→ Fully nonlinear equation F(D2u) = 0, F convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Two-player games ←→ Fully nonlinear equation F(D2u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' One could even consider the equations with x-dependence, or with lower order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' All equations studied in Chapters 4 and 5 have a probabilistic interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — — DRAFT — Appendix D Motivations and applications for the obstacle problem Here, we give a brief overview of the motivations and applications for the obstacle problem listed in Chapter 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer to the books [DL76, KS80, Rod87, Fri88, PSU12] for more details, as well as for further applications of obstacle-type problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Fluid filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Consider two reservoirs of water at different heights sep- arated by a porous dam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' For simplicity, we will assume a flat dam, with rectangular cross section, which yields a problem in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Alternatively, one could consider variable cross sections, which would yield an analogous ob- stacle problem in R3 instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The dam is permeable to the water, except in the base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, there is some flow of fluid between the two reservoirs across the dam, and some wet part of the cross section depending only on the relative distance to each of the two water sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us assume one reservoir has water at height 1, and the other has water at height 0 < h < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us denote by ϕ(x) the profile of the water through the dam cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' See Figure D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1 for a representation of the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us denote by u = u(x, y) : [0, 1] × [0, 1] → R+ the hydraulic piezo- metric head of the fluid, given by the sum between the pressure p(x, y) and 211 — DRAFT — 212 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Motivations and applications for the obstacle problem y x 1 1 h D y = ϕ(x) Figure D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Graphic representation of the cross section of a porous dam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' the elevation head (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', the potential energy of the fluid): u(x, y) = y + 1 γ p(x, y), where γ is a constant depending on the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The hydraulic head is defined where there is fluid, namely, in D := � (x, y) ∈ (0, 1) × (0, 1) : y < ϑ(x) � , and is such that u(0, y) = 1 for 0 ≤ y ≤ 1, and u(1, y) = h for 0 ≤ y ≤ h and u(1, y) = y for h ≤ y ≤ ϑ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Here, u itself is an unknown, but D is also to be determined (and there- fore, ϑ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In these circumstances we have that u(x, y) ≥ y in D, and if we define w(x, y) := � ϕ(x) y � u(x, ζ) − ζ � dζ for (x, y) ∈ D, and w(x, y) ≡ 0 for (x, y) ∈ [0, 1] × [0, 1] \\ D, then w fulfils the equation ∆w = χ{w>0} = χD in [0, 1] × [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' That is, w is a solution to the obstacle problem (see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6)) with f ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer to [Bai74] and the references therein for more details about the Dam problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The Stefan problem, dating back to the 19th century, is the most classical and important free boundary problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It aims to describe the temperature distribution in a homogeneous medium undergoing a phase change, such as ice melting to water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We denote by θ(x, t) the temperature (at position x and time t), and assume θ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The function θ satisfies the heat equation ∂tθ − ∆θ = 0 in the region {θ > 0}, while the evolution of the free boundary ∂{θ > 0} is — DRAFT — D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Motivations and applications for the obstacle problem 213 dictated by the Stefan condition ∂tθ = |∇xθ|2 on ∂{θ > 0} — where the gradient is computed from inside {θ > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' After the transformation u(x, t) := � t 0 θ(x, τ)dτ (see [Duv73, Fig18]), the problem is locally equivalent to � � � ∂tu − ∆u = −χ{u>0} in B1 × (0, T) ⊂ R3 × R u ≥ 0 ∂tu ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This is the parabolic version of the obstacle problem ∆u = χ{u>0} in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Hele-Shaw flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This model, dating back to 1898, describes a fluid flow between two flat parallel plates separated by a very thin gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Various prob- lems in fluid mechanics can be approximated to Hele-Shaw flows, and that is why understanding these flows is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A Hele-Shaw cell is an experimental device in which a viscous fluid is sandwiched in a narrow gap between two parallel plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In certain regions, the gap is filled with fluid while in others the gap is filled with air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' When liquid is injected inside the device through some sinks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' through a small hole on the top plate) the region filled with liquid grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We denote by p(x, t) the pressure of the fluid (at position x and time t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By definition, {p > 0} is the region filled with liquid, while in {p = 0} there is just air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The pressure p is harmonic in {p > 0}, and the evolution of the free boundary ∂{p > 0} is dictated by ∂tp = |∇xp|2 on ∂{p > 0} — where the gradient is computed from inside {p > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice the striking similarity to the Stefan problem — the only important difference here is that p is harmonic (and not caloric) in the region where it is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' After the transformation u(x, t) = � t 0 p(x, τ)dτ, it turns out that u solves locally (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=', outside the region where liquid is injected) � � � ∆u = χ{u>0} in B1 × (0, T) ⊂ R2 × R u ≥ 0 ∂tu ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This means that, for each fixed time t, u(·, t) is a solution to the (stationary) obstacle problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Optimal stopping, finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' As explained in Appendix C, the obstacle problem appears when considering optimal stopping problems for stochastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A typical example is the Black–Scholes model for pricing of American options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' An American option is a contract that entitles its owner to buy some financial asset (typically a share of some company) at some specified price (the “strike price”) at any time — often before some specified date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 214 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Motivations and applications for the obstacle problem This option has some value, since in case that the always fluctuating market price of the asset goes higher than the strike price then the option can be “exercised” to buy the asset at the lower price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The Black-Sholes model aims to calculate the rational price u = u(x, t) of an option at any time t prior to the maturity date and depending on the current price x of the financial asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Since the option can be exercised at any time, determining the “exercise region” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' the region in which it is better to exercise the option) is a part of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Interestingly, this problem leads to an obstacle problem (often parabolic) posed in Rn, where the dimension n is the number of assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer to [LS09] and the references therein for more details about such kind of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Interacting particle systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Large systems of interacting particles arise in several models in the natural sciences (one can think of physical particles in Physics or Biology, for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In such systems the discrete energy can be well approximated by the continuum interacting energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We denote µ the (probability) measure representing the particle density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In several models the particles attract each other when they are far, but experience a repulsive force when they are close [CDM16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, the interaction energy E associated to the interaction potential W ∈ L1 loc(R3), is given by E[µ] := 1 2 � R3 � R3 W(x − y)dµ(x) dµ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In general, the interaction potential can have very different structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It is common to assume a repulsive behaviour for particles that are very close (blowing up at zero distance), and attractive behaviour when they are far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A typical assumption is to have W(z) ∼ |z|−1 near the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In other models in statistical mechanics, the particles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' electrons) repel with a Coulomb force and one wants to understand their behaviour in presence of some external field that confines them [Ser18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In that case, the interaction energy associated with the system is given by E[µ] := 1 2 � R3 � R3 dµ(x) dµ(y) |x − y| + � R3 V dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' One of the main questions when dealing with these systems is to under- stand the “equilibrium configurations”, that is, minimizers of the energy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' It turns out that, in both cases, any minimizer µ◦ is given by µ◦ = −∆u, with u satisfying (locally) the obstacle problem min{−∆u, u − ϕ} = 0, — DRAFT — D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Motivations and applications for the obstacle problem 215 for some obstacle ϕ that depends on W (or on V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The free boundary corresponds to the boundary of the region in which the particles concentrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer to [CDM16, Ser18] and the references therein for a thorough study of these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Quasi-Steady Electrochemical Shaping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Electrochemical Machining (ECM) is an electrochemical method to remove metals (electroconductive) by plac- ing the material inside an electrolytic call as an anode, surrounded by a fixed cathode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then an electric potential is applied between a cathode and an anode, which is submerged in an appropriate electrolyte, thus producing a chemical reaction that removes the metal from the anode and gives rise to a moving boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' This method is used to shape extremely hard materials, to produce complicated shapes which are otherwise very difficult to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let us suppose we have cylindrical symmetry (that is, both anode and cathode are long cylindrical materials), so that we can work with the cross section and thus in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' A similar approach works in the three- dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Let Ω ⊂ R2 denote the domain enclosed by the cathode, and Λ(0) ⊂ Ω denote the anode at time t = 0 (an electric potential is applied between ∂Ω and ∂Λ(0), where the region Ω \\ Λ(0) contains the electrolyte).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Then, the metal starts to be removed, so that after a time t ≥ 0, we denote by Λ(t) the set defining the anode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' By this process we have that Λ(t) ⊂ Λ(t′) if t ≥ t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' The boundary Γ(t) = ∂Λ(t) is unknown, it is a free boundary, which we assume is represented by a function γ : Ω → R as Γ(t) = {(x, y) ∈ Ω : γ(x, y) = t}, for some function γ to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We assume that γ(x, y) = 0 in Ω \\ Λ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' If we denote by π = π(t) > 0 the potential difference at time t > 0 between anode and cathode, then the ECM problem is concerned with finding a function η(t, x, y) that solves ∆η(t, x, y) = 0 in Ω \\ Γ(t), η(t, x, y) = 0 on {t > 0} × ∂Ω, η(t, x, y) = π(t), ∇η(t, x, y) · ∇γ(x, y) = λ on {t > 0} × Γ(t) (with the convention that the gradient and the Laplacian are only taken in the spatial variables), for some constant λ > 0 (the ECM constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Notice that 0 ≤ η(t, x, y) ≤ π(t) in Λ(t) by the maximum principle, and let us extend η to Ω as η(t, x, y) = π(t) in Λ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Now, if we define u(t, x, y) = � t 0 (π(s) − η(s, x, y)) ds, then u ≥ 0 and in Λ(0), u fulfils ∆u(t, ·, ·) = λχ{u(t,·,·)>0} for any t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' — DRAFT — 216 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Motivations and applications for the obstacle problem That is, u fulfils an obstacle problem (compare with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content='6)) with f ≡ λ, for each time t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' We refer to [Rod87] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Heat control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Given a domain Ω and a temperature T◦, we have heating devices evenly distributed on Ω that need to ensure that the temperature u(x), x ∈ Ω, is as close as possible to T◦, by injecting flux proportional to the distance between u(x) and T◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Due to the limited power of the devices, the heat flux generated by them needs to remain in the interval (−q, 0] for q ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' Thus, the heat flux injected is Φ(u) = max{C(u − T◦)−, −q} for some constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfmf14/content/2301.01564v1.pdf'} +page_content=' In equilibrium, the temperature satisfies ∆u = Φ(u) in Ω, In particular, letting C → ∞, the previous equation becomes ∆u = −qχ{u, which should explain the observed increase in Tc +by 9 K during annealing. +Keywords: fluctuating conductivity, pseudogap, excess conductivity, annealing, HoBCO single crystals. + + +1. Introduction +In modern solid state physics, one of the urgent prob- +lems is the construction of a theory of high-temperature +superconductors (HTSCs), which is necessary to elucidate +the possibility of creating new superconductors with even +higher, preferably room, critical temperatures Tc of transi- +tion to the superconducting (SC) state. The solution of this +problem is complicated by the lack of a clear understand- +ing of the physics of internal interactions in such multicom- +ponent compounds as HTSCs, in particular, the mechanism +of SC pairing [1], which makes it possible to have a very +high critical temperature of the SC transition [2]. It is be- +lieved that the answer to the question of SC pairing, as +well as the possible role of the interplay between super- +conductivity and magnetism in the formation of paired +fermions at temperatures above 100 K [2] in HTSCs, can +be obtained by studying such an interesting phenomenon +as a pseudogap (PG), which opens in cuprate HTSCs, type +ReBa2Cu3O7–δ (Re = Y, Ho, Gd, Pr) at temperature T* >> Tc. +PG is a special state of matter, which is characterized by + +A. L. Solovjov, L. V. Omelchenko, E. V. Petrenko, et al. +116 +Low Temperature Physics/Fizyka Nyzkykh Temperatur, 2023, Vol. 49, No. 1 +a reduced (but not to zero) density of electronic states at the +Fermi level and a probable transformation of the Fermi sur- +face below T* [1–5]. However, the physics of PG and its role +in the formation of coupled electrons (fluctuating Cooper +pairs) (FKPs) above Tc are still unclear, despite the huge +number of theoretical and experimental works devoted to +this problem. +It has long been established that the behavior of HTSCs +in the normal state goes far beyond the standard Fermi +liquid approach [6–9]. As a result, a large number of non- +Fermi liquid models [10–12] as well as marginal Fermi +liquid models [13] have been proposed. All these models +largely explain various specific aspects of the behavior of +cuprates observed in the experiment. However, until now +there is no unified theory that would be able to describe all +the features of the behavior of HTSCs in the PG state. Re- +cently, there has been a noticeable increase in interest in +studying the problem of PG in HTSCs [3, 14–18]. More- +over, in addition to the already mentioned spin fluctuations +[1, 7–9], spin density waves (SDW) [12, 17], charge order- +ing (CO) [1, 17], charge density waves (CDW) [15–18] as +well as pair density waves (PDW) [19, 20] are proposed to +explain the PG physics. We share a different approach to +the problem of the PG appearance, which assumes the pos- +sibility of the formation of paired fermions, the so-called +local pairs (LPs), in HTSCs below the PG opening temper- +ature T* >> Tc [5, 21–23]. Moreover, it can be assumed that +SDW, CDW, CO, and PDW can be considered as different +possible mechanisms of LP pairing in cuprates in the PG state. +One of the most interesting materials for studying PG +are ReBa2Cu3O7–δ compounds, which is due to the possibil- +ity of wide variation of their composition by replacing yt- +trium with its isoelectronic analogues, or by changing the +degree of oxygen nonstoichiometry. As is known, in the +ground state all ReBa2Cu3O7–δ are Mott dielectrics with a +long-range antiferromagnetic (AF) order, in which the elec- +tron spins S = 1/2 are localized on copper ions Cu2+ [6, 24]. +The dielectric state is a consequence of strong electronic +(Hubbard) correlations [3, 4] (and references therein). +When charge carriers (holes) appear during doping, the +long-range AF order is rapidly violated. However, as shown +by neutron experiments with YBCO samples of different +doping levels less than optimal, well-developed short-range +order AF fluctuations [25] (and references therein) are re- +tained in the normal metal phase of cuprates, which are ob- +served up to very high doping levels (Tc ~ 85 K ) [5]. +Taking into account the presence of AF correlations in +cuprates, of particular interest are compounds with partial +replacement of yttrium (Y) by praseodymium (PrBCO with +a magnetic moment μeff ≈ 2μB) and especially with com- +plete replacement by holmium, HoBa2Cu3O7−δ (HoBCO), +which has a magnetic moment μeff ≈ 9.7μB, due to magnetic +moment of pure Ho which is μHo ≈ 10.6μB [5, 26]. However, +PrBCO, being a magnetic dielectric, where all electrons are +localized in the Fehrenbacher–Rice zone [27], quickly +suppresses superconductivity in YBCO — the so-called +“praseodymium anomaly”. Whereas, HoBCO shows al- +most the same high Tc values as YBCO. +Substitution of Y in single crystals of ReBa2Cu3O7–δ +(Re = Y, Ho, Dy, etc.) with Ho of a fairly large magnetic +moment suggests a change in behavior of the system due to +paramagnetic properties of HoBa2Cu3O7–δ in its normal +state [28]. Samples exhibiting oxygen nonstoichiometry +are of special interest. In this state the redistribution of +labile oxygen and structural relaxation take place simulta- +neously, thus significantly affecting electron transport pa- +rameters of the system [29, 30]. In this case rare alkaline +earth metal ion can serve as a sensor, sensitive to local +symmetry of its surroundings and charge density distribu- +tion, which can be affected by outside factors such as tem- +perature [29], high pressure [30], or room-temperature an- +nealing [31, 32]. +To study the physical nature of the interaction between +superconductivity and magnetism, single crystals were +chosen, which have the advantage that their properties can +noticeably change either during annealing of samples in an +oxygen atmosphere due to an increase in the density of +charge carriers nf, or due to the formation of additional +defects, as a result of rapid quenching from ~ 600 °C, fol- +lowed by improving of their parameters by holding the sam- +ples in air at room temperature (the so-called room- +temperature annealing) immediately after fabrication [31, 32]. +The effect of annealing at room temperature (hereinafter +simply annealing) on Тс, nf and the change in the lattice +parameters of oxygen-deficient ReBa2Cu3O7–δ (Re = Y, Ho) +single crystals after their quenching from T = 600 °C is ex- +plained by the ordering of their structure due to the ordering +of oxygen atoms in the CuO2 planes without a noticeable +change in the oxygen content in the sample [31, 32] (and +references therein). Nevertheless, in spite of a number of +studies on the relaxation processes in the 1–2–3 system +during annealing [31–33] or, for example, under high pres- +sure [34] (and references therein), many aspects, such as +the charge transfer and the nature of the redistribution of +the vacancy subsystem, still remain uncertain. The study of +the influence of intrinsic magnetism Ho on the fluctuation +conductivity (FLC) and PG in HoBa2Cu3O7-δ single crys- +tals upon annealing at room temperature is considered very +promising for elucidating the mechanisms of the mutual +influence of superconductivity and magnetism in HTSCs, +which is important for the final elucidation of the nature of +both PG and HTSC in general +This work is devoted to the study of the effect of an- +nealing on the temperature dependences of the resistivity +ρ(Т) of lightly doped HoBa2Cu3O7–δ quenched single crys- +tals, from which the temperature dependences of the excess +conductivity σ′(Т) and pseudogap Δ*(Т) are calculated in +the local pair model, as discussed in detail below. So far, +no such studies have been carried out. + +Effects of annealing on the fluctuation conductivity and pseudogap in slightly doped HoBa2Cu3O7–δ single crystals +Low Temperature Physics/Fizyka Nyzkykh Temperatur, 2023, Vol. 49, No. 1 +117 +2. Experiment +HoBa2Cu3O7−δ single crystals were grown with the so- +lution-melt technique in a gold crucible as described else- +where [33]. Rectangular crystals of about 1.9 × 1.5 × 0.2 +mm were selected to perform the resistivity measurements. +The smallest parameter of the crystal corresponds to the c +axis. A fully computerized setup utilizing the four-point +probe technique with stabilized measuring current of up to +10 mA was used to measure the ab plane resistivity, ρab(T) +[34]. Silver epoxy contacts were glued to the opposite +sides of the crystal in order to produce a uniform current +distribution in the central region where voltage probes in +the form of parallel silver stripes were placed. Contact re- +sistances below 1 Ω were obtained. Temperatures were meas- +ured with a Pt sensor having an accuracy of about 1 mK. +To reduce the oxygen content, the samples were an- +nealed at 600 °C in ambient atmosphere for 24 h. After the +annealing step, the crystals were quenched to room tem- +perature within 2 min to 3 min, mounted on the holder and +further cooled down to liquid-nitrogen temperature within +15 min. All measurements were conducted while warming +up the samples. Experimental runs were carried at rates of +about 0.1 K/min near Tc and 0.5 K/min at T >> Tc. +In order to determine the room-temperature annealing +effect on the electrical conductivity, after the first meas- +urement of ρ(T) (sample S1), the samples were kept at +room temperature for 20 h, after which the measurements +were repeated (sample S2). The last series of measurements +was done after the room-temperature annealing of the sam- +ples for about 5 days (120 h) (sample S3), depending on +completion of relaxation processes. Temperature dependen- +cies of resistivity ρ(T) = ρab(T) for HoBa2Cu3O6.65 single +crystal with initial Tc = 63.6 K and δ ≈ 0.35 (sample S1) +and measured after annealing during 20 h (sample S2) and +120 h (sample S3) are shown in Fig. 1. In the Insert to Fig. 1 +shows a special approach to a more accurate determination +of the PG opening temperature T* using the criterion +(ρ(T) – ρ0)/aT = 1, where a =dρ/dT [2]. +Thus, we have three curves or actually three samples. +The sample parameters are listed in the Tables. All curves +have an expected S-shape with positive thermally activated +bending, characteristic for slightly doped cuprates [23, 35]. +However, above T* ≈ 256.5 K (S1), ≈ 238 K (S2), and +≈ 235 K (S3) ρ(T) varies linearly with T at rates dρ/dT = += 2.48, 1.91, and 1.57 μΩ⋅cm⋅K‒1 for S1, S2, and S3, re- +spectively. The figure shows that after the first annealing, +the resistivity rapidly decreases, but reaches saturation, +when the annealing time increases up to 120 h [31]. +At the same time, during annealing Tc of the samples +increases (Table 1), while T* decreases (Table 2) in good +agreement with phase diagram of cuprates [1, 17, 35]. All +dependences also show a clear trend towards saturation +upon annealing (see the Tables). Interestingly, neither ρ(T) +nor Tc practically increase with an increase in the annealing +time above 120 h. Rapid cooling of crystals from about +600 °C (quenching) leads to the appearance of numerous +additional disordered defects in the form of oxygen vacan- +cies in CuO2 planes [31] (and references there). It is these +defects that are responsible for the increased resistivity of +sample S1. It can be concluded that the decrease in ρ(T) +and the increase in Tc observed during annealing are due to +the ordering of labile oxygen in the CuO2 planes due to +diffusion but without an increase in the oxygen content. +A decrease in the slope dρ/dT indeed indicates the ther- +mally activated character of the resistance relaxation pro- +cess. The fact that the relaxation activation energy ρ(T) co- +incides with the oxygen diffusion activation energy confirms +the possibility of oxygen ordering in the planes. It has been +shown that the oxygen diffusion, at least in YBCO single +crystals, can proceed at an average rate of 150 Å per day [31]. +Fig. 1. (Color online) Temperature dependences of the resistivity +ρab of single crystals of HoBa2Cu3O6.65 for different stages of +room temperature annealing: S1 — without annealing, S2 — annea- +ling for 20 h, S3 — annealing for five days (120 h). The straight +solid lines determine the normal-state resistivity ρN (see the text). +The Inset shows the method for determining Т* using the criterion +(ρ(T) – ρ0)/aT = 1 [2]. +Table 1. The resistivity and FLP parameters of the HoBa2Cu3O6.65 single crystal +Sample +ρ(300 K), µΩ⋅cm +ρ(100 K), µΩ⋅cm +Tc, K +mf +cT +, K +TG, K +T0, K +T01, K +Δln σ′ +d01, Å +ξc(0), Å +S1 +751.5 +192.1 +63.6 +70.65 +71.0 +74.8 +108.2 +0.68 +3.87 +2.82 +S2 +537.5 +130.1 +69.6 +73.88 +74.6 +77.5 +125.3 +1.0 +3.10 +2.60 +S3 +451.5 +112.1 +72.7 +74.35 +75.0 +78.9 +113.4 +0.60 +3.90 +2.89 + + +A. L. Solovjov, L. V. Omelchenko, E. V. Petrenko, et al. +118 +Low Temperature Physics/Fizyka Nyzkykh Temperatur, 2023, Vol. 49, No. 1 +All these facts confirm the conclusion made. It appears that +oxygen vacancies in slightly doped cuprates make the oxy- +gen rearrangement easier to achieve. +Figure 2 shows the resistivity curves near Tc for S1 (a) +and S3 (b), respectively, in which all representative tem- +peratures are designated. The observed stepwise resistive +transition is characteristic of lightly doped HTSC single +crystals, especially after quenching [31–34]. This is very +likely due to the fact that rapid quenching from 600 °C +also leads to a nonstoichiometric ratio of oxygen and va- +cancies, which, in turn, leads to the formation of separate +phases in the sample. These phases are characterized by +different oxygen content and its ordering and, accordingly, +have different Tc [31–33]. This is precisely what leads to +the experimentally observed stepwise superconducting +transitions (Fig. 2). The intersection of straight lines with +the x axis shows how Tc was determined. +From Fig. 2 it follows that Tc increases during annealing +from 63.6 K (S1) to 72.7 K (S3) (Table 1), while the width +of the resistive transition noticeably decreases by about +1.8 times. Accordingly, the two-step shape ρ(T) practically +disappears [Fig. 2(b)], which suggests the ordering of the +crystal structure due to the redistribution of oxygen, as +discussed above +The details of the superconducting transitions are best +seen in the temperature dependences of the derivatives +dρ(T)/dT obtained upon annealing at 20 °C, as was shown +in Refs. 32, 36. In this case, the SC transitions are charac- +terized by the low-temperature (LT) and high-temperature +(HT) dρ(T)/dT maxima, indicating the presence of two +superconducting phases with different Tc, as mentioned +above. Upon annealing, the low-temperature peak shifts +toward the high-temperature peak. After 120 h of anneal- +ing the LT peak becomes the highest and most uniform, +and the HT peak is practically suppressed. Moreover, the +distance between the peaks decreases as expected, which +indicates a decrease in the width of the SC transition dur- +ing annealing by about a factor of two (see Fig. 2) [32]. +Interestingly, high hydrostatic pressure also reduces the +resistance and increases Tc of lightly doped HoBCO single +crystals; however, in contrast to annealing, the width of the +superconducting transition increases in this case [34]. This +fact allows us to conclude that annealing, in contrast to +pressure, creates a different mechanism of oxygen redistri- +bution in a single crystal. +The effect of annealing on the parameters of lightly +doped samples ReВСО (Re = Y, Ho) is usually interpret- +ed by the ordering of oxygen atoms in the CuO2 plane +without changing the oxygen content in the sample (refer +to [31, 32] and references therein), as mentioned above. +However, an increase in Tc during annealing can be associ- +ated either with the local ordering of oxygen [37] or with +an increase in the density of charge carriers nf [38]. Another +reason for the increase in Tc can be a change in the parame- +ters of the crystal cell, for example, change in Сu–O and +Сu–Сu distances in the ab plane [39]. In turn, the electrical +resistivity decreases not only due to the ordering of oxygen +atoms during the holding of the sample at room tempera- +ture after quenching from a high temperature [32]. But, it +can also decrease due to an increase in nf or a decrease in +defects as a result of the ordering of oxygen vacancies +[31]. The possibility of increasing the charge carrier densi- +ty nf upon annealing is discussed below. +Thus, we can conclude that, strictly speaking, many as- +pects of lightly doped quenched HoBCO single crystals, +such as charge transfer and the nature of the redistribution +of the vacancy subsystem, especially with allowance for +Table 2. The pseudogap parameters of the HoBa2Cu3O6.65 single crystal +Sample +T*, K +α* +* +0 +cε + +C3D +A4 +D* +∆*(TG), K +∆*(Tmax), K +Tmax1, K +Tmax2, K +ΔTmax, K +S1 +256.5 +1.56 +0.64 +2.1 +26 +2.5 +156.1 +190.1 +199.7 +214.5 +14.8 +S2 +238 +1.4 +0.71 +1.35 +20 +2.5 +170.5 +221.9 +202.7 +217.5 +14.8 +S3 +235 +1.5 +0.67 +2.85 +40 +2.4 +189.3 +232.8 +200.2 +215.0 +14.8 + +Fig. 2. (Color online) Resistive transitions of single crystals of +HoBa2Cu3O6.65: (a) without annealing, sample S1 (turquoise dots), +(b) after annealing for five days (120 h), sample S3 (gray dots). + +Effects of annealing on the fluctuation conductivity and pseudogap in slightly doped HoBa2Cu3O7–δ single crystals +Low Temperature Physics/Fizyka Nyzkykh Temperatur, 2023, Vol. 49, No. 1 +119 +the high Ho magnetism, still remain undetermined. We +hoped to shed more light on this problem by studying the +effect of annealing on FLC and PS in lightly doped +quenched HoBCO single crystals. +3. Results and discussion +3.1 Fluctuation conductivity +As clearly seen from Fig. 1, below the PG opening tem- +peratures T* >> Tc the resistivity curves of HoBa2Cu3O6.65 +single crystal deviate downward from linear dependencies +at high temperatures. This resulting in appearance of the +excess conductivity + +( ) +( ) +( ) +( ) +( ) + +/ +N +N +T +T +T +T +T +′ +σ += ρ +−ρ +ρ +ρ + +  + + +  + , +(2) +as the difference between the experimentally measured +resistivity +( ) +T +ρ + and the linear resistivity of the normal +state +0 +( ) +N T +aT +ρ += ρ + +, extrapolated towards low tempe- +ratures. Here, +0 +ρ is the residual resistivity determined by +extrapolating +( ) +N T +ρ + towards 0 K, and +/ +a +d +dT += +ρ + is the +slope of the linear straight line. This procedure of the nor- +mal state resistivity determination is widely used in litera- +ture (see [2, 5, 26, 40, 41] and references therein) and has +been justified theoretically by the nearly antiferromagnetic +Fermi liquid (NAFL) model [8]. The T* temperature is taken +at the point where the experimental resistivity curve starts to +downturn from the high-temperature linear behavior (Fig. 1). +For a more accurate determination of T*, the criterion +0 +( +) +( ) – +/ +1 +T +aT +ρ +ρ += [2, 5, 40] was also used (see Insert in +Fig. 1). Both approaches give practically the same T*’s. +In the local pairs model, it is believed that PG and ex- +cess conductivity are due to the formation of LPs below +T* [3, 5, 14, 22]. The properties of the LPs are determined +by the coherence length along the c axis +( ) +c T +ξ += +1/2 +–1/2 +(0) +– +/ +(0) +[( +) +] +mf +mf +c +c +c +c +T +T +T +− += ξ += ξ +ε + [42, 43], where +) / +( +mf +mf +c +c +T +T +T +ε = +− + is the reduced temperature and +mf +c +T + is +the mean-field critical temperature, which separates the +range of SC fluctuations above Tc from the region of criti- +cal fluctuations around Tc, where the SC order parameter +Δ < kT [43, 44]. Hence, it is evident that the correct deter- +mination of +mf +c +T + is decisive in FLC and PG analysis. It is +well established [5, 23, 40, 45–47] that the small coherence +length in combination with the quasi-layered structure of +HTSCs leads to the formation of a noticeable, in compari- +son with conventional superconductors, range of SC fluc- +tuations, ΔTfl, above Tc. Near Tc, where +( ) +c T +ξ + > d = 11.68 Å +(d is the unit cell size along the c axis [48]), the LPs are +very large and interact throughout the entire volume of the +sample, forming a 3D state. In this case the FLC is always +described by the Aslamazov–Larkin equation for any 3D +system [49]: + +2 +1/2 +3 +3 +32 +(0) +DAL +D +c +e +C +h +− +′ +σ += +ε +ξ +, +(3) +where C3D is a numerical factor used to fit the data by the +theory [23, 44]. This means that the conventional 3D FLC +is realized in HTSCs as T → Tc [43, 50]. From Eq. (3), one +can easy obtain +2 ~ +~ ( +) / +mf +mf +c +c +T +T +T +− +σ′ +ε +− +. Obviously, +2 +΄ − +σ + = 0, when + +mf +c +T +T += +. This way of +mf +c +T + determination +was proposed by Beasley [44] and substantiated in various +FLC experiments [5, 26, 50]. Moreover, with the correct +choice of +mf +c +T +, the data in the three-dimensional fluctuation +region near Tc are always approximated by Eq. (3). +Figure 3 shows the +2 +΄ − +σ + vs T plot for samples S1 +[(a) turquoise dots] and S3 [(b) gray dots]. The interception +of the extrapolated linear +2 +΄ − +σ + with T axis determines +mf +c +T + = 70.65 K (S1) and = 74.35 K (S3). S2 (not shown) +demonstrates some intermediate +2 +΄ − +σ + on T dependence +with +mf +c +T + = 73.88 K given in Table 1. It should be noted, +that the revealed dependences of +2 +΄ − +σ + on T differ marked- +ly. When the measurements were carried out immediately +after quenching (S1), the separation between the LT and +HT phases is pronounced (see Fig. 2), and S1 exhibit a +dependence +2 +΄ − +σ + on T characteristic of most HTSCs. But +in this case, the LT phase, which is clearly seen in Fig. 2(a) +has virtually no effect on the definition of +mf +c +T +. However, +unlike Fig. 2(b), after annealing, both the LT and HT phas- +es plotted in these coordinates become quite pronounced. +But, fortunately and somewhat surprisingly, the approxi- +mation of both phases by straight lines gives the same +mf +c +T +. +Fig. 3. (Color online) Temperature dependences of the inverse +square of the excess conductivity +2( ) +΄ +T +− +σ + for sample S1 [(a), +turquoise dots] and S3 [(b), gray dots)]. The interception of the +extrapolated linear +2 +΄ − +σ + with T-axis determines +mf +cT +. Also shown +are Tc, the Ginsburg temperature TG and 3D–2D crossover tem- +perature T0. + +A. L. Solovjov, L. V. Omelchenko, E. V. Petrenko, et al. +120 +Low Temperature Physics/Fizyka Nyzkykh Temperatur, 2023, Vol. 49, No. 1 +Above the crossover temperature the data deviates right +from the line suggesting the 2D Maki–Thompson (MT) +[42, 51, 52] fluctuation contribution into FLC [23]. Obvi- +ously, at the crossover temperature +0 +0 +~ +T +ε the coherence +length +1/2 +0 +0 + +( +) +(0) +c +c +T +− +ξ += ξ +ε + is expected to amount to d +[9, 37] which yields + +0 +(0) +c +d +ξ += +ε +(4) +and allows the possibility of +(0) +c +ξ + determination. +(0) +c +ξ + +is one of the important parameters of the PG analysis. +Figure 4 shows the ln ′ +σ vs ln ε: (a) S1 (turquoise dots), +(b) S2 (yellow dots), and (c) S3 (gray dots) in comparison +with the fluctuation theories. As expected, all samples +demonstrate fairly good agreement with the AL theory +near Tc. For example, above the Ginzburg temperature +ln +( +5 3) +. +G +mf +c +G +T +T +> +ε += − + (refer to Fig. 3), down to which +the mean-field theory works [50], and up to T0 = 74.8 K +0 +(ln ε = –2.84) the data for sample S1 are well extrapola- +ted by the 3D fluctuation term (3) of the AL theory, +Fig. 4(a), solid red line with a slope –1/2) with +(0) +c +ξ + = += (2.82 ± 0.02) Å determined by Eq. (4) and C3D = 2.1 +(see Table 2). It should be noted that the same value of +(0) +c +ξ + was determined for a FeAs-based superconductor +ErFeAsO0.85F0.15 [53]. Samples S2 and S3 behave similarly +near Tc (Fig. 4). +Above T0, the data deviate sharply upwards from the +AL theory. This is due to the fact that at +0 +T +T +≥ +, where +( ) +c T +ξ + < d, the three-dimensional regime ends. However, it +is still +01 +( ) +c T +d +ξ +> +, which is the distance between conduct- +ing planes CuO2 [48], and +( ) +c T +ξ + connects the planes with +the Josephson interaction. This is a 2D fluctuation regime, +which is described by the MT term of the Hikani–Larkin +(HL) theory [42]: + +2 +1 +2 +1 +1 +1 +2 +ln ( / +) +8 +1 +/ +1 +1 +2 +MT +D +e +C +d +− + + ++ α + ++ α +′ +σ += +⋅ +⋅ +δ α ⋅ +ε + + + + +− α δ ++ δ + ++ δ + + + +. + + +(5) +These are the blue solid curves in the figure. In Eq. (5) +2 +1 +[ +(0) +] +2 +/ +c +d +− +α = +ξ +ε + is a coupling parameter, + +2 +(0) +16 +c +B +k T +h +d +ϕ +ξ + + +δ = β +τ + + +π  + + +(6) +is the pair-breaking parameter, and +ϕ +τ that is defined by +equation + +0 +0 +/ 8 +/ +B +T +h +k +A +ϕ +τ β += π +ε = +ε +(7) +is the phase relaxation time, where A = 2.998⋅1012 s·K. The +factor β = 1.203 (l/ξab), where l is the mean-free path and +ξab(0) is the coherence length in the ab plane considering +the clean limit approach l > ξ, which is always takes place +in HTSCs [5, 6, 26, 40, 43, 50]. +Above T01, indicated on all graphs as +01 +ln ε , the data of +all samples deviate definitively downward from the theory +(Fig. 4). Thus, T01 limits the range of SC fluctuations. +In this range, fluctuating pairs behave much like ordinary +Cooper pairs, but without long-range ordering (the so- +called short-range phase correlations [5, 22, 23, 44]). +Above T01, +( ) +c T +ξ + < d01 and LPs are confined within the +CuO2 planes, which are no longer connected by any cor- +relation interaction. Thus, it is clear that +01 +01 +( +) +c T +d +ξ += +. To +estimate +01 +d , we use the condition +( ) +0 +0 +c +d +ξ += +ε += +01 +01 +(2.82 +0.02) +d += +ε += +± + Å (S1). Since d = c = 11.68 Å and +01 +ln ε ≈ −0.63 ( 01 +ε = 0.532 and T01 ≈ 108.2 K), we obtain: +Fig. 4. (Color online) ln σ′ vs ln ε: (a) S1 (turquoise dots), (b) S2 (yel- +low dots), and (c) S3 (gray dots) compared with fluctuation theories: +3D-AL — red lines; 2D-MT — blue curves. Δln (σ′) designates the +maximal deviation of the data from the extrapolated 3D-AL lines. + +Effects of annealing on the fluctuation conductivity and pseudogap in slightly doped HoBa2Cu3O7–δ single crystals +Low Temperature Physics/Fizyka Nyzkykh Temperatur, 2023, Vol. 49, No. 1 +121 +01 +0 +01 +(3.87 +0.05) +d +d += +ε +ε += +± + Å for S1 in good agree- +ment with results of the structural studies [48]. Having +carried out a similar analysis for other samples, we obtain +the values of +(0) +c +ξ + and d01 for S2 and S3 (Table 1). Inter- +estingly, in contrast to S1 and S2, in the case of S3, both +the LT and HT phases are clearly visible on the plot of +ln ′ +σ vs ln ε below T0, but both fully follow the AL theo- +ry. However, to keep the logic with S1, we determined T0 +and other parameters from the high temperature phase. +Strictly speaking, extrapolation of 2D-MT data above +T0 is not entirely successful. This is due to the fact that +above T0 there is a sharp increase in data, leading to the ap- +pearance of enhanced 2D fluctuations. As a result, the max- +imal deviation of the data above the extrapolated 3D-AL +line Δ ln σ′ = 0.68 obtained for S1 [Fig. 4(a)] is approxi- +mately 3.4 times greater than that observed for YBCO, +where magnetism is not expected [5, 23, 47]. This enhanced +behavior of 2D fluctuation is typical of FeSe-based super- +conductors such as ErFeAsO0.85F0.15 [53] and SmFeAsO0.85 +[54], as well as superconductors with magnetic impurities +such as YBCO–PrBCO superlattices [50], suggesting a no- +ticeable influence of own HoBCO magnetism in our case. +The highest value Δ ln σ′ = 1.0, which is approximately +5 times greater than that observed for YBCO, was obtained +for S2 [Fig. 4(b)]. In this case, fitting the data by the MT +theory is completely impossible. However, to provide a +more informative analysis, we used the found fitting pa- +rameters (ξc(0), ε0, ε01) to derive theoretical 2D-MT curve +that would intersect the red 3D-AL line at ln ε0 (corre- +sponding temperature T0) and the 2D-data at ln ε01 (corre- +sponding temperature T01) [Fig. 4(b)]. We emphasize that, +despite the unsatisfactory description of the 2D data, all +temperatures T01 found in this way (indicated in the figure +as ln ε01) clearly correspond to the minima on the tempera- +ture dependences of the PG parameter Δ*(T), which fol- +lows from the theory (see Fig. 7), thereby confirming the +correctness of our analysis. The enhanced 2D fluctuations +found for S2 are reminiscent of these observed for magnet- +ic superconductors such as Dy0.6Y0.4Rh3.85Ru0.15B4, which +have an intrinsic magnetic moment μ ≈ 6.2μB per Dy3+ ion +[55]. Taking all above facts into account, we can conclude +that the observed anomalous 2D fluctuations in sample S2 +is most likely caused by the noncompensated magnetic +moments of Ho, which is thought to be responsible for +interplay between magnetic interaction and superconduc- +tivity [5, 26]. However, since the resistivity noticeably +decreases upon annealing, it can be concluded that mag- +netic interaction does not strongly affect the rate of +charge carrier scattering in HoBCO single crystals. At the +same time, S2 has the lowest values of ξc(0), d01 (Table 1) and +unexpectedly C3D = 1.35, confirming a strong influence of +oxygen diffusion on the sample structure [31, 32]. Recall +that the smaller the C3D, the smaller the effect of defects in +the sample, which is directly related to the decrease in re- +sistivity. +In turn, after five days of annealing, S3 demonstrates +the lowest value Δ ln σ′ = 0.6 and the shape of ln σ′ vs ln ε +resembling S1, except for the low temperature 3D-AL re- +gion [Fig. 4(c)]. This indicates the final ordering of defects +and the crystal structure as a whole, which leads to the +lowest resistivity (Fig. 1 and Table 1). It can also be as- +sumed that ordered oxygen somehow shielded the influ- +ence of Ho magnetic moments. However, despite the sup- +posed ordering of defects and oxygen, the values of ξc(0), +d01 (Table 1) and C3D = 2.85 are practically the same as in +S1, which indicates a nonmonotonic, in contrast to the re- +sistivity, effect of defects and magnetism on the FLC of the +sample upon annealing. We expected to obtain confirma- +tion or refutation of this conclusion by analyzing the +change in the temperature dependence of the pseudogap +during annealing. +3.2. Pseudogap analysis + As mentioned above, the number of the different non- +Fermi-liquid models proposed to explain the physics of PG +is quite large. However, a very large number of models +raises doubts about their correctness. In addition, none of +the mentioned models gives explicitly the temperature de- +pendence of PG, which could be verified experimentally. +Clearly, to attain information about the pseudogap we need +an equation which specifies a whole experimental curve, +from TG up to T*, and contains the PG parameter Δ*(T) in +an explicit form. The issue was resolved within the frame- +work of our LP model [23, 56], in which such an equation +was proposed for σ'(ε): + +* +2 +* +4 +0 +* +0 +1 +exp +( ) +16 +(0) 2 +sinh 2 +c +c +c +T +e +T +T +T +A + + +∆ + + +− +− + + + + + + + + +′ +σ += + + +ε +ξ +ε + + +ε + + + +, +(8) +where, for a correct description of the experiment, the dy- +namics of pair formation (1 – T/T*) and pair breaking +(exp(‒Δ*/T)) above Tc are taken into account. Solving +Eq. (8) for the pseudogap Δ*(T) one can readily obtain: +2 +* +4 +* +0 +*0 +1 +1 +( ) +ln +1 +. +( ) 16 +(0) +2 +sinh 2 +c +c +c +T +e +T +T +A +T + + + + + + + + +∆ += +− + + + + +′ +σ ε +ξ + + + + + + +ε +ε + + + + +ε + + + + + + + + +(9) +Here +( ) +′ +σ ε is the experimentally measured excess con- +ductivity over the whole temperature interval from T* +down to TG, and A4 is a numerical factor that has the +meaning of the C factor in the fluctuation conductivity +theory [42–44, 51, 56]. All other parameters, including +the coherence length along the c axis, +(0) +c +ξ +, [Eq. (4)] and +the theoretical parameter *0 +c +ε + [57], directly come from the + +A. L. Solovjov, L. V. Omelchenko, E. V. Petrenko, et al. +122 +Low Temperature Physics/Fizyka Nyzkykh Temperatur, 2023, Vol. 49, No. 1 +experiment [5, 23, 50, 54–56]. To find +*0 +c +ε + we use the +experimental fact that in some temperature range above +T01, namely +01 +02 +ln +ln +ln +c +c +ε +< +ε < +ε + (Fig. 5) or accordingly +01 +02 +c +c +ε +< ε < ε + (Insert in Fig. 5), +1 ~ e p( ) +x +−′ +σ +ε [23, 56, 57]. +As a result +1 +ln ( +) +−′ +σ + is a linear function of ε with a slope +* +α = 1.56 which determines parameter +* +*0 +1/ +cε += +α = 0.64 +for S1 (Insert in Fig. 5). To find A4, we calculate +( ) +′ +σ ε +from T* and down to TG using Eq. (8) and fit experiment in +the range of 3D-AL fluctuations near Tc (Fig. 5, red curve) +where ln ′ +σ on ln ε is a linear function of the reduced tem- +perature ε with a slope λ= ‒ 1/2. Besides, +*( +) +(0) +G +T +∆ += ∆ + is +assumed [58, 59]. To estimate +*( +) +G +T +∆ +, which we use in +Eq. (8), we plot ln ′ +σ as a function of 1/T for S1 (Fig. 6) +and S3 (Insert to Fig. 6). After annealing (Insert to Fig. 6) the +approximation, as expected, looks better, due to the ordering +of defects. In this case the slope of the theoretical curve +[Eq. (8)] turns out to be very sensitive to the value of +*( +) +G +T +∆ + [23, 50, 56]. For sample S1 the best fit is obtained +when +* +* +) +2 +/ +5 +( G +B c +D +T +k T += ∆ += and D* = 4.8 for S3 (Table 2). +Note, that D* = 5 is the typical value for cuprates [5, 60]. +Having determined all the necessary parameters (refer to +Tables 1, 2) we succeeded to plot the temperature depen- +dences PG, +*( ) +T +∆ + for all stages of annealing. Fig. 7(a) (tur- +quoise dots) displays +*( ) +T +∆ + for S1 calculated using Eq. (9) +with the following set of parameters derived from the exper- +iment within the LP model: T* = 256.5 K, +mf +c +T + = 70.65 K, +(0) +c +ξ + = 2.82 Å, *0 +c +ε + = 0.64, A4 = 26. The resulting form of +*( ) +T +∆ + with a high-temperature maximum at Tmax = 239.5 K, +followed by a section of the linear dependence +*( ) +T +∆ + +with a moderate positive slope αmax = 1.16 ± 0.01, is typical +of a lightly doped HTSC single crystals, containing various +defects, including tweens [61] (and references therein). +This confirms our assumption made above about numerous +defects in the quenched crystal. The low-temperature be- +havior of +*( ) +T +∆ + in S1 with minimum at T01, maximum at +about T0 and final minimum at TG is also characteristic of +all HTSCs (see Fig. 12 in [50]). The range of SC fluctua- +tions +fl +01 +G +T +T +T +∆ += +− + = 37.2 K is large but comparable with +fl +T +∆ + obtained for slightly doped YBCO single crystals +with TBs [61]. The only peculiarities are two small max- +ima at Tmax1 = 199.7 K and Tmax2 = 214.5 K which is a +feature of the PG behavior found only on HoBCO single +crystals [26, 34]. +Dependences +*( ) +T +∆ + constructed for the samples S2 and +S3 with the corresponding sets of parameters given in the +Tables, are displayed in Figs. 7(b) and 7(c), respectively. It +can be seen that the shape of +*( ) +T +∆ + noticeably changed +upon annealing. The linear section with a moderate posi- +tive slope disappeared, but maxima at Tmax1 and Tmax2 be- +came much more pronounced. It is very tempting to attrib- +ute these maxima to two phases with different Tc observed +at resistive transitions (Figs. 2 and 3). But upon annealing, +the separation into two phases gradually disappears, +whereas, despite a significant change in all parameters of +the sample during annealing, the distance between these +maxima remains constant at ΔTmax = 14.8 K (Table 2). +There have also been attempts to relate these maxima to a +process of the so-called ascending diffusion, which is be- +lieved to increase the separation of charge carriers between +tweens and TBS [34] (and references therein). But the as- +cending diffusion also changes during the annealing, +whereas the distance between these maxima remains con- +stant, as discussed above. We believe that these maxima are +somehow connected with enhanced magnetism of HoBCO, +but, strictly speaking, the appearance of these unusual +maxima is still in question. +Fig. 5. (Color online) ln ′ +σ vs lnε for S1 (turquoise dots) plotted +in the whole temperature range from T* down to TG. The solid +red curve is a fit to the data with Eq. (8). Insert: +1 +ln +− +σ as a func- +tion of ε. Solid line indicates the linear part of the curve between +01 +cε + = 0.20 and +02 +cε + = 1.34. Corresponding +01 +ln +cε + = − 1.59 and +02 +ln +cε + = − 0.29 are marked by arrows in the main panel. The slope +* +α = 1.6 determines the parameter * +* +0 + 1/ +cε += +α = 0.64 (Table 2). +Fig. 6. (Color online) ln σ vs 1/T for S1 (turquoise dots) and S3 +(Insert, gray dots) plotted in the whole temperature range from T* +down to TG . The red solid curves are fits to the data with Eq. (8). +The best fit is obtained when Eq. (8) is calculated with +* +* +) +2 +/ +5 +( G +B c +D +T +k T += ∆ += for S1 and D* = 4.8 for S3 (Table 2). + +Effects of annealing on the fluctuation conductivity and pseudogap in slightly doped HoBa2Cu3O7–δ single crystals +Low Temperature Physics/Fizyka Nyzkykh Temperatur, 2023, Vol. 49, No. 1 +123 +As seen in Fig. 7(b), S2 demonstrates a rather specific +*( ) +T +∆ + with pronounced wide minimum, as expected, at +T01 ≈ 125.3 K. This leads to an anomalously large range of +SC fluctuations, +fl +01 +G +T +T +T +∆ += +− +, about 50 K above Tc. In +addition, the two maxima look more pronounced and shift- +ed slightly towards higher temperatures. And, more im- +portantly, the data slope at high temperature is about 3.5 +times steeper than that of the S1. Taking into account the +results of the study of the FLC [Fig. 4(b)], we associate +this form of +*( ) +T +∆ + with an increased magnetism of the +uncompensated magnetic moments Ho. In turn, S3 [Fig. 7(c)] +demonstrates +*( ) +T +∆ + characteristic of HoBCO single crys- +tals with ordered defects [26, 34], which allows us to con- +clude that after five days of annealing, oxygen diffusion +almost ceased. Indeed, the minimum at T01 = 113.4 K is +noticeably smaller, and the high-temperature maxima are +not as pronounced as for S2. However, the range of SC +fluctuation +fl +01 +G +T +T +T +∆ += +− + ≈ 38 K and the temperatures of +these maxima in this case are almost the same as for the +quenched sample S1 [Fig. 7(a)]. This allows us to draw the +following conclusion that ordered oxygen somewhat +shields the magnetic interaction in the crystal. On the other +hand, in general, the shape of the dependence +*( ) +T +∆ + dif- +fers markedly from the shape of S1. The value of +*( +) +G +T +∆ +, +which gradually increases during annealing, reaches the +maximum value of 189.3 K for S3 (Table 2). In addition, +the minimum at T01 is still quite pronounced and, more im- +portantly, the slope of the data at high temperature αmax = 4.0, +marked in the figure by the red line, is the same as for S2. +Here we would like to emphasize that the same data +slope at high temperatures is observed for all magnetic su- +perconductors, including FeAs-based compounds (see [50], +Fig. 11). This is confirmed by Fig. 8, where we compare +our data for sample S3 with results obtained for the highly +magnetic superlattice 7YBCO×14PrBCO (sample SL3) +from Ref. 50. It can be seen that the slope at high T is ex- +actly the same. Moreover, the shape of both curves below +T01 is also almost the same. Interestingly, in 1111 FeAs-based +superconductors the maximum corresponds to the structur- +al transition from a tetragonal to an orthorhombic phase. +Accordingly, the temperature of data deviation from the +linear dependence corresponds to transition to the AF state +Fig. 7. (Color online) Temperature dependences of pseudogap +*( ) +T +∆ + (a) S1 (turquoise dots), (b) S2 (yellow dots), and (c) S3 +(gray dots), analyzed with Eq. (9). All characteristic temperatures +are marked with arrows. The red lines designate the data slope at +high temperatures, which is equal for S2 and S3. Solid black lines +are a guide for the eyes. +Fig. 8. (Color online) +* +* +max +/ +( ) +T +∆ +∆ + as a function of T/T* for stu- +died single crystals of HoBa2Cu3O6.65 after annealing for five +days (120 h), sample S3, and superlattice 7YBCO×14PrBCO, +sample SL3. All characteristic temperatures are marked with +arrows. The red lines designate the slope of the data at high tem- +peratures, which is the same for both samples. Solid black lines +are a guide for the eyes. + +A. L. Solovjov, L. V. Omelchenko, E. V. Petrenko, et al. +124 +Low Temperature Physics/Fizyka Nyzkykh Temperatur, 2023, Vol. 49, No. 1 +of spin–density–waves (SDW) [62–65]. Thus, an important +conclusion can be drawn that the AF interaction of the SDW +type should take place in lightly doped HoBCO single crys- +tals below the Tmax1 (Fig. 7) due to the large intrinsic mag- +netism of Ho. This interaction is believed to be responsible +for the formation of both PG and SDW state in such com- +pounds. The possibility of SDW state in lightly doped +YBCO compounds is discussed in Ref. 17. Thus, returning +to the question of possible models of SC pairing in HTSCs, +we can assume that the SDW model is the most probable +one, at least for HTSCs with a strong magnetic interaction. +To clarify the question of a possible increase in the den- +sity of charge carriers in a crystal during annealing, we +compare the pseudogap parameter +* +* +max +( ) / +T +∆ +∆ + of samples +S1, S2, and S3 near Tc with the Peters–Bauer (PB) theory +[3] (Fig. 9). In [3], the temperature dependences of the +local pairs density in HTSCs were theoretically +calculated within the framework of the three-dimensional +attractive Hubbard model for different temperatures T/W, +interactions U/W, and filling factor, where U is the activa- +tion energy and W is the band width. The shape of +*( ) +T +∆ + +for all cuprates, with a maximum near T0 followed by a +minimum at TG [50] (see Fig. 12 in [50]) and Fig. 7 in [66]), +resembles the shape of theoretical curves at low T/W +and U/W [3]. This fact should justify such an approach. +To carry out the analysis, we combine the measured +values of +* +* +max +( ) / +T +∆ +∆ + for S1 at ТG with the minimum, and +at T0 with the maximum of each theoretical curve calculat- +ed at different values of U/W, thus achieving the best +agreement between the experiment and theory in the wid- +est possible temperature range. It is important that the fit- +ting factors found for S1 are also used for the other two +samples [67, 68]. The fitting results for all three samples are +shown in Fig. 9. The best fit for S1 near Tc is obtained with +U/W = 0.2 curve indicating that in this case ≈ 0.3, +which is a typical value for various HTSCs [67, 68]. Fur- +ther, it was taken into account that +* +* +max +( +) / +G +T +∆ +∆ + = 0.82 for +S1, where +* +max +∆ + is taken at Tmax = 239.5 K, which is clearly +seen in Fig. 7(a). Unfortunately, due to the specific form of +*( ) +T +∆ + found for S2 and S3, leading to ambiguity in the +definition of Tmax, it was not possible to obtain reasonable +values of +* +* +max +( +) / +G +T +∆ +∆ + in this case. Therefore, we could +not compare them with that found for S1 in order to obtain +the corresponding fitting coefficients, as we did in our pre- +vious works [67, 68]. + However, both TG and +*( +) +G +T +∆ + are clearly defined for +all the studied samples (Fig. 7). Moreover, +*( +) +G +T +∆ + notice- +ably increases upon annealing (Table 2), most likely due to +an increase in the charge carrier density nf. This fact sug- +gests that nf must be proportional to the value of +*( +) +G +T +∆ + +[5, 23]. Taking into account that found for S1 +*( +) +G +T +∆ + cor- +responds to = 0.30195, the simple algebra yields: + (S2) = [ +* +* +, S2 / +( +) +( +) +, S1 +G +G +T +T +∆ +∆ +] +× (S1) = (170.5×0.30195)/156.1 ≈ 0.33, and + (S3) = +* +* +( +) +[ +, S3 / +( + 1) +, S +G +G +T +T += ∆ +∆ +] +× (S1) = (189.3×0.30195)/156.1 ≈ 0.366, +which gives the corresponding curves at the figure. This +means that the density of charge carriers in HoBCO single +crystals somewhat increases due to oxygen diffusion dur- +ing annealing, as it was assumed in Refs. 69, 70. It is very +likely that the observed slight increase in nf is quite suffi- +cient to explain the observed increase in Tc by ~ 9 K. This +may also be responsible to some extent for the observed +decrease in ρ(T) (Table 1). +It is also worth noting that the best agreement with the +PB theory among HTSCs in a wide temperature range was +obtained for non-twinned optimally doped YBCO single +crystals, naturally, without any magnetism [67]. In the pre- +sent case the data noticeably deviate downward from the +theoretical curves with increasing temperature, which is +most likely due to the enhancement of the magnetic interac- +tion in HoBCO. This conclusion is confirmed by the follow- +ing results. The quenched sample S1 shows the smallest +deviation, since the magnetic moments Ho are considered to +be randomly distributed due to multiple defects. The sample +S2 shows the largest deviation, as a result of influence of the +uncompensated magnetic moments, as discussed above. +In the case of sample S3, the magnetic interaction is some- +how compensated by the ordering of the distribution of oxy- +gen and crystal structure defects [31, 32]. As a result, despite +the rather complicated shape of the +* +* +max +( +) / +G +T +∆ +∆ + curve at +low T, the deviation from the theory is expectedly moderate +(Fig. 9). Moreover, above (T/W, T/T*) ≈ 0.25, the experi- +mental data deviate upward from the theory, which confirms +Fig. 9. (Color online) Curves of +* +* +max +/ +∆ ∆ + as functions T/T* for +samples S1 (turquoise dots), S2 (yellow dots), and S3 (gray +dots) in comparison with the theoretical curves of as functions +of T/W, at the corresponding interaction values U/W: 0.2 (black +curve), 0.4 (red curve). The arrows indicate the temperatures +T0 (▲) and TG (▼). + +Effects of annealing on the fluctuation conductivity and pseudogap in slightly doped HoBa2Cu3O7–δ single crystals +Low Temperature Physics/Fizyka Nyzkykh Temperatur, 2023, Vol. 49, No. 1 +125 +a fundamentally different mechanism of magnetic interac- +tion in the HoBCO single crystal after five days of anneal- +ing at room temperature. +Conclusion +The magnitude and temperature dependence of fluctua- +tion conductivity and pseudogap +*( ) +T +∆ + in lightly doped +HoBa2Cu3O7–δ single crystals rapidly quenched from 600 °С +were studied for the first time at different stages of anneal- +ing at room temperature. During annealing, a significant +decrease in the resistance of the samples, the width of re- +sistive transitions, and an increase in Tc were observed. +These observations are consistent with the processes of the +oxygen diffusion and structural relaxation in the volume of +experimental samples, leading to the appearance of phase +separation. However, in addition to the expected change in +oxygen distribution, several new interesting results were +revealed. +At all stages of annealing, the FLC near Tc is well de- +scribed by the 3D Aslamazov–Larkin fluctuation theory. +However, at the intermediate stage of annealing (sample S2), +an anomalous increase in 2D FLC was revealed, which is +associated with the influence of uncompensated magnetic +moments in HoBa2Cu3O7–δ, since μeff, Ho = 9.7 μB. As a result, +in this case, the 2D Maki–Thompson fluctuation theory +failed to describe the data. However, after five days of an- +nealing, the 2D-MT fit improved, since the magnetic inter- +action is somehow compensated by the ordering of the +distribution of oxygen and crystal structure defects. +For the quenched sample S1, the temperature depend- +ence of the PG has a shape typical of single crystals with a +large number of defects. However, +*( ) +T +∆ + has two small +additional maxima at high temperature, which is a feature +of HoBa2Cu3O7–δ single crystals with pronounced twins and +indicates the two-phase nature of the sample. Upon anneal- +ing, the shape of +*( ) +T +∆ + noticeably changes, very likely due +to an increase in the magnetic interaction (sample S2). +The two additional peaks became more pronounced. But, +more important is the change in the slope αmax of the data +at high temperatures, which has become about 3.5 times +steeper. The ordering of the oxygen distribution due to the +diffusion process during annealing somewhat compensates +for the influence of magnetic interaction. But the slope +does not change (sample S3). Moreover, the slope turns out +to be the same as for FeAs-based superconductors, sug- +gesting the possibility of the existence of spin density +waves in HoBa2Cu3O7–δ in the PG state. The comparison of +the pseudogap parameter +* +* +max +( +) / +G +T +∆ +∆ + near Tc with the +Peters–Bauer theory revealed a slight increase in the densi- +ty of local pairs , which should explain the ob- +served increase in Tc by 9 K during annealing. Interesting- +ly, despite the rather complicated shape of the +* +* +max +( +) / +G +T +∆ +∆ + +curve at low T (sample S3), the deviation from the PB the- +ory is expectedly moderate (Fig. 9). Moreover, above +(T/W, T/T*) ≈ 0.25, the experimental data deviate upward +from the theory, which confirms a fundamentally different, +compare with S1 and S2, mechanism of magnetic interac- +tion in the HoBCO single crystal after five days of anneal- +ing. Thus, the studies of FLC and PG turned out to be very +informative and made it possible to obtain new results, +which, in turn, are definitely a consequence of the expected +rearrangement of the oxygen distribution and the defect +structure during annealing at room temperature. +Acknowledgments +We thank support from the National Academy of Sci- +ences of Ukraine through Young Scientists Grant No. 1/N- +2021 (L. V. O. and E. V. P.). We acknowledge support +from the Ministry of Innovative Development of the Re- +public of Uzbekistan through Grant No. F-FА-2021-433 +(A. L. S., S. D., and R. V. V.). A. L. 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Solovjov, E. V. Petrenko, L. V. Omelchenko, R. V. +Vovk, I. L. Goulatis, and A. Chroneos, Sci. Rep. 9, 9274 +(2019). +68. A. L. Solovjov, E. V. Petrenko, L. V. Omelchenko, E. Nazarova, +K. Buchkov, and K. Rogacki, Fiz. Nizk. Temp. 46, 638 +(2020) [Low Temp. Phys. 46, 538 (2020)]. +69. J. Kircher, M. Cardona, A. Zibold, K. Widder, and H. P. +Getherich, Phys. Rev. B 48, 9684 (1993). +70. K. Widder, A. Zibold, M. Merz, H. P. Getherich, A. Erb, and +G. Müller-Vogt, Physica C 232, 82, (1994). + ___________________________ +Вплив відпалу на флуктуаційну провідність +та псевдощілину у слаболегованих монокристалах +HoBa2Cu3O7-δ +A. L. Solovjov, L. V. Omelchenko, E. V. Petrenko, +Yu. A. Kolesnichenko, A. S. Kolesnik, S. Dzhumanov, +R. V. Vovk +Вивчено вплив відпалу при кімнатній температурі на +флуктуаційну провідність (ФЛП) σ′(T) і псевдощілину (ПЩ) +Δ*(T) у базисній площині ab монокристалів ReBa2Cu3O7–δ +(Re = Ho) з нестачею кисню. Показано, що на всіх етапах +відпалу ФЛП поблизу Tc можна описати флуктуаційними +теоріями Асламазова–Ларкіна та Макі–Томпсона, де спосте- +рігається 3D–2D кросовер із підвищенням температури. +За температурою кросовера Т0 визначено довжину когерент- +ності вздовж осі c — ξс(0) = (2,82 ± 0,2) Å. На проміжному +етапі відпалу виявлено аномальне зростання 2D ФЛП, що +пов’язано з впливом некомпенсованих магнітних моментів +у HoBa2Cu3O7–δ (HoBCO): µeff, Ho = 9,7µB. Для загартованого +зразка S1 температурна залежність ПЩ має форму, типову +для монокристалів з великою кількістю дефектів. Проте Δ*(T) +має два невеликі додаткові максимуми при високих темпера- +турах, що є особливістю монокристалів HoBCO з виражени- +ми двійниками та вказує на двофазність зразка. Під час від- +палу форма Δ *(T) помітно змінюється, ймовірно, за рахунок +збільшення магнітної взаємодії (зразок S2). Більш важливою +є зміна нахилу даних при високих температурах, який став +приблизно в 3,5 рази крутішим. Упорядкування розподілу +кисню за рахунок процесу дифузії під час відпалу дещо ком- +пенсує вплив магнітної взаємодії, проте нахил не змінюється +(зразок S3). Цікаво, що нахил виявляється таким же, як і для +надпровідників на основі FeAs, що свідчить про можливість +існування хвиль спінової щільності в HoBCO в ПЩ стані. +Порівняння псевдощілинного параметра Δ*(T)/Δ* +max поблизу +Tc з теорією Пітерса–Бауера виявило незначне збільшення +щільності локальних пар , що має пояснювати спосте- +режене підвищення Tc на 9 K під час відпалу. +Ключові слова: флуктуаційна провідність, псевдощілина, +надлишкова провідність, відпал, монокрис- +тали HoBCO. + + diff --git a/YNE0T4oBgHgl3EQfmgH3/content/tmp_files/load_file.txt b/YNE0T4oBgHgl3EQfmgH3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..30f88f3789a6a91a47ff05e17e76584d49bddf7b --- /dev/null +++ b/YNE0T4oBgHgl3EQfmgH3/content/tmp_files/load_file.txt @@ -0,0 +1,1491 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf,len=1490 +page_content='© A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Kolesnichenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Kolesnik, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Dzhumanov, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Vovk, 2023 Low Temperature Physics/Fizyka Nyzkykh Temperatur, 2023, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 49, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 115–127 Effects of annealing on the fluctuation conductivity and pseudogap in slightly doped HoBa2Cu3O7–δ single crystals A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Solovjov1,2,3, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Omelchenko1, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Petrenko1, Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Kolesnichenko1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Kolesnik1, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Dzhumanov4, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Vovk3 1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Verkin Institute for Low Temperatures Physics and Engineering of the National Academy of Sciences of Ukraine Kharkiv 61103, Ukraine 2Institute for Low Temperatures and Structure Research, Polish Academy of Sciences, Wroclaw 50-422, Poland 3The Faculty of Physics, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Karazin Kharkiv National University, Kharkiv 61022, Ukraine 4Institute of Nuclear Physics, Uzbek Academy of Sciences, Ulugbek, Tashkent 100214, Uzbekistan E-mail: solovjov@ilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='kharkov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='ua Received September 11, 2022, published online November 21, 2022 The effect of annealing at room temperature on the fluctuation conductivity (FLC) σ′(T) and pseudogap (PG) Δ*(T) in the basal ab plane of ReBa2Cu3O7–δ (Re = Ho) single crystals with a lack of oxygen has been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' It is shown that at all stages of annealing, the FLC near Tc can be described by the Aslamazov–Larkin and Maki– Thompson fluctuation theories, demonstrating a 3D–2D crossover with increasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The crossover temperature Т0 was used to determine the coherence length along the с axis, ξс(0) = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='2) Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' At the inter- mediate stage of annealing, an anomalous increase in 2D FLC was revealed, which is associated with the influ- ence of uncompensated magnetic moments in HoBa2Cu3O7–δ (HoBCO): μeff, Ho = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='7μB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' For the quenched sample S1, the temperature dependence of the PG has a shape typical of single crystals with a large number of defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' However, Δ*(T) has two small additional maxima at high temperature, which is a feature of HoBCO single crys- tals with pronounced twins and indicates the two-phase nature of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Upon annealing, the shape of Δ*(T) noticeably changes, very likely due to an increase in the magnetic interaction (sample S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' More important is the change in the slope of the data at high temperatures, which has become about 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='5 times steeper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The ordering of the oxygen distribution due to the diffusion process during annealing somewhat compensates for the influence of magnetic interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' But the slope does not change (sample S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Interestingly, the slope turns out to be the same as for FeAs-based superconductors, suggesting the possibility of the existence of spin density waves in HoBCO in the PG state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The comparison of the pseudogap parameter Δ*(T)/Δ* max near Tc with the Peters–Bauer theory revealed a slight increase in the density of local pairs , which should explain the observed increase in Tc by 9 K during annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Keywords: fluctuating conductivity, pseudogap, excess conductivity, annealing, HoBCO single crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Introduction In modern solid state physics, one of the urgent prob- lems is the construction of a theory of high-temperature superconductors (HTSCs), which is necessary to elucidate the possibility of creating new superconductors with even higher, preferably room, critical temperatures Tc of transi- tion to the superconducting (SC) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The solution of this problem is complicated by the lack of a clear understand- ing of the physics of internal interactions in such multicom- ponent compounds as HTSCs, in particular, the mechanism of SC pairing [1], which makes it possible to have a very high critical temperature of the SC transition [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' It is be- lieved that the answer to the question of SC pairing, as well as the possible role of the interplay between super- conductivity and magnetism in the formation of paired fermions at temperatures above 100 K [2] in HTSCs, can be obtained by studying such an interesting phenomenon as a pseudogap (PG), which opens in cuprate HTSCs, type ReBa2Cu3O7–δ (Re = Y, Ho, Gd, Pr) at temperature T* >> Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' PG is a special state of matter, which is characterized by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Solovjov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Omelchenko, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Petrenko, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 116 Low Temperature Physics/Fizyka Nyzkykh Temperatur, 2023, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 49, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 1 a reduced (but not to zero) density of electronic states at the Fermi level and a probable transformation of the Fermi sur- face below T* [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' However, the physics of PG and its role in the formation of coupled electrons (fluctuating Cooper pairs) (FKPs) above Tc are still unclear, despite the huge number of theoretical and experimental works devoted to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' It has long been established that the behavior of HTSCs in the normal state goes far beyond the standard Fermi liquid approach [6–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' As a result, a large number of non- Fermi liquid models [10–12] as well as marginal Fermi liquid models [13] have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' All these models largely explain various specific aspects of the behavior of cuprates observed in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' However, until now there is no unified theory that would be able to describe all the features of the behavior of HTSCs in the PG state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Re- cently, there has been a noticeable increase in interest in studying the problem of PG in HTSCs [3, 14–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' More- over, in addition to the already mentioned spin fluctuations [1, 7–9], spin density waves (SDW) [12, 17], charge order- ing (CO) [1, 17], charge density waves (CDW) [15–18] as well as pair density waves (PDW) [19, 20] are proposed to explain the PG physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' We share a different approach to the problem of the PG appearance, which assumes the pos- sibility of the formation of paired fermions, the so-called local pairs (LPs), in HTSCs below the PG opening temper- ature T* >> Tc [5, 21–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Moreover, it can be assumed that SDW, CDW, CO, and PDW can be considered as different possible mechanisms of LP pairing in cuprates in the PG state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' One of the most interesting materials for studying PG are ReBa2Cu3O7–δ compounds, which is due to the possibil- ity of wide variation of their composition by replacing yt- trium with its isoelectronic analogues, or by changing the degree of oxygen nonstoichiometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' As is known, in the ground state all ReBa2Cu3O7–δ are Mott dielectrics with a long-range antiferromagnetic (AF) order, in which the elec- tron spins S = 1/2 are localized on copper ions Cu2+ [6, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The dielectric state is a consequence of strong electronic (Hubbard) correlations [3, 4] (and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' When charge carriers (holes) appear during doping, the long-range AF order is rapidly violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' However, as shown by neutron experiments with YBCO samples of different doping levels less than optimal, well-developed short-range order AF fluctuations [25] (and references therein) are re- tained in the normal metal phase of cuprates, which are ob- served up to very high doping levels (Tc ~ 85 K ) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Taking into account the presence of AF correlations in cuprates, of particular interest are compounds with partial replacement of yttrium (Y) by praseodymium (PrBCO with a magnetic moment μeff ≈ 2μB) and especially with com- plete replacement by holmium, HoBa2Cu3O7−δ (HoBCO), which has a magnetic moment μeff ≈ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='7μB, due to magnetic moment of pure Ho which is μHo ≈ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='6μB [5, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' However, PrBCO, being a magnetic dielectric, where all electrons are localized in the Fehrenbacher–Rice zone [27], quickly suppresses superconductivity in YBCO — the so-called “praseodymium anomaly”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Whereas, HoBCO shows al- most the same high Tc values as YBCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Substitution of Y in single crystals of ReBa2Cu3O7–δ (Re = Y, Ho, Dy, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=') with Ho of a fairly large magnetic moment suggests a change in behavior of the system due to paramagnetic properties of HoBa2Cu3O7–δ in its normal state [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Samples exhibiting oxygen nonstoichiometry are of special interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' In this state the redistribution of labile oxygen and structural relaxation take place simulta- neously, thus significantly affecting electron transport pa- rameters of the system [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' In this case rare alkaline earth metal ion can serve as a sensor, sensitive to local symmetry of its surroundings and charge density distribu- tion, which can be affected by outside factors such as tem- perature [29], high pressure [30], or room-temperature an- nealing [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' To study the physical nature of the interaction between superconductivity and magnetism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' single crystals were chosen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' which have the advantage that their properties can noticeably change either during annealing of samples in an oxygen atmosphere due to an increase in the density of charge carriers nf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' or due to the formation of additional defects,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' as a result of rapid quenching from ~ 600 °C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' fol- lowed by improving of their parameters by holding the sam- ples in air at room temperature (the so-called room- temperature annealing) immediately after fabrication [31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The effect of annealing at room temperature (hereinafter simply annealing) on Тс, nf and the change in the lattice parameters of oxygen-deficient ReBa2Cu3O7–δ (Re = Y, Ho) single crystals after their quenching from T = 600 °C is ex- plained by the ordering of their structure due to the ordering of oxygen atoms in the CuO2 planes without a noticeable change in the oxygen content in the sample [31, 32] (and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Nevertheless, in spite of a number of studies on the relaxation processes in the 1–2–3 system during annealing [31–33] or, for example, under high pres- sure [34] (and references therein), many aspects, such as the charge transfer and the nature of the redistribution of the vacancy subsystem, still remain uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The study of the influence of intrinsic magnetism Ho on the fluctuation conductivity (FLC) and PG in HoBa2Cu3O7-δ single crys- tals upon annealing at room temperature is considered very promising for elucidating the mechanisms of the mutual influence of superconductivity and magnetism in HTSCs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' which is important for the final elucidation of the nature of both PG and HTSC in general This work is devoted to the study of the effect of an- nealing on the temperature dependences of the resistivity ρ(Т) of lightly doped HoBa2Cu3O7–δ quenched single crys- tals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' from which the temperature dependences of the excess conductivity σ′(Т) and pseudogap Δ*(Т) are calculated in the local pair model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' as discussed in detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' So far, no such studies have been carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Effects of annealing on the fluctuation conductivity and pseudogap in slightly doped HoBa2Cu3O7–δ single crystals Low Temperature Physics/Fizyka Nyzkykh Temperatur, 2023, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 49, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 1 117 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Experiment HoBa2Cu3O7−δ single crystals were grown with the so- lution-melt technique in a gold crucible as described else- where [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Rectangular crystals of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='9 × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='5 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='2 mm were selected to perform the resistivity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The smallest parameter of the crystal corresponds to the c axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' A fully computerized setup utilizing the four-point probe technique with stabilized measuring current of up to 10 mA was used to measure the ab plane resistivity, ρab(T) [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Silver epoxy contacts were glued to the opposite sides of the crystal in order to produce a uniform current distribution in the central region where voltage probes in the form of parallel silver stripes were placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Contact re- sistances below 1 Ω were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Temperatures were meas- ured with a Pt sensor having an accuracy of about 1 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' To reduce the oxygen content, the samples were an- nealed at 600 °C in ambient atmosphere for 24 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' After the annealing step, the crystals were quenched to room tem- perature within 2 min to 3 min, mounted on the holder and further cooled down to liquid-nitrogen temperature within 15 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' All measurements were conducted while warming up the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Experimental runs were carried at rates of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='1 K/min near Tc and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='5 K/min at T >> Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' In order to determine the room-temperature annealing effect on the electrical conductivity, after the first meas- urement of ρ(T) (sample S1), the samples were kept at room temperature for 20 h, after which the measurements were repeated (sample S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The last series of measurements was done after the room-temperature annealing of the sam- ples for about 5 days (120 h) (sample S3), depending on completion of relaxation processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Temperature dependen- cies of resistivity ρ(T) = ρab(T) for HoBa2Cu3O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='65 single crystal with initial Tc = 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='6 K and δ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='35 (sample S1) and measured after annealing during 20 h (sample S2) and 120 h (sample S3) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' In the Insert to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 1 shows a special approach to a more accurate determination of the PG opening temperature T* using the criterion (ρ(T) – ρ0)/aT = 1, where a =dρ/dT [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Thus, we have three curves or actually three samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The sample parameters are listed in the Tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' All curves have an expected S-shape with positive thermally activated bending, characteristic for slightly doped cuprates [23, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' However, above T* ≈ 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='5 K (S1), ≈ 238 K (S2), and ≈ 235 K (S3) ρ(T) varies linearly with T at rates dρ/dT = = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='48, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='91, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='57 μΩ⋅cm⋅K‒1 for S1, S2, and S3, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The figure shows that after the first annealing, the resistivity rapidly decreases, but reaches saturation, when the annealing time increases up to 120 h [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' At the same time, during annealing Tc of the samples increases (Table 1), while T* decreases (Table 2) in good agreement with phase diagram of cuprates [1, 17, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' All dependences also show a clear trend towards saturation upon annealing (see the Tables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Interestingly, neither ρ(T) nor Tc practically increase with an increase in the annealing time above 120 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Rapid cooling of crystals from about 600 °C (quenching) leads to the appearance of numerous additional disordered defects in the form of oxygen vacan- cies in CuO2 planes [31] (and references there).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' It is these defects that are responsible for the increased resistivity of sample S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' It can be concluded that the decrease in ρ(T) and the increase in Tc observed during annealing are due to the ordering of labile oxygen in the CuO2 planes due to diffusion but without an increase in the oxygen content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' A decrease in the slope dρ/dT indeed indicates the ther- mally activated character of the resistance relaxation pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The fact that the relaxation activation energy ρ(T) co- incides with the oxygen diffusion activation energy confirms the possibility of oxygen ordering in the planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' It has been shown that the oxygen diffusion, at least in YBCO single crystals, can proceed at an average rate of 150 Å per day [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (Color online) Temperature dependences of the resistivity ρab of single crystals of HoBa2Cu3O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='65 for different stages of room temperature annealing: S1 — without annealing, S2 — annea- ling for 20 h, S3 — annealing for five days (120 h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The straight solid lines determine the normal-state resistivity ρN (see the text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The Inset shows the method for determining Т* using the criterion (ρ(T) – ρ0)/aT = 1 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The resistivity and FLP parameters of the HoBa2Cu3O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='65 single crystal Sample ρ(300 K), µΩ⋅cm ρ(100 K), µΩ⋅cm Tc, K mf cT , K TG, K T0, K T01, K Δln σ′ d01, Å ξc(0), Å S1 751.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='5 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='1 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='6 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='65 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='0 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='8 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='68 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='87 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='82 S2 537.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='5 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='1 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='6 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='88 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='6 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='5 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='60 S3 451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='5 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='1 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='7 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='35 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='0 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='9 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='60 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='90 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='89 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Solovjov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Omelchenko, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Petrenko, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 118 Low Temperature Physics/Fizyka Nyzkykh Temperatur, 2023, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 49, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 1 All these facts confirm the conclusion made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' It appears that oxygen vacancies in slightly doped cuprates make the oxy- gen rearrangement easier to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Figure 2 shows the resistivity curves near Tc for S1 (a) and S3 (b), respectively, in which all representative tem- peratures are designated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The observed stepwise resistive transition is characteristic of lightly doped HTSC single crystals, especially after quenching [31–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' This is very likely due to the fact that rapid quenching from 600 °C also leads to a nonstoichiometric ratio of oxygen and va- cancies, which, in turn, leads to the formation of separate phases in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' These phases are characterized by different oxygen content and its ordering and, accordingly, have different Tc [31–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' This is precisely what leads to the experimentally observed stepwise superconducting transitions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The intersection of straight lines with the x axis shows how Tc was determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 2 it follows that Tc increases during annealing from 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='6 K (S1) to 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='7 K (S3) (Table 1), while the width of the resistive transition noticeably decreases by about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='8 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Accordingly, the two-step shape ρ(T) practically disappears [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 2(b)], which suggests the ordering of the crystal structure due to the redistribution of oxygen, as discussed above The details of the superconducting transitions are best seen in the temperature dependences of the derivatives dρ(T)/dT obtained upon annealing at 20 °C, as was shown in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 32, 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' In this case, the SC transitions are charac- terized by the low-temperature (LT) and high-temperature (HT) dρ(T)/dT maxima, indicating the presence of two superconducting phases with different Tc, as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Upon annealing, the low-temperature peak shifts toward the high-temperature peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' After 120 h of anneal- ing the LT peak becomes the highest and most uniform, and the HT peak is practically suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Moreover, the distance between the peaks decreases as expected, which indicates a decrease in the width of the SC transition dur- ing annealing by about a factor of two (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 2) [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Interestingly, high hydrostatic pressure also reduces the resistance and increases Tc of lightly doped HoBCO single crystals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' however, in contrast to annealing, the width of the superconducting transition increases in this case [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' This fact allows us to conclude that annealing, in contrast to pressure, creates a different mechanism of oxygen redistri- bution in a single crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The effect of annealing on the parameters of lightly doped samples ReВСО (Re = Y, Ho) is usually interpret- ed by the ordering of oxygen atoms in the CuO2 plane without changing the oxygen content in the sample (refer to [31, 32] and references therein), as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' However, an increase in Tc during annealing can be associ- ated either with the local ordering of oxygen [37] or with an increase in the density of charge carriers nf [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Another reason for the increase in Tc can be a change in the parame- ters of the crystal cell, for example, change in Сu–O and Сu–Сu distances in the ab plane [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' In turn, the electrical resistivity decreases not only due to the ordering of oxygen atoms during the holding of the sample at room tempera- ture after quenching from a high temperature [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' But, it can also decrease due to an increase in nf or a decrease in defects as a result of the ordering of oxygen vacancies [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The possibility of increasing the charge carrier densi- ty nf upon annealing is discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Thus, we can conclude that, strictly speaking, many as- pects of lightly doped quenched HoBCO single crystals, such as charge transfer and the nature of the redistribution of the vacancy subsystem, especially with allowance for Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The pseudogap parameters of the HoBa2Cu3O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='65 single crystal Sample T*, K α* 0 cε C3D A4 D* ∆*(TG), K ∆*(Tmax), K Tmax1, K Tmax2, K ΔTmax, K S1 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='64 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='1 26 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='5 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='1 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='1 199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='7 214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='8 S2 238 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='71 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='35 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='5 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='5 221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='9 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='7 217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='8 S3 235 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='67 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='85 40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='4 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='3 232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='8 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='2 215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (Color online) Resistive transitions of single crystals of HoBa2Cu3O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='65: (a) without annealing, sample S1 (turquoise dots), (b) after annealing for five days (120 h), sample S3 (gray dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Effects of annealing on the fluctuation conductivity and pseudogap in slightly doped HoBa2Cu3O7–δ single crystals Low Temperature Physics/Fizyka Nyzkykh Temperatur, 2023, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 49, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 1 119 the high Ho magnetism, still remain undetermined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' We hoped to shed more light on this problem by studying the effect of annealing on FLC and PS in lightly doped quenched HoBCO single crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Results and discussion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='1 Fluctuation conductivity As clearly seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 1, below the PG opening tem- peratures T* >> Tc the resistivity curves of HoBa2Cu3O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='65 single crystal deviate downward from linear dependencies at high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' This resulting in appearance of the excess conductivity ( ) ( ) ( ) ( ) ( ) / N N T T T T T ′ σ = ρ −ρ ρ ρ \uf8ee \uf8f9 \uf8ee \uf8f9 \uf8f0 \uf8fb \uf8f0 \uf8fb , (2) as the difference between the experimentally measured resistivity ( ) T ρ and the linear resistivity of the normal state 0 ( ) N T aT ρ = ρ + , extrapolated towards low tempe- ratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Here, 0 ρ is the residual resistivity determined by extrapolating ( ) N T ρ towards 0 K, and / a d dT = ρ is the slope of the linear straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' This procedure of the nor- mal state resistivity determination is widely used in litera- ture (see [2, 5, 26, 40, 41] and references therein) and has been justified theoretically by the nearly antiferromagnetic Fermi liquid (NAFL) model [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The T* temperature is taken at the point where the experimental resistivity curve starts to downturn from the high-temperature linear behavior (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' For a more accurate determination of T*, the criterion 0 ( ) ( ) – / 1 T aT ρ ρ = [2, 5, 40] was also used (see Insert in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Both approaches give practically the same T*’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' In the local pairs model, it is believed that PG and ex- cess conductivity are due to the formation of LPs below T* [3, 5, 14, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The properties of the LPs are determined by the coherence length along the c axis ( ) c T ξ = 1/2 –1/2 (0) – / (0) [( ) ] mf mf c c c c T T T − = ξ = ξ ε [42, 43], where ) / ( mf mf c c T T T ε = − is the reduced temperature and mf c T is the mean-field critical temperature, which separates the range of SC fluctuations above Tc from the region of criti- cal fluctuations around Tc, where the SC order parameter Δ < kT [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Hence, it is evident that the correct deter- mination of mf c T is decisive in FLC and PG analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' It is well established [5, 23, 40, 45–47] that the small coherence length in combination with the quasi-layered structure of HTSCs leads to the formation of a noticeable, in compari- son with conventional superconductors, range of SC fluc- tuations, ΔTfl, above Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Near Tc, where ( ) c T ξ > d = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='68 Å (d is the unit cell size along the c axis [48]), the LPs are very large and interact throughout the entire volume of the sample, forming a 3D state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' In this case the FLC is always described by the Aslamazov–Larkin equation for any 3D system [49]: 2 1/2 3 3 32 (0) DAL D c e C h − ′ σ = ε ξ , (3) where C3D is a numerical factor used to fit the data by the theory [23, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' This means that the conventional 3D FLC is realized in HTSCs as T → Tc [43, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (3), one can easy obtain 2 ~ ~ ( ) / mf mf c c T T T − σ′ ε − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Obviously, 2 ΄ − σ = 0, when mf c T T = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' This way of mf c T determination was proposed by Beasley [44] and substantiated in various FLC experiments [5, 26, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Moreover, with the correct choice of mf c T , the data in the three-dimensional fluctuation region near Tc are always approximated by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Figure 3 shows the 2 ΄ − σ vs T plot for samples S1 [(a) turquoise dots] and S3 [(b) gray dots].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The interception of the extrapolated linear 2 ΄ − σ with T axis determines mf c T = 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='65 K (S1) and = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='35 K (S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' S2 (not shown) demonstrates some intermediate 2 ΄ − σ on T dependence with mf c T = 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='88 K given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' It should be noted, that the revealed dependences of 2 ΄ − σ on T differ marked- ly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' When the measurements were carried out immediately after quenching (S1), the separation between the LT and HT phases is pronounced (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 2), and S1 exhibit a dependence 2 ΄ − σ on T characteristic of most HTSCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' But in this case, the LT phase, which is clearly seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 2(a) has virtually no effect on the definition of mf c T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' However, unlike Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 2(b), after annealing, both the LT and HT phas- es plotted in these coordinates become quite pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' But, fortunately and somewhat surprisingly, the approxi- mation of both phases by straight lines gives the same mf c T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (Color online) Temperature dependences of the inverse square of the excess conductivity 2( ) ΄ T − σ for sample S1 [(a), turquoise dots] and S3 [(b), gray dots)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The interception of the extrapolated linear 2 ΄ − σ with T-axis determines mf cT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Also shown are Tc, the Ginsburg temperature TG and 3D–2D crossover tem- perature T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Solovjov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Omelchenko, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Petrenko, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 120 Low Temperature Physics/Fizyka Nyzkykh Temperatur, 2023, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 49, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 1 Above the crossover temperature the data deviates right from the line suggesting the 2D Maki–Thompson (MT) [42, 51, 52] fluctuation contribution into FLC [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Obvi- ously, at the crossover temperature 0 0 ~ T ε the coherence length 1/2 0 0 ( ) (0) c c T − ξ = ξ ε is expected to amount to d [9, 37] which yields 0 (0) c d ξ = ε (4) and allows the possibility of (0) c ξ determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (0) c ξ is one of the important parameters of the PG analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Figure 4 shows the ln ′ σ vs ln ε: (a) S1 (turquoise dots), (b) S2 (yellow dots), and (c) S3 (gray dots) in comparison with the fluctuation theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' As expected, all samples demonstrate fairly good agreement with the AL theory near Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' For example, above the Ginzburg temperature ln ( 5 3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' G mf c G T T > ε = − (refer to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 3), down to which the mean-field theory works [50], and up to T0 = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='8 K 0 (ln ε = –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='84) the data for sample S1 are well extrapola- ted by the 3D fluctuation term (3) of the AL theory, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 4(a), solid red line with a slope –1/2) with (0) c ξ = = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='02) Å determined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (4) and C3D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='1 (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' It should be noted that the same value of (0) c ξ was determined for a FeAs-based superconductor ErFeAsO0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='85F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='15 [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Samples S2 and S3 behave similarly near Tc (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Above T0, the data deviate sharply upwards from the AL theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' This is due to the fact that at 0 T T ≥ , where ( ) c T ξ < d, the three-dimensional regime ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' However, it is still 01 ( ) c T d ξ > , which is the distance between conduct- ing planes CuO2 [48], and ( ) c T ξ connects the planes with the Josephson interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' This is a 2D fluctuation regime, which is described by the MT term of the Hikani–Larkin (HL) theory [42]: 2 1 2 1 1 1 2 ln ( / ) 8 1 / 1 1 2 MT D e C d − \uf8eb \uf8f6 + α + + α ′ σ = ⋅ ⋅ δ α ⋅ ε \uf8ec \uf8f7 \uf8ec \uf8f7 − α δ + δ + + δ \uf8ed \uf8f8 \uf068 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (5) These are the blue solid curves in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (5) 2 1 [ (0) ] 2 / c d − α = ξ ε is a coupling parameter, 2 (0) 16 c B k T h d ϕ ξ \uf8ee \uf8f9 δ = β τ \uf8ef \uf8fa π \uf8f0 \uf8fb (6) is the pair-breaking parameter, and ϕ τ that is defined by equation 0 0 / 8 / B T h k A ϕ τ β = π ε = ε (7) is the phase relaxation time, where A = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='998⋅1012 s·K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The factor β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='203 (l/ξab), where l is the mean-free path and ξab(0) is the coherence length in the ab plane considering the clean limit approach l > ξ, which is always takes place in HTSCs [5, 6, 26, 40, 43, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Above T01, indicated on all graphs as 01 ln ε , the data of all samples deviate definitively downward from the theory (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Thus, T01 limits the range of SC fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' In this range, fluctuating pairs behave much like ordinary Cooper pairs, but without long-range ordering (the so- called short-range phase correlations [5, 22, 23, 44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Above T01, ( ) c T ξ < d01 and LPs are confined within the CuO2 planes, which are no longer connected by any cor- relation interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Thus, it is clear that 01 01 ( ) c T d ξ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' To estimate 01 d , we use the condition ( ) 0 0 c d ξ = ε = 01 01 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='02) d = ε = ± Å (S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Since d = c = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='68 Å and 01 ln ε ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='63 ( 01 ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='532 and T01 ≈ 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='2 K), we obtain: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (Color online) ln σ′ vs ln ε: (a) S1 (turquoise dots), (b) S2 (yel- low dots), and (c) S3 (gray dots) compared with fluctuation theories: 3D-AL — red lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 2D-MT — blue curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Δln (σ′) designates the maximal deviation of the data from the extrapolated 3D-AL lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Effects of annealing on the fluctuation conductivity and pseudogap in slightly doped HoBa2Cu3O7–δ single crystals Low Temperature Physics/Fizyka Nyzkykh Temperatur, 2023, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 49, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 1 121 01 0 01 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='05) d d = ε ε = ± Å for S1 in good agree- ment with results of the structural studies [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Having carried out a similar analysis for other samples, we obtain the values of (0) c ξ and d01 for S2 and S3 (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Inter- estingly, in contrast to S1 and S2, in the case of S3, both the LT and HT phases are clearly visible on the plot of ln ′ σ vs ln ε below T0, but both fully follow the AL theo- ry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' However, to keep the logic with S1, we determined T0 and other parameters from the high temperature phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Strictly speaking, extrapolation of 2D-MT data above T0 is not entirely successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' This is due to the fact that above T0 there is a sharp increase in data, leading to the ap- pearance of enhanced 2D fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' As a result, the max- imal deviation of the data above the extrapolated 3D-AL line Δ ln σ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='68 obtained for S1 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 4(a)] is approxi- mately 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='4 times greater than that observed for YBCO, where magnetism is not expected [5, 23, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' This enhanced behavior of 2D fluctuation is typical of FeSe-based super- conductors such as ErFeAsO0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='85F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='15 [53] and SmFeAsO0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='85 [54], as well as superconductors with magnetic impurities such as YBCO–PrBCO superlattices [50], suggesting a no- ticeable influence of own HoBCO magnetism in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The highest value Δ ln σ′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='0, which is approximately 5 times greater than that observed for YBCO, was obtained for S2 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 4(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' In this case, fitting the data by the MT theory is completely impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' However, to provide a more informative analysis, we used the found fitting pa- rameters (ξc(0), ε0, ε01) to derive theoretical 2D-MT curve that would intersect the red 3D-AL line at ln ε0 (corre- sponding temperature T0) and the 2D-data at ln ε01 (corre- sponding temperature T01) [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 4(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' We emphasize that, despite the unsatisfactory description of the 2D data, all temperatures T01 found in this way (indicated in the figure as ln ε01) clearly correspond to the minima on the tempera- ture dependences of the PG parameter Δ*(T), which fol- lows from the theory (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 7), thereby confirming the correctness of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The enhanced 2D fluctuations found for S2 are reminiscent of these observed for magnet- ic superconductors such as Dy0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='6Y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='4Rh3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='85Ru0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='15B4, which have an intrinsic magnetic moment μ ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='2μB per Dy3+ ion [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Taking all above facts into account, we can conclude that the observed anomalous 2D fluctuations in sample S2 is most likely caused by the noncompensated magnetic moments of Ho, which is thought to be responsible for interplay between magnetic interaction and superconduc- tivity [5, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' However, since the resistivity noticeably decreases upon annealing, it can be concluded that mag- netic interaction does not strongly affect the rate of charge carrier scattering in HoBCO single crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' At the same time, S2 has the lowest values of ξc(0), d01 (Table 1) and unexpectedly C3D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='35, confirming a strong influence of oxygen diffusion on the sample structure [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Recall that the smaller the C3D, the smaller the effect of defects in the sample, which is directly related to the decrease in re- sistivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' In turn, after five days of annealing, S3 demonstrates the lowest value Δ ln σ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='6 and the shape of ln σ′ vs ln ε resembling S1, except for the low temperature 3D-AL re- gion [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 4(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' This indicates the final ordering of defects and the crystal structure as a whole, which leads to the lowest resistivity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 1 and Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' It can also be as- sumed that ordered oxygen somehow shielded the influ- ence of Ho magnetic moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' However, despite the sup- posed ordering of defects and oxygen, the values of ξc(0), d01 (Table 1) and C3D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='85 are practically the same as in S1, which indicates a nonmonotonic, in contrast to the re- sistivity, effect of defects and magnetism on the FLC of the sample upon annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' We expected to obtain confirma- tion or refutation of this conclusion by analyzing the change in the temperature dependence of the pseudogap during annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Pseudogap analysis As mentioned above, the number of the different non- Fermi-liquid models proposed to explain the physics of PG is quite large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' However, a very large number of models raises doubts about their correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' In addition, none of the mentioned models gives explicitly the temperature de- pendence of PG, which could be verified experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Clearly, to attain information about the pseudogap we need an equation which specifies a whole experimental curve, from TG up to T*, and contains the PG parameter Δ*(T) in an explicit form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=" The issue was resolved within the frame- work of our LP model [23, 56], in which such an equation was proposed for σ'(ε): 2 4 0 0 1 exp ( ) 16 (0) 2 sinh 2 c c c T e T T T A \uf8eb \uf8f6 ∆ \uf8eb \uf8f6 − − \uf8ec \uf8f7 \uf8ec \uf8f7 \uf8ed \uf8f8 \uf8ed \uf8f8 ′ σ = \uf8eb \uf8f6 ε ξ ε \uf8ec \uf8f7 ε \uf8ed \uf8f8 \uf068 , (8) where, for a correct description of the experiment, the dy- namics of pair formation (1 – T/T*) and pair breaking (exp(‒Δ*/T)) above Tc are taken into account." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (8) for the pseudogap Δ*(T) one can readily obtain: 2 4 0 0 1 1 ( ) ln 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' ( ) 16 (0) 2 sinh 2 c c c T e T T A T \uf8ee \uf8f9 \uf8ef \uf8fa \uf8ef \uf8fa \uf8eb \uf8f6 ∆ = − \uf8ef \uf8fa \uf8ec \uf8f7 ′ σ ε ξ \uf8ed \uf8f8 \uf8eb \uf8f6 \uf8ef \uf8fa ε ε \uf8ec \uf8f7 \uf8ef \uf8fa ε \uf8ed \uf8f8 \uf8f0 \uf8fb \uf068 (9) Here ( ) ′ σ ε is the experimentally measured excess con- ductivity over the whole temperature interval from T* down to TG, and A4 is a numerical factor that has the meaning of the C factor in the fluctuation conductivity theory [42–44, 51, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' All other parameters, including the coherence length along the c axis, (0) c ξ , [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (4)] and the theoretical parameter *0 c ε [57], directly come from the A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Solovjov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Omelchenko, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Petrenko, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 122 Low Temperature Physics/Fizyka Nyzkykh Temperatur, 2023, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 49, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 1 experiment [5, 23, 50, 54–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' To find 0 c ε we use the experimental fact that in some temperature range above T01, namely 01 02 ln ln ln c c ε < ε < ε (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 5) or accordingly 01 02 c c ε < ε < ε (Insert in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 5), 1 ~ e p( ) x −′ σ ε [23, 56, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' As a result 1 ln ( ) −′ σ is a linear function of ε with a slope α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='56 which determines parameter 0 1/ cε = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='64 for S1 (Insert in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' To find A4, we calculate ( ) ′ σ ε from T* and down to TG using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (8) and fit experiment in the range of 3D-AL fluctuations near Tc (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 5, red curve) where ln ′ σ on ln ε is a linear function of the reduced tem- perature ε with a slope λ= ‒ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Besides, ( ) (0) G T ∆ = ∆ is assumed [58, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' To estimate ( ) G T ∆ , which we use in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (8), we plot ln ′ σ as a function of 1/T for S1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 6) and S3 (Insert to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' After annealing (Insert to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 6) the approximation, as expected, looks better, due to the ordering of defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' In this case the slope of the theoretical curve [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (8)] turns out to be very sensitive to the value of ( ) G T ∆ [23, 50, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' For sample S1 the best fit is obtained when ) 2 / 5 ( G B c D T k T = ∆ = and D* = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='8 for S3 (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Note, that D* = 5 is the typical value for cuprates [5, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Having determined all the necessary parameters (refer to Tables 1, 2) we succeeded to plot the temperature depen- dences PG, ( ) T ∆ for all stages of annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 7(a) (tur- quoise dots) displays ( ) T ∆ for S1 calculated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (9) with the following set of parameters derived from the exper- iment within the LP model: T* = 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='5 K, mf c T = 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='65 K, (0) c ξ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='82 Å, *0 c ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='64, A4 = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The resulting form of ( ) T ∆ with a high-temperature maximum at Tmax = 239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='5 K, followed by a section of the linear dependence ( ) T ∆ with a moderate positive slope αmax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='01, is typical of a lightly doped HTSC single crystals, containing various defects, including tweens [61] (and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' This confirms our assumption made above about numerous defects in the quenched crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The low-temperature be- havior of ( ) T ∆ in S1 with minimum at T01, maximum at about T0 and final minimum at TG is also characteristic of all HTSCs (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 12 in [50]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The range of SC fluctua- tions fl 01 G T T T ∆ = − = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='2 K is large but comparable with fl T ∆ obtained for slightly doped YBCO single crystals with TBs [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The only peculiarities are two small max- ima at Tmax1 = 199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='7 K and Tmax2 = 214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='5 K which is a feature of the PG behavior found only on HoBCO single crystals [26, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Dependences ( ) T ∆ constructed for the samples S2 and S3 with the corresponding sets of parameters given in the Tables, are displayed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 7(b) and 7(c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' It can be seen that the shape of ( ) T ∆ noticeably changed upon annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The linear section with a moderate posi- tive slope disappeared, but maxima at Tmax1 and Tmax2 be- came much more pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' It is very tempting to attrib- ute these maxima to two phases with different Tc observed at resistive transitions (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' But upon annealing, the separation into two phases gradually disappears, whereas, despite a significant change in all parameters of the sample during annealing, the distance between these maxima remains constant at ΔTmax = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='8 K (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' There have also been attempts to relate these maxima to a process of the so-called ascending diffusion, which is be- lieved to increase the separation of charge carriers between tweens and TBS [34] (and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' But the as- cending diffusion also changes during the annealing, whereas the distance between these maxima remains con- stant, as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' We believe that these maxima are somehow connected with enhanced magnetism of HoBCO, but, strictly speaking, the appearance of these unusual maxima is still in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (Color online) ln ′ σ vs lnε for S1 (turquoise dots) plotted in the whole temperature range from T* down to TG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The solid red curve is a fit to the data with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Insert: 1 ln − σ as a func- tion of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Solid line indicates the linear part of the curve between 01 cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='20 and 02 cε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Corresponding 01 ln cε = − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='59 and 02 ln cε = − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='29 are marked by arrows in the main panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The slope α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='6 determines the parameter * 0 1/ cε = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='64 (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (Color online) ln σ vs 1/T for S1 (turquoise dots) and S3 (Insert, gray dots) plotted in the whole temperature range from T* down to TG .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The red solid curves are fits to the data with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The best fit is obtained when Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (8) is calculated with ) 2 / 5 ( G B c D T k T = ∆ = for S1 and D* = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='8 for S3 (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Effects of annealing on the fluctuation conductivity and pseudogap in slightly doped HoBa2Cu3O7–δ single crystals Low Temperature Physics/Fizyka Nyzkykh Temperatur, 2023, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 49, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 1 123 As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 7(b), S2 demonstrates a rather specific ( ) T ∆ with pronounced wide minimum, as expected, at T01 ≈ 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='3 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' This leads to an anomalously large range of SC fluctuations, fl 01 G T T T ∆ = − , about 50 K above Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' In addition, the two maxima look more pronounced and shift- ed slightly towards higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' And, more im- portantly, the data slope at high temperature is about 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='5 times steeper than that of the S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Taking into account the results of the study of the FLC [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 4(b)], we associate this form of ( ) T ∆ with an increased magnetism of the uncompensated magnetic moments Ho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' In turn, S3 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 7(c)] demonstrates ( ) T ∆ characteristic of HoBCO single crys- tals with ordered defects [26, 34], which allows us to con- clude that after five days of annealing, oxygen diffusion almost ceased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Indeed, the minimum at T01 = 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='4 K is noticeably smaller, and the high-temperature maxima are not as pronounced as for S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' However, the range of SC fluctuation fl 01 G T T T ∆ = − ≈ 38 K and the temperatures of these maxima in this case are almost the same as for the quenched sample S1 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 7(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' This allows us to draw the following conclusion that ordered oxygen somewhat shields the magnetic interaction in the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' On the other hand, in general, the shape of the dependence ( ) T ∆ dif- fers markedly from the shape of S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The value of ( ) G T ∆ , which gradually increases during annealing, reaches the maximum value of 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='3 K for S3 (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' In addition, the minimum at T01 is still quite pronounced and, more im- portantly, the slope of the data at high temperature αmax = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='0, marked in the figure by the red line, is the same as for S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Here we would like to emphasize that the same data slope at high temperatures is observed for all magnetic su- perconductors, including FeAs-based compounds (see [50], Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' This is confirmed by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 8, where we compare our data for sample S3 with results obtained for the highly magnetic superlattice 7YBCO×14PrBCO (sample SL3) from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' It can be seen that the slope at high T is ex- actly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Moreover, the shape of both curves below T01 is also almost the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Interestingly, in 1111 FeAs-based superconductors the maximum corresponds to the structur- al transition from a tetragonal to an orthorhombic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Accordingly, the temperature of data deviation from the linear dependence corresponds to transition to the AF state Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (Color online) Temperature dependences of pseudogap ( ) T ∆ (a) S1 (turquoise dots), (b) S2 (yellow dots), and (c) S3 (gray dots), analyzed with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' All characteristic temperatures are marked with arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The red lines designate the data slope at high temperatures, which is equal for S2 and S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Solid black lines are a guide for the eyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (Color online) max / ( ) T ∆ ∆ as a function of T/T* for stu- died single crystals of HoBa2Cu3O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='65 after annealing for five days (120 h), sample S3, and superlattice 7YBCO×14PrBCO, sample SL3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' All characteristic temperatures are marked with arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The red lines designate the slope of the data at high tem- peratures, which is the same for both samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Solid black lines are a guide for the eyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Solovjov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Omelchenko, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Petrenko, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 124 Low Temperature Physics/Fizyka Nyzkykh Temperatur, 2023, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 49, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 1 of spin–density–waves (SDW) [62–65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Thus, an important conclusion can be drawn that the AF interaction of the SDW type should take place in lightly doped HoBCO single crys- tals below the Tmax1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 7) due to the large intrinsic mag- netism of Ho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' This interaction is believed to be responsible for the formation of both PG and SDW state in such com- pounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The possibility of SDW state in lightly doped YBCO compounds is discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Thus, returning to the question of possible models of SC pairing in HTSCs, we can assume that the SDW model is the most probable one, at least for HTSCs with a strong magnetic interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' To clarify the question of a possible increase in the den- sity of charge carriers in a crystal during annealing, we compare the pseudogap parameter max ( ) / T ∆ ∆ of samples S1, S2, and S3 near Tc with the Peters–Bauer (PB) theory [3] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' In [3], the temperature dependences of the local pairs density in HTSCs were theoretically calculated within the framework of the three-dimensional attractive Hubbard model for different temperatures T/W, interactions U/W, and filling factor, where U is the activa- tion energy and W is the band width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The shape of ( ) T ∆ for all cuprates, with a maximum near T0 followed by a minimum at TG [50] (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 12 in [50]) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 7 in [66]), resembles the shape of theoretical curves at low T/W and U/W [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' This fact should justify such an approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' To carry out the analysis, we combine the measured values of max ( ) / T ∆ ∆ for S1 at ТG with the minimum, and at T0 with the maximum of each theoretical curve calculat- ed at different values of U/W, thus achieving the best agreement between the experiment and theory in the wid- est possible temperature range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' It is important that the fit- ting factors found for S1 are also used for the other two samples [67, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The fitting results for all three samples are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The best fit for S1 near Tc is obtained with U/W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='2 curve indicating that in this case ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='3, which is a typical value for various HTSCs [67, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Fur- ther, it was taken into account that max ( ) / G T ∆ ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='82 for S1, where max ∆ is taken at Tmax = 239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='5 K, which is clearly seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Unfortunately, due to the specific form of ( ) T ∆ found for S2 and S3, leading to ambiguity in the definition of Tmax, it was not possible to obtain reasonable values of max ( ) / G T ∆ ∆ in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Therefore, we could not compare them with that found for S1 in order to obtain the corresponding fitting coefficients, as we did in our pre- vious works [67, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' However, both TG and ( ) G T ∆ are clearly defined for all the studied samples (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Moreover, ( ) G T ∆ notice- ably increases upon annealing (Table 2), most likely due to an increase in the charge carrier density nf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' This fact sug- gests that nf must be proportional to the value of ( ) G T ∆ [5, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Taking into account that found for S1 ( ) G T ∆ cor- responds to = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='30195, the simple algebra yields: (S2) = [ , S2 / ( ) ( ) , S1 G G T T ∆ ∆ ] × (S1) = (170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='5×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='30195)/156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='33, and (S3) = ( ) [ , S3 / ( 1) , S G G T T = ∆ ∆ ] × (S1) = (189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='3×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='30195)/156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='366, which gives the corresponding curves at the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' This means that the density of charge carriers in HoBCO single crystals somewhat increases due to oxygen diffusion dur- ing annealing, as it was assumed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 69, 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' It is very likely that the observed slight increase in nf is quite suffi- cient to explain the observed increase in Tc by ~ 9 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' This may also be responsible to some extent for the observed decrease in ρ(T) (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' It is also worth noting that the best agreement with the PB theory among HTSCs in a wide temperature range was obtained for non-twinned optimally doped YBCO single crystals, naturally, without any magnetism [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' In the pre- sent case the data noticeably deviate downward from the theoretical curves with increasing temperature, which is most likely due to the enhancement of the magnetic interac- tion in HoBCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' This conclusion is confirmed by the follow- ing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The quenched sample S1 shows the smallest deviation, since the magnetic moments Ho are considered to be randomly distributed due to multiple defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The sample S2 shows the largest deviation, as a result of influence of the uncompensated magnetic moments, as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' In the case of sample S3, the magnetic interaction is some- how compensated by the ordering of the distribution of oxy- gen and crystal structure defects [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' As a result, despite the rather complicated shape of the max ( ) / G T ∆ ∆ curve at low T, the deviation from the theory is expectedly moderate (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Moreover, above (T/W, T/T*) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='25, the experi- mental data deviate upward from the theory, which confirms Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' (Color online) Curves of max / ∆ ∆ as functions T/T* for samples S1 (turquoise dots), S2 (yellow dots), and S3 (gray dots) in comparison with the theoretical curves of as functions of T/W, at the corresponding interaction values U/W: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='2 (black curve), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='4 (red curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The arrows indicate the temperatures T0 (▲) and TG (▼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Effects of annealing on the fluctuation conductivity and pseudogap in slightly doped HoBa2Cu3O7–δ single crystals Low Temperature Physics/Fizyka Nyzkykh Temperatur, 2023, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 49, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 1 125 a fundamentally different mechanism of magnetic interac- tion in the HoBCO single crystal after five days of anneal- ing at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Conclusion The magnitude and temperature dependence of fluctua- tion conductivity and pseudogap ( ) T ∆ in lightly doped HoBa2Cu3O7–δ single crystals rapidly quenched from 600 °С were studied for the first time at different stages of anneal- ing at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' During annealing, a significant decrease in the resistance of the samples, the width of re- sistive transitions, and an increase in Tc were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' These observations are consistent with the processes of the oxygen diffusion and structural relaxation in the volume of experimental samples, leading to the appearance of phase separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' However, in addition to the expected change in oxygen distribution, several new interesting results were revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' At all stages of annealing, the FLC near Tc is well de- scribed by the 3D Aslamazov–Larkin fluctuation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' However, at the intermediate stage of annealing (sample S2), an anomalous increase in 2D FLC was revealed, which is associated with the influence of uncompensated magnetic moments in HoBa2Cu3O7–δ, since μeff, Ho = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='7 μB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' As a result, in this case, the 2D Maki–Thompson fluctuation theory failed to describe the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' However, after five days of an- nealing, the 2D-MT fit improved, since the magnetic inter- action is somehow compensated by the ordering of the distribution of oxygen and crystal structure defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' For the quenched sample S1, the temperature depend- ence of the PG has a shape typical of single crystals with a large number of defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' However, ( ) T ∆ has two small additional maxima at high temperature, which is a feature of HoBa2Cu3O7–δ single crystals with pronounced twins and indicates the two-phase nature of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Upon anneal- ing, the shape of ( ) T ∆ noticeably changes, very likely due to an increase in the magnetic interaction (sample S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The two additional peaks became more pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' But, more important is the change in the slope αmax of the data at high temperatures, which has become about 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='5 times steeper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The ordering of the oxygen distribution due to the diffusion process during annealing somewhat compensates for the influence of magnetic interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' But the slope does not change (sample S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Moreover, the slope turns out to be the same as for FeAs-based superconductors, sug- gesting the possibility of the existence of spin density waves in HoBa2Cu3O7–δ in the PG state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' The comparison of the pseudogap parameter max ( ) / G T ∆ ∆ near Tc with the Peters–Bauer theory revealed a slight increase in the densi- ty of local pairs , which should explain the ob- served increase in Tc by 9 K during annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Interesting- ly, despite the rather complicated shape of the max ( ) / G T ∆ ∆ curve at low T (sample S3), the deviation from the PB the- ory is expectedly moderate (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Moreover, above (T/W, T/T*) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content='25, the experimental data deviate upward from the theory, which confirms a fundamentally different, compare with S1 and S2, mechanism of magnetic interac- tion in the HoBCO single crystal after five days of anneal- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Thus, the studies of FLC and PG turned out to be very informative and made it possible to obtain new results, which, in turn, are definitely a consequence of the expected rearrangement of the oxygen distribution and the defect structure during annealing at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Acknowledgments We thank support from the National Academy of Sci- ences of Ukraine through Young Scientists Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 1/N- 2021 (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' We acknowledge support from the Ministry of Innovative Development of the Re- public of Uzbekistan through Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' F-FА-2021-433 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=', and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' A.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 44, 81 (2018)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Fehrenbacher and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Rice, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 70, 3471 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Goshchitskii, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Kozhevnikov, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Sadovskii, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 2, 1331 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Zavgorodniy, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Goulatis, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Beletskii, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Chroneos, Physica C 469, 203 (2009).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Zavgorodniy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Bondarenko, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Goulatis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Samoilov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Chroneos, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Alloys Compd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 453, 69 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Obolensky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Bondarenko, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Vovk, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Prodan, Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Nizk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 23, 1178 (1997) [Low Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 23, 882 (1997)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} 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Dobrovolskiy, Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' B 32, 1750367 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Vovk, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' B 84, 014522 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Grbić, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Požek, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Paar, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Hinkov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Raichle, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Haug, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Keimer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Barišić, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Dulčić, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' B 83, 144508 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Chryssikos, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Kamitsos, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Kapoutsis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Patsis, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Psycharis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Koufoudakis, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Mitros, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Kallias, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Gamari-Seale, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Niarchos, Physica C 254, 44 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Aslamazov and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Larkin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' A 26, 238 (1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Solovjov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Omelchenko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Stepanov, R.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' B 94, 224505 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Maki, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 39, 897, (1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Thompson, Microwave, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' B 1, 327 (1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Solovjov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Omelchenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Terekhov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Rogacki, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Vovk, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Khlybov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Chroneos, Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Express 3, 076001 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Solovjov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Svetlov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Stepanov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Sidorov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Tarenkov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Dyachenko, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Agafonov, Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Nizk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 37, 703 (2011) [Low Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 37, 557 (2011)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Solovjov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Terekhov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Petrenko, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Omelchenko, and Zhang Cuiping, Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Nizk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 45, 1403 (2019) [Low Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 45, 1193 (2019)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' A.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 35, 169 (2009)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Leridon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Défossez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Dumont, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Lesueur, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Contour, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 87, 197007 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Nizk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 46, 638 (2020) [Low Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 46, 538 (2020)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Getherich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Erb, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Müller-Vogt, Physica C 232, 82, (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' ___________________________ Вплив відпалу на флуктуаційну провідність та псевдощілину у слаболегованих монокристалах HoBa2Cu3O7-δ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Solovjov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Omelchenko, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Petrenko, Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Kolesnichenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Kolesnik, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Dzhumanov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Vovk Вивчено вплив відпалу при кімнатній температурі на флуктуаційну провідність (ФЛП) σ′(T) і псевдощілину (ПЩ) Δ*(T) у базисній площині ab монокристалів ReBa2Cu3O7–δ (Re = Ho) з нестачею кисню.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Показано, що на всіх етапах відпалу ФЛП поблизу Tc можна описати флуктуаційними теоріями Асламазова–Ларкіна та Макі–Томпсона, де спосте- рігається 3D–2D кросовер із підвищенням температури.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' За температурою кросовера Т0 визначено довжину когерент- ності вздовж осі c — ξс(0) = (2,82 ± 0,2) Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' На проміжному етапі відпалу виявлено аномальне зростання 2D ФЛП, що пов’язано з впливом некомпенсованих магнітних моментів у HoBa2Cu3O7–δ (HoBCO): µeff, Ho = 9,7µB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Для загартованого зразка S1 температурна залежність ПЩ має форму, типову для монокристалів з великою кількістю дефектів.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Проте Δ*(T) має два невеликі додаткові максимуми при високих темпера- турах, що є особливістю монокристалів HoBCO з виражени- ми двійниками та вказує на двофазність зразка.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Під час від- палу форма Δ *(T) помітно змінюється, ймовірно, за рахунок збільшення магнітної взаємодії (зразок S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Більш важливою є зміна нахилу даних при високих температурах, який став приблизно в 3,5 рази крутішим.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Упорядкування розподілу кисню за рахунок процесу дифузії під час відпалу дещо ком- пенсує вплив магнітної взаємодії, проте нахил не змінюється (зразок S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Цікаво, що нахил виявляється таким же, як і для надпровідників на основі FeAs, що свідчить про можливість існування хвиль спінової щільності в HoBCO в ПЩ стані.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Порівняння псевдощілинного параметра Δ*(T)/Δ* max поблизу Tc з теорією Пітерса–Бауера виявило незначне збільшення щільності локальних пар , що має пояснювати спосте- режене підвищення Tc на 9 K під час відпалу.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} +page_content=' Ключові слова: флуктуаційна провідність, псевдощілина, надлишкова провідність, відпал, монокрис- тали HoBCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE0T4oBgHgl3EQfmgH3/content/2301.02501v1.pdf'} diff --git a/Z9E2T4oBgHgl3EQfZgdy/content/tmp_files/2301.03865v1.pdf.txt b/Z9E2T4oBgHgl3EQfZgdy/content/tmp_files/2301.03865v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a1782ca6438b0de22e8f987c7ae41648526e480d --- /dev/null +++ b/Z9E2T4oBgHgl3EQfZgdy/content/tmp_files/2301.03865v1.pdf.txt @@ -0,0 +1,1055 @@ +Contact graphs of boxes with unidirectional contacts∗ +Daniel Gonçalvesa, Vincent Limouzyb, and Pascal Ochema +aLIRMM, Univ Montpellier, CNRS, Montpellier, France. +bUniversité Clermont Auvergne, Clermont Auvergne INP, CNRS, Mines +Saint-Etienne, Limos, F-63000 Clermont-Ferrand, France +Abstract +This paper is devoted to the study of particular classes of geometri- +cally defined intersection graphs. Those are contact graphs of axis parallel +boxes in Rd, where the intersection of any pair of boxes is parallel to a +given hyperplane (among the d possible ones). +C. Magnant and D.L. +Martin showed that these graphs (already for d = 3) have arbitrary large +chromatic number, while being triangle free [27]. We give several struc- +tural properties of these graphs, and those raise many questions. These +graphs have the particular feature to serve as a counter-example for a +conjecture in [2]. +1 +Introduction +A lot of graph classes studied in the literature are defined by a geometric mod- +els, where vertices are represented by geometric objects (e.g. intervals on a line, +disks in the plane, chords inscribed in a circle...) and the adjacency of two ver- +tices is determined according to the relation between the corresponding objects. +A large amount of graph classes consider the intersection relation (e.g. interval +graphs, disk graphs or circle graphs). However some other relations might be +considered such as the containment, the overlap or also the contact between +objects. Recently several groups of authors started to study graph classes de- +fined by contact models, as for example Contact of Paths in a grid (CPG) [11], +Contact of L shapes in R2 or even contact of triangles in the plane [10]. In +this note we consider a new class defined by a contact model. More precisely +we consider the class of graphs defined by contact of axis parallel boxes in Rd +where the contact occurs on (d − 1)-dimensional object in only one direction +(CBU) . +When considering graph defined by axis-parallel boxes in Rd and the ad- +jacency relation is given by the intersection it corresponds to the important +∗This research is partially supported by the ANR GATO, under contract ANR-16-CE40- +0009 and ANR project GRALMECO (ANR-21-CE48-0004-01) +1 +arXiv:2301.03865v1 [cs.DM] 10 Jan 2023 + +notion of boxicity introduced by Roberts [32], when the adjacency relation is +given by the containment relation it correspond to comparability graphs and it +is connected to the poset dimension introduced by Dushnik & Miller [13] +The motivation for this class of graph originate from an article of Magnant +and Martin [27] where an wireless channel assignment is considered. The prob- +lem consider rectangular rooms in a building and ask to find a channel assign- +ment for each room. In order to avoid interferences rooms sharing a same wall, +floor or ceiling need to use different channels. The question was to determine +whether a constant number of channel would suffice to answer this problem. A +first negative answer was provided by Reed and Allwright [31] that a constant +number of channels is not sufficient. Magnant and Martin strengthened their re- +sult that for any integer k there exist a building that require exactly k channels. +In addition their construction only require floor-ceiling contact. +We provide the first structural properties of this class. We first establish +some links between the well known concept of boxicity, then we consider the +recognition problem and we prove that it is actually NP-complete to determine +if a graph is d-CBU for any integer d ≥ 3. Then we provide a characterization +of the class of general CBU. This characterization is expressed in terms of an +acyclic orientation. Thanks to this characterization it is immediate to realize +that the class of CBU constitutes a proper sub-class of Hasse diagram graph (A +Hasse diagram graph, is the undirected graph obtained from a Hasse diagram +associated to a poset). Then we prove that several well studied optimization +problems remains NP-hard on either 2- or 3-CBU graphs. +2 +Preliminaries +We consider Rd and d orthogonal vectors e1, . . . , ed and we introduce a new +class of geometric intersection graphs. Here, the vertices correspond to interior +disjoint d-dimensional axis-parallel boxes in Rd, and two such boxes are only +allowed to intersect on a (d − 1)-dimensional box orthogonal to e1. This class +of graphs is denoted by d-CBU, for Contact graphs of d-dimensional Boxes with +Unidirectional contacts. We denote CBU the union of d-CBU for all d. +Note that 1-CBU correspond to the forests of paths. +Claim 1 For every d ≥ 1, d-CBU graphs are triangle free. +Indeed, note that orienting the edges according to vector e1 and labeling each +arc with the coordinate of the corresponding (d − 1)-hyperplane, one obtains +an acyclic orientation such that for every vertex, all the outgoing arcs have the +same label, all the ingoing arcs have the same label, and the label of ingoing arcs +is smaller than the label of outgoing arcs. We call such a labeling of the arcs +an homogeneous arc labeling. Note that an oriented cycle cannot admit such a +labeling. A triangle abc oriented acyclically is, up to automorphism, such that +d+(a) = 2, d+(b) = 1, and d+(c) = 0. Now ab and ac should have the same +label, such as ac and bc, but ab and bc should be distinct, a contradiction. Thus +2 + +1 +2 +3 +1 +3 +x +x +x +x +x +(i) +(ii) +(iii) +Figure 1: (i) Example of a good orientation of a C5 with some valid labels, (ii) +example of bad orientation. Once a label x is fixed for one arc, this label is +propagated to all the arcs leading to the conclusion that x < x. (iii) the two +valid orientations of a C4 +a triangle cannot admit an homogeneous arc labeling. This completes the proof +of the claim. +With similar arguments one obtains the following for short cycles. See Fig- +ure 1. +Claim 2 For any homogeneous arc labeling of a graph G, its restriction to a +short cycle is as follows. +• For a 4-cycle, the orientation is either such that there are two sources and +two sinks, or it is such that there is one source and one sink linked by two +oriented paths of length 2. +• For a 5-cycle, the orientation is such that there is one source and one sink +linked by two oriented paths, one of length 2 and one of length 3. +3 +Relation with Cover Graphs +An undirected graph is a cover graph if it is the underlying graph of the Hasse +diagram of some partial order. +It was shown by Brightwell [6] and also by +Nešetřil and Rödl [28, 29, 30] that deciding whether a graph is a cover graph is +NP-complete. However, they came up with a simple characterization in terms +of acyclic orientations. +Their characterization states that a graph is a cover graph if and only if +there exists an acyclic orientation without quasi-cycle. A quasi-cycle, being an +orientation of a cycle (v1, v2, . . . , vn) with the arcs (vi, vi+1) for all 1 ≤ n−1 plus +the arc (v1, vn). From an homogeneous arc labelling, the orientation provided +by the labelling clearly fulfills the above defined condition. +Claim 3 For any homogeneous arc labeling of a graph G, the orientation of G +does not contain any quasi-cycle. +Corollary 4 The class of CBU graphs is contained in the class of cover graphs. +3 + +From the previous result, it is natural to ask whether both classes are equiv- +alent. The following remark provides an answer. +Remark 5 The class of CBU graphs is strictly contained in the class of cover +graphs. In Lemma 16 we will exhibit a graph that is not CBU but is a cover +graph. +We will see in the following that an orientation of a graph G, fulfilling the con- +ditions of Claim 2 and of Claim 3 may not admit an homogeneous arc labeling. +4 +Boxicity +The boxicity box(G) of a graph G, is the minimum dimension d such that G +admits an intersection representation with axis-aligned boxes. Of course, the +graphs in d-CBU have boxicity at most d. The converse cannot hold for the +graphs containing a triangle, as those are not in CBU. However, some relations +hold for triangle-free graphs. Let us begin with bipartite graphs. +Theorem 6 Every bipartite graphs of boxicity b belongs to (b + 1)-CBU. +Proof. +Consider a bipartite graph G with vertex sets A and B. Consider a +boxicity b representation of G and slightly expand each box in such a way that +the intersection graph remains unchanged (we stop the expansion of the boxes +before creating new intersections). Now, any two intersecting boxes intersect +on a b-dimensional box. Assume that this representation is drawn in the space +spanned by e2, . . . , eb+1, and let us set for the first dimension (spanned by +e1) that the vertices of A and B, correspond to the intervals [0, 1] and [1, 2], +respectively. As A and B are independent sets, it is clear that the boxes in the +representation are interior disjoint and that any two intersecting boxes intersect +on a b-dimensional box orthogonal to e1. The obtained representation is thus a +(b + 1)-CBU representation of G. +2 +Theorem 6 does not extend to triangle-free graphs. We will see in the fol- +lowing section that there exists triangle-free graphs with bounded boxicity that +are not d-CBU, for any value d. Actually, Lemma 16 tells that there exists such +graphs with girth 5. In other words, for a 3 ≤ g ≤ 5, there is no function fg such +that every graph G, of girth at least g and of boxicity b belongs to (fg(b))-CBU. +Problem 7 For g ≥ 6, is there a function fg such that every graph G, of girth +at least g and of boxicity b belongs to (fg(b))-CBU? +By Theorem 12, we know that if f6 exists, then f6(2) is at least 3. Nevertheless, +the following theorem shows that subdividing the edges enables to consider +every triangle-free graph. An intersection representation is said proper if two +objects intersect if and only if some point of the representation belongs to these +2 objects, only. +4 + +Theorem 8 For every graph G having a proper intersection representation with +axis-parallel boxes in Rb, the 1-subdivision of G belongs to (b + 1)-CBU. +Proof. +Consider such a representation of G and slightly expand each box +in such a way that the intersection graph remains unchanged, and any two +intersecting boxes intersect on a b-dimensional box. Assume that this represen- +tation is drawn in the space spanned by e2, . . . , eb+1, and for the first dimension +(spanned by e1) let us consider any vertex ordering, v1, . . . , vn. For the first +dimension, a vertex vi, corresponds to the interval [2i, 2i + 1]. Clearly, none of +these boxes intersect. Let us now add the boxes for the vertices added by subdi- +viding the edges of G. For any edge vivj, in the space spanned by e2, . . . , eb+1, +the expansion ensured that the intersection of vi and vj contains a box Bij, that +does not intersect any other box of the representation. If i < j, the subdivision +vertex of vivj, is represented by [2i + 1, 2j] × Bi,j. The obtained representation +is clearly a (b + 1)-CBU representation of the subdivision of G. +2 +Corollary 9 For every triangle-free graphs G of boxicity b, the 1-subdivision of +G belongs to (b + 1)-CBU. +5 +Planar graphs +While planar graphs have boxicity at most 3 [36, 19, 5], many subclasses of pla- +nar graphs are known to have boxicity at most 2. This is the case for 4-connected +planar graphs [35], and their subgraphs. The subgraphs of 4-connected graphs +include every triangle-free planar graph (see Lemma 4.1 in [21]). As observed +earlier, for those the representation is necessarily proper. For general planar +graphs, the representation in R3 provided in [19] is clearly proper. So Theo- +rem 8 implies the following. +Corollary 10 For every planar graph G, the 1-subdivision of G belongs to 4- +CBU. Furthermore, if G is triangle-free then it even belongs to 3-CBU. +5.1 +2-CBU graphs +One can easily see that 2-CBU graphs are planar graphs, and that every forest +is a 2-CBU graph. Actually, this class contains every triangle-free outerplanar +graph. +Theorem 11 Every triangle-free outerplanar graph is 2-CBU. +Proof. +Let us prove that for any outerplanar graph G, and any facial walk +v1, v2, . . . , vk, vk+1 = v1 of the outerboundary of G (with separating vertices +appearing several times in this walk), there exists a 2-CBU representation of G +such that the higher rectangle with respect to e1 is successively v1, v2, . . . , vk +when increasing e2. We proceedd by induction on the number of vertices in G. +5 + +a +b +c +d +e +f +g +h +h +d +b +c +a +g +e +f +1 +2 +3 +4 +1 +1 +2 +2 +1 +1 +1 +3 +3 +Figure 2: An example of 2-CBU graph and its associated acyclic orientation +a +b +xi +yi +u +v +a +b +xi +yi +u +v +a +b +xi +yi +u +xi +Figure 3: The series-parallel graph G of Theorem 12. +This clearly holds if G has just one edge, or if G is a cycle. Otherwise, there +exists a degree one vertex u with u ̸= v1, vk, a path u1, . . . , ut of length at least +three (i.e. t ≥ 4), or a cycle u1, . . . , ut = u1 of length at least four (i.e. t ≥ 5). +In the first case we can add the box of u in the representation of G \ u obtained +by induction. In the second and third case we can add the boxes of u2, . . . , ut−1 +in the representation of G \ {u2, . . . , ut−1} obtained by induction. +2 +Theorem 12 +There are series-parallel graphs of girth 6 that are not 2-CBU. +Proof. +Consider a graph G with two vertices, a and b, linked by 9 disjoint +ab-paths of length three axiyib, for i ∈ {1, . . . , 9}. Then for each edge xiyi add a +length 5 path from xi to yi. The obtained graph has girth 6 and is series-parallel +(see Figure 3). +Note that in a 2-CBU representation of G, the box of a (resp. b) has at +most four neighbor such that their intersection contains a corner of a (resp. b). +Thus, there exists an i ∈ {1, . . . , 9} such that one side of xi is contained in one +side of a, and one side of yi is contained in one side of b. Now, whatever the +way xi and yi intersect (a side of xi may be contained in a side of yi, or the +other way around, or also their intersection may contain a corner of each box), +it is not possible to have the length 5 xiyi-path (see Figure 3, right). If a side +of xi is contained in a side of yi there is no place left around xi to draw a third +neighbor. If the intersection of xi and yi contains a corner of each xi and a +corner of yi, there is space to draw a third neighbor for these vertices, say u +6 + +W +′ +6 +a +b +c +d +e +f +x +y +a +b +c +d +e +f +x +Figure 4: The graph W ′ +6. +and v respectively, but in that case the uv-path should go around a or b, but it +would intersect the paths axjyjb with j ̸= i. Thus G does not admit a 2-CBU +representation. +2 +Problem 13 Is there a girth g such that every series parallel graph of girth at +least g belongs to 2-CBU ? +For planar graphs, the following theorem shows that such a bound on the +girth does not exist. Let us denote by W 2 +g the double wheel graph, obtained from +a cycle Cg by adding two non-adjacent vertices, each of them being adjacent to +every vertex of Cg. An edge incident to one of these two vertices (i.e., an edge +not contained in Cg) is called a ray. Now, let W ′ +g be the graph obtained from +W 2 +g by subdividing ⌊g/2⌋ times every ray (see Figure 4). This graph is planar +and has girth g. +Theorem 14 +The graph W ′ +g does not belong to 2-CBU. +Proof. +For any 2-CBU representation of the cycle C of length g there is a +rectangle R, for example the one with the leftmost right side, such that none of +the top or bottom side of R is incident to the inner region. It is thus impossible +to connect R with a ray in the inner region. On the other hand, there is no +planar embedding of W ′ +g where C bounds an inner face. +2 +5.2 +3-CBU graphs +Bipartite planar graphs are known to be contact graphs of axis-aligned segments +in R2 [4, 9], and their boxicity is thus at most two. By Theorem 6, we thus have +the following. +Corollary 15 Every bipartite planar graph belongs to 3-CBU. +7 + +a +b +c +d +e +f +a +b +c +d +e +f +g +h +g +h +a +b +c +d +e +f +G1 +G2 +a +b +c +d +e +f +G3 +g +h +i +k +j +a +b +c +d +e +f +g +h +i +k +j +∗ +∗ +a +b +c +d +e f +Hasse Diagram of G3 +g h +i +j +Figure 5: The graph G3 is planar and non CBU. It is however a cover graph. +The following lemma tells us that this property does not generalize, in a strong +sense, to triangle-free planar graphs. +Lemma 16 +There exists a girth 4 planar graph that is not CBU. +Proof. +Let us consider the graph G1 represented in Figure 5. One can show +that this graph does not admit any valid CBU orientation where in the C4 +induced by a, b, c and d, a is a source and d is a sink (nor the converse). Let us +assume that there exist an CBU orientation such that a is a source and d is a +sink. By fixing the orientation of edges a, b and a, c from a to b and from a to c +respectively. It forces to orient the edges b, e from b to e and the edge c, f from +c to f. The edge e, f is oriented in any direction, w.l.o.g. let us say from e to +f. But in that case, in the C5 induced by b, e, f, c and d it contains two sources +and two sinks which is not a valid CBU orientation. +By adding a path of length 3 between a and d, we obtain the graph G2. +8 + +From the previous observation, we can conclude the same property for vertices +b and c. Hence, in the valid orientation of G2 a and d are sources and b and +c are sinks (or the converse). Let us now consider the graph G3 obtained by +gluing two copies of G2 in a special manner (see Figure 5). Let us remark that +the graph obtained is planar. Let us consider, w.l.o.g., that a and d are sources +and b and c are sinks in the C4 induced by a, b, c and d. From the fact that a +C5, in a valid orientation, only admit one source and one sink. We can deduce +that for the C5 induced by the vertices a, c, d, k and j the c has to be a sink +from the already fixed orientation. Hence, it forces the edge a, j to be directed +from j to a and the edge d, k from k to d (the edge k, j can be oriented in any +direction), since the length of a path from a source to a sink in an orientation +is exactly 2 for one path and 3 for the other. +Then in the partial orientation obtained, we can conclude that in the C4 +induced by a, c, i and j. The vertex c will be a sink and vertex j will be a +source. However, as mentioned in the beginning of this proof, this orientation +will not lead to valid orientation, since j and c plays the same role as a and d +in G1. Hence, G3 does admit a valid CBU orientation. +2 +Remark 17 The graph G3 used in the proof of Lemma 16 is actually a cover +graph. In Figure 5, the bottom picture depicts its Hasse diagram. +Since every planar graph with girth at least 10 has circular chromatic num- +ber at most 5/2 [15], the forthcoming Theorem 35 implies that such a graph +necessarily belongs to CBU. +Problem 18 What is the lowest g ∈ [5, . . . , 10] such that every planar graph G +with girth at least g belongs to CBU. +6 +Structural properties of d-CBU and CBU +Theorem 19 For every d ≥ 1, the class of d-CBU graphs is strictly contained +in the class of (d + 1)-CBU graphs. +For d = 1, the theorem follows from the earlier observation that 1-CBU corre- +spond to forests of paths, and from the many examples of 2-CBU graphs pro- +vided above. For d ≥ 2, the following structural lemma allows us to translate +the strict containment of boxicity b bipartite graphs, to the strict containment of +d-CBU. Indeed, it is known that the graph obtained from the complete bipartite +graph K2b,2b by removing a perfect matching has boxicity exactly b [34]. +Lemma 20 Given a connected bipartite graph B, with parts X and Y , let B′ +be the graph obtained from B, by adding a path xzy and by connecting x and y +to every vertex in X and Y , respectively. Then, B has boxicity at most d if and +only if B′ belongs to (d + 1)-CBU. +9 + +Proof. +Let us begin with the simpler "only if" part. We proceed as in the +proof of Theorem 6 in order to obtain (d + 1)-CBU representation of B such +that every vertex of X (resp. +Y ) corresponds to [0, 1] (resp. +[1, 2]) in the +space spanned by e1. Then it suffices to add the boxes for x, y and z. For a +sufficiently large Ω, x is represented by [−1, 0] × [−Ω, +Ω] × . . . × [−Ω, +Ω], y +is represented by [2, 3] × [−Ω, +Ω] × . . . × [−Ω, +Ω], and z is represented by +[0, 2] × [Ω − 1, Ω] × . . . × [Ω − 1, Ω]. +For the "if" part, consider a (d + 1)-CBU representation of B′, and the +homogeneous arc labeling of B′ induced by this representation. We first prove +that all the arcs between X and Y are oriented in the same direction. Towards +a contradiction, consider a path x1y2x3 with x1, x3 ∈ X and y2 ∈ Y , and such +that the edges are oriented from x1 to y2, and from y2 to x3. This forces the +remaining edges of the 4-cycle yx1y2x3 to be oriented from x1 to y, and from y +to x3 (see Claim 2). Now we cannot orient the edge y2x, xz, zy in such a way to +fulfill Claim 2 for the 5-cycles xzyx1y2 and xzyx3y2. Indeed for the first one, yz +should be oriented from y to z, while for the second one it should be oriented +from z to y, a contradiction. +This orientation ensures that the labels of all the arcs is the same. This +implies that there is an hyperplane H orthogonal to e1 such that for any pair +of intersecting boxes x′ ∈ X and y′ ∈ Y , their intersection belongs to H. This +implies that projecting the (d + 1)-CBU representation (restricted to B) along +e1 leads to a boxicity d representation of B. +2 +It is clear that CBU is hereditary (i.e. closed under induced subgraphs) but +actually it is also closed under subgraphs. +Theorem 21 +For any subgraph H of G, G ∈ CBU implies that H ∈ CBU. +More precisely, if there is a complete bipartite graph Ka,b such that V (Ka,b) ⊆ +V (G), and such that E(H) = E(G) \ E(Ka,b), then if G belongs to d-CBU then +H belongs to (d + 1)-CBU. +Proof. +Let A, B be the parts of Ka,b. Given a CBU representation of G in Rd +we are going to build a CBU representation of H in Rd+1. For this, the first d +intervals defining each d-box remain unchanged while the last interval is [0, 1] +for the vertices in A, [2, 3] for the vertices in B, and [0, 3] for the remaining +vertices. It is now easy to check that two boxes intersect if and only if they +intersect in G and if they are not adjacent in Ka,b. It is also clear that the +intersections occur on planes orthogonal to e1. +2 +The graph class CBU is also closed by the addition of false twins. +Theorem 22 +For any graph G and any vertex v of G, consider the graph Gv +obtained from G by adding a new vertex v′ such that N(v′) = N(v). Then G ∈ +CBU if and only if Gv ∈ CBU. Furthermore, if G ∈ d-CBU then Gv ∈ (d + 1)- +CBU. +Proof. +The "if" part is obvious as G is an induced subgraph of Gv. For the +"only if" part, given a CBU representation of G in Rd we are going to build a +10 + +CBU representation of Gv in Rd+1. For this, the first d intervals defining each +d-box remain unchanged, and those of v′ are the same as those of v. The last +interval is [0, 1] for v, [2, 3] for v′, and [0, 3] for all the remaining vertices. It is +now easy to check that two boxes intersect if and only if they intersected and if +one of them is distinct from v or v′. It is also clear that the intersections occur +on planes orthogonal to e1. +2 +Shift graphs were introduced by P. Erdős and A. Hajnal in [18] (see Theorem +6 therein). Those are the graphs Hm whose vertices are the ordered pairs (i, j) +satisfying 1 ≤ i < j ≤ m, and where two pairs (i, j) and (k, l) form an edge if and +only if j = k or l = i. Note that such graphs admit a homogeneous arc labeling +ℓ defined by ℓ({(i, j), (j, k)}) = j, and by orienting any edge {(i, j), (j, k)} from +(i, j) to (j, k). +Theorem 23 +The graph Hm belongs to (m−1)-CBU. Furthermore, Hm has a +CBU representation such that in the first dimension the vertex (i, j) corresponds +to interval [i, j]. +Proof. +This clearly holds for the one vertex graph H2. By induction on m +consider a representation of Hm−1, add a false twin for every vertex (i, m − 1) +and modify the first interval of these new twins, so that the interval [i, m − 1] +becomes [i, m]. These boxes correspond to the vertices (i, m) with i < m−1. For +the vertex (m−1, m), one should add a box [m−1, m]×[−Ω, +Ω]×. . .×[−Ω, +Ω], +for a sufficiently large Ω. To deal with the intersections between this box and +the boxes of the other vertices (i, m), we add a new dimension such that vertex +(m − 1, m) has interval [1, 2], the vertices (i, m − 1) have interval [1, 2], the +vertices (i, m) with i < m− 1 have interval [3, 4], and all the other vertices have +interval [1, 4]. +2 +Theorem 24 +For every n-vertex graph G the following properties are equiva- +lent. +a) G belongs to CBU. +b) G admits an homogeneous arc labeling. +c) G is the subgraph of a graph Ht +m, obtained from the shift graph Hm by +iteratively adding t false twins, for some values m, t such that m+t ≤ n+1. +d) G belongs to (2n − 1)-CBU. +Proof. +We have already seen that a) ⇒ b). Let us show b) ⇒ c). Consider an +homogeneous arc labeling of G, with labels in [2, m−1], for the minimum m. By +minimality of m, note that all the labels are used, and thus m − 2 ≤ n − 1. Let +Ht +m be the graph obtained from the shift graph Hm by adding ti,j false twins +of vertex (i, j) if there are ti,j + 1 vertices of G whose incoming arcs are labeled +i, and whose outgoing arcs are labeled j. For the vertices without incoming +(resp. +outgoing) arcs assume that those are labeled 1 (resp. +m). +Consider +11 + +now an injective mapping γ +: +V (G) −→ V (Ht +m), such that any vertex with +incoming and outgoing arcs labeled i, j is mapped to (i, j) or one of its twins. +This mapping ensures us that G is a subgraph of Ht +m. Indeed, for any two +adjacent vertices u, v of G linked by an edge labelled j oriented from u to v, +their incoming and outgoing arcs are labeled i, j and j, k respectively, for some +i < j < k, and thus the vertices γ(u) and γ(v) of Ht +m are adjacent, as they +correspond to or are twins of (i, j) and (j, k). +We now show c) ⇒ d). Consider a graph Ht +m containing G as a subgraph, +for some m, t such that m+t ≤ n+1. By Theorem 23 and Theorem 22 we have +that Ht +m belongs to (m − 1 + t)-CBU, and so to n-CBU. Starting from Ht +m one +can obtain G by successively deleting n − 1 stars K1,b, so by Theorem 21, we +have that G belongs to (2n − 1)-CBU. Finally, d) ⇒ a) is obvious. +2 +It is easy to see that every complete bipartite graph belongs to 3-CBU. By +Theorem 21, removing stars K1,b centered on the smallest part, one obtains +that every n-vertex bipartite graph belongs to (⌊n/2⌋ + 3)-CBU. One can reach +a slightly better bound from Theorem 6, and the fact that for every graph G, +box(G) ≤ ⌊n/2⌋ [32]. +Corollary 25 Every bipartite graph G belongs to CBU. Furthemore, if |V (G)| = +n then G belongs to (⌊n/2⌋ + 1)-CBU. +As already mentioned, some bipartite graphs have arbitrary large boxicity, and +thus there is no fixed d such that every bipartite graph belongs to d-CBU. For +large girth graphs it is a different. +Theorem 26 For any g ≥ 3, there exist graphs of girth g not contained in +CBU. +Proof. +Indeed, for any g ≥ 3 there exist graphs of girth g with fractional +chromatic number at least 4 [17]. (Actually, their fractional chromatic number +is arbitrarily large). By Theorem 39, such graphs cannot belong to CBU. +2 +Nevertheless, the following remains open. +Problem 27 Are there integers d, g such that every girth g graph G of CBU, +belongs to d-CBU? +The remarks above imply that testing if a bipartite graph belongs to CBU is +obvious (computable in constant time), while for girth g graphs the question is +more involved, as CBU has such graphs included and some other excluded. The +following section treats the computational problem of recognizing CBU graphs. +7 +Recognition +Computing the boxicity of a bipartite graph is a difficult problem. It is known +that deciding whether a bipartite graph has boxicity two in NP-complete [25]. +Furthermore, it is proven in [1] that it is not possible to approximate the boxicity +12 + +of bipartite graph within a O(n0.5−ε)-factor in polynomial time, unless NP = +ZPP. By Lemma 20, for every bipartite graph B there is a graph B′ (obtained +in polynomial time) such that the minimum value d such that B′ belongs to +d-CBU, is exactly d = box(B) + 1. +Corollary 28 It is NP-complete to decide whether a graph belongs to 3-CBU. +Furthermore, unless NP = ZPP, one cannot approximate in polynomial time +and within a O(n0.5−ε)-factor, the minimum value d for which an input graph +G belongs to d-CBU. +This implies that for most values d the problem of deciding whether an input +graph belongs to d-CBU, cannot be computed in polynomial time, unless NP = +ZPP. The hypothesis NP = P being stronger than NP = ZPP, it would be +stronger to know that it is NP-complete to decide if an input graph belongs to +d-CBU. +Problem 29 For which values d, is it NP-complete to decide whether a graph +belongs to d-CBU? Are there values d, in particular for d = 2, for which the +problem is polynomial? +By Lemma 20, this problem would be solved, for d ≥ 3, if the following problem +admits a positive answer. +Problem 30 For any d ≥ 3, is it NP-complete to decide whether a bipartite +graph B has boxicity at most d? +Another computational problem is testing the membership in CBU. +Problem 31 Is it polynomial to decide whether a graph belongs to CBU? +We have seen that some triangle-free planar graphs, or some graphs with +arbitrary large girth, are not in CBU. We can thus restrict the problem. +Problem 32 Is it polynomial to decide whether a planar graph G belongs to +CBU? For some g ≥ 3, is it polynomial to decide whether a graph G of girth at +least g belongs to CBU? +7.1 +Recognition through forbidden induced subgraphs +As CBU and d-CBU are closed under induced subgraphs, they are characterized +by a set of minimal excluded induced subgraphs, FCBU and Fd−CBU. If one of +these sets is finite, then recognizing the corresponding class becomes polynomial- +time tractable (and this would also contradict Conjecture 2 of [2]). Thus by +Corollary 28, the set F3−CBU (resp. Fd−CBU for d ≥ 4) is not finite, unless +P = NP (resp. unless NP = ZPP). For the set F2−CBU (resp. FCBU), we +are sure that it is infinite. Indeed, Theorem 14 (resp. Theorem 26) provides an +infinite sequence of graphs (Gi)i≥0 not in 2-CBU (resp. not in CBU) such that +the girth of Gi is at least i. If there was an n such that every graph in F2−CBU +(resp. FCBU) has at most n vertices, then to exclude Gn+1 one would need to +have a tree in F2−CBU (resp. FCBU). This is not the case as for every tree T, +we have that T ∈ 2-CBU ⊆ CBU. +13 + +7.2 +Recognition through homogeneous arc labelings +By Theorem 24, a graph G belongs to CBU if and only if it admits an homo- +geneous arc labeling. If we are given an orientation of a graph G it is simple +to check whether this orientation admits such labeling. For example, one can +use linear programming. For each arc uv, set a variable ℓuv corresponding to a +label, and for any two incident arcs, add a constraint. For two arcs uv and uw +(resp. uv and wv), the constraint is ℓuv = ℓuw (resp. ℓuv = ℓwv). For two arcs +uv and vw, the constraint is ℓuv < ℓvw. Problem 31 thus reduces to deciding +whether a graph G admits an orientation that is homogeneously labelable. In +the following we characterize such orientations. +A cycle (v0, v1, . . . , vn−1) is said badly oriented if there is a vertex vi whose +incident arcs are vi−1vi and vivi+1, and if there is no vertex vj whose incident +arcs are vj+1vj and vjvj−1 (indices being considered modn). +Theorem 33 An orientation of a graph G admits an homogeneous labeling if +and only if there is no badly oriented cycle. +Proof. +For the "only if" part, consider a badly oriented cycle (v0, v1, . . . , vn−1) +with arcs vn−1v0 and v0v1, but with no vertex vj ̸= v0 whose incident arcs are +vj+1vj and vjvj−1. This latter condition implies that in any homogeneous label- +ing the sequence of labels for the edges (without considering their orientation) +v0v1, v1v2, . . . , vn−2vn−1, vn−1v0 is non-decreasing, while the former condition +implies that the label of v0v1 is greater than the one of vn−1v0, a contradiction. +Thus this orientation of G does not allows any homogeneous labeling. +For the "if" part, consider a graph G oriented without badly oriented cycle, +and consider a source u, and let us denote v1, . . . , vn its out-neighbors. If for +every vertex vi, u is its unique in-neighbor, then by recurrence on the number +of vertices we assume that G \ {u} has a homogeneous labeling, and we label +the arcs incident to u with a sufficiently small value, say −Ω. In that case it is +easy to check that this labeling is homogeneous. +Otherwise, let vi and u′ be vertices such that G has arcs from both u and +u′ toward vertex vi. In that case, consider the oriented graph G′ obtained from +G \ {u} by adding the arcs u′v1, . . . , u′vn, if missing. +Claim 34 G′ has no badly oriented cycle. +Proof. +If G′ had a badly oriented cycle C, this one should go through a newly +added arc u′vj. If vi /∈ C, by replacing the arc u′vj by the path (u′, vi, u, vj) one +would obtain a badly oriented cycle in G, a contradiction. We thus assume that +vi /∈ C, and now by replacing the arc u′vj by the path (u′, vi, u, vj) we obtain +a badly oriented closed walk W (that is a walk where there are consecutive +"forward" arcs, but no consecutive "backward" arcs). +Let us denote P and +P ′ the sub-paths of C \ {u′vj} ⊊ G linking vi and vj, and linking u′ and vi, +respectively. +Let us show that if the edge incident to vi in P ′ is oriented from vi to the +other end, denoted v, then this arc is backward with respect to C. Indeed, the +cycle CP ′ of G formed by P ′ and the arc u′vi, has consecutive arcs oriented in +14 + +the same direction, u′vi and viv, and (as G contains no badly oriented cycles) +has consecutive arcs oriented in the other direction. +The latter pair of arcs +belonging both to P ′ ⊂ C, they are forward with respect to C, thus viv is +backward. +Similarly, let us show that if the edge incident to vi in P is oriented from +vi to the other end, denoted w, then this arc is backward with respect to C. +Indeed, they cycle CP of G formed by P and the arcs uviand uvj, has consecutive +arcs oriented in the same direction, uvi and viw, and (as G contains no badly +oriented cycles) has consecutive arcs oriented in the other direction. The latter +pair of arcs belong both to P ⊂ C, or they are the arcs incident to vj. In the +former case, these arcs are forward with respect to C, thus viv is backward. In +the latter case, replacing uvj with u′vj, one has that the incident arcs of vj in C +are oriented in the same direction. this direction is thus the forward direction, +and in that case also viv is backward. +We thus have that the arcs incident to vi cannot be oriented in the same di- +rection (they would form consecutive backward arcs in C), and they are not both +oriented from vi to the other end (they would be both backwards although they +have distinct directions). Now we distinguish cases according to the position of +the consecutive forward arcs in C. We have that: +a) there are two consecutive forward arcs in P ∪ {u′vj}, or +b) there are two consecutive forward arcs in P ′ ∪ {u′vj}. +In case a), the cycle CP of G has consecutive forward arcs (by replacing if +necessary the arc u′vj with uvj). Since this cycle is not badly oriented it also +contains consecutive backward arcs. According to the orientation of the arcs, +those backwards arcs cannot be the arcs incident to u, or those incident to vi. +Thus they belong both to P ∪ {uvj}, but this would imply that C also contains +consecutive backward arcs, a contradiction. +In case b), the cycle CP ′ of G has consecutive forward arcs (by replacing if +necessary the arc u′vj with u′vi). Since this cycle is not badly oriented it also +contains consecutive backward arcs. According to the orientation of the arcs, +those backwards arcs cannot be the arcs incident to vi. This would imply that +C also contains consecutive backward arcs, a contradiction. +This concludes the proof of the claim +2 +So now, by recurrence on the number of vertices we can assume that G′ has +a homogeneous labeling, and let ℓ be the label of the arcs outgoing from u′. In +that case one can derive a labeling of G by keeping the same labels, and by +setting the label ℓ for the arcs outgoing from u. It is easy to check that this +labeling is homogeneous. +2 +Note that Theorem 33 provides another proof that CBU contains every bi- +partite graph. Indeed, orienting all the edges from one part toward the other, +the direction of the arcs alternate along any cycle, and so there is no badly +oriented cycle. Actually, we can go a little further. +15 + +Theorem 35 Every graph G with circular chromatic number χc(G) ≤ 5/2 be- +longs to CBU. +Proof. +A graph G with circular chromatic number χc(G) ≤ 5/2 has a ho- +momorphism into the circular complete graph K5/2 that is the 5-cycle. As this +graph belongs to CBU the theorem follows from Theorem 37. +2 +Note that we cannot replace 5/2 by 8/3 in Theorem 35, as one can easily +check that every orientation of K8/3 contains a badly oriented cycle. +Problem 36 What is the largest c such that every graph G with χc(G) ≤ c (or +with χc(G) < c) belongs to CBU. +Theorem 37 Given two graphs G, H such that there is an homomorphism γ : +V (G) −→ V (H), then if H ∈ CBU we have that G ∈ CBU. +Proof. +By Theorem 24, the graph H admits an homogeneous arc labeling, +ℓH. Orient the edges of G in such a way that uv ∈ E(G) is oriented as the +edge γ(u)γ(v) ∈ E(H), that is from u to v if and only if γ(u)γ(v) is oriented +from γ(u) to γ(v) in H. Similarly we copy the labeling of H’s arcs by setting +ℓG(uv) = ℓH(γ(u)γ(v)). One can easily check that this is an homogeneous arc +labeling of G, and thus that G belongs to CBU. +2 +8 +Chromatic Number and Independent Sets +While 2-CBU graphs have chromatic number at most 3 (by Grötzsch’s theorem), +3-CBU graphs have unbounded chromatic number. +Theorem 38 (Magnant and Martin [27]) For any χ ≥ 1, there exist a +graph in 3-CBU, with chromatic number χ. +However, these graphs have bounded fractional chromatic number, and thus +have linear independent sets. Indeed, G. Simonyi and G. Tardos [33] showed +that shift graphs have fractional chromatic number less than 4. As such a bound +extends by adding a false twin and by taking a subgraph, we have the following. +Theorem 39 +For any graph G ∈ CBU, χf(G) < 4, and α(G) > |V (G)|/4. +For planar graphs in CBU, this bound on χf can be improved by one, but +not more. +Theorem 40 +For every planar graph G in CBU we have χf(G) ≤ χ(G) ≤ 3. +On the other hand, for every n ≡ 2 (mod 3) there is a n-vertex planar graph G +in CBU such that α(G) = (n + 1)/3, and thus χf(G) ≥ n/α(G) = 3 − +3 +n+1. +Proof. +The first statement follows from Grötzsch’s theorem. +The second +statement follows from graphs constructed by Jones [23], which were proved +to have independence number α(G) = (n + 1)/3. +Those graphs form a se- +quence J1, J2, . . . such that J1 is the 5-cycle (a1, b1, c1, d, e), and such that +16 + +J1 +Ji+1 +d +b1 +Ji +ci +bi+1 +ci+1 +c1 +a1 +e +0 +2 +1 +0 +2i +2i +2i + 2 +2 +2i +2i + 1 +2i +2i + 2 +ai+1 +ai +bi +Figure 6: The Jones graphs J1 and Ji+1, with a homogeneous arc labeling. For +every i ≥ 1, this embedding is such that the path aibici is on the outer-boundary. +Thus, adding vertices ai+1, bi+1, ci+1 does not break planarity. +Ji+1 is obtained from Ji by adding three vertices ai+1, bi+1, ci+1 such that +N(ai+1) = {bi, bi+1}, N(bi+1) = {ai+1, ci+1}, and N(ci+1) = {ai, ci, bi+1} (see +Figure 6). It is already known that those graphs are planar, and it does only +remain to show that they belong to CBU. Let us do so by exhibiting a homoge- +neous arc labeling ℓ. This labeling is such that for any i ≥ 1 we orient the edges +aibi and bici toward bi, we orient the edges xiyi+1, for x, y ∈ {a, b, c}, from xi +towards yi+1, and we set ℓ(aibi) = ℓ(aici+1) = 2i, ℓ(cibi) = ℓ(cici+1) = 2i, and +ℓ(biai+1) = 2i + 1. By examining Figure 6 it is clear that this is a homogeneous +arc labeling. +2 +Although 2-CBU lies in the intersection of CBU and planar graphs, it might +be the case that the fractional chromatic number of graphs in 2-CBU is bounded +by some c < 3. Indeed, Jones graphs Ji, for a sufficiently large i, seem to not +be in 2-CBU. +Problem 41 Is there a c < 3 such that every graph G in 2-CBU has fractional +chromatic number χf(G) ≤ c ? +A positive answer to this question, would give support to two conjectures. Let +Pg≥5 be the set of planar graph with girth at least five, and let Pf +g≥4 be the +set of planar graph with girth at least four, where every 4-cycle bounds a face. +Clearly Pg≥5 ⊊ Pf +g≥4, since these classes avoid Jones graphs it is conjectured +that graphs in Pg≥5, or more generally graphs in Pf +g≥4, have fractional chromatic +number at most c, for some c < 3 [14, 16]. However, our problem is not a sub- +case of these conjectures (as K2,t belongs to 2-CBU \ Pf +g≥4), nor a super-case +(as Pg≥5 \ 2-CBU is not empty, by Theorem 14). +9 +Computational hardness for many problems +We have seen (c.f. Theorem 8, Corollary 9, and Corollary 10) that many 1- +subdivided graphs belong to CBU, or even to 3- or 4-CBU. For (≥ 2)-subdivided +graphs, the picture is even simpler. +17 + +... +v1 +v2 +v3 +vn +e2 +e3 +e1 +Figure 7: Construction of a 3-CBU representation of a 2-subdivision of a graph. +Theorem 42 For every graph G, if we subdivide every edge at least twice, the +obtained graph belongs to 3-CBU. +Proof. +Let us denote v1, . . . , vn the vertices of G, and let m = |E(G)|. To +construct a CBU representation for any (≥ 2)-subdivision, we start by assigning +each vertex vi to the box [3i, 3i + 1] × [n − i, n − i + 1] × [0, 2m]. Then consider +each edge e of G in any given order. For the kth edge e assume it links vi and +vj, for some i < j, and assume e is replaced by the path (vi, u1, . . . , ur, vj) for +some r ≥ 2. Here, u1 is assigned to [3i+1, 3i+2]×[n−j, n−i+1]×[2k−1, 2k], +while the vertices uℓ with 2 ≤ ℓ ≤ r are assigned to [3i + 2 + (ℓ − 2)(3j − 3i − +2)/(r−1), 3i+2+(ℓ−1)(3j −3i−2)/(r−1)]×[n−j, n−j +1]×[2k−1, 2k] (see +Figure 7). One can easily check that the obtained representation is a 3-CBU +representation of the subdivided graph. +2 +Corollary 43 The problems of Minimum Feedback Vertex Set and Cutwidth +are NP-hard even when restricted to 3-CBU graphs. The problems Maximum +Cut, Minimum Vertex Cover, Minimum Dominating Set, and Minimum +Independent Dominating Set are APX-hard even when restricted to 3-CBU +graphs. +Proof. +For Minimum Feedback Vertex Set and Cutwidth, this follows +from the fact that these problems are NP-hard, and that for any instance, +subdividing an edge does not change the solution. For Maximum Cut, it follows +from its APX-hardness and the fact that the maximum cut of a graph G and +its 2-subdivision G2-sub verify mc(G) = mc(G2-sub) − 2|E(G)| and 3|E(G)|/2 = +|E(G2-sub)|/2 ≤ mc(G2-sub) ≤ |E(G2-sub)| = 3|E(G)|. The other problems are +shown APX-hard even when restricted to 6-subdivided graphs [7]. +2 +18 + +(1,1) +(1,2) +(2,1) +Figure 8: 2-CBU representation of the 4 × 4 grid. +When restricted to 2-CBU some of these problems become simpler to han- +dle, as every graph in 2-CBU is planar. Indeed, the Maximum Cut problem +turns out to be polynomial time solvable [12], while Minimum Vertex Cover, +Minimum Dominating Set, and Minimum Independent Dominating Set +admit PTAS [3, 26] (with standard techniques), such as Minimum Feedback +Vertex Set [24]. However, many problems remain NP-hard when restricted +to 2-CBU. +Theorem 44 The problems Maximum Independent Set, Minimum Ver- +tex Cover, Minimum Dominating Set, Hamiltonian Path, and Hamil- +tonian cycle are NP-complete, even when restricted to 2-CBU graphs. +Proof. +As these problems belong to NP, it remains to show that they are +NP-hard for 2-CBU graphs. Let us first show that the induced subgraphs of +grids (so called grid graphs) belong to 2-CBU. Consider the n × n grid G such +that V (G) = {1, . . . , n} × {1, . . . , n}, and such that the neighbors of any vertex +(i, j) are {(i, j −1)(i−1, j), (i, j +1), (i+1, j)}∩{1, . . . , n}×{1, . . . , n}. Since it +suffices to delete some boxes to obtain an induced subgraph, the claim follows by +constructing a 2-CBU representation for any such grid G. This construction is +obtained by mapping any vertex (i, j) to the box [i+j−1, i+j]×[2i−2j, 2i−2j+3] +(see Figure 8). As Domination [8], Hamiltonian Path, and Hamiltonian +cycle [22] are NP-hard for grid graphs, those problems are NP-hard for 2-CBU +graphs. +For the problems Maximum Independent Set and Minimum Vertex +Cover, we have to consider a variant of grid graphs, the graph R′(n1, n2) +depicted in Figure 9, and it is easy to see how to modify the construction above +in order to obtain a 2-CBU representation of this type of graphs. 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B, 40:9–20, 1986. +22 + diff --git a/Z9E2T4oBgHgl3EQfZgdy/content/tmp_files/load_file.txt b/Z9E2T4oBgHgl3EQfZgdy/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e1adcd3f6763fa354b039ac918289754b4b8b08d --- /dev/null +++ b/Z9E2T4oBgHgl3EQfZgdy/content/tmp_files/load_file.txt @@ -0,0 +1,710 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf,len=709 +page_content='Contact graphs of boxes with unidirectional contacts∗ Daniel Gonçalvesa, Vincent Limouzyb, and Pascal Ochema aLIRMM, Univ Montpellier, CNRS, Montpellier, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' bUniversité Clermont Auvergne, Clermont Auvergne INP, CNRS, Mines Saint-Etienne, Limos, F-63000 Clermont-Ferrand, France Abstract This paper is devoted to the study of particular classes of geometri- cally defined intersection graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Those are contact graphs of axis parallel boxes in Rd, where the intersection of any pair of boxes is parallel to a given hyperplane (among the d possible ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Magnant and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Martin showed that these graphs (already for d = 3) have arbitrary large chromatic number, while being triangle free [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' We give several struc- tural properties of these graphs, and those raise many questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' These graphs have the particular feature to serve as a counter-example for a conjecture in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 1 Introduction A lot of graph classes studied in the literature are defined by a geometric mod- els, where vertices are represented by geometric objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' intervals on a line, disks in the plane, chords inscribed in a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=') and the adjacency of two ver- tices is determined according to the relation between the corresponding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' A large amount of graph classes consider the intersection relation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' interval graphs, disk graphs or circle graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' However some other relations might be considered such as the containment, the overlap or also the contact between objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Recently several groups of authors started to study graph classes de- fined by contact models, as for example Contact of Paths in a grid (CPG) [11], Contact of L shapes in R2 or even contact of triangles in the plane [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' In this note we consider a new class defined by a contact model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' More precisely we consider the class of graphs defined by contact of axis parallel boxes in Rd where the contact occurs on (d − 1)-dimensional object in only one direction (CBU) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' When considering graph defined by axis-parallel boxes in Rd and the ad- jacency relation is given by the intersection it corresponds to the important ∗This research is partially supported by the ANR GATO, under contract ANR-16-CE40- 0009 and ANR project GRALMECO (ANR-21-CE48-0004-01) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='03865v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='DM] 10 Jan 2023 notion of boxicity introduced by Roberts [32], when the adjacency relation is given by the containment relation it correspond to comparability graphs and it is connected to the poset dimension introduced by Dushnik & Miller [13] The motivation for this class of graph originate from an article of Magnant and Martin [27] where an wireless channel assignment is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' The prob- lem consider rectangular rooms in a building and ask to find a channel assign- ment for each room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' In order to avoid interferences rooms sharing a same wall, floor or ceiling need to use different channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' The question was to determine whether a constant number of channel would suffice to answer this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' A first negative answer was provided by Reed and Allwright [31] that a constant number of channels is not sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Magnant and Martin strengthened their re- sult that for any integer k there exist a building that require exactly k channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' In addition their construction only require floor-ceiling contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' We provide the first structural properties of this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' We first establish some links between the well known concept of boxicity, then we consider the recognition problem and we prove that it is actually NP-complete to determine if a graph is d-CBU for any integer d ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Then we provide a characterization of the class of general CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' This characterization is expressed in terms of an acyclic orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Thanks to this characterization it is immediate to realize that the class of CBU constitutes a proper sub-class of Hasse diagram graph (A Hasse diagram graph, is the undirected graph obtained from a Hasse diagram associated to a poset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Then we prove that several well studied optimization problems remains NP-hard on either 2- or 3-CBU graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 2 Preliminaries We consider Rd and d orthogonal vectors e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , ed and we introduce a new class of geometric intersection graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Here, the vertices correspond to interior disjoint d-dimensional axis-parallel boxes in Rd, and two such boxes are only allowed to intersect on a (d − 1)-dimensional box orthogonal to e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' This class of graphs is denoted by d-CBU, for Contact graphs of d-dimensional Boxes with Unidirectional contacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' We denote CBU the union of d-CBU for all d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Note that 1-CBU correspond to the forests of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Claim 1 For every d ≥ 1, d-CBU graphs are triangle free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Indeed, note that orienting the edges according to vector e1 and labeling each arc with the coordinate of the corresponding (d − 1)-hyperplane, one obtains an acyclic orientation such that for every vertex, all the outgoing arcs have the same label, all the ingoing arcs have the same label, and the label of ingoing arcs is smaller than the label of outgoing arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' We call such a labeling of the arcs an homogeneous arc labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Note that an oriented cycle cannot admit such a labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' A triangle abc oriented acyclically is, up to automorphism, such that d+(a) = 2, d+(b) = 1, and d+(c) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Now ab and ac should have the same label, such as ac and bc, but ab and bc should be distinct, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Thus 2 1 2 3 1 3 x x x x x (i) (ii) (iii) Figure 1: (i) Example of a good orientation of a C5 with some valid labels, (ii) example of bad orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Once a label x is fixed for one arc, this label is propagated to all the arcs leading to the conclusion that x < x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' (iii) the two valid orientations of a C4 a triangle cannot admit an homogeneous arc labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' This completes the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' With similar arguments one obtains the following for short cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' See Fig- ure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Claim 2 For any homogeneous arc labeling of a graph G, its restriction to a short cycle is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For a 4-cycle, the orientation is either such that there are two sources and two sinks, or it is such that there is one source and one sink linked by two oriented paths of length 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For a 5-cycle, the orientation is such that there is one source and one sink linked by two oriented paths, one of length 2 and one of length 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 3 Relation with Cover Graphs An undirected graph is a cover graph if it is the underlying graph of the Hasse diagram of some partial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' It was shown by Brightwell [6] and also by Nešetřil and Rödl [28, 29, 30] that deciding whether a graph is a cover graph is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' However, they came up with a simple characterization in terms of acyclic orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Their characterization states that a graph is a cover graph if and only if there exists an acyclic orientation without quasi-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' A quasi-cycle, being an orientation of a cycle (v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , vn) with the arcs (vi, vi+1) for all 1 ≤ n−1 plus the arc (v1, vn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' From an homogeneous arc labelling, the orientation provided by the labelling clearly fulfills the above defined condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Claim 3 For any homogeneous arc labeling of a graph G, the orientation of G does not contain any quasi-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Corollary 4 The class of CBU graphs is contained in the class of cover graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 3 From the previous result, it is natural to ask whether both classes are equiv- alent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' The following remark provides an answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Remark 5 The class of CBU graphs is strictly contained in the class of cover graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' In Lemma 16 we will exhibit a graph that is not CBU but is a cover graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' We will see in the following that an orientation of a graph G, fulfilling the con- ditions of Claim 2 and of Claim 3 may not admit an homogeneous arc labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 4 Boxicity The boxicity box(G) of a graph G, is the minimum dimension d such that G admits an intersection representation with axis-aligned boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Of course, the graphs in d-CBU have boxicity at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' The converse cannot hold for the graphs containing a triangle, as those are not in CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' However, some relations hold for triangle-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Let us begin with bipartite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Theorem 6 Every bipartite graphs of boxicity b belongs to (b + 1)-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Consider a bipartite graph G with vertex sets A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Consider a boxicity b representation of G and slightly expand each box in such a way that the intersection graph remains unchanged (we stop the expansion of the boxes before creating new intersections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Now, any two intersecting boxes intersect on a b-dimensional box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Assume that this representation is drawn in the space spanned by e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , eb+1, and let us set for the first dimension (spanned by e1) that the vertices of A and B, correspond to the intervals [0, 1] and [1, 2], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' As A and B are independent sets, it is clear that the boxes in the representation are interior disjoint and that any two intersecting boxes intersect on a b-dimensional box orthogonal to e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' The obtained representation is thus a (b + 1)-CBU representation of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 2 Theorem 6 does not extend to triangle-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' We will see in the fol- lowing section that there exists triangle-free graphs with bounded boxicity that are not d-CBU, for any value d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Actually, Lemma 16 tells that there exists such graphs with girth 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' In other words, for a 3 ≤ g ≤ 5, there is no function fg such that every graph G, of girth at least g and of boxicity b belongs to (fg(b))-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Problem 7 For g ≥ 6, is there a function fg such that every graph G, of girth at least g and of boxicity b belongs to (fg(b))-CBU?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' By Theorem 12, we know that if f6 exists, then f6(2) is at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Nevertheless, the following theorem shows that subdividing the edges enables to consider every triangle-free graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' An intersection representation is said proper if two objects intersect if and only if some point of the representation belongs to these 2 objects, only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 4 Theorem 8 For every graph G having a proper intersection representation with axis-parallel boxes in Rb, the 1-subdivision of G belongs to (b + 1)-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Consider such a representation of G and slightly expand each box in such a way that the intersection graph remains unchanged, and any two intersecting boxes intersect on a b-dimensional box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Assume that this represen- tation is drawn in the space spanned by e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , eb+1, and for the first dimension (spanned by e1) let us consider any vertex ordering, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For the first dimension, a vertex vi, corresponds to the interval [2i, 2i + 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Clearly, none of these boxes intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Let us now add the boxes for the vertices added by subdi- viding the edges of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For any edge vivj, in the space spanned by e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , eb+1, the expansion ensured that the intersection of vi and vj contains a box Bij, that does not intersect any other box of the representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' If i < j, the subdivision vertex of vivj, is represented by [2i + 1, 2j] × Bi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' The obtained representation is clearly a (b + 1)-CBU representation of the subdivision of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 2 Corollary 9 For every triangle-free graphs G of boxicity b, the 1-subdivision of G belongs to (b + 1)-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 5 Planar graphs While planar graphs have boxicity at most 3 [36, 19, 5], many subclasses of pla- nar graphs are known to have boxicity at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' This is the case for 4-connected planar graphs [35], and their subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' The subgraphs of 4-connected graphs include every triangle-free planar graph (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='1 in [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' As observed earlier, for those the representation is necessarily proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For general planar graphs, the representation in R3 provided in [19] is clearly proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' So Theo- rem 8 implies the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Corollary 10 For every planar graph G, the 1-subdivision of G belongs to 4- CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Furthermore, if G is triangle-free then it even belongs to 3-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='1 2-CBU graphs One can easily see that 2-CBU graphs are planar graphs, and that every forest is a 2-CBU graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Actually, this class contains every triangle-free outerplanar graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Theorem 11 Every triangle-free outerplanar graph is 2-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Let us prove that for any outerplanar graph G, and any facial walk v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , vk, vk+1 = v1 of the outerboundary of G (with separating vertices appearing several times in this walk), there exists a 2-CBU representation of G such that the higher rectangle with respect to e1 is successively v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , vk when increasing e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' We proceedd by induction on the number of vertices in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 5 a b c d e f g h h d b c a g e f 1 2 3 4 1 1 2 2 1 1 1 3 3 Figure 2: An example of 2-CBU graph and its associated acyclic orientation a b xi yi u v a b xi yi u v a b xi yi u xi Figure 3: The series-parallel graph G of Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' This clearly holds if G has just one edge, or if G is a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Otherwise, there exists a degree one vertex u with u ̸= v1, vk, a path u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , ut of length at least three (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' t ≥ 4), or a cycle u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , ut = u1 of length at least four (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' t ≥ 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' In the first case we can add the box of u in the representation of G \\ u obtained by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' In the second and third case we can add the boxes of u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , ut−1 in the representation of G \\ {u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , ut−1} obtained by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 2 Theorem 12 There are series-parallel graphs of girth 6 that are not 2-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Consider a graph G with two vertices, a and b, linked by 9 disjoint ab-paths of length three axiyib, for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , 9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Then for each edge xiyi add a length 5 path from xi to yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' The obtained graph has girth 6 and is series-parallel (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Note that in a 2-CBU representation of G, the box of a (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' b) has at most four neighbor such that their intersection contains a corner of a (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Thus, there exists an i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , 9} such that one side of xi is contained in one side of a, and one side of yi is contained in one side of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Now, whatever the way xi and yi intersect (a side of xi may be contained in a side of yi, or the other way around, or also their intersection may contain a corner of each box), it is not possible to have the length 5 xiyi-path (see Figure 3, right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' If a side of xi is contained in a side of yi there is no place left around xi to draw a third neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' If the intersection of xi and yi contains a corner of each xi and a corner of yi, there is space to draw a third neighbor for these vertices, say u 6 W ′ 6 a b c d e f x y a b c d e f x Figure 4: The graph W ′ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' and v respectively, but in that case the uv-path should go around a or b, but it would intersect the paths axjyjb with j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Thus G does not admit a 2-CBU representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 2 Problem 13 Is there a girth g such that every series parallel graph of girth at least g belongs to 2-CBU ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For planar graphs, the following theorem shows that such a bound on the girth does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Let us denote by W 2 g the double wheel graph, obtained from a cycle Cg by adding two non-adjacent vertices, each of them being adjacent to every vertex of Cg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' An edge incident to one of these two vertices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=', an edge not contained in Cg) is called a ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Now, let W ′ g be the graph obtained from W 2 g by subdividing ⌊g/2⌋ times every ray (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' This graph is planar and has girth g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Theorem 14 The graph W ′ g does not belong to 2-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For any 2-CBU representation of the cycle C of length g there is a rectangle R, for example the one with the leftmost right side, such that none of the top or bottom side of R is incident to the inner region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' It is thus impossible to connect R with a ray in the inner region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' On the other hand, there is no planar embedding of W ′ g where C bounds an inner face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='2 3-CBU graphs Bipartite planar graphs are known to be contact graphs of axis-aligned segments in R2 [4, 9], and their boxicity is thus at most two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' By Theorem 6, we thus have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Corollary 15 Every bipartite planar graph belongs to 3-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 7 a b c d e f a b c d e f g h g h a b c d e f G1 G2 a b c d e f G3 g h i k j a b c d e f g h i k j ∗ ∗ a b c d e f Hasse Diagram of G3 g h i j Figure 5: The graph G3 is planar and non CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' It is however a cover graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' The following lemma tells us that this property does not generalize, in a strong sense, to triangle-free planar graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Lemma 16 There exists a girth 4 planar graph that is not CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Let us consider the graph G1 represented in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' One can show that this graph does not admit any valid CBU orientation where in the C4 induced by a, b, c and d, a is a source and d is a sink (nor the converse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Let us assume that there exist an CBU orientation such that a is a source and d is a sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' By fixing the orientation of edges a, b and a, c from a to b and from a to c respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' It forces to orient the edges b, e from b to e and the edge c, f from c to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' The edge e, f is oriented in any direction, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' let us say from e to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' But in that case, in the C5 induced by b, e, f, c and d it contains two sources and two sinks which is not a valid CBU orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' By adding a path of length 3 between a and d, we obtain the graph G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 8 From the previous observation, we can conclude the same property for vertices b and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Hence, in the valid orientation of G2 a and d are sources and b and c are sinks (or the converse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Let us now consider the graph G3 obtained by gluing two copies of G2 in a special manner (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Let us remark that the graph obtained is planar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Let us consider, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=', that a and d are sources and b and c are sinks in the C4 induced by a, b, c and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' From the fact that a C5, in a valid orientation, only admit one source and one sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' We can deduce that for the C5 induced by the vertices a, c, d, k and j the c has to be a sink from the already fixed orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Hence, it forces the edge a, j to be directed from j to a and the edge d, k from k to d (the edge k, j can be oriented in any direction), since the length of a path from a source to a sink in an orientation is exactly 2 for one path and 3 for the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Then in the partial orientation obtained, we can conclude that in the C4 induced by a, c, i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' The vertex c will be a sink and vertex j will be a source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' However, as mentioned in the beginning of this proof, this orientation will not lead to valid orientation, since j and c plays the same role as a and d in G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Hence, G3 does admit a valid CBU orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 2 Remark 17 The graph G3 used in the proof of Lemma 16 is actually a cover graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' In Figure 5, the bottom picture depicts its Hasse diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Since every planar graph with girth at least 10 has circular chromatic num- ber at most 5/2 [15], the forthcoming Theorem 35 implies that such a graph necessarily belongs to CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Problem 18 What is the lowest g ∈ [5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , 10] such that every planar graph G with girth at least g belongs to CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 6 Structural properties of d-CBU and CBU Theorem 19 For every d ≥ 1, the class of d-CBU graphs is strictly contained in the class of (d + 1)-CBU graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For d = 1, the theorem follows from the earlier observation that 1-CBU corre- spond to forests of paths, and from the many examples of 2-CBU graphs pro- vided above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For d ≥ 2, the following structural lemma allows us to translate the strict containment of boxicity b bipartite graphs, to the strict containment of d-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Indeed, it is known that the graph obtained from the complete bipartite graph K2b,2b by removing a perfect matching has boxicity exactly b [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Lemma 20 Given a connected bipartite graph B, with parts X and Y , let B′ be the graph obtained from B, by adding a path xzy and by connecting x and y to every vertex in X and Y , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Then, B has boxicity at most d if and only if B′ belongs to (d + 1)-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Let us begin with the simpler "only if" part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' We proceed as in the proof of Theorem 6 in order to obtain (d + 1)-CBU representation of B such that every vertex of X (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Y ) corresponds to [0, 1] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' [1, 2]) in the space spanned by e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Then it suffices to add the boxes for x, y and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For a sufficiently large Ω, x is represented by [−1, 0] × [−Ω, +Ω] × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' × [−Ω, +Ω], y is represented by [2, 3] × [−Ω, +Ω] × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' × [−Ω, +Ω], and z is represented by [0, 2] × [Ω − 1, Ω] × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' × [Ω − 1, Ω].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For the "if" part, consider a (d + 1)-CBU representation of B′, and the homogeneous arc labeling of B′ induced by this representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' We first prove that all the arcs between X and Y are oriented in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Towards a contradiction, consider a path x1y2x3 with x1, x3 ∈ X and y2 ∈ Y , and such that the edges are oriented from x1 to y2, and from y2 to x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' This forces the remaining edges of the 4-cycle yx1y2x3 to be oriented from x1 to y, and from y to x3 (see Claim 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Now we cannot orient the edge y2x, xz, zy in such a way to fulfill Claim 2 for the 5-cycles xzyx1y2 and xzyx3y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Indeed for the first one, yz should be oriented from y to z, while for the second one it should be oriented from z to y, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' This orientation ensures that the labels of all the arcs is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' This implies that there is an hyperplane H orthogonal to e1 such that for any pair of intersecting boxes x′ ∈ X and y′ ∈ Y , their intersection belongs to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' This implies that projecting the (d + 1)-CBU representation (restricted to B) along e1 leads to a boxicity d representation of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 2 It is clear that CBU is hereditary (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' closed under induced subgraphs) but actually it is also closed under subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Theorem 21 For any subgraph H of G, G ∈ CBU implies that H ∈ CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' More precisely, if there is a complete bipartite graph Ka,b such that V (Ka,b) ⊆ V (G), and such that E(H) = E(G) \\ E(Ka,b), then if G belongs to d-CBU then H belongs to (d + 1)-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Let A, B be the parts of Ka,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Given a CBU representation of G in Rd we are going to build a CBU representation of H in Rd+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For this, the first d intervals defining each d-box remain unchanged while the last interval is [0, 1] for the vertices in A, [2, 3] for the vertices in B, and [0, 3] for the remaining vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' It is now easy to check that two boxes intersect if and only if they intersect in G and if they are not adjacent in Ka,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' It is also clear that the intersections occur on planes orthogonal to e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 2 The graph class CBU is also closed by the addition of false twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Theorem 22 For any graph G and any vertex v of G, consider the graph Gv obtained from G by adding a new vertex v′ such that N(v′) = N(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Then G ∈ CBU if and only if Gv ∈ CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Furthermore, if G ∈ d-CBU then Gv ∈ (d + 1)- CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' The "if" part is obvious as G is an induced subgraph of Gv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For the "only if" part, given a CBU representation of G in Rd we are going to build a 10 CBU representation of Gv in Rd+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For this, the first d intervals defining each d-box remain unchanged, and those of v′ are the same as those of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' The last interval is [0, 1] for v, [2, 3] for v′, and [0, 3] for all the remaining vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' It is now easy to check that two boxes intersect if and only if they intersected and if one of them is distinct from v or v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' It is also clear that the intersections occur on planes orthogonal to e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 2 Shift graphs were introduced by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Erdős and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Hajnal in [18] (see Theorem 6 therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Those are the graphs Hm whose vertices are the ordered pairs (i, j) satisfying 1 ≤ i < j ≤ m, and where two pairs (i, j) and (k, l) form an edge if and only if j = k or l = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Note that such graphs admit a homogeneous arc labeling ℓ defined by ℓ({(i, j), (j, k)}) = j, and by orienting any edge {(i, j), (j, k)} from (i, j) to (j, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Theorem 23 The graph Hm belongs to (m−1)-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Furthermore, Hm has a CBU representation such that in the first dimension the vertex (i, j) corresponds to interval [i, j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' This clearly holds for the one vertex graph H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' By induction on m consider a representation of Hm−1, add a false twin for every vertex (i, m − 1) and modify the first interval of these new twins, so that the interval [i, m − 1] becomes [i, m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' These boxes correspond to the vertices (i, m) with i < m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For the vertex (m−1, m), one should add a box [m−1, m]×[−Ω, +Ω]×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='×[−Ω, +Ω], for a sufficiently large Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' To deal with the intersections between this box and the boxes of the other vertices (i, m), we add a new dimension such that vertex (m − 1, m) has interval [1, 2], the vertices (i, m − 1) have interval [1, 2], the vertices (i, m) with i < m− 1 have interval [3, 4], and all the other vertices have interval [1, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 2 Theorem 24 For every n-vertex graph G the following properties are equiva- lent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' a) G belongs to CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' b) G admits an homogeneous arc labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' c) G is the subgraph of a graph Ht m, obtained from the shift graph Hm by iteratively adding t false twins, for some values m, t such that m+t ≤ n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' d) G belongs to (2n − 1)-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' We have already seen that a) ⇒ b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Let us show b) ⇒ c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Consider an homogeneous arc labeling of G, with labels in [2, m−1], for the minimum m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' By minimality of m, note that all the labels are used, and thus m − 2 ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Let Ht m be the graph obtained from the shift graph Hm by adding ti,j false twins of vertex (i, j) if there are ti,j + 1 vertices of G whose incoming arcs are labeled i, and whose outgoing arcs are labeled j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For the vertices without incoming (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' outgoing) arcs assume that those are labeled 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Consider 11 now an injective mapping γ : V (G) −→ V (Ht m), such that any vertex with incoming and outgoing arcs labeled i, j is mapped to (i, j) or one of its twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' This mapping ensures us that G is a subgraph of Ht m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Indeed, for any two adjacent vertices u, v of G linked by an edge labelled j oriented from u to v, their incoming and outgoing arcs are labeled i, j and j, k respectively, for some i < j < k, and thus the vertices γ(u) and γ(v) of Ht m are adjacent, as they correspond to or are twins of (i, j) and (j, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' We now show c) ⇒ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Consider a graph Ht m containing G as a subgraph, for some m, t such that m+t ≤ n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' By Theorem 23 and Theorem 22 we have that Ht m belongs to (m − 1 + t)-CBU, and so to n-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Starting from Ht m one can obtain G by successively deleting n − 1 stars K1,b, so by Theorem 21, we have that G belongs to (2n − 1)-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Finally, d) ⇒ a) is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 2 It is easy to see that every complete bipartite graph belongs to 3-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' By Theorem 21, removing stars K1,b centered on the smallest part, one obtains that every n-vertex bipartite graph belongs to (⌊n/2⌋ + 3)-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' One can reach a slightly better bound from Theorem 6, and the fact that for every graph G, box(G) ≤ ⌊n/2⌋ [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Corollary 25 Every bipartite graph G belongs to CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Furthemore, if |V (G)| = n then G belongs to (⌊n/2⌋ + 1)-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' As already mentioned, some bipartite graphs have arbitrary large boxicity, and thus there is no fixed d such that every bipartite graph belongs to d-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For large girth graphs it is a different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Theorem 26 For any g ≥ 3, there exist graphs of girth g not contained in CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Indeed, for any g ≥ 3 there exist graphs of girth g with fractional chromatic number at least 4 [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' (Actually, their fractional chromatic number is arbitrarily large).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' By Theorem 39, such graphs cannot belong to CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 2 Nevertheless, the following remains open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Problem 27 Are there integers d, g such that every girth g graph G of CBU, belongs to d-CBU?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' The remarks above imply that testing if a bipartite graph belongs to CBU is obvious (computable in constant time), while for girth g graphs the question is more involved, as CBU has such graphs included and some other excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' The following section treats the computational problem of recognizing CBU graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 7 Recognition Computing the boxicity of a bipartite graph is a difficult problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' It is known that deciding whether a bipartite graph has boxicity two in NP-complete [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Furthermore, it is proven in [1] that it is not possible to approximate the boxicity 12 of bipartite graph within a O(n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='5−ε)-factor in polynomial time, unless NP = ZPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' By Lemma 20, for every bipartite graph B there is a graph B′ (obtained in polynomial time) such that the minimum value d such that B′ belongs to d-CBU, is exactly d = box(B) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Corollary 28 It is NP-complete to decide whether a graph belongs to 3-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Furthermore, unless NP = ZPP, one cannot approximate in polynomial time and within a O(n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='5−ε)-factor, the minimum value d for which an input graph G belongs to d-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' This implies that for most values d the problem of deciding whether an input graph belongs to d-CBU, cannot be computed in polynomial time, unless NP = ZPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' The hypothesis NP = P being stronger than NP = ZPP, it would be stronger to know that it is NP-complete to decide if an input graph belongs to d-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Problem 29 For which values d, is it NP-complete to decide whether a graph belongs to d-CBU?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Are there values d, in particular for d = 2, for which the problem is polynomial?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' By Lemma 20, this problem would be solved, for d ≥ 3, if the following problem admits a positive answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Problem 30 For any d ≥ 3, is it NP-complete to decide whether a bipartite graph B has boxicity at most d?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Another computational problem is testing the membership in CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Problem 31 Is it polynomial to decide whether a graph belongs to CBU?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' We have seen that some triangle-free planar graphs, or some graphs with arbitrary large girth, are not in CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' We can thus restrict the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Problem 32 Is it polynomial to decide whether a planar graph G belongs to CBU?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For some g ≥ 3, is it polynomial to decide whether a graph G of girth at least g belongs to CBU?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='1 Recognition through forbidden induced subgraphs As CBU and d-CBU are closed under induced subgraphs, they are characterized by a set of minimal excluded induced subgraphs, FCBU and Fd−CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' If one of these sets is finite, then recognizing the corresponding class becomes polynomial- time tractable (and this would also contradict Conjecture 2 of [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Thus by Corollary 28, the set F3−CBU (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Fd−CBU for d ≥ 4) is not finite, unless P = NP (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' unless NP = ZPP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For the set F2−CBU (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' FCBU), we are sure that it is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Indeed, Theorem 14 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Theorem 26) provides an infinite sequence of graphs (Gi)i≥0 not in 2-CBU (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' not in CBU) such that the girth of Gi is at least i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' If there was an n such that every graph in F2−CBU (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' FCBU) has at most n vertices, then to exclude Gn+1 one would need to have a tree in F2−CBU (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' FCBU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' This is not the case as for every tree T, we have that T ∈ 2-CBU ⊆ CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 13 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='2 Recognition through homogeneous arc labelings By Theorem 24, a graph G belongs to CBU if and only if it admits an homo- geneous arc labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' If we are given an orientation of a graph G it is simple to check whether this orientation admits such labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For example, one can use linear programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For each arc uv, set a variable ℓuv corresponding to a label, and for any two incident arcs, add a constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For two arcs uv and uw (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' uv and wv), the constraint is ℓuv = ℓuw (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' ℓuv = ℓwv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For two arcs uv and vw, the constraint is ℓuv < ℓvw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Problem 31 thus reduces to deciding whether a graph G admits an orientation that is homogeneously labelable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' In the following we characterize such orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' A cycle (v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , vn−1) is said badly oriented if there is a vertex vi whose incident arcs are vi−1vi and vivi+1, and if there is no vertex vj whose incident arcs are vj+1vj and vjvj−1 (indices being considered modn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Theorem 33 An orientation of a graph G admits an homogeneous labeling if and only if there is no badly oriented cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For the "only if" part, consider a badly oriented cycle (v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , vn−1) with arcs vn−1v0 and v0v1, but with no vertex vj ̸= v0 whose incident arcs are vj+1vj and vjvj−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' This latter condition implies that in any homogeneous label- ing the sequence of labels for the edges (without considering their orientation) v0v1, v1v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , vn−2vn−1, vn−1v0 is non-decreasing, while the former condition implies that the label of v0v1 is greater than the one of vn−1v0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Thus this orientation of G does not allows any homogeneous labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For the "if" part, consider a graph G oriented without badly oriented cycle, and consider a source u, and let us denote v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , vn its out-neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' If for every vertex vi, u is its unique in-neighbor, then by recurrence on the number of vertices we assume that G \\ {u} has a homogeneous labeling, and we label the arcs incident to u with a sufficiently small value, say −Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' In that case it is easy to check that this labeling is homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Otherwise, let vi and u′ be vertices such that G has arcs from both u and u′ toward vertex vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' In that case, consider the oriented graph G′ obtained from G \\ {u} by adding the arcs u′v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , u′vn, if missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Claim 34 G′ has no badly oriented cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' If G′ had a badly oriented cycle C, this one should go through a newly added arc u′vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' If vi /∈ C, by replacing the arc u′vj by the path (u′, vi, u, vj) one would obtain a badly oriented cycle in G, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' We thus assume that vi /∈ C, and now by replacing the arc u′vj by the path (u′, vi, u, vj) we obtain a badly oriented closed walk W (that is a walk where there are consecutive "forward" arcs, but no consecutive "backward" arcs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Let us denote P and P ′ the sub-paths of C \\ {u′vj} ⊊ G linking vi and vj, and linking u′ and vi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Let us show that if the edge incident to vi in P ′ is oriented from vi to the other end, denoted v, then this arc is backward with respect to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Indeed, the cycle CP ′ of G formed by P ′ and the arc u′vi, has consecutive arcs oriented in 14 the same direction, u′vi and viv, and (as G contains no badly oriented cycles) has consecutive arcs oriented in the other direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' The latter pair of arcs belonging both to P ′ ⊂ C, they are forward with respect to C, thus viv is backward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Similarly, let us show that if the edge incident to vi in P is oriented from vi to the other end, denoted w, then this arc is backward with respect to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Indeed, they cycle CP of G formed by P and the arcs uviand uvj, has consecutive arcs oriented in the same direction, uvi and viw, and (as G contains no badly oriented cycles) has consecutive arcs oriented in the other direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' The latter pair of arcs belong both to P ⊂ C, or they are the arcs incident to vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' In the former case, these arcs are forward with respect to C, thus viv is backward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' In the latter case, replacing uvj with u′vj, one has that the incident arcs of vj in C are oriented in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' this direction is thus the forward direction, and in that case also viv is backward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' We thus have that the arcs incident to vi cannot be oriented in the same di- rection (they would form consecutive backward arcs in C), and they are not both oriented from vi to the other end (they would be both backwards although they have distinct directions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Now we distinguish cases according to the position of the consecutive forward arcs in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' We have that: a) there are two consecutive forward arcs in P ∪ {u′vj}, or b) there are two consecutive forward arcs in P ′ ∪ {u′vj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' In case a), the cycle CP of G has consecutive forward arcs (by replacing if necessary the arc u′vj with uvj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Since this cycle is not badly oriented it also contains consecutive backward arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' According to the orientation of the arcs, those backwards arcs cannot be the arcs incident to u, or those incident to vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Thus they belong both to P ∪ {uvj}, but this would imply that C also contains consecutive backward arcs, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' In case b), the cycle CP ′ of G has consecutive forward arcs (by replacing if necessary the arc u′vj with u′vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Since this cycle is not badly oriented it also contains consecutive backward arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' According to the orientation of the arcs, those backwards arcs cannot be the arcs incident to vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' This would imply that C also contains consecutive backward arcs, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' This concludes the proof of the claim 2 So now, by recurrence on the number of vertices we can assume that G′ has a homogeneous labeling, and let ℓ be the label of the arcs outgoing from u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' In that case one can derive a labeling of G by keeping the same labels, and by setting the label ℓ for the arcs outgoing from u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' It is easy to check that this labeling is homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 2 Note that Theorem 33 provides another proof that CBU contains every bi- partite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Indeed, orienting all the edges from one part toward the other, the direction of the arcs alternate along any cycle, and so there is no badly oriented cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Actually, we can go a little further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 15 Theorem 35 Every graph G with circular chromatic number χc(G) ≤ 5/2 be- longs to CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' A graph G with circular chromatic number χc(G) ≤ 5/2 has a ho- momorphism into the circular complete graph K5/2 that is the 5-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' As this graph belongs to CBU the theorem follows from Theorem 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 2 Note that we cannot replace 5/2 by 8/3 in Theorem 35, as one can easily check that every orientation of K8/3 contains a badly oriented cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Problem 36 What is the largest c such that every graph G with χc(G) ≤ c (or with χc(G) < c) belongs to CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Theorem 37 Given two graphs G, H such that there is an homomorphism γ : V (G) −→ V (H), then if H ∈ CBU we have that G ∈ CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' By Theorem 24, the graph H admits an homogeneous arc labeling, ℓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Orient the edges of G in such a way that uv ∈ E(G) is oriented as the edge γ(u)γ(v) ∈ E(H), that is from u to v if and only if γ(u)γ(v) is oriented from γ(u) to γ(v) in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Similarly we copy the labeling of H’s arcs by setting ℓG(uv) = ℓH(γ(u)γ(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' One can easily check that this is an homogeneous arc labeling of G, and thus that G belongs to CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 2 8 Chromatic Number and Independent Sets While 2-CBU graphs have chromatic number at most 3 (by Grötzsch’s theorem), 3-CBU graphs have unbounded chromatic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Theorem 38 (Magnant and Martin [27]) For any χ ≥ 1, there exist a graph in 3-CBU, with chromatic number χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' However, these graphs have bounded fractional chromatic number, and thus have linear independent sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Indeed, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Simonyi and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Tardos [33] showed that shift graphs have fractional chromatic number less than 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' As such a bound extends by adding a false twin and by taking a subgraph, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Theorem 39 For any graph G ∈ CBU, χf(G) < 4, and α(G) > |V (G)|/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For planar graphs in CBU, this bound on χf can be improved by one, but not more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Theorem 40 For every planar graph G in CBU we have χf(G) ≤ χ(G) ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' On the other hand, for every n ≡ 2 (mod 3) there is a n-vertex planar graph G in CBU such that α(G) = (n + 1)/3, and thus χf(G) ≥ n/α(G) = 3 − 3 n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' The first statement follows from Grötzsch’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' The second statement follows from graphs constructed by Jones [23], which were proved to have independence number α(G) = (n + 1)/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Those graphs form a se- quence J1, J2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' such that J1 is the 5-cycle (a1, b1, c1, d, e), and such that 16 J1 Ji+1 d b1 Ji ci bi+1 ci+1 c1 a1 e 0 2 1 0 2i 2i 2i + 2 2 2i 2i + 1 2i 2i + 2 ai+1 ai bi Figure 6: The Jones graphs J1 and Ji+1, with a homogeneous arc labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For every i ≥ 1, this embedding is such that the path aibici is on the outer-boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Thus, adding vertices ai+1, bi+1, ci+1 does not break planarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Ji+1 is obtained from Ji by adding three vertices ai+1, bi+1, ci+1 such that N(ai+1) = {bi, bi+1}, N(bi+1) = {ai+1, ci+1}, and N(ci+1) = {ai, ci, bi+1} (see Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' It is already known that those graphs are planar, and it does only remain to show that they belong to CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Let us do so by exhibiting a homoge- neous arc labeling ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' This labeling is such that for any i ≥ 1 we orient the edges aibi and bici toward bi, we orient the edges xiyi+1, for x, y ∈ {a, b, c}, from xi towards yi+1, and we set ℓ(aibi) = ℓ(aici+1) = 2i, ℓ(cibi) = ℓ(cici+1) = 2i, and ℓ(biai+1) = 2i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' By examining Figure 6 it is clear that this is a homogeneous arc labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 2 Although 2-CBU lies in the intersection of CBU and planar graphs, it might be the case that the fractional chromatic number of graphs in 2-CBU is bounded by some c < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Indeed, Jones graphs Ji, for a sufficiently large i, seem to not be in 2-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Problem 41 Is there a c < 3 such that every graph G in 2-CBU has fractional chromatic number χf(G) ≤ c ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' A positive answer to this question, would give support to two conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Let Pg≥5 be the set of planar graph with girth at least five, and let Pf g≥4 be the set of planar graph with girth at least four, where every 4-cycle bounds a face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Clearly Pg≥5 ⊊ Pf g≥4, since these classes avoid Jones graphs it is conjectured that graphs in Pg≥5, or more generally graphs in Pf g≥4, have fractional chromatic number at most c, for some c < 3 [14, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' However, our problem is not a sub- case of these conjectures (as K2,t belongs to 2-CBU \\ Pf g≥4), nor a super-case (as Pg≥5 \\ 2-CBU is not empty, by Theorem 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 9 Computational hardness for many problems We have seen (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Theorem 8, Corollary 9, and Corollary 10) that many 1- subdivided graphs belong to CBU, or even to 3- or 4-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For (≥ 2)-subdivided graphs, the picture is even simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 17 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' v1 v2 v3 vn e2 e3 e1 Figure 7: Construction of a 3-CBU representation of a 2-subdivision of a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Theorem 42 For every graph G, if we subdivide every edge at least twice, the obtained graph belongs to 3-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Let us denote v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , vn the vertices of G, and let m = |E(G)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' To construct a CBU representation for any (≥ 2)-subdivision, we start by assigning each vertex vi to the box [3i, 3i + 1] × [n − i, n − i + 1] × [0, 2m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Then consider each edge e of G in any given order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For the kth edge e assume it links vi and vj, for some i < j, and assume e is replaced by the path (vi, u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , ur, vj) for some r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Here, u1 is assigned to [3i+1, 3i+2]×[n−j, n−i+1]×[2k−1, 2k], while the vertices uℓ with 2 ≤ ℓ ≤ r are assigned to [3i + 2 + (ℓ − 2)(3j − 3i − 2)/(r−1), 3i+2+(ℓ−1)(3j −3i−2)/(r−1)]×[n−j, n−j +1]×[2k−1, 2k] (see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' One can easily check that the obtained representation is a 3-CBU representation of the subdivided graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 2 Corollary 43 The problems of Minimum Feedback Vertex Set and Cutwidth are NP-hard even when restricted to 3-CBU graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' The problems Maximum Cut, Minimum Vertex Cover, Minimum Dominating Set, and Minimum Independent Dominating Set are APX-hard even when restricted to 3-CBU graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For Minimum Feedback Vertex Set and Cutwidth, this follows from the fact that these problems are NP-hard, and that for any instance, subdividing an edge does not change the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For Maximum Cut, it follows from its APX-hardness and the fact that the maximum cut of a graph G and its 2-subdivision G2-sub verify mc(G) = mc(G2-sub) − 2|E(G)| and 3|E(G)|/2 = |E(G2-sub)|/2 ≤ mc(G2-sub) ≤ |E(G2-sub)| = 3|E(G)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' The other problems are shown APX-hard even when restricted to 6-subdivided graphs [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 2 18 (1,1) (1,2) (2,1) Figure 8: 2-CBU representation of the 4 × 4 grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' When restricted to 2-CBU some of these problems become simpler to han- dle, as every graph in 2-CBU is planar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Indeed, the Maximum Cut problem turns out to be polynomial time solvable [12], while Minimum Vertex Cover, Minimum Dominating Set, and Minimum Independent Dominating Set admit PTAS [3, 26] (with standard techniques), such as Minimum Feedback Vertex Set [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' However, many problems remain NP-hard when restricted to 2-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Theorem 44 The problems Maximum Independent Set, Minimum Ver- tex Cover, Minimum Dominating Set, Hamiltonian Path, and Hamil- tonian cycle are NP-complete, even when restricted to 2-CBU graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' As these problems belong to NP, it remains to show that they are NP-hard for 2-CBU graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Let us first show that the induced subgraphs of grids (so called grid graphs) belong to 2-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Consider the n × n grid G such that V (G) = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , n} × {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , n}, and such that the neighbors of any vertex (i, j) are {(i, j −1)(i−1, j), (i, j +1), (i+1, j)}∩{1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , n}×{1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Since it suffices to delete some boxes to obtain an induced subgraph, the claim follows by constructing a 2-CBU representation for any such grid G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' This construction is obtained by mapping any vertex (i, j) to the box [i+j−1, i+j]×[2i−2j, 2i−2j+3] (see Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' As Domination [8], Hamiltonian Path, and Hamiltonian cycle [22] are NP-hard for grid graphs, those problems are NP-hard for 2-CBU graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' For the problems Maximum Independent Set and Minimum Vertex Cover, we have to consider a variant of grid graphs, the graph R′(n1, n2) depicted in Figure 9, and it is easy to see how to modify the construction above in order to obtain a 2-CBU representation of this type of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Again, this implies that every induced subgraph of such a graph belongs to 2-CBU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' As the problems Maximum Independent Set and Minimum Vertex Cover are NP-hard for this class (see the proof of Theorem 10 in [20]), those problems are NP-hard for 2-CBU graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 2 References [1] Abhijin Adiga, Diptendu Bhowmick, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' Sunil Chandran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' The hardness of approximating the boxicity, cubicity and threshold dimension of a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' 19 (i − 1, j) (i − 1, j + 1) (i, j + 1) (i, j − 1) (i + 1, j) (i + 1, j − 1) Figure 9: The graph R′(n1, n2) and the local modification to obtain its 2-CBU representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} +page_content=' From the 2-CBU representation of the grid given above, one has to delete the box of every vertex (i, j), where i and j are even, and if i + j ≡ 2 mod 4 one has to replace the box by 4 smaller boxes.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E2T4oBgHgl3EQfZgdy/content/2301.03865v1.pdf'} diff --git a/_9FJT4oBgHgl3EQfqyzS/content/2301.11606v1.pdf b/_9FJT4oBgHgl3EQfqyzS/content/2301.11606v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..cd10d7d9ec381d4515d5dd690ec173ee51cde890 --- /dev/null +++ b/_9FJT4oBgHgl3EQfqyzS/content/2301.11606v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ee84e6705669e55e99c729c479506b91e5c63142fa9a05a6b0a369cd8b789f48 +size 1596764 diff --git a/_9FJT4oBgHgl3EQfqyzS/vector_store/index.pkl b/_9FJT4oBgHgl3EQfqyzS/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..e837f10090042b1534cd5e40ee6398fafba4af4f --- /dev/null +++ b/_9FJT4oBgHgl3EQfqyzS/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4876638d0499b76dc2d768c0b7d839c064cf642276a968e5a033739f343d9d53 +size 931173 diff --git a/b9AyT4oBgHgl3EQfi_hp/content/tmp_files/2301.00406v1.pdf.txt b/b9AyT4oBgHgl3EQfi_hp/content/tmp_files/2301.00406v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..febde1dda01fd6730ccb8af96bda11548f539109 --- /dev/null +++ b/b9AyT4oBgHgl3EQfi_hp/content/tmp_files/2301.00406v1.pdf.txt @@ -0,0 +1,1686 @@ +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +1 +Curvature regularization for Non-line-of-sight +Imaging from Under-sampled Data +Rui Ding, Juntian Ye, Qifeng Gao, Feihu Xu, and Yuping Duan +Abstract—Non-line-of-sight (NLOS) imaging aims to reconstruct the three-dimensional hidden scenes from the data measured in the +line-of-sight, which uses photon time-of-flight information encoded in light after multiple diffuse reflections. The under-sampled +scanning data can facilitate fast imaging. However, the resulting reconstruction problem becomes a serious ill-posed inverse problem, +the solution of which is of high possibility to be degraded due to noises and distortions. In this paper, we propose two novel NLOS +reconstruction models based on curvature regularization, i.e., the object-domain curvature regularization model and the dual (i.e., +signal and object)-domain curvature regularization model. Fast numerical optimization algorithms are developed relying on the +alternating direction method of multipliers (ADMM) with the backtracking stepsize rule, which are further accelerated by GPU +implementation. We evaluate the proposed algorithms on both synthetic and real datasets, which achieve state-of-the-art performance, +especially in the compressed sensing setting. All our codes and data are available at https://github.com/Duanlab123/CurvNLOS. +Index Terms—Non-line-of-sight, under-sampled scanning, curvature regularization, dual-domain reconstruction, GPU implementation +! +1 +INTRODUCTION +N +ON-LINE-OF-SIGHT +(NLOS) +imaging +uses +time- +resolved measurements to recover the 3D shape and +visual appearance of hidden objects beyond the direct line +of sight of sensors [1], [2], which has various applications +such as autonomous driving [3], 3D human pose estimation +[4], sensor system [5], [6], and many other domains. Recent +advances in single-photon detector technology and compu- +tational algorithms for solving large-scale inverse problems +make NLOS imaging feasible in different conditions. For +instance, NLOS imaging and real-time tracking of hidden +objects have been demonstrated over a distance of 1.43 km +[7] and at a resolution of 0.6 mm at a distance of 0.55 m [8], +[9], respectively. +The inverse problem of reconstructing the 3D shape and +appearance of the hidden object is also very challenging [10], +[11]. Roughly speaking the NLOS imaging reconstruction +methods can be divided into three categories, i.e., direct re- +construction methods, iterative reconstruction methods, and +deep learning-based reconstruction methods. The filtered +back-projection algorithm is a kind of fast direct method by +filtering the data and performing the back-projection opera- +tion, where the data is painted back in the image along the +direction it being measured [12], [13], [14], [15]. Other direct +reconstruction methods also contain the phasor field [16], +frequency-domain method [17], Fermat flow method [18], +etc, which are proposed by modeling the physical process +of imaging. These direct reconstruction methods are fast and +efficient but are sensitive to noises and measurement distor- +R. +Ding, +Q. +Gao, +and +Y. +Duan +are +with +Center +for +Applied +Mathematics, +Tianjin +University, +Tianjin +300072, +China. +E-mail: +{rding,gaoqifeng 98,yuping.duan}@tju.edu.cn +J. Ye and F. Xu are with Hefei National Laboratory for Physical Sciences at Mi- +croscale and Department of Modern Physics, University of Science and Tech- +nology of China, Hefei 230026, China. E-mail: jt141884@mail.ustc.edu.cn; fei- +huxu@ustc.edu.cn +K. Chen is with Department of Mathematical Sciences, University of Liver- +pool, United Kingdom. +Manuscript received April 19, 2005; revised August 26, 2015. +tions. The iterative reconstruction methods have been used +by introducing priors to regularize the NLOS reconstruction +problem and improve the image quality [19], [20], [21], +[22], [23]. Although iterative reconstruction methods can +provide high-quality reconstruction images, they also con- +sume much more computational time than direct methods. +Due to the development of deep learning, convolutional +neural network models have been developed for solving the +NLOS reconstruction problem [24], [25], [26], [27]. However, +deep learning methods are highly dependent on the training +datasets, which makes them possible to lose their impact on +real measurement data and spatial/temporal degradation +data [24]. +Indeed, the dense raster scanning used in the afore- +mentioned methods is detrimental to high-speed NLOS +applications. Thus, different strategies have been employed +to reduce the acquisition time. One strategy is to use the +multi-pixel time-of-flight NLOS system. Nam et al. [28] used +the specifically designed single photon avalanche diode +(SPAD) array detectors to realize a multi-pixel NLOS imag- +ing method together with a fast reconstruction algorithm +that can capture and reconstruct live low-latency videos of +NLOS scenes. Pei et al. [29] used the SPAD array and an +optimization-based computational method to achieve NLOS +reconstruction of 20 frames per second. Another strategy +is to use fewer scanning points to reconstruct the scene. +Isogawa et al. [30] proposed a circular and confocal non- +line-of-sight scan, for which the scanning involves sampling +points forming a circle on a visible wall to reduce the +dimension of transient measurements. Ye et al. [22] explored +compressed sensing in active NLOS imaging to reduce the +required number of scanning points for fast implementation. +As illustrated in previous works [22], [30], NLOS recon- +struction can be realized by under-sampled measurements +to facilitate high-speed acquisition. However, sparse mea- +surements may lead to the degradation of reconstructed +images. Curvature regularization is an important technique +arXiv:2301.00406v1 [cs.CV] 1 Jan 2023 + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +2 +(a) Non-confocal NLOS +(b) Confocal NLOS +(c) Under-sampled Confocal NLOS +Fig. 1: Illustration of transient NLOS imaging, where (a) non-confocal NLOS; (b) confocal NLOS with full scanning, and (c) +confocal NLOS with under-sampled scanning. +for various shape and image processing tasks [31], [32], +[33], which is well-known for its ability in modeling the +continuities of edges and surfaces. Initially, curvature regu- +larization was applied to the problem of image inpainting +to restore satisfactory results meeting human perception +[34], [35], [36]. Due to its superiority in recovering missing +data, curvature regularity has also been used for surface +reconstruction [37], [38], sparse image reconstruction [39], +[40]. Since the curvature regularization is capable to capture +the tiny but elongated structures in images, it has also +been used for image segmentation [41], tubular structure +tracking [42], etc. Obviously, curvature regularization is a +good choice for under-sampled NLOS imaging problems to +obtain smooth and satisfied reconstructed surfaces. +In this paper, we study the curvature regularization +reconstruction model for under-sampled NLOS imaging +problems, called curvNLOS, which can reconstruct surfaces +with good quality through as few measurements as possible. +We also develop an effective numerical algorithm based on +the alternating direction methods of multipliers (ADMM). +Comprehensive experiments are conducted on both syn- +thetic and real data. The numerical results demonstrate +that curvature regularization can effectively guarantee the +smoothness of the object and estimated signals. The main +contribution of this work can be summarized as follows: +• +We propose novel curvature regularization mod- +els for solving the NLOS imaging problem, where +the curvature regularization is used to restore the +smooth surface of hidden objects and fill in the sparse +measured signals. +• +We present fast iterative optimization algorithms for +solving the curvature minimization problems, where +the high-order curvature is regarded as the adaptive +weight for total variation and the acceleration tech- +nique is used to obtain a faster convergence. +• +We develop a GPU-based NLOS reconstruction pack- +age by utilizing the parallel computation ability of a +GPU card, which is desirable for high-speed NLOS +applications. +• +By comparing with the state-of-the-art NLOS meth- +ods, our curvature regularization models can ro- +bustly restore estimated signals and the three- +dimensional scene points, especially when the mea- +surements are under-sampled measured data. +The roadmap of this paper is as follows. In Section 2, we +briefly describe NLOS physics and mathematical formula- +tion. Section 3 presents the object-domain curvature regular- +ization method and the ADMM-based algorithm. Section 4 +proposes the dual-domain curvature regularization method +for NLOS. Numerical experiments are evaluated on both +synthetic and real measured data by comparing with other +established methods in Section 5. Finally, we conclude our +paper in Section 6 and present some discussions. +2 +FORWARD PROPAGATION MODEL +In transient imaging, a time-resolved detector is used to +measure the incident flux of photons as a function of +emitted light impulses. Let x = (x, y, z) be the three- +dimensional scene coordinates, and x′ +i = (x′ +i, y′ +i, z = 0), +x′ +d = (x′ +d, y′ +d, z = 0) be the illumination and detection +coordinates on the visual wall, respectively. As shown in +Fig. 1, the light emitted by the laser passes through the +scanning galvanometer and hits the illumination point x′ +i, +and diffuses towards the target point x. Then, the photons +reflect on the target and propagate to the detection point x′ +d. +Finally, the photons diffuse back and enter the single-pixel +detectors such as SPAD. The NLOS reconstruction aims to +recover the location, shape, albedo, and normal of the target +from the detected number of photons. +The +general +non-confocal +direction-albedo +forward +propagation model for NLOS imaging can be described +below +τ(x′ +i, x′ +d, t) = +��� +Ω +(x′ +i − x) · n(x) +d(x′ +i, x)3 +· (x′ +d − x) · n(x) +d(x′ +d, x)3 +u(x) +· δ +� +d(x′ +i, x) + d(x′ +d, x) − tc +� +dx, +(1) +where τ is the recorded transient image, u is the albedo of +the hidden scene at each point x with z > 0 in the 3D half- +space Ω, c is the speed of the light, n(x) = (nx, ny, nz)(x) +denotes the surface normal, and δ represents the surface of +a spatio-temporal four-dimensional hypercone defined by +x2 + y2 + z2 − (tc/2)2 = 0 modeling light propagation from +the wall to the object and back to the wall. Studies on non- +confocal NLOS imaging can be found in [43], [44]. The dis- + +Visible +Wall +SPAD +Wall +Laser +Hidden ObjectVisible +Wall +SPAD +Wall +Hidden ObjectVisible +Wall +SPAD +Wall +Hidden ObjectJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +3 +tances between the reconstructed point and the illumination +point and detection point are defined as +d(x′ +i, x) = +� +(x′ +i − x)2 + (y′ +i − y)2 + z2, +and +d(x′ +d, x) = +� +(x′ +d − x)2 + (y′ +d − y)2 + z2, +respectively. When the illumination point and detection +point locate at the same position, i.e., x′ = x′ +i = x′ +d, we +have the confocal direction and albedo NLOS reconstruction +model as given below +τ(x′, t) = +��� +Ω +u(x)n(x) +d(x′, x)4 ·n(x)·δ +� +2d(x′, x)−tc +� +dx. (2) +Methods to localize the three-dimensional scene points and +estimate their surface normals of hidden objects have been +established for the NLOS imaging [21], [45], [46]. As long +as ⟨n(x), n(x)⟩ = 1, the direction-albedo forward model +reduces into the volumetric albedo model +τ(x′, t) = +��� +Ω +u(x) +d(x′, x)4 · δ +� +2d(x′, x) − tc +� +dx. +(3) +The corresponding discrete image formation of the NLOS +model (3) is given as +τ = Au = R−1 +t HRzu, +(4) +where the matrix A = R−1 +t HRz is the light transport matrix, +H represents the shift-invariant 3D convolution, Rt and +Rz represent the transformation operations applied to the +temporal and spatial dimensions, respectively. +The task of recovering the object u from the measured +signal τ is a typical ill-posed inverse problem. The regular- +ization method is a good choice to solve ill-posed inverse +problems. In [22], the sparsity regularization and a non- +negative constraint were used to restore the surface of +objects. Although the algorithm can reconstruct a three- +dimensional hidden image of 64×64 spatial resolution with +5 × 5 scanning points, the results greatly depend on the +post-processing step to smooth out the noises and outliers. +The collaborative signal and objective regularization was +used in [21], which is shown effective on both confocal and +non-confocal NLOS imaging reconstruction. However, such +complex regularization makes the computational costs in- +crease extremely, which is unfavorable for real applications. +Thus, finding a regularization method suitable for NLOS +imaging algorithms can give consideration to both imaging +quality and speed, which is still a challenging problem. +3 +INVERSE PROBLEM BY CURVATURE REGULAR- +IZATION +The curvature regularization can naturally fill the missing +information and obtain a smooth surface, which motivates +us to use it to approximate the oracle signal corresponding +to the real hidden scene. Thus, we propose to reconstruct +3D objects by employing three-dimensional curvature regu- +larization. To be specific, we aim to minimize the following +curvature-related energy for the NLOS imaging problem +min +u +� +E(u) = 1 +2∥Au − τ0∥2 +Ω\X + R(κ(u)) +� +, +(5) +where X ⊂ Ω denotes the missing region, and R(·) denotes +the curvature regularization such as +R(κ(u)) = +� +x +φ(κ(u(x)))|∇u(x)| +with φ(·) being the function of curvature and |∇u| being +the total variation of u. In order to efficiently solve the +curvature regularization model (14), we reformulate it into +a constrained minimization problem +min +u,v +1 +2∥Au − τ0∥2 +Ω\X + R(κ(u)), +s.t. +v = ∇u. +Then the associated augmented Lagrangian functional can +be defined as follows +L(u, v; Λ) =1 +2∥Au − τ0∥2 +Ω\X + +� +x +φ(κ(u(x)))|v(x)| ++ ⟨Λ, v − ∇u⟩ + µ +2 ∥v − ∇u∥2 +2, +(6) +where Λ is the Lagrange multiplier, and µ is the positive +parameter. We use the Alternating Direction Method of +Multipliers (ADMM) to iteratively and alternatively solve +the sub-minimization problems. +3.1 +Computation of curvature +As shown in Fig. 2, the 3D transient measurement τ(x, y, t) +is a time-continuous function, and u(x, y, z) is a spatial +continuous function. The curvature regularization term can +provide strong prior information on the continuity of the +image. Therefore, we use the curvature regularization to +both hidden objects and the signals, which are defined as +κ(u) := ∇ · ∇u(x, y, z) +|∇u(x, y, z)|. +The function of the curvature can be defined in different +forms similar to [35], where we use the total squared curva- +ture as follows +φ(κ) = a + b|κ|2 +with a, b ∈ R. +In order to ease the computation, we regard curvature terms +as the weights of the total variation, which are computed +explicitly as long as u being updated. +3.2 +Sub-minimization problems and the solutions +Now we discuss the sub-minimization problems for solving +(4.1) and the corresponding solutions. +3.2.1 +Sub-minimization problem w.r.t. v +The sub-minimization problem w.r.t. v is defined as +min +v +� +x +φ(κ(uk(x)))|v(x)| + µ +2 +��v − (∇uk − Λk +µ ) +��2 +2. +(7) +Since φ(κ(uk(x))) is known in advance, the above mini- +mization problem becomes the weighted ℓ1 and ℓ2 mini- +mization problem, which can be solved by the shrinkage +operator as follows +vk+1 = shrinkage +� +∇uk − Λk +µ , φ(κ(u)) +µ +� +, +(8) + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +4 +(a) Ground truth +(b) τ(x, y, t) +(c) τ(32, 32, :) +Fig. 2: Overview of NLOS imaging measurements, where (a) The ground truth image of the hidden objects; (b) The +measurements of the wall τ attenuated along the time axis; and (c) A histogram measured at the scanned point indexed by +(32,32) on the visible wall. +with the shrinkage operator being defined as +shrinkage(a, b) = max{|a| − b, 0} ◦ a +|a| +and ◦ being the element-wise multiplication. +3.2.2 +Sub-minimization problem w.r.t. u +The u sub-minimization problem can be formulated as a +quadratic minimization problem +min +u +1 +2 +��ADu − τ0 +��2 +2 + µ +2 +��∇u − (vk+1 + Λk +µ ) +��2 +2, +(9) +where AD = DA with D being the sampling operator to +define the missing data domain. For simplicity, we denote +f(u) = 1 +2 +��ADu − τ0 +��2 +2, g(u) = µ +2 +��∇u − (vk+1 + Λk +µ ) +��2 +2. +Then we reformulate the minimization problem (9) using +a quadratic approximation of f(u) at a given point uk as +follows +min +u +f(uk) + ⟨u − uk, ∇f(uk)⟩ + L +2 +��u − uk +��2 +2 + g(u), (10) +which is equivalent to +min +u +g(u) + L +2 +��u − (uk − 1 +L∇f(uk) +��2 +2, +(11) +with L being 2∥AD∥2 +2. The optimal value of the (11) becomes +the solution to the following partial differential equation +(PDE) +(LI + µ∇∗∇)u = Luk − ∇f(uk) + µ∇∗(vk+1 + Λk +µ ), +where ∇f(uk) = AT +D(ADuk − τ0) and ∇∗ is the adjoint +operator of gradient. The above PDE can be solved by the +Fast Fourier Transform (FFT) as follows +uk+1 = F−1 +�F +�Luk − ∇f(uk) + µ∇∗(vk+1 + Λk +µ +� +LI + µF(∇∗∇) +� +, +(12) +where F and F−1 denote the forward and inverse FFT +operation, respectively. +3.2.3 +Update of Lagrange multipliers +Finally, we update the Lagrange multipliers by the gradient +ascend method as follows +Λk+1 = Λk + µ(vk+1 − ∇uk+1). +(13) +3.3 +Our algorithm +We summarize the ADMM-based algorithm for solving the +object-domain curvature regularization model as Algorithm +1. As seen, the acceleration technique in [47] is applied to +obtain a faster convergence rate, where the specific linear +combination of the previous two points {uk−1, uk} is used +in the computation. +Remark 1. Due to the non-convexity of the curvature regulariza- +tion in our model (5), the convergence of Algorithm 1 is difficult +to obtain theoretically. A partial convergence result can be found +in our previous work [35]. From observing the numerical energy +decay, Algorithm 1 numerically converges quite stable; see Fig. 5. +4 +DUAL-DOMAIN CURVATURE METHOD AND GPU +IMPLEMENTATION +4.1 +Dual-domain reconstruction method +Since the under-sampled signal can be regarded as the +data inpainting problem, we also propose a dual-domain +reconstruction model, where the curvature regularization is +employed for both the signal domain and object domain. +Mathematically, we present the dual-domain NLOS imaging +reconstruction model as follows +min +u,τ +� +E(u, τ) = 1 +2∥Au−τ∥2 +2+λ +2 ∥τ−τ0∥2 +Ω\X+R(u)+R(τ) +� +, +(14) +where λ is the positive parameter, R(τ) is with the same +formulation as R(u) and κ(τ) is calculated as follows +κ(τ) := ∇ · ∇τ(x, y, t) +|∇τ(x, y, t)|. +Note that κ(u) calculates the spatial curvature of the hidden +object, while κ(τ) is used to measure the curvature of + +Photon +20 +0 +0 +200 +4 +Time Bil00 +600 +n60 +Count +40JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +5 +Algorithm 1: The ADMM algorithm for solving the +object-domain curvature regularization model (5) +Input: Raw data τ0 and parameters Λ, a, b, µ, ϵ, +¯u0 = u0 = 0, Λ0 = 0, Tmax,t0 = 1; +Output: uk+1 +1 for k = 0, 1, 2, . . . do +/* Solve the saddle-point problem +*/ +2 +(uk+1, vk+1; Λk+1) = max +Λ +min +u,v L(¯uk, vk; Λk); +/* Compute tk+1 and ¯uk+1 from +*/ +3 +tk+1 = 1 + +� +1 + 4(tk)2 +2 +; +¯uk+1 = uk+1 + tk − 1 +tk+1 (uk+1 − uk); +/* Update φ(κu) +*/ +4 +φ(κ(u)) = a + b(∇ · ∇u +|∇u|)2; +/* Stopping condition +*/ +5 +k ≥ Tmax or en+1 = +��E(uk)−E(uk+1) +�� +��E(uk+1) +�� +≤ ϵ; +6 end +the time-dependent signal. Similarly, we rewrite the above +model into the following constrained minimization problem +min +u,τ,v,w,f +1 +2∥Au − τ∥2 +2 + λ +2 ∥f − τ0∥2 +Ω\X + R(u) + R(τ), +s.t. +v = ∇u, w = ∇τ, f = τ. +Then the associated augmented Lagrangian functional can +be defined as follows +L(u, τ, v,w, f; Λ1, Λ2, Λ3) = 1 +2∥Au − τ∥2 +2 + λ +2 ∥f − τ0∥2 +Ω\X ++ +� +x +φ(κ(u(x)))|v(x)| + +� +x +φ(κ(τ(x)))|w(x)| ++ ⟨Λ1, v − ∇u⟩ + µ1 +2 ∥v − ∇u∥2 +2 + ⟨Λ2, w − ∇τ⟩ ++ µ2 +2 ∥w − ∇τ∥2 +2 + ⟨Λ3, f − τ⟩ + µ3 +2 ∥f − τ∥2 +2, +where Λ1, Λ2, Λ3 are the Lagrange multipliers, and µ1, +µ2, µ3 are the positive parameters. Then the Alternat- +ing Direction Method of Multipliers (ADMM) can be im- +plemented to iteratively and alternatively solve the sub- +minimization problems. The algorithm and solutions to the +sub-minimization problems can be generalized from the +object-domain cases. Both the object-domain reconstruction +algorithm and dual-domain reconstruction algorithm are +provided at https://github.com/Duanlab123/CurvNLOS. +More details can be found in our public codes. +Remark 2. Note that we initialize the variable u by Algorithm 1 +to obtain better convergence and high-quality reconstructions. +4.2 +GPU implementation +The GPU has a distinct advantage in parallel computing, +consisting of thousands of smaller, more efficient cores +designed for multitasking. The GPU-based image recon- +struction allows for the use of more complex models and +maintains reasonable execution time. Thus, we implement +both Algorithm 1 and Algorithm 2 on the GPU to reduce +the computational time. We utilized one RTX 2080 graphics +card to run our algorithms. For 64 × 64 × 512 data, each +iteration takes about 0.1 seconds, and for 128 × 128 × 512 +data, each iteration takes about 0.2 seconds. For data with +higher dimensions, the advantage of GPU over CPU is more +obvious. +5 +NUMERICAL RESULTS +In this section, we discuss the performance of our dual- +domain curvature method on both synthetic and real imag- +ing data. We use the accuracy, RMSE, PSNR, and SSIM +to evaluate the reconstruction performance. The accuracy +refers to the foreground/background classification accuracy, +which is defined as +Accuracy = +TP + TN +TP + TN + FP + FN , +where TP and TN are correct foreground(true positives) +and correct background(true negatives), FP and FN are +excess(false positives) and missing geometry (false nega- +tives). After dividing the foreground and background, the +depth error of the foreground is calculated by RMSE. Both +PSNR and SSIM are common evaluation indicators used in +image processing. Since we want to achieve the purpose +of fast reconstruction by quickly collecting information, +reconstruction time is also an important evaluation index. +We will compare the time used for reconstruction by each +method. +5.1 +Comparison methods +We evaluate the performance of our curvature regulariza- +tion methods by comparing them with the following state- +of-the-art approaches +• +LCT: The light cone transform was proposed in [10] +for confocal NLOS imaging, which is a parameter- +free back-projection-based reconstruction method. +• +Phasor field: The phasor field [16] formulates the +NLOS imaging problem as a wave imaging problem +and uses the techniques of classic optics for the +NLOS imaging, which is also a parameter-free direct +reconstruction method. +• +F-K migration: The frequency-domain method pro- +posed in [17] can handle both confocal and non- +confocal NLOS imaging problems. The F-K migra- +tion is robust to objects with complex reflective prop- +erties and easy to implement with no parameters to +tune. +• +SPIRAL+∥ · ∥1 + R+: The modified sparse Poisson +intensity reconstruction algorithm proposed in [22], +where the non-negativity prior and sparsity prior +of the hidden scene are used as the regularization + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +6 +Algorithm 2: Our ADMM algorithm for solving the dual-domain curvature regularization model (14) +Input: Raw data τ0, and parameters au, bu, aτ, bτ, Λ1, Λ2, Λ3, µ1, µ2, µ3, ϵ, ¯u0 = u0, ¯τ = τ = 0, Λ0 +1 = Λ0 +2 = Λ0 +3 = 0, +Tmax, t0 = 1; +Output: uk+1 +1 for k = 0, 1, 2, . . . do +/* Compute the following v subproblem by the soft shrinkage operator +*/ +2 +min +v +� +x +φ(κ(uk(x)))|v(x)| + µ1 +2 +��v − (∇uk − Λk +1 +µ1 +) +��2 +2; +/* Compute the following w subproblem by the soft shrinkage operator +*/ +3 +min +w +� +x +φ(κ(τ k(x)))|w(x)| + µ2 +2 +��w − (∇τ k − Λk +2 +µ2 +) +��2 +2; +/* Compute the following f subproblem by the Fast Fourier Transform +*/ +4 +min +f +λ +2 +��f − τ0 +��2 +Ω\X + µ3 +2 +��f − (τ k − Λk +3 +µ3 +) +��2 +2; +/* Compute the following τ subproblem by the Fast Fourier Transform +*/ +5 +min +τ +1 +2 +��Auk − τ∥2 +2 + µ2 +2 +��∇τ − (wk+1 + Λk +2 +µ2 +) +��2 +2 + µ3 +2 +��τ − (f k+1 + Λk +3 +µ3 +) +��2 +2; +/* Compute the following u subproblem by the Fast Fourier Transform +*/ +6 +min +u +1 +2 +��Au − τ k+1∥2 +2 + µ1 +2 +��∇u − (vk+1 + Λk +1 +µ1 +) +��2 +2; +/* Compute tk+1 and ¯uk+1 from +*/ +7 +tk+1 = 1 + +� +1 + 4(tk)2 +2 +; +¯uk+1 = uk+1 + tk − 1 +tk+1 (uk+1 − uk); +¯τ k+1 = τ k+1 + tk − 1 +tk+1 (τ k+1 − τ k); +/* Update φ(κu) and φ(κτ) by +*/ +8 +φ(κu) = au + bu(∇ · ∇u +|∇u|)2; +φ(κτ) = aτ + bτ(∇ · ∇τ +|∇τ|)2; +/* Stopping condition +*/ +9 +k ≥ Tmax +or +en+1 = +��E(uk) − E(uk+1) +��/ +��E(uk+1) +�� ≤ ϵ; +10 end +terms. There is one regularization parameter that is +required to adaptive adjust in different scenes. +• +SOCR: The signal–object collaborative regularization +proposed in [21], which incorporated sparseness and +non-local self-similarity of the hidden objects and the +smoothness of the measured data. There are eight pa- +rameters in SOCR, which are regularization param- +eters su, λu, λpu, λd, λpd, λsd, σ, and split Bregman +algorithm parameter µ. Among these parameters, λd, +λpd and λsd are fixed as λd = 1, λpd = 16, λsd = 0.25 +respectively. And σ ranges from 20 to 80. Other +parameters need to choose adaptive according to the +raw measurements. In particular, for the scenes used +in [21], we directly reconstruct the images using the +provided codes without adjusting any parameters. +For the Bowling scene, we have fine-tuned the pa- +rameters according to the supplementary materials. +5.2 +Parameter discussing +For the object-domain Algorithm 1, there are three param- +eters that need to be adjusted, namely µ, a, b. The penalty +parameter µ controls the convergence of the algorithm. Too +small µ will lead to non-convergence of the algorithm, and +too large µ will reduce the quality of the reconstructed +image. The two parameters a and b are the regularization +parameters used to control the smoothness of the solution. +The larger the values of a and b are, the smoother the +surfaces will be. The specific values of the three parameters +are provided in each scene. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +7 +(a)Ground Truth +(e)SOCR +(b)LCT +(f)SPIRAL+∥ · ∥1 + R+ +(c)Phasor field +(f)Algorithm 1 +(d)F-K +(g)Algorithm 2 +Fig. 3: The visual comparison of the comparison reconstruction methods under full sampling on Bowling, where the +parameters of our methods are set as: a = 5 × 10−5, b = 5 × 10−5, µ = 0.1 for Algorithm 1; λ = 400, aτ = 1 × 10−4, bτ = +3 × 10−2, au = 1 × 10−4, bu = 1 × 10−4 for Algorithm 2. +TABLE 1: The comparison of the Accuracy, RMSE, PSNR, SSIM, and computational time among different methods on +image Bowling with scanning points of 64 × 64. +Image ID +Scan points +Reconstruction Method +Accuracy +RMSE +PSNR +SSIM +Time (s) +Bowling +64×64 +LCT +0.8818 +0.1977 +13.9723 +0.2385 +0.28 +Phasor field +0.9333 +0.1712 +13.9628 +0.2344 +0.35 +F-K +0.8594 +0.2055 +10.9895 +0.1826 +0.84 +SOCR +0.8889 +0.5770 +11.1099 +0.2311 +876 +SPIRAL+∥ · ∥1 + R+ +0.9470 +0.1368 +13.9735 +0.4321 +136 +Object-domain Alg 1 +0.9558 +0.1069 +16.2712 +0.4473 +13 +Dual-domain Alg 2 +0.9585 +0.0947 +16.1162 +0.4746 +21 +On the other hand, there are total eight parameters +λ, µ1, µ2, µ3, au, bu, aτ, bτ in the dual-domain Algorithm +2. Similarly, the penalty parameters µ1, µ2, µ3 affect the +stability of the algorithm, which are fixed µ1 = 1, µ2 = +800, µ3 = 2 in all experiments. Varying these values in a +small range will not affect the quality of the reconstructed +image. The other five arguments are used to balance the data +fidelity and curvature regularity. More specifically, au and +bu control the curvature regularization of the object domain, +while aτ and bτ control the curvature regularization of the +measured signals. Likewise, we provide the specific values +in each experiment. +In addition, we need to determine the termination con- +dition for Algorithm 1 and Algorithm 2, which are termi- +nated by both the number of iterations and the relative +error bound. Because Algorithm 1 converges faster than +Algorithm 2, the number of iterations of Algorithm 1 is set +to Tmax = 200, and the number of iterations of Algorithm +2 is set to Tmax = 300. In order to ensure the quality of +the reconstructed image, we set the relative error bound ϵ as +1 × 10−6. +5.3 +Experiments on synthetic data +We use two synthetic data to verify the reconstruction +performance of our curvNLOS methods w.r.t. different sam- +pling rates. The bowling scene was generated in [22], where +the LCT model was used to generate the transient image. +There are 64 ×64 points on the visible wall with a spatial +resolution of 1m×1m. The time resolution is 256 and each +time bin spans 32 ps. +Firstly, we compare our reconstruction results with other +methods using the full sampling data; see Fig. 1 for a visual +comparison. It should be noted that since the dimensions +of the reconstructed results of SOCR are different from +those of other methods, we fill the result with 0 for the +comparison. As can be observed, different methods can +produce meaningful reconstruction results on the full sam- +pling data. And our reconstructions achieve the best visual +quality, where the reconstructed scenes are quite close to +the ground truth with fine structures and deails. Table1 +records the evaluation indicators for all comparison meth- +ods, where our curvNLOS gives the best accuracy. Although +the object-domain reconstruction (Algorithm 1) and dual- +domain reconstruction (Algorithm 2) can obtain similar re- + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +8 +8×8 +6×6 +4×4 +(a) LCT +(b) SPIRAL w/o smoothing +(c) SPIRAL with smoothing +(d) Algorithm 1 +(e) Algorithm 2 +Fig. 4: Reconstruction results using different numbers of scanning points from up to down the scanning points are of 8 × 8, +6 × 6, and 4 × 4, respectively. The hidden scene is processed at a 64 × 64 spatial resolution and 256 temporal resolution. +The parameters of our methods are set as: µ = 0.1, a = 9 × 10−5, b = 2 × 10−5 (8 × 8 ), a = 1 × 10−6, b = 4 × 10−5 (6 × 6), +a = 1 × 10−5, b = 1 × 10−5 (4 × 4) for Algorithm 1; λ = 235, aτ = 4 × 10−4, bτ = 3 × 10−2, au = 2 × 10−4, bu = 4.5 × 10−4 +(8 × 8), λ = 260, aτ = 2 × 10−4, bτ = 3.5 × 10−2, au = 8 × 10−4, bu = 3 × 10−4 (6 × 6), λ = 300, aτ = 1 × 10−4, bτ = +1.2 × 10−2, au = 6 × 10−4, bu = 3 × 10−4 (4 × 4) for Algorithm 2. +TABLE 2: The comparison among SPIRAL, our Algorithm 1 and Algorithm 2 in terms of Accuracy, RMSE, PSNR, SSIM +and computational time for under-sampled data. +Image ID +Scan points +Reconstruction Method +Accuracy +RMSE +PSNR +SSIM +Time (s) +Bowling +8×8 +SPIRAL+∥ · ∥1 + R+ +0.9177 +0.1581 +12.9125 +0.2561 +85 +Object-domain Alg 1 +0.9287 +0.1510 +14.7216 +0.3112 +13 +Dual-domain Alg 2 +0.9431 +0.1479 +15.3208 +0.3113 +21 +6×6 +SPIRAL+∥ · ∥1 + R+ +0.9063 +0.1690 +12.8557 +0.2059 +102 +Object-domain Alg 1 +0.9224 +0.1503 +14.6288 +0.2728 +13 +Dual-domain Alg 2 +0.9338 +0.1501 +15.0831 +0.2910 +21 +4×4 +SPIRAL+∥ · ∥1 + R+ +0.8608 +0.2076 +12.3797 +0.1671 +62 +Object-domain Alg 1 +0.8784 +0.1982 +14.5066 +0.2219 +13 +Dual-domain Alg 2 +0.8979 +0.1956 +14.5718 +0.2273 +21 + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +9 +(a) SPIRAL+∥ · ∥1 + R+ +(b) Object-domain Algorithm 1 +(c) Dual-domain Algorithm 2 +Fig. 5: The comparison of energy decays between SPIRAL in [22] and our Algorithm 1, Algorithm 2. +(a)Ground truth +(e)SPIRAL+∥ · ∥1 + R+ +(b)LCT +(f)SOCR +(c)Phasor field +(g)Algorithm 1 +(d)F-K +(h)Algorithm 2 +Fig. 6: Comparison for reconstruction results of the full-sampling Stanford bunny scene, where the scanning points are +of resolution 64×64. The parameters are set as: µ = 0.1, a = 1 × 10−4, b = 1 × 10−4 for Algorithm 1; λ = 300, aτ = +1 × 10−3, bτ = 1 × 10−3, au = 1 × 10−4, bu = 1 × 10−5 for Algorithm 2. +construction results, the quantitative indexes indicate dual- +domain Algorithm 2 gives the best scores. We can see that +the iterative reconstruction methods, i.e., SPIRAL+∥·∥1+R+, +SOCR, and our curvNLOS, consume more time than the +direct reconstruction methods, i.e., LCT, Phasor field and F- +K. Among all reconstruction methods, SOCR consumes the +most computational time even if parallel computing is used. +Thanks to the GPU implementation, our curvNLOS is much +faster than the other two iterative reconstruction methods. +Secondly, we compare the performance for under- +sampled sparse reconstruction problems. In the case of +under-sampled scanning, all comparison methods use the +measurement data obtained through linear interpolation +filling. When the number of scanning points is set as 8 × 8 +or less, the image quality reconstructed by LCT, Phasor +field, and F-K is significantly degraded. We use LCT as the +representative of the direct methods, which gives the best +visual reconstruction. The first column of Fig. 4 displays +the reconstruction results of LCT using 8 × 8, 6 × 6, and +4 × 4 (from top to bottom) scanning points. It is difficult to +identify the meaningful scene information from the recon- +structed images. The same is true for Phasor field and F-K, + +8 +6 +4 +E +2 +0 +0 +20 +40 +60 +Iteration8 +6 +4 +E +2 +0 +0 +50 +100 +150 +200 +Iteration2000 +1500 +F +1000 +E +500 +0 +0 +100 +200 +300 +Iteration0.00 +1.000.01 +1.000.00 +1.000.01 +1.000.00 +1.000.00 +1.000.00 +1.00JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +10 +8×8 +6×6 +4×4 +(a)SPIRAL+∥ · ∥1 + R+ +(b)SOCR +(c)Algorithm 1 +(d)Algorithm 2 +Fig. 7: Comparison for the reconstruction results of under-sampled Stanford bunny, where scanning points are of resolution +4×4, 6×6, 8×8, respectively. The parameters are set as: a = 1×10−4, b = 4×10−5 (8×8), a = 2×10−4, b = 2×10−5 (6×6), +a = 1 × 10−4, b = 1.5 × 10−5 (4 × 4) for Algorithm 1; λ = 150, aτ = 5 × 10−3, bτ = 1 × 10−3, au = 5 × 10−4, bu = 3 × 10−4 +(8 × 8), λ = 180, aτ = 1 × 10−3, bτ = 1 × 10−3, au = 2 × 10−4, bu = 2 × 10−4 (6 × 6), λ = 250, aτ = 1 × 10−3, bτ = +1 × 10−3, au = 5 × 10−5, bu = 2 × 10−4 (4 × 4) for Algorithm 2. + +0.00 +1.000.00 +1.000.00 +1.000.00 +1.000.00 +1.000.00 +1.000.00 +1.000.00 +1.000.00 +1.000.00 +1.000.00 +1.000.00 +1.00JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +11 +(a)Ground truth +(b)LCT +(c)Phasor field +(d)F-K +SPIRAL+∥ · ∥1 + R+ +Algorithm 1 +Algorithm 2 +(e) 4 × 4 +(f) 6 × 6 +(g) 8 × 8 +(h) 64 × 64 +Fig. 8: Comparison for reconstruction results of the SU scene, where the parameters are set as: a = 1.5×10−4, b = 4×10−5 +(64 × 64), a = 6 × 10−5, b = 3 × 10−5 (8 × 8), a = 5 × 10−5, b = 3 × 10−5 (6 × 6), a = 7 × 10−5, b = 1.5 × 10−5 (4 × 4) for +Algorithm 1; λ = 200, aτ = 3×10−3, bτ = 5×10−4, au = 5×10−4, bu = 5×10−5 (64×64), λ = 100, aτ = 3×10−3, bτ = +3 × 10−3, au = 8 × 10−4, bu = 3 × 10−5 (8 × 8), λ = 120, aτ = 1 × 10−3, bτ = 1 × 10−3, au = 1 × 10−4, bu = 2 × 10−5 +(6 × 6), λ = 150, aτ = 1 × 10−3, bτ = 1 × 10−3, au = 3 × 10−4, bu = 1 × 10−5 (4 × 4) for Algorithm 2. + +SSJSJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +12 +(a)Ground truth +(e)SPIRAL+∥ · ∥1 + R+ +(b)LCT +(f)SOCR +(c)Phasor field +(g)Algorithm 1 +(d)F-K +(h)Algorithm 2 +Fig. 9: Comparison for reconstruction results of the outdoor scene (10 min), where the scanning points are of resolution +16 × 16. The parameters are set as: µ = 1, a = 5 × 10−5, b = 2 × 10−4 for Algorithm 1; λ = 50, aτ = 1 × 10−4, bτ = +1 × 10−4, au = 5 × 10−4, bu = 1 × 10−4 for Algorithm 2. +(a)Ground truth +(b)LCT +(e)SPIRAL+∥ · ∥1 + R+ +(c)Phasor field +(f)Algorithm 1 +(d)F-K +(g)Algorithm 2 +Fig. 10: Comparison for reconstruction results of the bike (10 min), where the scanning points are of resolution 16 × 16. +The parameters are set as: µ = 1, a = 2 × 10−4, b = 5 × 10−5 for Algorithm 1; λ = 35, aτ = 1 × 10−4, bτ = 1 × 10−4, au = +6 × 10−4, bu = 2 × 10−4 for Algorithm 2. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +13 +(a) Ground truth +(b) LCT +(e) SPIRAL+∥ · ∥1 + R+ +(h) SPIRAL+∥ · ∥1 + R+ +(c) Phasor field +(f) Algorithm 1 +(i) Algorithm 1 +(d) F-K +(g) Algorithm 2 +(j) Algorithm 2 +Fig. 11: Comparison for reconstruction results of the teaser scene (180 min), where the scanning points are of resolution +16 × 16. The parameters are set as: µ = 1, a = 5 × 10−5, b = 1 × 10−5 for Algorithm 1; λ = 100, aτ = 1 × 10−4, bτ = +1 × 10−4, au = 2 × 10−4, bu = 1.2 × 10−4 for Algorithm 2. +which we have not shown here. Therefore, we can conclude +that the direct reconstruction methods can not deal with the +compressed reconstruction scenarios. +The second column and third column of Fig. 4 are +the reconstruction results of SPIRAL+∥ · ∥1 + R+ without +smoothing and after smoothing. As can be seen, smoothing +plays a very important role for SPIRAL+∥ · ∥1 + R+, while +our approaches do not involve any post-processing step. +The last two columns are the reconstruction obtained by +our object-domain Algorithm 1 and dual-domain Algorithm +2, respectively. We can observe that the image contrast is +significantly improved, and the structural information is +much clearer than both LCT and SPIRAL, especially for +4 × 4 scanning points. The dual-domain Algorithm 2 tends +to produce reconstruction results with better smoothness +due to the introduction of the curvature regularization for +measured signals. When the scanning points become more +and more sparse, the advantages of the dual domain algo- +rithm become more and more obvious. Furthermore, Table +2 exhibits the metrics estimated by SPIRAL+∥·∥1 +R+ with +smoothing and our methods, which further convinced the +advantages of our methods over SPIRAL. +In what follows, we examine the energy decay of both +SPIRAL+∥ · ∥1 + R+ and our methods. As shown in Fig.5, +although SPIRAL+∥·∥1 +R+ converges quickly, the numer- +ical energy fluctuates greatly in the early stage, while our +methods converge much more stable. +Another synthetic data is Stanford bunny from the +Zaragoza NLOS synthetic dataset. The total 64×64 scanning +points occupy an area of 0.6 × 0.6 m2 on the wall. The data +has 512 time bins and the photon travels 0.0025 m in each +bin. The reconstruction results of each method in the case of +full sampling are shown in Fig. 6. In the scene, our methods +still maintain obvious advantages, which can preserve the +structural details, especially in the rabbit ear region. Fig. 7 +shows the reconstruction result with sparse scanning points, + +1 +33JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +14 +15s +(a) LCT +(b) Phasor field +(c) F-K +(d) SPIRAL+∥·∥1 +R+ +(e) SOCR +(f) Algorithm 1 +60min +(A) LCT +(B) Phasor field +(C) F-K +(D) SPIRAL+∥·∥1+R+ +(E) SOCR +(F) Algorithm 1 +Fig. 12: Comparison for reconstruction results of the dragon with an exposure time of 15s and 60min, where the scanning +points are of resolution 16 × 16. The parameters are set as: µ = 1, a = 8 × 10−4, b = 5 × 10−5 for data of exposure time 15s; +µ = 1, a = 1 × 10−3, b = 2.5 × 10−4 for data of exposure time 60min. +where the sampling points of 8×8, 6×6, 4×4 were used for +reconstruction. As can be observed, both SPIRAL+∥·∥1+R+ +and our methods can estimate the shape of the bunny +even when there are only 4 × 4 scanning points. And +our curvNLOS obviously gives better reconstruction quality +with much smoother surfaces and fewer artifacts. Although +we use the interpolated data for SOCR, it still fails to +obtain meaningful reconstruction results, which reveals its +limitation to deal with compressed sensing reconstruction +scenarios. +5.4 +Experiments on measured data +To further prove the effectiveness of our curvNLOS meth- +ods, we evaluate them on measured data of the real scenes +in [10] and [17]. We first verify our methods under multiple +sampling rates in the ”SU” scene, the results of which are +shown in Fig. 8. The scene consists of two letter planes, with +the front ’S’ obscuring the back ’U’, which was sampled +at 64 × 64 locations on the wall of size 0.7 × 0.7 m2. The +time resolution is 512 and each time bin spans 16 ps. The +first row lists the ground truth image and the reconstruction +results of LCT, Phasor field, and F-K using the full sampling +data, and the rest three rows are the reconstruction results of +SPIRAL+∥·∥1+R+ and our two algorithms under the sparse +sampling data. As can be observed, our curvNLOS methods +can produce satisfactory reconstruction results even when +the number of scanning points is reduced to 4×4. Moreover, +compared to the SPIRAL+∥ · ∥1 + R+, our results preserve +the structure of the two letters, which are visually clearer +and more continuous. +In addition, we also apply our methods to another three +scenes in the Stanford dataset, which are the outdoor, bike, +and teaser, respectively. The size of raw measurement data +is 512 × 512 × 2048, and the wall size is 2 × 2 m2. The time +resolution is cropped to 512 and each time bin spans 32 ps, +while the spatial resolution is 64 × 64 for outdoor and bike +and 128 × 128 for the teaser. On this basis, we uniformly +sample 16 × 16 scanning points to reconstruct the scenes. +The comparison results are displayed in Fig. 9, Fig. 10, and +Fig. 11, respectively. Similar to the previous experiments, +the qualities of the images reconstructed by our methods are +much better than other comparison methods. Although the +difference between the images reconstructed by Algorithm 1 +and Algorithm 2 is visually negligible for the outdoor scene, +we can observe the advantages of dual-domain curvature +regularization on both bike and teaser scene, where the +results of Algorithm 2 have fewer outliers. +5.5 +Experiments on data with different exposure time +In this subsection, we verify the performance of our curvN- +LOS method on the dragon scene of spatial resolution +64×64, where two measurements were captured with dif- +ferent exposure times, i.e., 15s and 60min respectively. The +shorter the total exposure time is, the fewer photons are +captured at each point, and the data is more affected by +noises. The reconstruction results are shown in Fig. 12. It +can be seen that except for our curvNLOS, all the com- +parison methods cannot reconstruct satisfactory images for +the data estimated by short exposure time. The images +reconstructed by Phasor field and F-K methods are blurry +and unrecognizable. The images reconstructed by LCT and +SPIRAL+∥ · ∥1 + R+ methods can be roughly identified, but +contain a large amount of noise. The reconstruction of the +SOCR is also greatly affected by noise and structural loss. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +15 +(a)bunny +(b)SU +(c)outdoor +(d)dragon15 +(e)dragon60 +(f)bike +(g)teaser +Fig. 13: The comparison of computational time among SPIRAL, SOCR, and our Algorithm 1, Algorithm 2 on the test data. +Although two legs of the dragon are connected together, +our object-domain Algorithm 1 still produces reconstruction +results with much better quality. On the other hand, the +reconstruction results of all comparison methods are sig- +nificantly improved on the long exposure time data. We can +observe the rough shape of the dragon in the reconstructed +images of LCT, Phasor field, F-K, SPIRAL+∥ · ∥1 + R+ and +SOCR, but they are still greatly affected by noise. Obviously, +the result of our method is the best. By increasing the time +resolution, the reconstruction quality is improved, especially +the edge information. +5.6 +Computational time comparison +Finally, we compare the computational time among the +three iterative reconstruction algorithms on different scenes, +which are exhibited in Fig.13. By looking into Fig. 13 (b), +(f) and (g), it can be seen that the computational time of +SPIRAL+∥·∥1+R+ varies with data dimensions and scenes. +It converges fast in the bike scene, while it consumes much +more time to reconstruct the teaser scene. Observing Fig.13 +(a), (c), (d), and (e), reveals that the computational cost +of SOCR is much high due to the signal-object collabo- +rative regularization. Although we implement the parallel +codes with 12 workers, the reconstruction time is still very +long. For our curvNlOS, due to the GPU computation, it +consumes the least computational time among the three +iterative methods. Furthermore, since there is no inner itera- +tion in our approaches, the computational time is relatively +stable without varying too much for measured data of the +same dimensions. +6 +CONCLUSION AND FUTURE WORKS +In this paper, we introduced the curvature regularization +models for the under-sampled sparse NLOS reconstruction +problem. The sparse scanning can effectively shorten the +acquisition time, but it also leads to the failure of reconstruc- +tion methods. The curvature regularization used in the ob- +ject domain and original signal domain can not only restore +the smooth surface of the hidden objects, but also the contin- +uous signals. Fast numerical algorithms were proposed for +solving the high-order curvature minimization problems, +where the curvature function was regarded as the adaptive +weight for the total variation to ease the computation, and +the linearization technique was used to accelerate the con- +vergence. Extensive numerical experiments were conducted +on both synthetic and real data. Compared to state-of-the- +art direct and iterative NLOS reconstruction methods, our +curvNLOS was shown with better reconstruction qualities +for different scenes, demonstrating the effectiveness of the +curvature in recovering the three-dimensional surfaces. The +results showed our curvNLOS can reconstruct the 3D hid- +den scenes with 64 × 64 spatial resolution by the measure- +ments of 4 × 4 sampling points. Besides that, thanks to +the GPU implementation, our curvNLOS performed much +faster than other iterative reconstruction methods, which +can facilitate its use in real applications. +Although this work improves both reconstruction qual- +ity and computational efficiency, NLOS imaging reconstruc- +tion is still a very challenging problem. The existing re- +construction methods suffer from poor spatial resolution, +noise sensitivity, and poor real-time performance. Thus, our +future work includes using the super-resolution method +and deep learning technique to achieve better NLOS recon- +struction methods. Currently, uniform sampling has been +successfully used to reduce the number of scanning points. +Such a sampling method lacks integration with the physical +process of imaging, which needs to be optimized in the +future. + +3000 +S +2000 +Time( +1000 +0 +SPIRAL SOCR +Alg1 +Alg2100 +Time(s +50 +0 +SPIRAL +Alg1 +Alg23000 +2000 +S +Time( +1000 +0 +SPIRAL SOCR +Alg1 +Alg24000 +3000 +Time(s) +2000 +1000 +0 +SPIRAL SOCR +Alg12000 +1500 +S +Time( +1000 +500 +0 +SPIRAL SOCR +Alg140 +30 +s +Time( +20 +10 +0 +SPIRAL +Alg1 +Alg2400 +300 +Time(s) +200 +100 +0 +SPIRAL +Alg1 +Alg2JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +16 +ACKNOWLEDGMENTS +The work was supported by the National Natural Science +Foundation of China (NSFC 12071345, 11701418). +REFERENCES +[1] +D. Faccio, A. Velten, and G. 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Teboulle, “A fast iterative shrinkage-thresholding +algorithm for linear inverse problems,” SIAM Journal on Imaging +Sciences, vol. 2, no. 1, pp. 183–202, 2009. + diff --git a/b9AyT4oBgHgl3EQfi_hp/content/tmp_files/load_file.txt b/b9AyT4oBgHgl3EQfi_hp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..25201f3a57f6e67237b1a19e4a2c10b518408597 --- /dev/null +++ b/b9AyT4oBgHgl3EQfi_hp/content/tmp_files/load_file.txt @@ -0,0 +1,1010 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf,len=1009 +page_content='JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 8, AUGUST 2015 1 Curvature regularization for Non-line-of-sight Imaging from Under-sampled Data Rui Ding, Juntian Ye, Qifeng Gao, Feihu Xu, and Yuping Duan Abstract—Non-line-of-sight (NLOS) imaging aims to reconstruct the three-dimensional hidden scenes from the data measured in the line-of-sight, which uses photon time-of-flight information encoded in light after multiple diffuse reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The under-sampled scanning data can facilitate fast imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' However, the resulting reconstruction problem becomes a serious ill-posed inverse problem, the solution of which is of high possibility to be degraded due to noises and distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' In this paper, we propose two novel NLOS reconstruction models based on curvature regularization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=', the object-domain curvature regularization model and the dual (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=', signal and object)-domain curvature regularization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Fast numerical optimization algorithms are developed relying on the alternating direction method of multipliers (ADMM) with the backtracking stepsize rule, which are further accelerated by GPU implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' We evaluate the proposed algorithms on both synthetic and real datasets, which achieve state-of-the-art performance, especially in the compressed sensing setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' All our codes and data are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='com/Duanlab123/CurvNLOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Index Terms—Non-line-of-sight, under-sampled scanning, curvature regularization, dual-domain reconstruction, GPU implementation !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 1 INTRODUCTION N ON-LINE-OF-SIGHT (NLOS) imaging uses time- resolved measurements to recover the 3D shape and visual appearance of hidden objects beyond the direct line of sight of sensors [1], [2], which has various applications such as autonomous driving [3], 3D human pose estimation [4], sensor system [5], [6], and many other domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Recent advances in single-photon detector technology and compu- tational algorithms for solving large-scale inverse problems make NLOS imaging feasible in different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' For instance, NLOS imaging and real-time tracking of hidden objects have been demonstrated over a distance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='43 km [7] and at a resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='6 mm at a distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='55 m [8], [9], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The inverse problem of reconstructing the 3D shape and appearance of the hidden object is also very challenging [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Roughly speaking the NLOS imaging reconstruction methods can be divided into three categories, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=', direct re- construction methods, iterative reconstruction methods, and deep learning-based reconstruction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The filtered back-projection algorithm is a kind of fast direct method by filtering the data and performing the back-projection opera- tion, where the data is painted back in the image along the direction it being measured [12], [13], [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Other direct reconstruction methods also contain the phasor field [16], frequency-domain method [17], Fermat flow method [18], etc, which are proposed by modeling the physical process of imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' These direct reconstruction methods are fast and efficient but are sensitive to noises and measurement distor- R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Ding, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Gao, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Duan are with Center for Applied Mathematics, Tianjin University, Tianjin 300072, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' E-mail: {rding,gaoqifeng 98,yuping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='duan}@tju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='cn J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Ye and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Xu are with Hefei National Laboratory for Physical Sciences at Mi- croscale and Department of Modern Physics, University of Science and Tech- nology of China, Hefei 230026, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' E-mail: jt141884@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' fei- huxu@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='cn K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Chen is with Department of Mathematical Sciences, University of Liver- pool, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Manuscript received April 19, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' revised August 26, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The iterative reconstruction methods have been used by introducing priors to regularize the NLOS reconstruction problem and improve the image quality [19], [20], [21], [22], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Although iterative reconstruction methods can provide high-quality reconstruction images, they also con- sume much more computational time than direct methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Due to the development of deep learning, convolutional neural network models have been developed for solving the NLOS reconstruction problem [24], [25], [26], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' However, deep learning methods are highly dependent on the training datasets, which makes them possible to lose their impact on real measurement data and spatial/temporal degradation data [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Indeed, the dense raster scanning used in the afore- mentioned methods is detrimental to high-speed NLOS applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Thus, different strategies have been employed to reduce the acquisition time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' One strategy is to use the multi-pixel time-of-flight NLOS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Nam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' [28] used the specifically designed single photon avalanche diode (SPAD) array detectors to realize a multi-pixel NLOS imag- ing method together with a fast reconstruction algorithm that can capture and reconstruct live low-latency videos of NLOS scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Pei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' [29] used the SPAD array and an optimization-based computational method to achieve NLOS reconstruction of 20 frames per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Another strategy is to use fewer scanning points to reconstruct the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Isogawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' [30] proposed a circular and confocal non- line-of-sight scan, for which the scanning involves sampling points forming a circle on a visible wall to reduce the dimension of transient measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' [22] explored compressed sensing in active NLOS imaging to reduce the required number of scanning points for fast implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' As illustrated in previous works [22], [30], NLOS recon- struction can be realized by under-sampled measurements to facilitate high-speed acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' However, sparse mea- surements may lead to the degradation of reconstructed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Curvature regularization is an important technique arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='00406v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='CV] 1 Jan 2023 JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 8, AUGUST 2015 2 (a) Non-confocal NLOS (b) Confocal NLOS (c) Under-sampled Confocal NLOS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 1: Illustration of transient NLOS imaging, where (a) non-confocal NLOS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' (b) confocal NLOS with full scanning, and (c) confocal NLOS with under-sampled scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' for various shape and image processing tasks [31], [32], [33], which is well-known for its ability in modeling the continuities of edges and surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Initially, curvature regu- larization was applied to the problem of image inpainting to restore satisfactory results meeting human perception [34], [35], [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Due to its superiority in recovering missing data, curvature regularity has also been used for surface reconstruction [37], [38], sparse image reconstruction [39], [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Since the curvature regularization is capable to capture the tiny but elongated structures in images, it has also been used for image segmentation [41], tubular structure tracking [42], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Obviously, curvature regularization is a good choice for under-sampled NLOS imaging problems to obtain smooth and satisfied reconstructed surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' In this paper, we study the curvature regularization reconstruction model for under-sampled NLOS imaging problems, called curvNLOS, which can reconstruct surfaces with good quality through as few measurements as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' We also develop an effective numerical algorithm based on the alternating direction methods of multipliers (ADMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Comprehensive experiments are conducted on both syn- thetic and real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The numerical results demonstrate that curvature regularization can effectively guarantee the smoothness of the object and estimated signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The main contribution of this work can be summarized as follows: We propose novel curvature regularization mod- els for solving the NLOS imaging problem, where the curvature regularization is used to restore the smooth surface of hidden objects and fill in the sparse measured signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' We present fast iterative optimization algorithms for solving the curvature minimization problems, where the high-order curvature is regarded as the adaptive weight for total variation and the acceleration tech- nique is used to obtain a faster convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' We develop a GPU-based NLOS reconstruction pack- age by utilizing the parallel computation ability of a GPU card, which is desirable for high-speed NLOS applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' By comparing with the state-of-the-art NLOS meth- ods, our curvature regularization models can ro- bustly restore estimated signals and the three- dimensional scene points, especially when the mea- surements are under-sampled measured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The roadmap of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' In Section 2, we briefly describe NLOS physics and mathematical formula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Section 3 presents the object-domain curvature regular- ization method and the ADMM-based algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Section 4 proposes the dual-domain curvature regularization method for NLOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Numerical experiments are evaluated on both synthetic and real measured data by comparing with other established methods in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Finally, we conclude our paper in Section 6 and present some discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 2 FORWARD PROPAGATION MODEL In transient imaging, a time-resolved detector is used to measure the incident flux of photons as a function of emitted light impulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Let x = (x, y, z) be the three- dimensional scene coordinates, and x′ i = (x′ i, y′ i, z = 0), x′ d = (x′ d, y′ d, z = 0) be the illumination and detection coordinates on the visual wall, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 1, the light emitted by the laser passes through the scanning galvanometer and hits the illumination point x′ i, and diffuses towards the target point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Then, the photons reflect on the target and propagate to the detection point x′ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Finally, the photons diffuse back and enter the single-pixel detectors such as SPAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The NLOS reconstruction aims to recover the location, shape, albedo, and normal of the target from the detected number of photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The general non-confocal direction-albedo forward propagation model for NLOS imaging can be described below τ(x′ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' x′ d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' t) = ��� Ω (x′ i − x) · n(x) d(x′ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' x)3 (x′ d − x) · n(x) d(x′ d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' x)3 u(x) δ � d(x′ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' x) + d(x′ d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' x) − tc � dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' (1) where τ is the recorded transient image,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' u is the albedo of the hidden scene at each point x with z > 0 in the 3D half- space Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' c is the speed of the light,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' n(x) = (nx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' ny,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' nz)(x) denotes the surface normal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' and δ represents the surface of a spatio-temporal four-dimensional hypercone defined by x2 + y2 + z2 − (tc/2)2 = 0 modeling light propagation from the wall to the object and back to the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Studies on non- confocal NLOS imaging can be found in [43], [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The dis- Visible Wall SPAD Wall Laser Hidden ObjectVisible Wall SPAD Wall Hidden ObjectVisible Wall SPAD Wall Hidden ObjectJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 8, AUGUST 2015 3 tances between the reconstructed point and the illumination point and detection point are defined as d(x′ i, x) = � (x′ i − x)2 + (y′ i − y)2 + z2, and d(x′ d, x) = � (x′ d − x)2 + (y′ d − y)2 + z2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' When the illumination point and detection point locate at the same position, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=', x′ = x′ i = x′ d, we have the confocal direction and albedo NLOS reconstruction model as given below τ(x′, t) = ��� Ω u(x)n(x) d(x′, x)4 ·n(x)·δ � 2d(x′, x)−tc � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' (2) Methods to localize the three-dimensional scene points and estimate their surface normals of hidden objects have been established for the NLOS imaging [21], [45], [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' As long as ⟨n(x), n(x)⟩ = 1, the direction-albedo forward model reduces into the volumetric albedo model τ(x′, t) = ��� Ω u(x) d(x′, x)4 · δ � 2d(x′, x) − tc � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' (3) The corresponding discrete image formation of the NLOS model (3) is given as τ = Au = R−1 t HRzu, (4) where the matrix A = R−1 t HRz is the light transport matrix, H represents the shift-invariant 3D convolution, Rt and Rz represent the transformation operations applied to the temporal and spatial dimensions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The task of recovering the object u from the measured signal τ is a typical ill-posed inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The regular- ization method is a good choice to solve ill-posed inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' In [22], the sparsity regularization and a non- negative constraint were used to restore the surface of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Although the algorithm can reconstruct a three- dimensional hidden image of 64×64 spatial resolution with 5 × 5 scanning points, the results greatly depend on the post-processing step to smooth out the noises and outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The collaborative signal and objective regularization was used in [21], which is shown effective on both confocal and non-confocal NLOS imaging reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' However, such complex regularization makes the computational costs in- crease extremely, which is unfavorable for real applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Thus, finding a regularization method suitable for NLOS imaging algorithms can give consideration to both imaging quality and speed, which is still a challenging problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 3 INVERSE PROBLEM BY CURVATURE REGULAR- IZATION The curvature regularization can naturally fill the missing information and obtain a smooth surface, which motivates us to use it to approximate the oracle signal corresponding to the real hidden scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Thus, we propose to reconstruct 3D objects by employing three-dimensional curvature regu- larization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' To be specific, we aim to minimize the following curvature-related energy for the NLOS imaging problem min u � E(u) = 1 2∥Au − τ0∥2 Ω\\X + R(κ(u)) � , (5) where X ⊂ Ω denotes the missing region, and R(·) denotes the curvature regularization such as R(κ(u)) = � x φ(κ(u(x)))|∇u(x)| with φ(·) being the function of curvature and |∇u| being the total variation of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' In order to efficiently solve the curvature regularization model (14), we reformulate it into a constrained minimization problem min u,v 1 2∥Au − τ0∥2 Ω\\X + R(κ(u)), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' v = ∇u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Then the associated augmented Lagrangian functional can be defined as follows L(u, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Λ) =1 2∥Au − τ0∥2 Ω\\X + � x φ(κ(u(x)))|v(x)| + ⟨Λ, v − ∇u⟩ + µ 2 ∥v − ∇u∥2 2, (6) where Λ is the Lagrange multiplier, and µ is the positive parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' We use the Alternating Direction Method of Multipliers (ADMM) to iteratively and alternatively solve the sub-minimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1 Computation of curvature As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 2, the 3D transient measurement τ(x, y, t) is a time-continuous function, and u(x, y, z) is a spatial continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The curvature regularization term can provide strong prior information on the continuity of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Therefore, we use the curvature regularization to both hidden objects and the signals, which are defined as κ(u) := ∇ · ∇u(x, y, z) |∇u(x, y, z)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The function of the curvature can be defined in different forms similar to [35], where we use the total squared curva- ture as follows φ(κ) = a + b|κ|2 with a, b ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' In order to ease the computation, we regard curvature terms as the weights of the total variation, which are computed explicitly as long as u being updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='2 Sub-minimization problems and the solutions Now we discuss the sub-minimization problems for solving (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1) and the corresponding solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1 Sub-minimization problem w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' v The sub-minimization problem w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' v is defined as min v � x φ(κ(uk(x)))|v(x)| + µ 2 ��v − (∇uk − Λk µ ) ��2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' (7) Since φ(κ(uk(x))) is known in advance, the above mini- mization problem becomes the weighted ℓ1 and ℓ2 mini- mization problem, which can be solved by the shrinkage operator as follows vk+1 = shrinkage � ∇uk − Λk µ , φ(κ(u)) µ � , (8) JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 8, AUGUST 2015 4 (a) Ground truth (b) τ(x, y, t) (c) τ(32, 32, :) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 2: Overview of NLOS imaging measurements, where (a) The ground truth image of the hidden objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' (b) The measurements of the wall τ attenuated along the time axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' and (c) A histogram measured at the scanned point indexed by (32,32) on the visible wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' with the shrinkage operator being defined as shrinkage(a, b) = max{|a| − b, 0} ◦ a |a| and ◦ being the element-wise multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='2 Sub-minimization problem w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' u The u sub-minimization problem can be formulated as a quadratic minimization problem min u 1 2 ��ADu − τ0 ��2 2 + µ 2 ��∇u − (vk+1 + Λk µ ) ��2 2, (9) where AD = DA with D being the sampling operator to define the missing data domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' For simplicity, we denote f(u) = 1 2 ��ADu − τ0 ��2 2, g(u) = µ 2 ��∇u − (vk+1 + Λk µ ) ��2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Then we reformulate the minimization problem (9) using a quadratic approximation of f(u) at a given point uk as follows min u f(uk) + ⟨u − uk, ∇f(uk)⟩ + L 2 ��u − uk ��2 2 + g(u), (10) which is equivalent to min u g(u) + L 2 ��u − (uk − 1 L∇f(uk) ��2 2, (11) with L being 2∥AD∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The optimal value of the (11) becomes the solution to the following partial differential equation (PDE) (LI + µ∇∗∇)u = Luk − ∇f(uk) + µ∇∗(vk+1 + Λk µ ), where ∇f(uk) = AT D(ADuk − τ0) and ∇∗ is the adjoint operator of gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The above PDE can be solved by the Fast Fourier Transform (FFT) as follows uk+1 = F−1 �F �Luk − ∇f(uk) + µ∇∗(vk+1 + Λk µ � LI + µF(∇∗∇) � , (12) where F and F−1 denote the forward and inverse FFT operation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='3 Update of Lagrange multipliers Finally, we update the Lagrange multipliers by the gradient ascend method as follows Λk+1 = Λk + µ(vk+1 − ∇uk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' (13) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='3 Our algorithm We summarize the ADMM-based algorithm for solving the object-domain curvature regularization model as Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' As seen, the acceleration technique in [47] is applied to obtain a faster convergence rate, where the specific linear combination of the previous two points {uk−1, uk} is used in the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Due to the non-convexity of the curvature regulariza- tion in our model (5), the convergence of Algorithm 1 is difficult to obtain theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' A partial convergence result can be found in our previous work [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' From observing the numerical energy decay, Algorithm 1 numerically converges quite stable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 4 DUAL-DOMAIN CURVATURE METHOD AND GPU IMPLEMENTATION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1 Dual-domain reconstruction method Since the under-sampled signal can be regarded as the data inpainting problem, we also propose a dual-domain reconstruction model, where the curvature regularization is employed for both the signal domain and object domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Mathematically, we present the dual-domain NLOS imaging reconstruction model as follows min u,τ � E(u, τ) = 1 2∥Au−τ∥2 2+λ 2 ∥τ−τ0∥2 Ω\\X+R(u)+R(τ) � , (14) where λ is the positive parameter, R(τ) is with the same formulation as R(u) and κ(τ) is calculated as follows κ(τ) := ∇ · ∇τ(x, y, t) |∇τ(x, y, t)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Note that κ(u) calculates the spatial curvature of the hidden object, while κ(τ) is used to measure the curvature of Photon 20 0 0 200 4 Time Bil00 600 n60 Count 40JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 8, AUGUST 2015 5 Algorithm 1: The ADMM algorithm for solving the object-domain curvature regularization model (5) Input: Raw data τ0 and parameters Λ, a, b, µ, ϵ, ¯u0 = u0 = 0, Λ0 = 0, Tmax,t0 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Output: uk+1 1 for k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' do /* Solve the saddle-point problem / 2 (uk+1, vk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Λk+1) = max Λ min u,v L(¯uk, vk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Λk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' /* Compute tk+1 and ¯uk+1 from / 3 tk+1 = 1 + � 1 + 4(tk)2 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' ¯uk+1 = uk+1 + tk − 1 tk+1 (uk+1 − uk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' /* Update φ(κu) / 4 φ(κ(u)) = a + b(∇ · ∇u |∇u|)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' /* Stopping condition / 5 k ≥ Tmax or en+1 = ��E(uk)−E(uk+1) �� ��E(uk+1) �� ≤ ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 6 end the time-dependent signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Similarly, we rewrite the above model into the following constrained minimization problem min u,τ,v,w,f 1 2∥Au − τ∥2 2 + λ 2 ∥f − τ0∥2 Ω\\X + R(u) + R(τ), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' v = ∇u, w = ∇τ, f = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Then the associated augmented Lagrangian functional can be defined as follows L(u, τ, v,w, f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Λ1, Λ2, Λ3) = 1 2∥Au − τ∥2 2 + λ 2 ∥f − τ0∥2 Ω\\X + � x φ(κ(u(x)))|v(x)| + � x φ(κ(τ(x)))|w(x)| + ⟨Λ1, v − ∇u⟩ + µ1 2 ∥v − ∇u∥2 2 + ⟨Λ2, w − ∇τ⟩ + µ2 2 ∥w − ∇τ∥2 2 + ⟨Λ3, f − τ⟩ + µ3 2 ∥f − τ∥2 2, where Λ1, Λ2, Λ3 are the Lagrange multipliers, and µ1, µ2, µ3 are the positive parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Then the Alternat- ing Direction Method of Multipliers (ADMM) can be im- plemented to iteratively and alternatively solve the sub- minimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The algorithm and solutions to the sub-minimization problems can be generalized from the object-domain cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Both the object-domain reconstruction algorithm and dual-domain reconstruction algorithm are provided at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='com/Duanlab123/CurvNLOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' More details can be found in our public codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Note that we initialize the variable u by Algorithm 1 to obtain better convergence and high-quality reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='2 GPU implementation The GPU has a distinct advantage in parallel computing, consisting of thousands of smaller, more efficient cores designed for multitasking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The GPU-based image recon- struction allows for the use of more complex models and maintains reasonable execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Thus, we implement both Algorithm 1 and Algorithm 2 on the GPU to reduce the computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' We utilized one RTX 2080 graphics card to run our algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' For 64 × 64 × 512 data, each iteration takes about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1 seconds, and for 128 × 128 × 512 data, each iteration takes about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='2 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' For data with higher dimensions, the advantage of GPU over CPU is more obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 5 NUMERICAL RESULTS In this section, we discuss the performance of our dual- domain curvature method on both synthetic and real imag- ing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' We use the accuracy, RMSE, PSNR, and SSIM to evaluate the reconstruction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The accuracy refers to the foreground/background classification accuracy, which is defined as Accuracy = TP + TN TP + TN + FP + FN , where TP and TN are correct foreground(true positives) and correct background(true negatives), FP and FN are excess(false positives) and missing geometry (false nega- tives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' After dividing the foreground and background, the depth error of the foreground is calculated by RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Both PSNR and SSIM are common evaluation indicators used in image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Since we want to achieve the purpose of fast reconstruction by quickly collecting information, reconstruction time is also an important evaluation index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' We will compare the time used for reconstruction by each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1 Comparison methods We evaluate the performance of our curvature regulariza- tion methods by comparing them with the following state- of-the-art approaches LCT: The light cone transform was proposed in [10] for confocal NLOS imaging, which is a parameter- free back-projection-based reconstruction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Phasor field: The phasor field [16] formulates the NLOS imaging problem as a wave imaging problem and uses the techniques of classic optics for the NLOS imaging, which is also a parameter-free direct reconstruction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' F-K migration: The frequency-domain method pro- posed in [17] can handle both confocal and non- confocal NLOS imaging problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The F-K migra- tion is robust to objects with complex reflective prop- erties and easy to implement with no parameters to tune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' SPIRAL+∥ · ∥1 + R+: The modified sparse Poisson intensity reconstruction algorithm proposed in [22], where the non-negativity prior and sparsity prior of the hidden scene are used as the regularization JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 8, AUGUST 2015 6 Algorithm 2: Our ADMM algorithm for solving the dual-domain curvature regularization model (14) Input: Raw data τ0, and parameters au, bu, aτ, bτ, Λ1, Λ2, Λ3, µ1, µ2, µ3, ϵ, ¯u0 = u0, ¯τ = τ = 0, Λ0 1 = Λ0 2 = Λ0 3 = 0, Tmax, t0 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Output: uk+1 1 for k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' do /* Compute the following v subproblem by the soft shrinkage operator / 2 min v � x φ(κ(uk(x)))|v(x)| + µ1 2 ��v − (∇uk − Λk 1 µ1 ) ��2 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' /* Compute the following w subproblem by the soft shrinkage operator / 3 min w � x φ(κ(τ k(x)))|w(x)| + µ2 2 ��w − (∇τ k − Λk 2 µ2 ) ��2 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' /* Compute the following f subproblem by the Fast Fourier Transform / 4 min f λ 2 ��f − τ0 ��2 Ω\\X + µ3 2 ��f − (τ k − Λk 3 µ3 ) ��2 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' /* Compute the following τ subproblem by the Fast Fourier Transform / 5 min τ 1 2 ��Auk − τ∥2 2 + µ2 2 ��∇τ − (wk+1 + Λk 2 µ2 ) ��2 2 + µ3 2 ��τ − (f k+1 + Λk 3 µ3 ) ��2 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' /* Compute the following u subproblem by the Fast Fourier Transform / 6 min u 1 2 ��Au − τ k+1∥2 2 + µ1 2 ��∇u − (vk+1 + Λk 1 µ1 ) ��2 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' /* Compute tk+1 and ¯uk+1 from / 7 tk+1 = 1 + � 1 + 4(tk)2 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' ¯uk+1 = uk+1 + tk − 1 tk+1 (uk+1 − uk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' ¯τ k+1 = τ k+1 + tk − 1 tk+1 (τ k+1 − τ k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' /* Update φ(κu) and φ(κτ) by / 8 φ(κu) = au + bu(∇ · ∇u |∇u|)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' φ(κτ) = aτ + bτ(∇ · ∇τ |∇τ|)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' /* Stopping condition / 9 k ≥ Tmax or en+1 = ��E(uk) − E(uk+1) ��/ ��E(uk+1) �� ≤ ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 10 end terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' There is one regularization parameter that is required to adaptive adjust in different scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' SOCR: The signal–object collaborative regularization proposed in [21], which incorporated sparseness and non-local self-similarity of the hidden objects and the smoothness of the measured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' There are eight pa- rameters in SOCR, which are regularization param- eters su, λu, λpu, λd, λpd, λsd, σ, and split Bregman algorithm parameter µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Among these parameters, λd, λpd and λsd are fixed as λd = 1, λpd = 16, λsd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='25 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' And σ ranges from 20 to 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Other parameters need to choose adaptive according to the raw measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' In particular, for the scenes used in [21], we directly reconstruct the images using the provided codes without adjusting any parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' For the Bowling scene, we have fine-tuned the pa- rameters according to the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='2 Parameter discussing For the object-domain Algorithm 1, there are three param- eters that need to be adjusted, namely µ, a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The penalty parameter µ controls the convergence of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Too small µ will lead to non-convergence of the algorithm, and too large µ will reduce the quality of the reconstructed image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The two parameters a and b are the regularization parameters used to control the smoothness of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The larger the values of a and b are, the smoother the surfaces will be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The specific values of the three parameters are provided in each scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 8, AUGUST 2015 7 (a)Ground Truth (e)SOCR (b)LCT (f)SPIRAL+∥ · ∥1 + R+ (c)Phasor field (f)Algorithm 1 (d)F-K (g)Algorithm 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 3: The visual comparison of the comparison reconstruction methods under full sampling on Bowling, where the parameters of our methods are set as: a = 5 × 10−5, b = 5 × 10−5, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1 for Algorithm 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' λ = 400, aτ = 1 × 10−4, bτ = 3 × 10−2, au = 1 × 10−4, bu = 1 × 10−4 for Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' TABLE 1: The comparison of the Accuracy, RMSE, PSNR, SSIM, and computational time among different methods on image Bowling with scanning points of 64 × 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Image ID Scan points Reconstruction Method Accuracy RMSE PSNR SSIM Time (s) Bowling 64×64 LCT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='8818 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1977 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='9723 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='2385 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='28 Phasor field 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='9333 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1712 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='9628 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='2344 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='35 F-K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='8594 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='2055 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='9895 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1826 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='84 SOCR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='8889 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='5770 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='2311 876 SPIRAL+∥ · ∥1 + R+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='9470 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1368 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='9735 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='4321 136 Object-domain Alg 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='9558 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1069 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='2712 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='4473 13 Dual-domain Alg 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='9585 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='0947 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1162 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='4746 21 On the other hand, there are total eight parameters λ, µ1, µ2, µ3, au, bu, aτ, bτ in the dual-domain Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Similarly, the penalty parameters µ1, µ2, µ3 affect the stability of the algorithm, which are fixed µ1 = 1, µ2 = 800, µ3 = 2 in all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Varying these values in a small range will not affect the quality of the reconstructed image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The other five arguments are used to balance the data fidelity and curvature regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' More specifically, au and bu control the curvature regularization of the object domain, while aτ and bτ control the curvature regularization of the measured signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Likewise, we provide the specific values in each experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' In addition, we need to determine the termination con- dition for Algorithm 1 and Algorithm 2, which are termi- nated by both the number of iterations and the relative error bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Because Algorithm 1 converges faster than Algorithm 2, the number of iterations of Algorithm 1 is set to Tmax = 200, and the number of iterations of Algorithm 2 is set to Tmax = 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' In order to ensure the quality of the reconstructed image, we set the relative error bound ϵ as 1 × 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='3 Experiments on synthetic data We use two synthetic data to verify the reconstruction performance of our curvNLOS methods w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' different sam- pling rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The bowling scene was generated in [22], where the LCT model was used to generate the transient image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' There are 64 ×64 points on the visible wall with a spatial resolution of 1m×1m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The time resolution is 256 and each time bin spans 32 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Firstly, we compare our reconstruction results with other methods using the full sampling data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 1 for a visual comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' It should be noted that since the dimensions of the reconstructed results of SOCR are different from those of other methods, we fill the result with 0 for the comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' As can be observed, different methods can produce meaningful reconstruction results on the full sam- pling data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' And our reconstructions achieve the best visual quality, where the reconstructed scenes are quite close to the ground truth with fine structures and deails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Table1 records the evaluation indicators for all comparison meth- ods, where our curvNLOS gives the best accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Although the object-domain reconstruction (Algorithm 1) and dual- domain reconstruction (Algorithm 2) can obtain similar re- JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 8, AUGUST 2015 8 8×8 6×6 4×4 (a) LCT (b) SPIRAL w/o smoothing (c) SPIRAL with smoothing (d) Algorithm 1 (e) Algorithm 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 4: Reconstruction results using different numbers of scanning points from up to down the scanning points are of 8 × 8, 6 × 6, and 4 × 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The hidden scene is processed at a 64 × 64 spatial resolution and 256 temporal resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The parameters of our methods are set as: µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1, a = 9 × 10−5, b = 2 × 10−5 (8 × 8 ), a = 1 × 10−6, b = 4 × 10−5 (6 × 6), a = 1 × 10−5, b = 1 × 10−5 (4 × 4) for Algorithm 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' λ = 235, aτ = 4 × 10−4, bτ = 3 × 10−2, au = 2 × 10−4, bu = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='5 × 10−4 (8 × 8), λ = 260, aτ = 2 × 10−4, bτ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='5 × 10−2, au = 8 × 10−4, bu = 3 × 10−4 (6 × 6), λ = 300, aτ = 1 × 10−4, bτ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='2 × 10−2, au = 6 × 10−4, bu = 3 × 10−4 (4 × 4) for Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' TABLE 2: The comparison among SPIRAL, our Algorithm 1 and Algorithm 2 in terms of Accuracy, RMSE, PSNR, SSIM and computational time for under-sampled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Image ID Scan points Reconstruction Method Accuracy RMSE PSNR SSIM Time (s) Bowling 8×8 SPIRAL+∥ · ∥1 + R+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='9177 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1581 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='9125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='2561 85 Object-domain Alg 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='9287 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1510 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='7216 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='3112 13 Dual-domain Alg 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='9431 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1479 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='3208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='3113 21 6×6 SPIRAL+∥ · ∥1 + R+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='9063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1690 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='8557 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='2059 102 Object-domain Alg 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='9224 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1503 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='6288 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='2728 13 Dual-domain Alg 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='9338 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1501 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='0831 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='2910 21 4×4 SPIRAL+∥ · ∥1 + R+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='8608 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='2076 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='3797 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1671 62 Object-domain Alg 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='8784 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1982 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='5066 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='2219 13 Dual-domain Alg 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='8979 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1956 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='5718 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='2273 21 JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 8, AUGUST 2015 9 (a) SPIRAL+∥ · ∥1 + R+ (b) Object-domain Algorithm 1 (c) Dual-domain Algorithm 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 5: The comparison of energy decays between SPIRAL in [22] and our Algorithm 1, Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' (a)Ground truth (e)SPIRAL+∥ · ∥1 + R+ (b)LCT (f)SOCR (c)Phasor field (g)Algorithm 1 (d)F-K (h)Algorithm 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 6: Comparison for reconstruction results of the full-sampling Stanford bunny scene, where the scanning points are of resolution 64×64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The parameters are set as: µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='1, a = 1 × 10−4, b = 1 × 10−4 for Algorithm 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' λ = 300, aτ = 1 × 10−3, bτ = 1 × 10−3, au = 1 × 10−4, bu = 1 × 10−5 for Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' construction results, the quantitative indexes indicate dual- domain Algorithm 2 gives the best scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' We can see that the iterative reconstruction methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=', SPIRAL+∥·∥1+R+, SOCR, and our curvNLOS, consume more time than the direct reconstruction methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=', LCT, Phasor field and F- K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Among all reconstruction methods, SOCR consumes the most computational time even if parallel computing is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Thanks to the GPU implementation, our curvNLOS is much faster than the other two iterative reconstruction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Secondly, we compare the performance for under- sampled sparse reconstruction problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' In the case of under-sampled scanning, all comparison methods use the measurement data obtained through linear interpolation filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' When the number of scanning points is set as 8 × 8 or less, the image quality reconstructed by LCT, Phasor field, and F-K is significantly degraded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' We use LCT as the representative of the direct methods, which gives the best visual reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The first column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 4 displays the reconstruction results of LCT using 8 × 8, 6 × 6, and 4 × 4 (from top to bottom) scanning points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' It is difficult to identify the meaningful scene information from the recon- structed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The same is true for Phasor field and F-K, 8 6 4 E 2 0 0 20 40 60 Iteration8 6 4 E 2 0 0 50 100 150 200 Iteration2000 1500 F 1000 E 500 0 0 100 200 300 Iteration0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='00JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 8, AUGUST 2015 10 8×8 6×6 4×4 (a)SPIRAL+∥ · ∥1 + R+ (b)SOCR (c)Algorithm 1 (d)Algorithm 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 7: Comparison for the reconstruction results of under-sampled Stanford bunny, where scanning points are of resolution 4×4, 6×6, 8×8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The parameters are set as: a = 1×10−4, b = 4×10−5 (8×8), a = 2×10−4, b = 2×10−5 (6×6), a = 1 × 10−4, b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='5 × 10−5 (4 × 4) for Algorithm 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' λ = 150, aτ = 5 × 10−3, bτ = 1 × 10−3, au = 5 × 10−4, bu = 3 × 10−4 (8 × 8), λ = 180, aτ = 1 × 10−3, bτ = 1 × 10−3, au = 2 × 10−4, bu = 2 × 10−4 (6 × 6), λ = 250, aτ = 1 × 10−3, bτ = 1 × 10−3, au = 5 × 10−5, bu = 2 × 10−4 (4 × 4) for Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='00JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 8, AUGUST 2015 11 (a)Ground truth (b)LCT (c)Phasor field (d)F-K SPIRAL+∥ · ∥1 + R+ Algorithm 1 Algorithm 2 (e) 4 × 4 (f) 6 × 6 (g) 8 × 8 (h) 64 × 64 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 8: Comparison for reconstruction results of the SU scene, where the parameters are set as: a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='5×10−4, b = 4×10−5 (64 × 64), a = 6 × 10−5, b = 3 × 10−5 (8 × 8), a = 5 × 10−5, b = 3 × 10−5 (6 × 6), a = 7 × 10−5, b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='5 × 10−5 (4 × 4) for Algorithm 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' λ = 200, aτ = 3×10−3, bτ = 5×10−4, au = 5×10−4, bu = 5×10−5 (64×64), λ = 100, aτ = 3×10−3, bτ = 3 × 10−3, au = 8 × 10−4, bu = 3 × 10−5 (8 × 8), λ = 120, aτ = 1 × 10−3, bτ = 1 × 10−3, au = 1 × 10−4, bu = 2 × 10−5 (6 × 6), λ = 150, aτ = 1 × 10−3, bτ = 1 × 10−3, au = 3 × 10−4, bu = 1 × 10−5 (4 × 4) for Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' SSJSJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 8, AUGUST 2015 12 (a)Ground truth (e)SPIRAL+∥ · ∥1 + R+ (b)LCT (f)SOCR (c)Phasor field (g)Algorithm 1 (d)F-K (h)Algorithm 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 9: Comparison for reconstruction results of the outdoor scene (10 min), where the scanning points are of resolution 16 × 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The parameters are set as: µ = 1, a = 5 × 10−5, b = 2 × 10−4 for Algorithm 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' λ = 50, aτ = 1 × 10−4, bτ = 1 × 10−4, au = 5 × 10−4, bu = 1 × 10−4 for Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' (a)Ground truth (b)LCT (e)SPIRAL+∥ · ∥1 + R+ (c)Phasor field (f)Algorithm 1 (d)F-K (g)Algorithm 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 10: Comparison for reconstruction results of the bike (10 min), where the scanning points are of resolution 16 × 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The parameters are set as: µ = 1, a = 2 × 10−4, b = 5 × 10−5 for Algorithm 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' λ = 35, aτ = 1 × 10−4, bτ = 1 × 10−4, au = 6 × 10−4, bu = 2 × 10−4 for Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 8, AUGUST 2015 13 (a) Ground truth (b) LCT (e) SPIRAL+∥ · ∥1 + R+ (h) SPIRAL+∥ · ∥1 + R+ (c) Phasor field (f) Algorithm 1 (i) Algorithm 1 (d) F-K (g) Algorithm 2 (j) Algorithm 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 11: Comparison for reconstruction results of the teaser scene (180 min), where the scanning points are of resolution 16 × 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The parameters are set as: µ = 1, a = 5 × 10−5, b = 1 × 10−5 for Algorithm 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' λ = 100, aτ = 1 × 10−4, bτ = 1 × 10−4, au = 2 × 10−4, bu = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='2 × 10−4 for Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' which we have not shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Therefore, we can conclude that the direct reconstruction methods can not deal with the compressed reconstruction scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The second column and third column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 4 are the reconstruction results of SPIRAL+∥ · ∥1 + R+ without smoothing and after smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' As can be seen, smoothing plays a very important role for SPIRAL+∥ · ∥1 + R+, while our approaches do not involve any post-processing step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The last two columns are the reconstruction obtained by our object-domain Algorithm 1 and dual-domain Algorithm 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' We can observe that the image contrast is significantly improved, and the structural information is much clearer than both LCT and SPIRAL, especially for 4 × 4 scanning points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The dual-domain Algorithm 2 tends to produce reconstruction results with better smoothness due to the introduction of the curvature regularization for measured signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' When the scanning points become more and more sparse, the advantages of the dual domain algo- rithm become more and more obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Furthermore, Table 2 exhibits the metrics estimated by SPIRAL+∥·∥1 +R+ with smoothing and our methods, which further convinced the advantages of our methods over SPIRAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' In what follows, we examine the energy decay of both SPIRAL+∥ · ∥1 + R+ and our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='5, although SPIRAL+∥·∥1 +R+ converges quickly, the numer- ical energy fluctuates greatly in the early stage, while our methods converge much more stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Another synthetic data is Stanford bunny from the Zaragoza NLOS synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The total 64×64 scanning points occupy an area of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='6 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='6 m2 on the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The data has 512 time bins and the photon travels 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='0025 m in each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The reconstruction results of each method in the case of full sampling are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' In the scene, our methods still maintain obvious advantages, which can preserve the structural details, especially in the rabbit ear region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 7 shows the reconstruction result with sparse scanning points, 1 33JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 8, AUGUST 2015 14 15s (a) LCT (b) Phasor field (c) F-K (d) SPIRAL+∥·∥1 +R+ (e) SOCR (f) Algorithm 1 60min (A) LCT (B) Phasor field (C) F-K (D) SPIRAL+∥·∥1+R+ (E) SOCR (F) Algorithm 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 12: Comparison for reconstruction results of the dragon with an exposure time of 15s and 60min, where the scanning points are of resolution 16 × 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The parameters are set as: µ = 1, a = 8 × 10−4, b = 5 × 10−5 for data of exposure time 15s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' µ = 1, a = 1 × 10−3, b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='5 × 10−4 for data of exposure time 60min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' where the sampling points of 8×8, 6×6, 4×4 were used for reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' As can be observed, both SPIRAL+∥·∥1+R+ and our methods can estimate the shape of the bunny even when there are only 4 × 4 scanning points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' And our curvNLOS obviously gives better reconstruction quality with much smoother surfaces and fewer artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Although we use the interpolated data for SOCR, it still fails to obtain meaningful reconstruction results, which reveals its limitation to deal with compressed sensing reconstruction scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='4 Experiments on measured data To further prove the effectiveness of our curvNLOS meth- ods, we evaluate them on measured data of the real scenes in [10] and [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' We first verify our methods under multiple sampling rates in the ”SU” scene, the results of which are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The scene consists of two letter planes, with the front ’S’ obscuring the back ’U’, which was sampled at 64 × 64 locations on the wall of size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='7 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='7 m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The time resolution is 512 and each time bin spans 16 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The first row lists the ground truth image and the reconstruction results of LCT, Phasor field, and F-K using the full sampling data, and the rest three rows are the reconstruction results of SPIRAL+∥·∥1+R+ and our two algorithms under the sparse sampling data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' As can be observed, our curvNLOS methods can produce satisfactory reconstruction results even when the number of scanning points is reduced to 4×4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Moreover, compared to the SPIRAL+∥ · ∥1 + R+, our results preserve the structure of the two letters, which are visually clearer and more continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' In addition, we also apply our methods to another three scenes in the Stanford dataset, which are the outdoor, bike, and teaser, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The size of raw measurement data is 512 × 512 × 2048, and the wall size is 2 × 2 m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The time resolution is cropped to 512 and each time bin spans 32 ps, while the spatial resolution is 64 × 64 for outdoor and bike and 128 × 128 for the teaser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' On this basis, we uniformly sample 16 × 16 scanning points to reconstruct the scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The comparison results are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 9, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 10, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 11, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Similar to the previous experiments, the qualities of the images reconstructed by our methods are much better than other comparison methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Although the difference between the images reconstructed by Algorithm 1 and Algorithm 2 is visually negligible for the outdoor scene, we can observe the advantages of dual-domain curvature regularization on both bike and teaser scene, where the results of Algorithm 2 have fewer outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='5 Experiments on data with different exposure time In this subsection, we verify the performance of our curvN- LOS method on the dragon scene of spatial resolution 64×64, where two measurements were captured with dif- ferent exposure times, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=', 15s and 60min respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The shorter the total exposure time is, the fewer photons are captured at each point, and the data is more affected by noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The reconstruction results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' It can be seen that except for our curvNLOS, all the com- parison methods cannot reconstruct satisfactory images for the data estimated by short exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The images reconstructed by Phasor field and F-K methods are blurry and unrecognizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The images reconstructed by LCT and SPIRAL+∥ · ∥1 + R+ methods can be roughly identified, but contain a large amount of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The reconstruction of the SOCR is also greatly affected by noise and structural loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 8, AUGUST 2015 15 (a)bunny (b)SU (c)outdoor (d)dragon15 (e)dragon60 (f)bike (g)teaser Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 13: The comparison of computational time among SPIRAL, SOCR, and our Algorithm 1, Algorithm 2 on the test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Although two legs of the dragon are connected together, our object-domain Algorithm 1 still produces reconstruction results with much better quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' On the other hand, the reconstruction results of all comparison methods are sig- nificantly improved on the long exposure time data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' We can observe the rough shape of the dragon in the reconstructed images of LCT, Phasor field, F-K, SPIRAL+∥ · ∥1 + R+ and SOCR, but they are still greatly affected by noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Obviously, the result of our method is the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' By increasing the time resolution, the reconstruction quality is improved, especially the edge information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='6 Computational time comparison Finally, we compare the computational time among the three iterative reconstruction algorithms on different scenes, which are exhibited in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' By looking into Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 13 (b), (f) and (g), it can be seen that the computational time of SPIRAL+∥·∥1+R+ varies with data dimensions and scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' It converges fast in the bike scene, while it consumes much more time to reconstruct the teaser scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Observing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content='13 (a), (c), (d), and (e), reveals that the computational cost of SOCR is much high due to the signal-object collabo- rative regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Although we implement the parallel codes with 12 workers, the reconstruction time is still very long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' For our curvNlOS, due to the GPU computation, it consumes the least computational time among the three iterative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Furthermore, since there is no inner itera- tion in our approaches, the computational time is relatively stable without varying too much for measured data of the same dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 6 CONCLUSION AND FUTURE WORKS In this paper, we introduced the curvature regularization models for the under-sampled sparse NLOS reconstruction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The sparse scanning can effectively shorten the acquisition time, but it also leads to the failure of reconstruc- tion methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The curvature regularization used in the ob- ject domain and original signal domain can not only restore the smooth surface of the hidden objects, but also the contin- uous signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Fast numerical algorithms were proposed for solving the high-order curvature minimization problems, where the curvature function was regarded as the adaptive weight for the total variation to ease the computation, and the linearization technique was used to accelerate the con- vergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Extensive numerical experiments were conducted on both synthetic and real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Compared to state-of-the- art direct and iterative NLOS reconstruction methods, our curvNLOS was shown with better reconstruction qualities for different scenes, demonstrating the effectiveness of the curvature in recovering the three-dimensional surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The results showed our curvNLOS can reconstruct the 3D hid- den scenes with 64 × 64 spatial resolution by the measure- ments of 4 × 4 sampling points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Besides that, thanks to the GPU implementation, our curvNLOS performed much faster than other iterative reconstruction methods, which can facilitate its use in real applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Although this work improves both reconstruction qual- ity and computational efficiency, NLOS imaging reconstruc- tion is still a very challenging problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' The existing re- construction methods suffer from poor spatial resolution, noise sensitivity, and poor real-time performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Thus, our future work includes using the super-resolution method and deep learning technique to achieve better NLOS recon- struction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Currently, uniform sampling has been successfully used to reduce the number of scanning points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' Such a sampling method lacks integration with the physical process of imaging, which needs to be optimized in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 3000 S 2000 Time( 1000 0 SPIRAL SOCR Alg1 Alg2100 Time(s 50 0 SPIRAL Alg1 Alg23000 2000 S Time( 1000 0 SPIRAL SOCR Alg1 Alg24000 3000 Time(s) 2000 1000 0 SPIRAL SOCR Alg12000 1500 S Time( 1000 500 0 SPIRAL SOCR Alg140 30 s Time( 20 10 0 SPIRAL Alg1 Alg2400 300 Time(s) 200 100 0 SPIRAL Alg1 Alg2JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfi_hp/content/2301.00406v1.pdf'} +page_content=' 8, AUGUST 2015 16 ACKNOWLEDGMENTS The work was supported by the National Natural Science Foundation of China (NSFC 12071345, 11701418).' 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b/ddFPT4oBgHgl3EQfDDRR/content/tmp_files/2301.12991v1.pdf.txt @@ -0,0 +1,808 @@ +1 +Vulnerability to Parameter Spread in +Josephson Traveling-Wave Parametric Amplifiers +C. Kissling, V. Gaydamachenko, F. Kaap, M. Khabipov, R. Dolata, A. B. Zorin, L. Gr¨unhaupt +Abstract—We analyze the effect of circuit parameter varia- +tion on the performance of Josephson traveling-wave paramet- +ric amplifiers (JTWPAs). Specifically, the JTWPA concept we +investigate is using flux-biased nonhysteretic rf-SQUIDs in a +transmission line configuration, which harnesses the three-wave +mixing (3WM) regime. Dispersion engineering enables phase- +matching to achieve power gain of ∼20 dB, while suppressing +the generation of unwanted mixing processes. Two dispersion +engineering concepts using a 3WM-JTWPA circuit model, i.e., +resonant phase-matching (RPM) and periodic capacitance mod- +ulation (PCM), are discussed, with results potentially also appli- +cable to four-wave-mixing (4WM) JTWPAs. We propose suitable +circuit parameter sets and evaluate amplifier performance with +and without circuit parameter variance using transient circuit +simulations. This approach inherently takes into account mi- +crowave reflections, unwanted mixing products, imperfect phase- +matching, pump depletion, etc. In the case of RPM the resonance +frequency spread is critical, while PCM is much less sensitive to +parameter spread. We discuss degrees of freedom to make the +JTWPA circuits more tolerant to parameter spread. Finally, our +analysis shows that the flux-bias point where rf-SQUIDs exhibit +Kerr-free nonlinearity is close to the sweet spot regarding critical +current spread. +Index Terms—Superconducting electronics, circuit analysis, +parametric amplifiers, SQUIDs +I. INTRODUCTION +J +OSEPHSON +traveling-wave +parametric +amplifiers +(JTWPAs) hold great promise for wideband amplification +of few-photon-level signals at microwave frequencies. Their +key characteristics are bandwidths of several GHz, added noise +close to the quantum-limit, and power-handling capabilities +above -100 dBm. These advantages are particularly important +for simultaneous readout of many qubits [1], [2], or for +reading out large arrays of kinetic inductance detectors [3]. +JTWPAs utilize the nonlinear inductance of Josephson junc- +tions, arranged as transmission line arrays of either single +Josephson junctions [4]–[7], dc superconducting quantum in- +terference devices (SQUIDs) [8], rf-SQUIDs [9], or supercon- +ducting nonlinear asymmetric inductive elements (SNAILs) +[10]–[13]. Parametric amplification occurs, depending on the +order of nonlinearity, in the three-wave-mixing (3WM) or +four-wave-mixing (4WM) regime, with fp = fs + fi and +Manuscript received November 18, 2022; accepted January 18, 2023. +C.K. and F.K. gratefully acknowledge the support of the Braunschweig +International Graduate School of Metrology B-IGSM and the DFG Research +Training Group 1952 Metrology for Complex Nanosystems. This work was +also supported by the German Federal Ministry of Education and Research +(BMBF) within the framework programme “Quantum technologies – from +basic research to market” (Grant No. 13N15949). +The authors are with Physikalisch-Technische Bundesanstalt, Bundesallee +100, 38116 Braunschweig, Germany (e-mail: christoph.kissling@ptb.de). +2fp = fs+fi, respectively, where fp, fs, and fi are the pump, +signal, and idler frequencies [14]. If their respective wave +numbers kp, ks, and ki fulfill the phase-matching criterion, +∆k = kp − ks − ki = 0 for 3WM, high signal gain can be +achieved, scaling exponentially with the circuit length [14]. +In the design of a 3WM JTWPA a major challenge is +to ensure phase-matching along the whole length of the +JTWPA. This has been achieved in both 3WM and 4WM +JTWPAs either by the resonant-phase-matching (RPM) tech- +nique [15] with lumped element resonators [5]–[7] or dis- +tributed resonators [4], [13], or by the periodic variation of the +transmission line parameters [8], [16]–[19], e.g., by periodic +capacitance modulation (PCM). Both dispersion engineering +methods create a narrow-band nonmonotonic feature in the +dispersion relation k(ω), which is used to compensate the +phase-mismatch ∆k by an appropriate choice of fp. An addi- +tional challenge is to suppress unwanted parametric processes +generating higher-frequency modes, which has been solved +by increased dispersion of the transmission line for higher +frequencies f > fp [13], [18]. +Aside from these design challenges, any practical JTWPA +implementation, made up of thousands of circuit elements, +must be robust enough not to be affected by the expected +parameter spread considering a typical fabrication process. Pa- +rameter variation impacts JTWPA performance in many ways, +including phase-matching, impedance-matching, nonlinearity, +etc. Previous works analyzed the effect of deteriorated phase- +matching by resonance frequency spread in RPM-JTWPAs +both analytically [15] and via Monte Carlo simulations using +a linearized model [6], the tolerable critical current spread +using analytical approaches [20], [21] and via quantum input- +output theory [22], and microwave reflections due to pa- +rameter spread and point defects in a JTWPA using circuit +simulations [23]. In this paper, we investigate both dispersion +engineering concepts, RPM and PCM, and rigorously analyze +their parameter spread sensitivity, employing transient circuit +simulations in WRspice as described in [24]. While most of +the aforementioned works treated only one aspect of parameter +spread isolatedly, our approach inherently includes the effect +of spread on phase-matching, pump depletion, deviations from +optimal bias point, microwave reflections, etc., and therefore +gives a more realistic picture. +II. CIRCUIT DESCRIPTION AND ANALYSIS +We investigate a JTWPA architecture with rf-SQUIDs as +nonlinear inductive elements in a transmission-line arrange- +ment [9]. The rf-SQUIDs are nonhysteretic, i.e., having +arXiv:2301.12991v1 [cond-mat.supr-con] 30 Jan 2023 + +2 +(a) +Cc +Lr +Cr +Cc +λ/4 +1 +2 +3 +4 +in +1 +2 +3 +4 +(c) +in +(b) +Cn/2 +Ic +Cj +Lg +B +Cn/2 +C0/2 +Ic +Cj +Lg +B +C0/2 +out +out +0 +25 +50 +75 +Unit cell index n +0 +20 +40 +Cn (fF) +0 +2 +4 +6 +8 +10 +12 +Signal frequency (GHz) +0 +10 +20 +30 +Signal gain (dB) +Cj =40 fF +Cj =40 fF, RPM +Cj =250 fF, RPM +Cj =100 fF, PCM +Fig. 1. JTWPA circuit model and effect of phase-matching on signal gain. +(a) Circuit schematic of the RPM-JTWPA. It shows one section of m = 4 +identical unit cells and one phase-matching resonator attached in the center. +The blue-shaded panel shows the unit cell, containing a flux-biased rf-SQUID +and two capacitances C0/2 to ground. RPM is implemented using either LC- +resonators or λ/4-resonators, each capacitively coupled to the transmission +line (brown-shaded panels). The resonators create a nonmonotonic feature in +the dispersion relation ω(k) to achieve phase-matching. (b) Gain vs frequency +profiles, obtained by transient circuit simulations for no dispersion engineering +(dash-dotted), RPM (dashed line), RPM with increased effective SQUID +capacitance Cj (brown), and PCM using periodic capacitance modulation +[gold, see panel (c)]. Phase-sensitive amplification at half of the pump +frequency causes the distinctive peaks in the centers of all gain profiles. (c) +Circuit schematic for periodic capacitance modulation (PCM). The values of +the ground capacitances Cn/2, where n is the unit cell index, are periodically +varied. This opens a stopband in ω(k), which is used for phase-matching. +Table I summarizes the circuit parameters. Simulations were performed with +pump current of 6.59 µA for the RPM-JTPWA and 4.12 µA for the PCM- +JTWPA, see main text, pump frequencies 13.3, 13.3, 13.12 and 13.92 GHz in +the order of the legend, bias current of 7.5 µA, and signal current of 0.01 µA. +screening parameter βL = 2πLgIc/Φ0 < 1, where Lg is +the geometrical inductance, Ic is the critical current of the +Josephson junction, and Φ0 is the flux quantum. Each rf- +SQUID forms a unit cell with two capacitors C0/2 to ground, +see Fig. 1, yielding impedance Z0 = +� +Lg/C0 = 50 Ω. +A magnetic field B, or alternatively a dc current Idc, flux- +biases the rf-SQUIDs such that the noncentrosymmetric χ(2)- +type nonlinearity of the circuit is large and the χ(3)-type Kerr +nonlinearity vanishes [9]. At this flux bias, the effective linear +rf-SQUID inductance is LS = Lg. The χ(2)-type nonlinearity +facilitates parametric amplification in the 3WM regime when +driven by a strong pump wave. Taking into account the +Josephson junction capacitance Cj, the resulting dispersion +relation [21], +k(ω) = 2 +a arcsin +� +ω +� +LgC0 +2 +� +1 − ω2LgCj +� +, +(1) +where a is the physical length of a unit cell, causes a nonzero +phase mismatch ∆k > 0. A possible way to compensate +∆k is lowering kp with the help of either RPM or PCM. +The effect of adding RPM elements to the JTWPA circuit is +illustrated by the dashed and the dash-dotted lines in Fig. 1b, +corresponding to the JTWPA with and without RPM and oth- +erwise identical parameters. Further improvement aims at sup- +pressing unwanted parametric processes. To achieve this, the +effective SQUID capacitance Cj is enlarged, e.g., by adding +a capacitor in parallel to the Josephson junction, lowering +the SQUID plasma frequency and increasing the dispersion, +eq. (1), which causes phase mismatch for the higher-frequency +mixing processes [25]. This also increases ∆k, which can +be compensated by RPM, however. The effectiveness of this +approach is demonstrated by the solid brown line in Fig. 1b, +indicating almost pure 3WM and yielding > 20 dB gain in +a 3-dB-bandwidth of more than 5 GHz. Increasing Cj above +250 fF does not result in significantly higher gain but causes +stronger gain ripple due to the alteration of the line impedance +[26, eq. (C8)]. The circuit parameters used in the simulations +are given in Table I. +To implement PCM, we periodically vary the ground ca- +pacitances Cn with a modulation depth ζ and a period m, +Cn = C0 (1 + ζ sin(2πn/m)) , +(2) +where C0 is the mean value, see Fig. 1c. The PCM opens a +stopband in k(ω) at ω1 = π/(m +� +LgC0) and fulfills the same +purpose as the resonant feature in RPM. The combination +of PCM and the SQUID plasma resonance leads to effec- +tive suppression of unwanted modes [13] and simultaneously +TABLE I +CIRCUIT PARAMETERS USED FOR TRANSIENT CIRCUIT SIMULATIONS +JTWPA with RPM +JTWPA with PCM +Inductance +Lg = 80 pH +Lg = 80 pH +Critical current +Ic = 1.03 µA +Ic = 1.03 µA +Junction capacitance +Cj = 250 fF +Cj = 100 fF +Screening parameter +βL = 0.25 +βL = 0.25 +Resonator capacitance +Cr = 1 pF +Resonator inductance +Lr = 150 pH +Coupling capacitance +Cc = 20 fF +Unit cells per resonator +m = 4 +Ground capacitance +C0 = 32 fF +C0 = 32 fF +Modulation depth +ζ = 0.3 +Unit cells per period +m = 24 +Number of unit cells +N = 1000 +N = 1488 + +3 +10 +−3 +10 +−2 +Resonance frequency spread σfr/ +­ +fr +® +0 +5 +10 +15 +20 +25 +Signal gain (dB) +(a) +10 +20 +30 +Cc (fF) +0.00 +0.01 +0.02 +Spread tolerance +Cc (fF) +10 +15 +20 +25 +30 +12.5 +13.0 +13.5 +Frequency (GHz) +0.12 +0.14 +0.16 +Wave num. (a +−1) +(d) +12.5 +13.0 +13.5 +Frequency (GHz) +(e) +−10 +−5 +0 +|S21| (dB) +(b) +Cc =15 fF +(c) +Cc =30 fF +Fig. 2. Sensitivity of the RPM-JTWPA to resonance frequency spread. (a) Power gain as a function of normalized resonance frequency standard deviation +for five coupling capacitances. The resonance frequencies of all 250 individual resonators are drawn from a normal distribution with standard deviation σfr +and mean value ⟨fr⟩. For each σ the maximum gain within a range of pump frequencies is plotted, see main text. As σfr increases, the gain drops due to +impaired phase-matching along the transmission line. Solid lines are moving-averages of 20 samples for illustrative purpose. The inset shows the value of +σfr/⟨fr⟩, where the moving average gain curve decreases by 1 dB. Larger coupling capacitance Cc results in greater robustness against resonator frequency +spread. (b)–(e) Small-signal transmission of the RPM-JTWPA in the vicinity of the resonance frequency with and without resonator spread. Dashed blue lines +depict the zero-spread cases, and the blue-shaded regions indicate the corresponding resonance dips, where |S21| < −1 dB. The dash-dotted green lines in +(d) and (e) show the dispersion-free wave-number. Phase-matching is achieved when the dashed and the dash-dotted lines intersect outside the resonance dip, +avoiding severe pump depletion. This is possible for both shown values of Cc at zero spread. The solid black lines are typical examples with σfr/⟨fr⟩=1.1%, +illustrating the cases with resonance frequency spread. The resonance frequency spread results in a widened resonance dip, shown by the grey-shaded area, and +also in weakened modification of the wave number. (d) In the case of the lower coupling strength there is no intersection of the solid line with the dash-dotted +line, resulting in reduced gain, see (a). (e) For stronger coupling, phase-matching is still possible and the frequency of zero phase-mismatch lies outside the +resonance dip. All simulations of (a) were performed with an identical pump current of Ip = 6.59 µA for comparability, and with a signal frequency of +7.7 GHz and pump frequencies in the range of 12.98–14.0 GHz. The small-signal simulations of (b)–(e) were carried out with a current of 0.1 µA << Ip. +provides phase-matching for 3WM. In contrast to JTWPAs +based on single Josephson junctions, where Ip < Ic, no such +limit exists for rf-SQUID based JTPWAs, where the Josephson +junction is shunted by a superconducting inductance. Instead, +the phase drop across a Josephson junction should be kept +below ∼1 rad [9], [25], above which the circuit may behave +unpredictable. Therefore, we chose a pump amplitude of +0.8 rad for the RPM-JTWPA, which translates to a pump +current Ip = φpΦ0/2πLg = 6.59 µA. Due to the superposed +forth- and back-propagating components of the pump wave +close to the stopband [8], for the PCM-JTWPA we chose +0.5 rad for the pump amplitude, translating to Ip = 4.12 µA, +and increase the number of unit cells to achieve the same gain. +The solid ocher line in Fig. 1b shows the gain profile of the +PCM-JTWPA, where the increased ripple stems from sidelobes +of the stopband and is an unwanted side effect of PCM [24]. +A higher-order stopband created by PCM causes the two +distinctive peaks at both edges of the gain profile, confining the +usable range of signal frequencies. The peaks in the center of +each gain profile indicate phase-sensitive amplification [13], +[24], an interesting regime at fs = fi = fp/2 which is +experimentally accessible in 3WM-JTWPAs. +III. PARAMETER SPREAD SENSITIVITY +Next, we modify the JTWPA circuit models to incorporate +unit cell to unit cell variation to analyze the effect of variation +of the parameters Ic, Lg, Cn,j,c, and fr on the performance of +the JTWPA. In the case of the resonance frequency spread, +we vary all capacitances Cr in the RPM-JTWPA circuit, +having a normal distribution with a mean value ⟨Cr⟩ equal +to the nominal value, and with a standard deviation σCr being +increased in small steps. Keeping inductance Lr constant, +σLr = 0, the resonance frequency spread is [27] +σfr +⟨fr⟩ = 1 +2 +� +σ2 +Lrσ2 +Cr +⟨Lr⟩2⟨Cr⟩2 + σ2 +Lr +⟨Lr⟩2 + σ2 +Cr +⟨Cr⟩2 = 1 +2 +σCr +⟨Cr⟩. +(3) +For each value of σCr, in the following referred to as sample, +a set of simulations is carried out for a range of pump +frequencies fp. This is necessary due to the stochastic nature +of phase-matching in a circuit with parameter spread; the +frequency where the phases are matched differs from sample +to sample, c.f. Fig. 2e. Each data point in Fig. 2a is the highest +achieved gain of one sample, and all samples in the ensemble +are statistically independent. To visualize the trend of the +achievable gain depending on σCr, we calculate a moving +average with a window size of 20 samples. +To define spread tolerance, we take the values of σfr/⟨fr⟩ +for which the moving average gain drops by 1 dB from its +zero-spread value. The inset in Fig. 2a shows the spread +tolerance for five different coupling capacitances and indicates +generally increased robustness for larger Cc. To illustrate this, +Fig. 2b–e present the small-signal transmission in the vicinity +of the resonance dip of samples having 2% capacitance +spread and 1% inductance spread, yielding σfr/⟨fr⟩ = 1.1% +using (3), and reflecting realistic values for lumped element +resonators [28]–[31]. Resonance frequency spread results in +a widened resonance dip and in reduced modification of the +wave number. Whether both phase-matching and avoiding + +4 +0 +10 +20 +30 +Signal gain (dB) +(a) +βL = 0.25 +βL = 0.55 +βL = 0.85 +(b) +(c) +10 +−2 +10 +−1 +Inductance spread σLg/ +­ +Lg +® +0 +10 +20 +30 +Signal gain (dB) +(d) +10 +−2 +10 +−1 +Critical current spread σIc/ +­ +Ic +® +(e) +10 +−2 +10 +−1 +Capacitance spread σC/ +­ +C +® +(f) +Fig. 3. +Spread sensitivity of the RPM-JTWPA (a)–(c) and the PCM-JTWPA (d)–(f) for three screening parameter values βL. In each panel, the respective +parameter is varied according to a normal distribution, while the remaining parameters are kept constant. Capacitance spread refers to the simultaneous variation +of the capacitances Cn, Cj, and Cc. Solid lines are moving-averages of 20 adjacent samples to illustrate the trends. Increasing parameter spread leads to a +drop of the gain, and to microwave reflections, visible as strongly diverging gain values higher than the zero-spread gain. Even though the PCM-JTWPA is +more sensitive to parameter spread than the RPM-JTWPA, it does not contain a critical element like the resonators in RPM. Both RPM and PCM are more +sensitive to inductance spread than to critical current spread or capacitance spread, and lower values of βL are favourable. The simulations of (a)-(c) were +performed with a signal frequency of 7 GHz, and with pump frequencies in the range of 12.98–13.1 GHz and pump currents of 6.59, 2.8, 1.8 µA for βL = +0.25, 0.55, 0.85, respectively. The simulations of (d)-(f) were performed with a signal frequency of 7.7 GHz, and with pump frequencies in the range of +12.98–13.1 GHz and pump currents of 3.84, 1.63, 1.07 µA for βL = 0.25, 0.55, 0.85, respectively. +severe pump depletion are simultaneously possible depends +on the coupling strength. Apart from stronger coupling, the +resonator spread tolerance can be increased by increasing the +number of resonators in the circuit, i.e., to decrease period m. +For m = 1, resonator spread tolerances of > 2% were reported +[6], [15]. This comes at the cost of larger footprint on chip and +higher complexity. Furthermore, instead of lumped element +resonators, distributed resonators can be used, exhibiting much +lower spread < 0.1% [32], also at the cost of larger footprint, +c.f. [4]. Note that 3WM, when compared to 4WM, requires +only half of the length of RPM resonators due to the twice +higher pump frequency. +Although the RPM-JTWPA is vulnerable to resonance fre- +quency spread, the spreads of the other parameters, Lg, Ic, and +C0,c,j, play a minor role, see Fig. 3a–c, because they do not +directly impact phase-matching. This is different in the case +of PCM, see Fig. 3d–f. The stopband, providing the means +for phase-matching in PCM, is facilitated by the spatially +periodic form of quantities like the wave impedance or the +wave velocity. Consequently, stochastic spatial variations to +that periodic shape impact the characteristics of the stopband +and, hence, phase-matching. On the other hand, within one +period, consisting of many unit cells, deviations cancel each +other out, which is in great contrast to RPM, where each +individual resonator directly impacts phase-matching. Leaving +the spread of the resonance frequencies aside, PCM is 2–4 +times more sensitive to the spread of Lg, Ic, Cn,j than RPM. +However, the spread sensitivity analysis shows that typical +spread values, being on the order of < 2% for capacitances and +inductances [29]–[31], and 1–8% for the critical currents of +both Nb and Al Josephson junctions [33]–[37], are tolerable. +Taking this into account, we conclude that PCM does not have +a critical circuit element as the resonators in RPM. +Among the parameters Lg, Ic, C0,c,j, the impact of induc- +tance spread is the strongest, since Lg directly influences the +flux biasing, inductance, and nonlinearity of the rf-SQUIDs +and, consequently, the local impedance and wave number +of the line. This does not only deteriorate phase-matching +and reduce the signal gain, but also causes increased mi- +crowave reflection [23] and gain ripple. Microwave reflections +at inhomogeneities in the circuit interfere either destructively +or constructively, leading to a dip below or a peak beyond +the zero-spread gain. Such diverging gain values are most +pronounced in Fig. 3d, but can be also seen in the other panels +for large spread values. Moreover, Fig. 3a,d show that the +inductance spread tolerance strongly depends on parameter +βL ∝ 2LgIc, which is a degree of freedom in rf-SQUIDs. +To vary βL, we keep Lg constant and increase Ic. Since the +exponential gain coefficient of the JTPWA is proportional to +βL [9], we reduce the pump amplitude in the simulation to +achieve similar gain for all βL. +Interestingly, the rf-SQUID based JTWPA is quite insensi- +tive to critical current spread up to spreads of 10–20%, see +Fig. 3b,e, in contrast to 3% for the SNAIL-based reversed- +Kerr JTWPA [20]. This is because the flux bias where the +Kerr-like nonlinearity of the rf-SQUID vanishes [9] is close +to the sweet spot of critical current spread tolerance. At that +point, the linear inductance of the Josephson junction goes +to infinity, so the effective linear rf-SQUID inductance is +LS = Lg. Hence, as a first-order approximation, the Josephson +junctions contribute only to the nonlinear but not to the linear +characteristics of the circuit. Using βL = 0.25 and taking +into account an inductance spread of 1% and a critical current +spread of 5%, the resulting SQUID inductance spread, see (11) + +5 +in appendix, is 1.4%, and is determined almost entirely (98%) +by inductance spread. +IV. CONCLUSION +In summary, we analyzed the sensitivity of JTWPA gain +on the spread of critical currents, inductances, capacitances, +and resonance frequencies, using transient circuit simulations. +In the case of a JTWPA with RPM, the resonance frequency +spread is critical to the performance of the amplifier; how +much resonance frequency spread can be tolerated is deter- +mined by the resonator coupling strength. Apart from the +resonator spread, the RPM-JTWPA is more tolerant than the +PCM-JTWPA to the spread of the other circuit parameters. +Among these, the inductance spread has the largest impact +on JTWPA performance, depending on the SQUID screening +parameter. The Kerr-free rf-SQUID based JTWPA is rather +insensitive to Josephson junction critical current spread. Our +analysis can be employed also to other JTWPA architectures, +and the results should also be applicable to SNAIL-based +JTWPAs in particular. We think that understanding the sen- +sitivity of a JTWPA concept to parameter spread is key to its +optimization and practical realization, and we believe that our +results help paving the way to a practical implementation of +rf-SQUID based JTWPAs. +APPENDIX +The linear inductance of an rf-SQUID is given by +LS = +Lg +1 + βL cos φdc +(4) +and the dc phase φdc is the root of equation [38] +φdc + βL sin φdc − φe = 0, +(5) +with the phase φe = 2πLgIdc/Φ0, given by a dc bias current +Idc. The latter is chosen such that φe = βL + π/2 and φdc = +π/2, yielding a zero Kerr nonlinearity coefficient [9]. +To show why the rf-SQUID inductance LS is relatively in- +sensitive to critical current spread, we propagate the variances +σ2 +Lg and σ2 +Ic to the variance of LS, +σ2 +LS ≈ +���� +∂LS +∂Lg +���� +2 +σ2 +Lg + +���� +∂LS +∂Ic +���� +2 +σ2 +Ic . +(6) +Bearing in mind βL = 2πLgIc/Φ0, and taking the implicit +derivatives of the transcendental equation (5), +∂φdc +∂Lg += 1 +Lg +φe − βL sin φdc +1 + βL cos φdc +, +(7) +∂φdc +∂Ic += − 1 +Ic +βL sin φdc +1 + βL cos φdc +, +(8) +we get the derivatives +∂LS +∂Lg += +(1 + βL cos φdc) − βL +� +cos φdc − sin φdc +φe−βL sin φdc +1+βL cos φdc +� +(1 + βL cos φdc)2 +(9) +and +∂LS +∂Ic += −LgβL +Ic +� +cos φdc + sin φdc +βL sin φdc +1+βL cos φdc +� +(1 + βL cos φdc)2 +. +(10) +Evaluating (9) and (10) at φdc = π/2 and using (6) yields the +normalized standard deviation of the rf-SQUID inductance +σLS +⟨LS⟩ = +�� +1 + π +2 βL +�2 σ2 +Lg +⟨Lg⟩2 + β4 +L +σ2 +Ic +⟨Ic⟩2 . +(11) +ACKNOWLEDGMENT +The authors would like to thank D. 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Bengtsson, S. Kosen, P. Krantz, D. P. Lozano, +M. Scigliuzzo, P. Delsing, J. Bylander, and A. Fadavi Roudsari, “Sim- +plified Josephson-junction fabrication process for reproducibly high- +performance superconducting qubits,” Appl. Phys. Lett., vol. 118, no. 6, +p. 064002, Feb. 2021. +[38] K. K. Likharev, Dynamics of Josephson Junctions and Circuits. +New +York: Gordon and Breach, 1986. + diff --git a/ddFPT4oBgHgl3EQfDDRR/content/tmp_files/load_file.txt b/ddFPT4oBgHgl3EQfDDRR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9e8855d5050c9ae5e24709a09ce98b0288597865 --- /dev/null +++ b/ddFPT4oBgHgl3EQfDDRR/content/tmp_files/load_file.txt @@ -0,0 +1,810 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf,len=809 +page_content='1 Vulnerability to Parameter Spread in Josephson Traveling-Wave Parametric Amplifiers C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Kissling, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Gaydamachenko, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Kaap, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Khabipov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Dolata, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Zorin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Gr¨unhaupt Abstract—We analyze the effect of circuit parameter varia- tion on the performance of Josephson traveling-wave paramet- ric amplifiers (JTWPAs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Specifically, the JTWPA concept we investigate is using flux-biased nonhysteretic rf-SQUIDs in a transmission line configuration, which harnesses the three-wave mixing (3WM) regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Dispersion engineering enables phase- matching to achieve power gain of ∼20 dB, while suppressing the generation of unwanted mixing processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Two dispersion engineering concepts using a 3WM-JTWPA circuit model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=', resonant phase-matching (RPM) and periodic capacitance mod- ulation (PCM), are discussed, with results potentially also appli- cable to four-wave-mixing (4WM) JTWPAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' We propose suitable circuit parameter sets and evaluate amplifier performance with and without circuit parameter variance using transient circuit simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' This approach inherently takes into account mi- crowave reflections, unwanted mixing products, imperfect phase- matching, pump depletion, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' In the case of RPM the resonance frequency spread is critical, while PCM is much less sensitive to parameter spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' We discuss degrees of freedom to make the JTWPA circuits more tolerant to parameter spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Finally, our analysis shows that the flux-bias point where rf-SQUIDs exhibit Kerr-free nonlinearity is close to the sweet spot regarding critical current spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Index Terms—Superconducting electronics, circuit analysis, parametric amplifiers, SQUIDs I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' INTRODUCTION J OSEPHSON traveling-wave parametric amplifiers (JTWPAs) hold great promise for wideband amplification of few-photon-level signals at microwave frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Their key characteristics are bandwidths of several GHz, added noise close to the quantum-limit, and power-handling capabilities above -100 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' These advantages are particularly important for simultaneous readout of many qubits [1], [2], or for reading out large arrays of kinetic inductance detectors [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' JTWPAs utilize the nonlinear inductance of Josephson junc- tions, arranged as transmission line arrays of either single Josephson junctions [4]–[7], dc superconducting quantum in- terference devices (SQUIDs) [8], rf-SQUIDs [9], or supercon- ducting nonlinear asymmetric inductive elements (SNAILs) [10]–[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Parametric amplification occurs, depending on the order of nonlinearity, in the three-wave-mixing (3WM) or four-wave-mixing (4WM) regime, with fp = fs + fi and Manuscript received November 18, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' accepted January 18, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' gratefully acknowledge the support of the Braunschweig International Graduate School of Metrology B-IGSM and the DFG Research Training Group 1952 Metrology for Complex Nanosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' This work was also supported by the German Federal Ministry of Education and Research (BMBF) within the framework programme “Quantum technologies – from basic research to market” (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' 13N15949).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The authors are with Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany (e-mail: christoph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='kissling@ptb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='de).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' 2fp = fs+fi, respectively, where fp, fs, and fi are the pump, signal, and idler frequencies [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' If their respective wave numbers kp, ks, and ki fulfill the phase-matching criterion, ∆k = kp − ks − ki = 0 for 3WM, high signal gain can be achieved, scaling exponentially with the circuit length [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' In the design of a 3WM JTWPA a major challenge is to ensure phase-matching along the whole length of the JTWPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' This has been achieved in both 3WM and 4WM JTWPAs either by the resonant-phase-matching (RPM) tech- nique [15] with lumped element resonators [5]–[7] or dis- tributed resonators [4], [13], or by the periodic variation of the transmission line parameters [8], [16]–[19], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=', by periodic capacitance modulation (PCM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Both dispersion engineering methods create a narrow-band nonmonotonic feature in the dispersion relation k(ω), which is used to compensate the phase-mismatch ∆k by an appropriate choice of fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' An addi- tional challenge is to suppress unwanted parametric processes generating higher-frequency modes, which has been solved by increased dispersion of the transmission line for higher frequencies f > fp [13], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Aside from these design challenges, any practical JTWPA implementation, made up of thousands of circuit elements, must be robust enough not to be affected by the expected parameter spread considering a typical fabrication process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Pa- rameter variation impacts JTWPA performance in many ways, including phase-matching, impedance-matching, nonlinearity, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Previous works analyzed the effect of deteriorated phase- matching by resonance frequency spread in RPM-JTWPAs both analytically [15] and via Monte Carlo simulations using a linearized model [6], the tolerable critical current spread using analytical approaches [20], [21] and via quantum input- output theory [22], and microwave reflections due to pa- rameter spread and point defects in a JTWPA using circuit simulations [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' In this paper, we investigate both dispersion engineering concepts, RPM and PCM, and rigorously analyze their parameter spread sensitivity, employing transient circuit simulations in WRspice as described in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' While most of the aforementioned works treated only one aspect of parameter spread isolatedly, our approach inherently includes the effect of spread on phase-matching, pump depletion, deviations from optimal bias point, microwave reflections, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=', and therefore gives a more realistic picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' CIRCUIT DESCRIPTION AND ANALYSIS We investigate a JTWPA architecture with rf-SQUIDs as nonlinear inductive elements in a transmission-line arrange- ment [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The rf-SQUIDs are nonhysteretic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=', having arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='12991v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='supr-con] 30 Jan 2023 2 (a) Cc Lr Cr Cc λ/4 1 2 3 4 in 1 2 3 4 (c) in (b) Cn/2 Ic Cj Lg B Cn/2 C0/2 Ic Cj Lg B C0/2 out out 0 25 50 75 Unit cell index n 0 20 40 Cn (fF) 0 2 4 6 8 10 12 Signal frequency (GHz) 0 10 20 30 Signal gain (dB) Cj =40 fF Cj =40 fF, RPM Cj =250 fF, RPM Cj =100 fF, PCM Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' JTWPA circuit model and effect of phase-matching on signal gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' (a) Circuit schematic of the RPM-JTWPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' It shows one section of m = 4 identical unit cells and one phase-matching resonator attached in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The blue-shaded panel shows the unit cell, containing a flux-biased rf-SQUID and two capacitances C0/2 to ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' RPM is implemented using either LC- resonators or λ/4-resonators, each capacitively coupled to the transmission line (brown-shaded panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The resonators create a nonmonotonic feature in the dispersion relation ω(k) to achieve phase-matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' (b) Gain vs frequency profiles, obtained by transient circuit simulations for no dispersion engineering (dash-dotted), RPM (dashed line), RPM with increased effective SQUID capacitance Cj (brown), and PCM using periodic capacitance modulation [gold, see panel (c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Phase-sensitive amplification at half of the pump frequency causes the distinctive peaks in the centers of all gain profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' (c) Circuit schematic for periodic capacitance modulation (PCM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The values of the ground capacitances Cn/2, where n is the unit cell index, are periodically varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' This opens a stopband in ω(k), which is used for phase-matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Table I summarizes the circuit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Simulations were performed with pump current of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='59 µA for the RPM-JTPWA and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='12 µA for the PCM- JTWPA, see main text, pump frequencies 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='3, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='3, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='12 and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='92 GHz in the order of the legend, bias current of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='5 µA, and signal current of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='01 µA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' screening parameter βL = 2πLgIc/Φ0 < 1, where Lg is the geometrical inductance, Ic is the critical current of the Josephson junction, and Φ0 is the flux quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Each rf- SQUID forms a unit cell with two capacitors C0/2 to ground, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' 1, yielding impedance Z0 = � Lg/C0 = 50 Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' A magnetic field B, or alternatively a dc current Idc, flux- biases the rf-SQUIDs such that the noncentrosymmetric χ(2)- type nonlinearity of the circuit is large and the χ(3)-type Kerr nonlinearity vanishes [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' At this flux bias, the effective linear rf-SQUID inductance is LS = Lg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The χ(2)-type nonlinearity facilitates parametric amplification in the 3WM regime when driven by a strong pump wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Taking into account the Josephson junction capacitance Cj, the resulting dispersion relation [21], k(ω) = 2 a arcsin � ω � LgC0 2 � 1 − ω2LgCj � , (1) where a is the physical length of a unit cell, causes a nonzero phase mismatch ∆k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' A possible way to compensate ∆k is lowering kp with the help of either RPM or PCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The effect of adding RPM elements to the JTWPA circuit is illustrated by the dashed and the dash-dotted lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' 1b, corresponding to the JTWPA with and without RPM and oth- erwise identical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Further improvement aims at sup- pressing unwanted parametric processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' To achieve this, the effective SQUID capacitance Cj is enlarged, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=', by adding a capacitor in parallel to the Josephson junction, lowering the SQUID plasma frequency and increasing the dispersion, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' (1), which causes phase mismatch for the higher-frequency mixing processes [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' This also increases ∆k, which can be compensated by RPM, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The effectiveness of this approach is demonstrated by the solid brown line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' 1b, indicating almost pure 3WM and yielding > 20 dB gain in a 3-dB-bandwidth of more than 5 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Increasing Cj above 250 fF does not result in significantly higher gain but causes stronger gain ripple due to the alteration of the line impedance [26, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' (C8)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The circuit parameters used in the simulations are given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' To implement PCM, we periodically vary the ground ca- pacitances Cn with a modulation depth ζ and a period m, Cn = C0 (1 + ζ sin(2πn/m)) , (2) where C0 is the mean value, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The PCM opens a stopband in k(ω) at ω1 = π/(m � LgC0) and fulfills the same purpose as the resonant feature in RPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The combination of PCM and the SQUID plasma resonance leads to effec- tive suppression of unwanted modes [13] and simultaneously TABLE I CIRCUIT PARAMETERS USED FOR TRANSIENT CIRCUIT SIMULATIONS JTWPA with RPM JTWPA with PCM Inductance Lg = 80 pH Lg = 80 pH Critical current Ic = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='03 µA Ic = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='03 µA Junction capacitance Cj = 250 fF Cj = 100 fF Screening parameter βL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='25 βL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='25 Resonator capacitance Cr = 1 pF Resonator inductance Lr = 150 pH Coupling capacitance Cc = 20 fF Unit cells per resonator m = 4 Ground capacitance C0 = 32 fF C0 = 32 fF Modulation depth ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='3 Unit cells per period m = 24 Number of unit cells N = 1000 N = 1488 3 10 −3 10 −2 Resonance frequency spread σfr/ \xad fr ® 0 5 10 15 20 25 Signal gain (dB) (a) 10 20 30 Cc (fF) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='02 Spread tolerance Cc (fF) 10 15 20 25 30 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='5 Frequency (GHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='16 Wave num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' (a −1) (d) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='5 Frequency (GHz) (e) −10 −5 0 |S21| (dB) (b) Cc =15 fF (c) Cc =30 fF Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Sensitivity of the RPM-JTWPA to resonance frequency spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' (a) Power gain as a function of normalized resonance frequency standard deviation for five coupling capacitances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The resonance frequencies of all 250 individual resonators are drawn from a normal distribution with standard deviation σfr and mean value ⟨fr⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' For each σ the maximum gain within a range of pump frequencies is plotted, see main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' As σfr increases, the gain drops due to impaired phase-matching along the transmission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Solid lines are moving-averages of 20 samples for illustrative purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The inset shows the value of σfr/⟨fr⟩, where the moving average gain curve decreases by 1 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Larger coupling capacitance Cc results in greater robustness against resonator frequency spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' (b)–(e) Small-signal transmission of the RPM-JTWPA in the vicinity of the resonance frequency with and without resonator spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Dashed blue lines depict the zero-spread cases, and the blue-shaded regions indicate the corresponding resonance dips, where |S21| < −1 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The dash-dotted green lines in (d) and (e) show the dispersion-free wave-number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Phase-matching is achieved when the dashed and the dash-dotted lines intersect outside the resonance dip, avoiding severe pump depletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' This is possible for both shown values of Cc at zero spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The solid black lines are typical examples with σfr/⟨fr⟩=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='1%, illustrating the cases with resonance frequency spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The resonance frequency spread results in a widened resonance dip, shown by the grey-shaded area, and also in weakened modification of the wave number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' (d) In the case of the lower coupling strength there is no intersection of the solid line with the dash-dotted line, resulting in reduced gain, see (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' (e) For stronger coupling, phase-matching is still possible and the frequency of zero phase-mismatch lies outside the resonance dip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' All simulations of (a) were performed with an identical pump current of Ip = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='59 µA for comparability, and with a signal frequency of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='7 GHz and pump frequencies in the range of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='98–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='0 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The small-signal simulations of (b)–(e) were carried out with a current of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='1 µA << Ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' provides phase-matching for 3WM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' In contrast to JTWPAs based on single Josephson junctions, where Ip < Ic, no such limit exists for rf-SQUID based JTPWAs, where the Josephson junction is shunted by a superconducting inductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Instead, the phase drop across a Josephson junction should be kept below ∼1 rad [9], [25], above which the circuit may behave unpredictable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Therefore, we chose a pump amplitude of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='8 rad for the RPM-JTWPA, which translates to a pump current Ip = φpΦ0/2πLg = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='59 µA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Due to the superposed forth- and back-propagating components of the pump wave close to the stopband [8], for the PCM-JTWPA we chose 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='5 rad for the pump amplitude, translating to Ip = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='12 µA, and increase the number of unit cells to achieve the same gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The solid ocher line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' 1b shows the gain profile of the PCM-JTWPA, where the increased ripple stems from sidelobes of the stopband and is an unwanted side effect of PCM [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' A higher-order stopband created by PCM causes the two distinctive peaks at both edges of the gain profile, confining the usable range of signal frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The peaks in the center of each gain profile indicate phase-sensitive amplification [13], [24], an interesting regime at fs = fi = fp/2 which is experimentally accessible in 3WM-JTWPAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' PARAMETER SPREAD SENSITIVITY Next, we modify the JTWPA circuit models to incorporate unit cell to unit cell variation to analyze the effect of variation of the parameters Ic, Lg, Cn,j,c, and fr on the performance of the JTWPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' In the case of the resonance frequency spread, we vary all capacitances Cr in the RPM-JTWPA circuit, having a normal distribution with a mean value ⟨Cr⟩ equal to the nominal value, and with a standard deviation σCr being increased in small steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Keeping inductance Lr constant, σLr = 0, the resonance frequency spread is [27] σfr ⟨fr⟩ = 1 2 � σ2 Lrσ2 Cr ⟨Lr⟩2⟨Cr⟩2 + σ2 Lr ⟨Lr⟩2 + σ2 Cr ⟨Cr⟩2 = 1 2 σCr ⟨Cr⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' (3) For each value of σCr, in the following referred to as sample, a set of simulations is carried out for a range of pump frequencies fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' This is necessary due to the stochastic nature of phase-matching in a circuit with parameter spread;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' the frequency where the phases are matched differs from sample to sample, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' 2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Each data point in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' 2a is the highest achieved gain of one sample, and all samples in the ensemble are statistically independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' To visualize the trend of the achievable gain depending on σCr, we calculate a moving average with a window size of 20 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' To define spread tolerance, we take the values of σfr/⟨fr⟩ for which the moving average gain drops by 1 dB from its zero-spread value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' 2a shows the spread tolerance for five different coupling capacitances and indicates generally increased robustness for larger Cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' To illustrate this, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' 2b–e present the small-signal transmission in the vicinity of the resonance dip of samples having 2% capacitance spread and 1% inductance spread, yielding σfr/⟨fr⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='1% using (3), and reflecting realistic values for lumped element resonators [28]–[31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Resonance frequency spread results in a widened resonance dip and in reduced modification of the wave number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Whether both phase-matching and avoiding 4 0 10 20 30 Signal gain (dB) (a) βL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='25 βL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='55 βL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='85 (b) (c) 10 −2 10 −1 Inductance spread σLg/ \xad Lg ® 0 10 20 30 Signal gain (dB) (d) 10 −2 10 −1 Critical current spread σIc/ \xad Ic ® (e) 10 −2 10 −1 Capacitance spread σC/ \xad C ® (f) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Spread sensitivity of the RPM-JTWPA (a)–(c) and the PCM-JTWPA (d)–(f) for three screening parameter values βL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' In each panel, the respective parameter is varied according to a normal distribution, while the remaining parameters are kept constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Capacitance spread refers to the simultaneous variation of the capacitances Cn, Cj, and Cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Solid lines are moving-averages of 20 adjacent samples to illustrate the trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Increasing parameter spread leads to a drop of the gain, and to microwave reflections, visible as strongly diverging gain values higher than the zero-spread gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Even though the PCM-JTWPA is more sensitive to parameter spread than the RPM-JTWPA, it does not contain a critical element like the resonators in RPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Both RPM and PCM are more sensitive to inductance spread than to critical current spread or capacitance spread, and lower values of βL are favourable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The simulations of (a)-(c) were performed with a signal frequency of 7 GHz, and with pump frequencies in the range of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='98–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='1 GHz and pump currents of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='59, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='8 µA for βL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='55, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='85, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The simulations of (d)-(f) were performed with a signal frequency of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='7 GHz, and with pump frequencies in the range of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='98–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='1 GHz and pump currents of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='84, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='63, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='07 µA for βL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='55, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='85, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' severe pump depletion are simultaneously possible depends on the coupling strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Apart from stronger coupling, the resonator spread tolerance can be increased by increasing the number of resonators in the circuit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=', to decrease period m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' For m = 1, resonator spread tolerances of > 2% were reported [6], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' This comes at the cost of larger footprint on chip and higher complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Furthermore, instead of lumped element resonators, distributed resonators can be used, exhibiting much lower spread < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='1% [32], also at the cost of larger footprint, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Note that 3WM, when compared to 4WM, requires only half of the length of RPM resonators due to the twice higher pump frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Although the RPM-JTWPA is vulnerable to resonance fre- quency spread, the spreads of the other parameters, Lg, Ic, and C0,c,j, play a minor role, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' 3a–c, because they do not directly impact phase-matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' This is different in the case of PCM, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' 3d–f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The stopband, providing the means for phase-matching in PCM, is facilitated by the spatially periodic form of quantities like the wave impedance or the wave velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Consequently, stochastic spatial variations to that periodic shape impact the characteristics of the stopband and, hence, phase-matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' On the other hand, within one period, consisting of many unit cells, deviations cancel each other out, which is in great contrast to RPM, where each individual resonator directly impacts phase-matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Leaving the spread of the resonance frequencies aside, PCM is 2–4 times more sensitive to the spread of Lg, Ic, Cn,j than RPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' However, the spread sensitivity analysis shows that typical spread values, being on the order of < 2% for capacitances and inductances [29]–[31], and 1–8% for the critical currents of both Nb and Al Josephson junctions [33]–[37], are tolerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Taking this into account, we conclude that PCM does not have a critical circuit element as the resonators in RPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Among the parameters Lg, Ic, C0,c,j, the impact of induc- tance spread is the strongest, since Lg directly influences the flux biasing, inductance, and nonlinearity of the rf-SQUIDs and, consequently, the local impedance and wave number of the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' This does not only deteriorate phase-matching and reduce the signal gain, but also causes increased mi- crowave reflection [23] and gain ripple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Microwave reflections at inhomogeneities in the circuit interfere either destructively or constructively, leading to a dip below or a peak beyond the zero-spread gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Such diverging gain values are most pronounced in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' 3d, but can be also seen in the other panels for large spread values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Moreover, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' 3a,d show that the inductance spread tolerance strongly depends on parameter βL ∝ 2LgIc, which is a degree of freedom in rf-SQUIDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' To vary βL, we keep Lg constant and increase Ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Since the exponential gain coefficient of the JTPWA is proportional to βL [9], we reduce the pump amplitude in the simulation to achieve similar gain for all βL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Interestingly, the rf-SQUID based JTWPA is quite insensi- tive to critical current spread up to spreads of 10–20%, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' 3b,e, in contrast to 3% for the SNAIL-based reversed- Kerr JTWPA [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' This is because the flux bias where the Kerr-like nonlinearity of the rf-SQUID vanishes [9] is close to the sweet spot of critical current spread tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' At that point, the linear inductance of the Josephson junction goes to infinity, so the effective linear rf-SQUID inductance is LS = Lg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Hence, as a first-order approximation, the Josephson junctions contribute only to the nonlinear but not to the linear characteristics of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Using βL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='25 and taking into account an inductance spread of 1% and a critical current spread of 5%, the resulting SQUID inductance spread, see (11) 5 in appendix, is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content='4%, and is determined almost entirely (98%) by inductance spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' CONCLUSION In summary, we analyzed the sensitivity of JTWPA gain on the spread of critical currents, inductances, capacitances, and resonance frequencies, using transient circuit simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' In the case of a JTWPA with RPM, the resonance frequency spread is critical to the performance of the amplifier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' how much resonance frequency spread can be tolerated is deter- mined by the resonator coupling strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Apart from the resonator spread, the RPM-JTWPA is more tolerant than the PCM-JTWPA to the spread of the other circuit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Among these, the inductance spread has the largest impact on JTWPA performance, depending on the SQUID screening parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The Kerr-free rf-SQUID based JTWPA is rather insensitive to Josephson junction critical current spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Our analysis can be employed also to other JTWPA architectures, and the results should also be applicable to SNAIL-based JTWPAs in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' We think that understanding the sen- sitivity of a JTWPA concept to parameter spread is key to its optimization and practical realization, and we believe that our results help paving the way to a practical implementation of rf-SQUID based JTWPAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' APPENDIX The linear inductance of an rf-SQUID is given by LS = Lg 1 + βL cos φdc (4) and the dc phase φdc is the root of equation [38] φdc + βL sin φdc − φe = 0, (5) with the phase φe = 2πLgIdc/Φ0, given by a dc bias current Idc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' The latter is chosen such that φe = βL + π/2 and φdc = π/2, yielding a zero Kerr nonlinearity coefficient [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' To show why the rf-SQUID inductance LS is relatively in- sensitive to critical current spread, we propagate the variances σ2 Lg and σ2 Ic to the variance of LS, σ2 LS ≈ ���� ∂LS ∂Lg ���� 2 σ2 Lg + ���� ∂LS ∂Ic ���� 2 σ2 Ic .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' (6) Bearing in mind βL = 2πLgIc/Φ0, and taking the implicit derivatives of the transcendental equation (5), ∂φdc ∂Lg = 1 Lg φe − βL sin φdc 1 + βL cos φdc , (7) ∂φdc ∂Ic = − 1 Ic βL sin φdc 1 + βL cos φdc , (8) we get the derivatives ∂LS ∂Lg = (1 + βL cos φdc) − βL � cos φdc − sin φdc φe−βL sin φdc 1+βL cos φdc � (1 + βL cos φdc)2 (9) and ∂LS ∂Ic = −LgβL Ic � cos φdc + sin φdc βL sin φdc 1+βL cos φdc � (1 + βL cos φdc)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' (10) Evaluating (9) and (10) at φdc = π/2 and using (6) yields the normalized standard deviation of the rf-SQUID inductance σLS ⟨LS⟩ = �� 1 + π 2 βL �2 σ2 Lg ⟨Lg⟩2 + β4 L σ2 Ic ⟨Ic⟩2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' (11) ACKNOWLEDGMENT The authors would like to thank D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Hanisch and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Feldhoff for fruitful discussions, and 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Melville, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Niedzielski, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Yoder, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' Schwartz, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' P.' 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of Josephson Junctions and Circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} +page_content=' New York: Gordon and Breach, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFPT4oBgHgl3EQfDDRR/content/2301.12991v1.pdf'} diff --git a/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf b/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..138d7876cb99c189843dbc106689e7df6cf8880e --- /dev/null +++ b/dtE4T4oBgHgl3EQfpg2t/content/2301.05193v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:07732e13ef16324cd0b84a7d49b5aa003d8565bb3aca33383c0974430e248ef6 +size 4388392 diff --git a/e9E3T4oBgHgl3EQf3AuS/content/tmp_files/2301.04760v1.pdf.txt b/e9E3T4oBgHgl3EQf3AuS/content/tmp_files/2301.04760v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ac2aa4c8fd60cd52dd74ae82f928569836f32b53 --- /dev/null +++ b/e9E3T4oBgHgl3EQf3AuS/content/tmp_files/2301.04760v1.pdf.txt @@ -0,0 +1,608 @@ +Page 1 of 14 + +Rev: 1/10/2023 +Pragmatic Estimation of Sample Size for Number of Interviews for PRO development in the 2009 FDA +PRO guidance + +Affiliation +Chris Barker, Ph.D. +Adjunct Associate Professor of Biostatistics +University of Illinois Chicago - School of Public Health +Chicago, Illinois + +And +Chris Barker, Ph.D. +CEO +Chris Barker Statistical Planning and Analysis Services Inc. +www.barkerstats.com + +Abbreviations +CRC – Capture-Recapture +KM – Kaplan Meier +PRO – Patient Reported Outcome +FDA-Food and Drug Administration +Abstract +Patient Reported Outcomes developed according to the 2009 FDA PRO guidance require an initial step +of structured patient interviews or focus groups. The guidance does not provide sample size suggestions +or methodology. This paper proposes statistical methodology and sample size guidance that address this +gap in the FDA PRO guidance. This paper also appears to be the first to provide a definition of Type I +error in interviews for a PRO methods for assessing sample size and a new definition of saturation based +on a probability distribution to confirm whether enough interviews have been prepared. Type I error is +declaring saturation when it has not been achieved during the interviews. Two worked examples +applied to actual interview data and published data, are presented. Guidelines are proposed for the +estimation of sample size useful for PRO experts conducting interviews for a PRO. These methods are +applied in the setting of qualitative research. Future research for other methods of interviewing and +determining saturation is required. + +Page 2 of 14 + +Background + +This paper and methodology arose because of a project progress teleconference with the CEO and CMO +for a small biotech, me, a PRO developer vendor and additional clinical operations staff. The vendor was +responsible for activities including interviewing patients, preparing and validating a PRO instrument for +use as a primary endpoint in a Phase III randomized trial according to the FDA PRO guidance (FDA, +2009). Due to confidentiality agreements, the names of the biotech, vendor, drug, and indication are +withheld. +The CEO asked the vendor how many interviews would be required to develop the instrument. The +vendor was processing the interviews using Qualitative Research and stated that interviews would be +complete “once saturation was achieved”. The CEO requested I explain the definition of “saturation”. +The vendor defined ‘saturation’ as the first interview that elicited no new concepts. I asked a follow-up +question “Is that the first occurrence of saturation or do you conduct several, perhaps 2 or more +additional consecutive interviews with no new concepts?”. The vendor replied “no, only one interview +with zero new codes”. I also asked if a single interview could be statistical noise and possibly a type I +error (declaring saturation when it had not occurred). The vendor was unable to answer that question +about Type I error. +Excerpting from the PRO guidance (page 13), “We cannot provide recommendations for the number +or size of the individual patient interviews or focus groups for establishing content validity. The +sample size depends on the completeness of the information obtained from analysis of the +transcripts. Generally, the number of patients is not as critical as interview quality and patient +diversity included in the sample in relation to intended clinical trial population characteristics.”. +The methodology of this paper addresses the number of patients but does not address the interview +quality or patient diversity. +Qualitative research is one of several methodologies for preparing and coding interviews used to +develop PRO’s. An overview of other methods is described in a paper sponsored by ISPOR, and includes, +phenomenology, Grounded Theory, etc. (Patrick 2011). The implementation of the interview methods is +not reviewed in this paper. This paper considers only qualitative research implements the empirical +concept of ‘saturation’ to determine when interviews are complete. In reviewing the literature about +the concept of saturation there does not appear to be a formal statistical method for determining either +saturation or statistical guidance for sample size. Note the applicability of the methods of this paper to +other methods of conducting interviews must be assessed on a case-by-case basis. Aside from the case- +by-case assessment a common property of the other methods is the range of number of interviews with +“zero new codes”. +This paper does not provide a complete literature review of the concept of saturation, an ambitious goal +well beyond the scope of this manuscript. The term “saturation” is frequently applied in grounded +theory research and its application beyond that theory is debated (Oreilly 2013). Theoretical saturation +means that researchers reach a point in their analysis of data that sampling more data will not lead to more +information related to their research questions (Seale,1999). I mention a small number of arbitrarily +selected papers have reviewed interview methodology, definition of saturation and saturation and the +number of interviews (Francis, 2010, Fusch 2015, Mason 2010). The term “saturation” is defined as + +Page 3 of 14 + +zero new concepts (or themes) elicited at an interview. Alternate definitions appearing in the literature +are: first occurrence of saturation, three consecutive interviews with saturation, expert Judgement +“additional interviews would be counter-productive”, minimum 10 interviews and 3 with zero new +codes (“10+3”), (Francis, 2010). Several deterministic recommendations for number of interviews +(sample size) appear in the literature. +Table 1 +Ethnography/ethnoscience + +“30 -60 interviews” +Grounded Theory + + +“30-50 interviews” +Phenomenology + + +“5-25 interviews” +All Qualitative research + +“at least 15 interviews” +Funded ($) research ‘time limited’ interviews ranged + +from 1 – 95 +Tesch (1990) enumerates 23 “qualitative research types” for which the methods of this paper may also +be applicable. +Methods +Assumptions for application of the methodology of Kaplan-Meier. + +Throughout the discussion I use the term “sample size” as synonymous with number of interviews. I +assume that there is one subject per interview (subjects do not give more than one interview). The +Kaplan Meier methodology adopted here for qualitative research is descriptive (no “p-values”). One +major property of the Kaplan Meier as defined here, the KM probability of not being saturated is +unchanged or declines with each successive interview. The KM probability estimates can decline and +exactly equal zero. A further characteristic is that at and after saturation, additional interviews are +redundant and do not provide new codes. +This paper assumes there is a fixed but unknown number of “codes”. One must assume that each +interview is conducted in a repeatable reproducible manner. Each interview results in none, one or +more new codes vs. preceding interview(s). The Kaplan-Meier assumes that each outcome or event +“new codes” or “no new codes”, in this example each interview, is statistically independent. This is +unlikely to be true for interviews. The fixed unknown total number of codes induces a correlation +among pairs of interviews. Modelling this correlation is outside the scope of this paper. The Kaplan- +Meier estimate of probability of saturation is unbiased. Due to the induced correlations, the estimates +of variability may be biased, and may tend to be too small relative to the independence case. +Alternative estimates of variability such as by using a Bootstrap might be applicable and are not further +examined in this manuscript. +Adaptation of KM methodology to other interview methodologies. +The KM methodology must be adapted for each interview methodology. + +Page 4 of 14 + +Example data sets and availability +These datasets below are available on request. Informed consent was obtained from the subjects +providing data for peach of the datasets. The Guest data used below was collected as part of a larger +study (Guest 2006). Informed consent for the subjects was obtained in the larger study (Bernard, +Personal communication, 2020). The patients for the second dataset gave informed consent (Revicki, +Personal communication, 2017). Table 1 presents a “flat file” structure for organizing the data. I +recommend that the interview datasets are included in the regulatory submission as part of +standardized ADaM and SDTM datasets. The FDA currently requires clinical trial data submitted in this +standardize format (Wood, 2008). +Data set I – anonymized interview data +I obtained an anonymized dataset from an expert in PRO’s (Kleinman, 2012) from a set of twenty-one +interviews processed using qualitative research. The anonymization replaced the code description with +a letter (A, B…). Interviews were in chronological order and a 1 represents the code observed at the +interview 0 otherwise. The interviews resulted in 20 separate codes. +The dataset may be represented in a tabular format in table 2. Aligning with conventional terminology +for capture-recapture discussed below the “codes” are termed “marked” (M) when first elicited, and +when a code is elicited a second or later interview, it is labelled a “recapture” (R). For each, code the 0’s +represent code not elicited in interview n and 1=code elicited in interview j. Table 3 summarizes the +data. + +Table 2- Organization for Interview Dataset + + +Anonymized Codes +(M=Elicited, R= code Elicited again) +Interview +ID +Interview +Sequence # + Code A, +Code B, +Code C, +… +Code K, + + +(M,R) +(M,R) +(M,R) +(M,R) (M,R) +aaaa +1 +(1, 0) +(0 ,0) +(0 ,0) + +0 ,0) +bbbb +2 +(0 ,0) +(0 ,0) +(1,0) + +(1,1) +cccc +3 +(1 ,1) +(1,0) +(0 ,0) + +(0 ,0) +dddd +4 +(1 ,1) +(1,1) +(0 ,0) + +(0 ,0) +eeee +5 +(1 ,1) +(1 ,1) +(0 ,0) + +(1 ,1) +… +… + + + + + + +The data set has the following raw and derived variables, +Let j = interview chronological sequence number, j=1,2,3…. J where J is the total number of +interviews +Note that total number of interviews, J is unknown at the start of the interviews. +Code -k-, (E-)licited at interview j, E j k: 1=yes, 0=no +Where k=1,2, 3… K, elicited from interview, j=1,2, 3… J + +Page 5 of 14 + +Note that the total number of codes K is unknown at the start the interviews. +Let Rj, k indicate Ej,k elicited again “recaptured” at interview, j ,(1=yes, 0=no k=1,2, 3… K ), +total codes elicited at interview j, Nj = ∑ (Ej,k ) and summation over k +cumulative elicitations of code k, Mk= ∑ Ej, k summation over j. + + +Table 3 interview descriptive statistics +N +marked +codes +Mean +marked +per +interview +Median +marked +per +interview +STD +marked +N +recap +Mean +recapture +Median +recapture +STD +recapture +47 +4.19 +4.00 +2.34 +47 +3.81 +3.00 +2.44 + + + + + + +Page 6 of 14 + + +Table 4 interview descriptive statistics recaptured +Recapture N +0 +2 +1 +6 +2 +8 +3 11 +4 +3 +5 +4 +6 +5 +7 +5 +8 +1 +9 +1 +10 +1 + + + +Data set II – using a Published figure from Guest et al + +I conducted a search in google and located a paper by Guest et al with codes elicited from interviews +then used to develop an instrument about HIV in subjects in Africa (Guest . 2006). +I contacted Dr. Guest and the raw data for his paper are not available. Figure 1 below reproduces figure +1 of the Guest paper. Guest summarized their interviews and reported number of codes elicited in +groups of 6 consecutive interviews and for interviews from Ghana then from Nigeria. For the estimate +of probability of saturation presented below a simple ad hoc imputation of codes to interviews was +implemented. When #codes > 6 (interviews) then, it was assumed every interview had at least 1 code. +When #codes <6, for example 4, it was assumed that four interviews resulted in 1 code and remaining 2 +randomly selected interviews had zero codes. +Figure 1 +Guest, reproduced with permission + +Page 7 of 14 + + +Reproduced from Guest et al with permission +Note numbers at top of bars are number of codes elicited in 6 interviews + + +A Type I error for PRO development may be defined as terminating interviews and incorrectly claiming +saturation is achieved when more interviews are required. Consider the deterministic rule that +saturation is the first occurrence of an interview with zero new codes. The data from Guest indicate that +for interview 13-18, there were 5 codes in 6 interviews, therefore there was at least one interview with +zero new codes. Note had interviews stopped at 13-18 would be a Type I error (saturation did not occur) +, and as many as 45 more interviews were required. + + +Figure 1 +(Guest, reproduced with permission) +80 +8 +20 +5 +2 +1-6 +13-18 +25-30 +37-42 +49-54 +7-12 +19-24 +31-36 +43-48 +55-60 +Interview #Page 8 of 14 + +Selecting a probability distribution to fit the decline in the number of “new codes” to zero can be based +on a goodness of fit. By a serendipitous choice I adopted the non-parametric Kaplan Meier (“K-M”) +(Kaplan, Meier, 1958) . The KM does not require an assumption of a distribution. The KM is zero at an +“interview” where saturation occurs. Other distributions, exponential, gamma etc. may be considered, +however those distributions can equal zero. I recommend selection of a distribution that can equal +zero. Consideration of adaptation of a distribution such as a truncated or triangular distribution is +beyond the scope of this article. + +I consider a simple application of the non-parametric Kaplan Meier estimate of the distribution of +elicited codes. For the K-M “0 new codes” is treated as an “event” and >=1 code as “censored”. That +curve and the 95% confidence interval is presented in Figure 2. Extracting data from the graph uses the +simplifying assumptions. For each group of 6 interviews, for example resulting in 5 codes, I use a +simplifying assumption that one interview had zero codes and I randomly select one interview, the +remaining interviews are imputed by one code per interview. +I fit a regression line to the upper 95% C.I. limits and extrapolate that to the x-axis for dataset II, +presented in Figure 2. The extrapolated regression crosses the axis at approximately 70 interviews. The +median is the statistic reported from the Kaplan-Meier. For saturation, the interest is the probability +estimates. The KM estimate starts at 1 (100%) because no interviews have occurred - the probability +that saturation has not occurred is 100%. The K-M probability estimate declines to 0.0%, at about 55 +interviews, interpreted as probability 0.0% that saturation has not occurred. +I do not recommend nor adopt a hypothesis testing framework for interpreting saturation. The Kaplan +Meier provides a confidence interval associated with the probability distribution to incorporate +variability. As above, it may not be reasonable to assume individual interviews and elicited codes are not +statistically independent. Future research in this area may consider use of bootstrap resampling or other +methods that account for the potential non-independence. +Results +Figure 3 present a Kaplan-Meier estimate of the probability of saturation, based on anonymized +interview data provided by Revicki. The x-axis is the interview in chronological sequence and y axis KM +probability. An “event” is an interview with zero new codes otherwise censored. The upper confidence +interval is extrapolated to approximately 55 using a linear fit. The KM drops to 0 at approximately 48 + + + +Page 9 of 14 + + +Figure 2 Anonymized Data: Probability of Saturation With Extrapolation + + + +The statistical methodology presented here, fits a probability distribution to the elicited code data from +Guest. The example is simplified in part because the raw data is not available, and I extract the data +from Figure 1 of Guest et all as appears in Figure 3. + + + +Anonymized data: Probability of saturation with extrapolation +0 +0 +0 +0 +10 +20 +30 +40 +50 +60 +Interview numberPage 10 of 14 + + +Figure 3 Guest Probability of Saturation + + + +Pragmatic Sample Size estimation +There are well established methods for estimating sample size for survival “time to event” endpoint +(Wu, 2016). When sample sizes (number of interviews) are on the order of 10 to as much as 50 +interviews, then a simple pragmatic “trial and error” type method may be used estimating sample size. +The Kaplan Meier can be estimated in excel or in an open source language such as R (R core team, +2020). Note a reason for estimating sample size (number of interviews) could be for estimating a +budget for an instrument development project. In a pharmaceutical Phase III clinical trial sample size + +Guest: Probability of saturation +0 +0 +6 +0 +4 +0 +2 +0 +0 +0 +0 +10 +20 +30 +40 +Interview numberPage 11 of 14 + +must be completely specified prior to enrolling the first patient. For instrument development the sample +size (number of interviews) should be a “very good guess” and allow for the possibility to be revised +during the interview process. +I present several sample size planning scenarios in Table 5, to illustrate a pragmatic approach to +estimating the number of interviews. Use best available methods, such as expert judgement for an +initial estimate. Then generate hypothetical sequences of interviews where each element is either 0 new +codes, or 1 or more new codes. +Scenario I +As an example, suppose hypothetically the initial expert judgement is that 10 interviews are needed. +Below I denote “1”= interview with new codes , “0”: interview with zero (0) new codes. Note ten +interviews is completely hypothetical. +Scenario1: All new codes elicited in early interviews : + + +(1,1,1,1,1,0,0,0,0,0) +Scenario2: All new codes elicited in the later interviews; + +(0,0,0,0,0,1,1,1,1,1) +Scenario3: Interviews eliciting new codes uniformly distributed: + +(1,0,1,0,0,1,1,0,1,0) +Table 5 +Pragmatic Sample size (number of interviews) determination +Assume base scenario of 10 interviews +Scenario +Interview Sequence 1=new +code 0= zero new codes +KM Probability of +saturation (95% CI) +Additional +interviews with +zero new codes +required for +KMprob=0 +1: All new codes elicited +in early interviews +(1,1,1,1,1,0,0,0,0,0) +0.5 (0.269 ,0.929) +3 +2: all new codes elicited +in the later interviews +(0,0,0,0,0,1,1,1,1,1) +0.0 ( na, na) +0 +3: Interviews eliciting +new codes uniformly +distributed: +(1,0,1,0,0,1,1,0,1,0) +0.24 (0.0479 ,1) +3 + + + + +Na: not estimated +Table 5, scenario 1, presents a scenario of ten interviews, where the KM probability of (not being +saturated) at the last interview is larger than zero. By a pragmatic trial and error, adding three +additional interviews with zero new code gives an estimate of probability of (not being ) saturated of +zero. +Discussion and Conclusions +This paper presents a probability foundation for the definition of saturation and provides a methodology +for defining Sample size in the context of the FDA PRO guidance. + +Page 12 of 14 + +PRO experts planning and conducting interviews for development of PRO’s may use standard one +sample statistical methods (Wu, 2016) for estimation of say, sample size required for estimation for a +survival distribution. +There do not appear to be any publications that provide any summary of interview level data in the +literature which could be used for sample size planning purposes. +Probability Framework for interviews +Guest (2020) argues that there may be no probability model for saturation and cites Galvin , Fugard +&Potts, and others as examples of probability-based models. Limiting the discussion to qualitative +research, I propose a probability framework for saturation. The probability assumption is based on a +hypergeometric distribution arising in an estimation setting called “capture recapture” (CRC) (Tilling, +2001). Capture-recapture estimates the total population (N) under the assumption there is a fixed and +unknown number of “elements”. Common examples of capture-recapture are in estimation of total +population (Wittes, 1974) , and completeness of a disease registry, cancer. Homelessness, mental +health, drug use, and in software development number of undetected software errors ( “bugs”) (Chun +2006). A key assumption of capture-recapture is that codes have an equal probability of elicitation. +Again this assumption may not be reasonable if successive interview questions change or evolve based +on prior interview questions and answers. +I used Capture-recapture (CRC) methods to estimate the total number of “codes” for an instrument for +an “indirect estimate” of sample size. The CRC would give an estimate of the number of codes. Consider +an instrument such as the SF36 where there are 36 codes. An estimate at say, the second or third +interview that the total number of codes is approximately 36 but at the hypothetical second interview +only two 2 codes suggests that approximately 34 (36-2) codes remain to be elicited. The estimate of +total number of codes is intended for use in arranging and budgeting for the remaining number of +interviews. The estimate is not intended to be the definitive statistical estimate of number of codes. +One key assumption for both Kaplan-Meier and capture-recapture is independence. This is unlikely to +be true with items (codes) for a patient reported outcome where items are routinely assessed for their +correlation. Preliminary assessments, using capture-recapture methods such as the Lincoln Peterson, or +Chapman estimator were prepared but not presented here. The estimators were computed at each +interview. Lincoln-Peterson and Chapman tended to underestimate the true number of codes. The +Good-Turing (GT) is simple to compute and underestimated the true number for the initial 5- interviews, +then overestimated. For example, while the true known number of codes is 19, after 5 interviews the +GT estimator was 19.5 and after 6 interviews was 28.4. Hypothetically, If the estimator at 5 interviews +was correct, then approximately 14 (19-5) or fewer interviews remain. Again the discrepancies are likely +due to the induced dependence among codes elicited at interviews + + +References + + +Page 13 of 14 + +Food and Drug Administration, 2009 Guidance for Industry, Patient-Reported Outcome Measures: Use +in Medical Product Development to Support Labeling Claims, +https://www.fda.gov/media/77832/download, accessed 2020. +Bernard, Russel, editor field methods, email of Feb 8, 2020. +Brittain, Sarah, and Dankmar Böhning. "Estimators in capture–recapture studies with two sources." AStA +Advances in Statistical Analysis 93.1 (2009): 23-47. +Bowen GA. Naturalistic inquiry and the saturation concept: a research note. Qualitative Research +2008;8(1):137-52. +Brédart, Anne, et al. "Interviewing to develop Patient-Reported Outcome (PRO) measures for clinical +research: eliciting patients’ experience." Health and quality of life outcomes 12.1 (2014): 15. +Chun, Y.H., 2006. 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The Quality of Qualitative Research.London: SAGE +Publications Ltd; 1999. p. 87-105. +Stolper , http://www.gutfeelings.eu/glossary/saturation-2/ accessed 2017. +Kate Tilling, Capture-recapture methods—useful or misleading?, International Journal of Epidemiology, +Volume 30, Issue 1, February 2001, Pages 12–14, +van Rijnsoever FJ (2017) (I Can’t Get No) Saturation: A simulation and guidelines for sample sizes in +qualitative research. PLoS ONE12(7): e0181689. https://doi.org/10.1371/journal.pone.0181689 + +Wittes, J.T., Colton, T. and Sidel, V.W., 1974. Capture-recapture methods for assessing the completeness +of case ascertainment when using multiple information sources. Journal of chronic diseases, 27(1), +pp.25-36. + +Wood, Fred, and Tom Guinter. "Evolution and implementation of the CDISC study data tabulation model +(SDTM)." Pharmaceutical Programming 1.1 (2008): 20-27. + +Wu, Jianrong. "Single-arm phase II cancer survival trial designs." Journal of biopharmaceutical +statistics 26.4 (2016): 644-656. + diff --git a/e9E3T4oBgHgl3EQf3AuS/content/tmp_files/load_file.txt b/e9E3T4oBgHgl3EQf3AuS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..159bf15ce3f8b5c3a74d1515bb5894f3d08ac4a2 --- /dev/null +++ b/e9E3T4oBgHgl3EQf3AuS/content/tmp_files/load_file.txt @@ -0,0 +1,346 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf,len=345 +page_content='Page 1 of 14 Rev: 1/10/2023 Pragmatic Estimation of Sample Size for Number of Interviews for PRO development in the 2009 FDA PRO guidance Affiliation Chris Barker, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Adjunct Associate Professor of Biostatistics University of Illinois Chicago - School of Public Health Chicago, Illinois And Chris Barker, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' CEO Chris Barker Statistical Planning and Analysis Services Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='barkerstats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='com Abbreviations CRC – Capture-Recapture KM – Kaplan Meier PRO – Patient Reported Outcome FDA-Food and Drug Administration Abstract Patient Reported Outcomes developed according to the 2009 FDA PRO guidance require an initial step of structured patient interviews or focus groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The guidance does not provide sample size suggestions or methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' This paper proposes statistical methodology and sample size guidance that address this gap in the FDA PRO guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' This paper also appears to be the first to provide a definition of Type I error in interviews for a PRO methods for assessing sample size and a new definition of saturation based on a probability distribution to confirm whether enough interviews have been prepared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Type I error is declaring saturation when it has not been achieved during the interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Two worked examples applied to actual interview data and published data, are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Guidelines are proposed for the estimation of sample size useful for PRO experts conducting interviews for a PRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' These methods are applied in the setting of qualitative research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Future research for other methods of interviewing and determining saturation is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Page 2 of 14 Background This paper and methodology arose because of a project progress teleconference with the CEO and CMO for a small biotech, me, a PRO developer vendor and additional clinical operations staff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The vendor was responsible for activities including interviewing patients, preparing and validating a PRO instrument for use as a primary endpoint in a Phase III randomized trial according to the FDA PRO guidance (FDA, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Due to confidentiality agreements, the names of the biotech, vendor, drug, and indication are withheld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The CEO asked the vendor how many interviews would be required to develop the instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The vendor was processing the interviews using Qualitative Research and stated that interviews would be complete “once saturation was achieved”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The CEO requested I explain the definition of “saturation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The vendor defined ‘saturation’ as the first interview that elicited no new concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' I asked a follow-up question “Is that the first occurrence of saturation or do you conduct several, perhaps 2 or more additional consecutive interviews with no new concepts?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The vendor replied “no, only one interview with zero new codes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' I also asked if a single interview could be statistical noise and possibly a type I error (declaring saturation when it had not occurred).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The vendor was unable to answer that question about Type I error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Excerpting from the PRO guidance (page 13), “We cannot provide recommendations for the number or size of the individual patient interviews or focus groups for establishing content validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The sample size depends on the completeness of the information obtained from analysis of the transcripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Generally, the number of patients is not as critical as interview quality and patient diversity included in the sample in relation to intended clinical trial population characteristics.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The methodology of this paper addresses the number of patients but does not address the interview quality or patient diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Qualitative research is one of several methodologies for preparing and coding interviews used to develop PRO’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' An overview of other methods is described in a paper sponsored by ISPOR, and includes, phenomenology, Grounded Theory, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' (Patrick 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The implementation of the interview methods is not reviewed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' This paper considers only qualitative research implements the empirical concept of ‘saturation’ to determine when interviews are complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' In reviewing the literature about the concept of saturation there does not appear to be a formal statistical method for determining either saturation or statistical guidance for sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Note the applicability of the methods of this paper to other methods of conducting interviews must be assessed on a case-by-case basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Aside from the case- by-case assessment a common property of the other methods is the range of number of interviews with “zero new codes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' This paper does not provide a complete literature review of the concept of saturation, an ambitious goal well beyond the scope of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The term “saturation” is frequently applied in grounded theory research and its application beyond that theory is debated (Oreilly 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Theoretical saturation means that researchers reach a point in their analysis of data that sampling more data will not lead to more information related to their research questions (Seale,1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' I mention a small number of arbitrarily selected papers have reviewed interview methodology, definition of saturation and saturation and the number of interviews (Francis, 2010, Fusch 2015, Mason 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The term “saturation” is defined as Page 3 of 14 zero new concepts (or themes) elicited at an interview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Alternate definitions appearing in the literature are: first occurrence of saturation, three consecutive interviews with saturation, expert Judgement “additional interviews would be counter-productive”, minimum 10 interviews and 3 with zero new codes (“10+3”), (Francis, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Several deterministic recommendations for number of interviews (sample size) appear in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Table 1 Ethnography/ethnoscience “30 60 interviews” Grounded Theory “30 50 interviews” Phenomenology “5 25 interviews” All Qualitative research “at least 15 interviews” Funded ($) research ‘time limited’ interviews ranged from 1 – 95 Tesch (1990) enumerates 23 “qualitative research types” for which the methods of this paper may also be applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Methods Assumptions for application of the methodology of Kaplan-Meier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Throughout the discussion I use the term “sample size” as synonymous with number of interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' I assume that there is one subject per interview (subjects do not give more than one interview).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The Kaplan Meier methodology adopted here for qualitative research is descriptive (no “p-values”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' One major property of the Kaplan Meier as defined here, the KM probability of not being saturated is unchanged or declines with each successive interview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The KM probability estimates can decline and exactly equal zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' A further characteristic is that at and after saturation, additional interviews are redundant and do not provide new codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' This paper assumes there is a fixed but unknown number of “codes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' One must assume that each interview is conducted in a repeatable reproducible manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Each interview results in none, one or more new codes vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' preceding interview(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The Kaplan-Meier assumes that each outcome or event “new codes” or “no new codes”, in this example each interview, is statistically independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' This is unlikely to be true for interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The fixed unknown total number of codes induces a correlation among pairs of interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Modelling this correlation is outside the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The Kaplan- Meier estimate of probability of saturation is unbiased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Due to the induced correlations, the estimates of variability may be biased, and may tend to be too small relative to the independence case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Alternative estimates of variability such as by using a Bootstrap might be applicable and are not further examined in this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Adaptation of KM methodology to other interview methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The KM methodology must be adapted for each interview methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Page 4 of 14 Example data sets and availability These datasets below are available on request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Informed consent was obtained from the subjects providing data for peach of the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The Guest data used below was collected as part of a larger study (Guest 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Informed consent for the subjects was obtained in the larger study (Bernard, Personal communication, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The patients for the second dataset gave informed consent (Revicki, Personal communication, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Table 1 presents a “flat file” structure for organizing the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' I recommend that the interview datasets are included in the regulatory submission as part of standardized ADaM and SDTM datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The FDA currently requires clinical trial data submitted in this standardize format (Wood, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Data set I – anonymized interview data I obtained an anonymized dataset from an expert in PRO’s (Kleinman, 2012) from a set of twenty-one interviews processed using qualitative research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The anonymization replaced the code description with a letter (A, B…).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Interviews were in chronological order and a 1 represents the code observed at the interview 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The interviews resulted in 20 separate codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The dataset may be represented in a tabular format in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Aligning with conventional terminology for capture-recapture discussed below the “codes” are termed “marked” (M) when first elicited, and when a code is elicited a second or later interview, it is labelled a “recapture” (R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' For each, code the 0’s represent code not elicited in interview n and 1=code elicited in interview j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Table 3 summarizes the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Table 2 Organization for Interview Dataset Anonymized Codes (M=Elicited,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' R= code Elicited again) Interview ID Interview Sequence # Code A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Code B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Code C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' … Code K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='R) (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='R) (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='R) (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='R) (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='R) aaaa 1 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' 0) (0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='0) (0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='0) 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='0) bbbb 2 (0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='0) (0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='0) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='0) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='1) cccc 3 (1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='0) (0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='0) (0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='0) dddd 4 (1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='1) (0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='0) (0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='0) eeee 5 (1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='1) (1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='1) (0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='0) (1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='1) … … The data set has the following raw and derived variables,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Let j = interview chronological sequence number,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' j=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='3….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' J where J is the total number of interviews Note that total number of interviews, J is unknown at the start of the interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Code -k-, (E-)licited at interview j, E j k: 1=yes, 0=no Where k=1,2, 3… K, elicited from interview, j=1,2, 3… J Page 5 of 14 Note that the total number of codes K is unknown at the start the interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Let Rj, k indicate Ej,k elicited again “recaptured” at interview, j ,(1=yes, 0=no k=1,2, 3… K ), total codes elicited at interview j, Nj = ∑ (Ej,k ) and summation over k cumulative elicitations of code k, Mk= ∑ Ej, k summation over j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Table 3 interview descriptive statistics N marked codes Mean marked per interview Median marked per interview STD marked N recap Mean recapture Median recapture STD recapture 47 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='34 47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='81 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='44 Page 6 of 14 Table 4 interview descriptive statistics recaptured Recapture N 0 2 1 6 2 8 3 11 4 3 5 4 6 5 7 5 8 1 9 1 10 1 Data set II – using a Published figure from Guest et al I conducted a search in google and located a paper by Guest et al with codes elicited from interviews then used to develop an instrument about HIV in subjects in Africa (Guest .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' I contacted Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Guest and the raw data for his paper are not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Figure 1 below reproduces figure 1 of the Guest paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Guest summarized their interviews and reported number of codes elicited in groups of 6 consecutive interviews and for interviews from Ghana then from Nigeria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' For the estimate of probability of saturation presented below a simple ad hoc imputation of codes to interviews was implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' When #codes > 6 (interviews) then, it was assumed every interview had at least 1 code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' When #codes <6, for example 4, it was assumed that four interviews resulted in 1 code and remaining 2 randomly selected interviews had zero codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Figure 1 Guest, reproduced with permission Page 7 of 14 Reproduced from Guest et al with permission Note numbers at top of bars are number of codes elicited in 6 interviews A Type I error for PRO development may be defined as terminating interviews and incorrectly claiming saturation is achieved when more interviews are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Consider the deterministic rule that saturation is the first occurrence of an interview with zero new codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The data from Guest indicate that for interview 13-18, there were 5 codes in 6 interviews, therefore there was at least one interview with zero new codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Note had interviews stopped at 13-18 would be a Type I error (saturation did not occur) , and as many as 45 more interviews were required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Figure 1 (Guest, reproduced with permission) 80 8 20 5 2 1-6 13-18 25-30 37-42 49-54 7-12 19-24 31-36 43-48 55-60 Interview #Page 8 of 14 Selecting a probability distribution to fit the decline in the number of “new codes” to zero can be based on a goodness of fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' By a serendipitous choice I adopted the non-parametric Kaplan Meier (“K-M”) (Kaplan, Meier, 1958) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The KM does not require an assumption of a distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The KM is zero at an “interview” where saturation occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Other distributions, exponential, gamma etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' may be considered, however those distributions can equal zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' I recommend selection of a distribution that can equal zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Consideration of adaptation of a distribution such as a truncated or triangular distribution is beyond the scope of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' I consider a simple application of the non-parametric Kaplan Meier estimate of the distribution of elicited codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' For the K-M “0 new codes” is treated as an “event” and >=1 code as “censored”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' That curve and the 95% confidence interval is presented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Extracting data from the graph uses the simplifying assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' For each group of 6 interviews, for example resulting in 5 codes, I use a simplifying assumption that one interview had zero codes and I randomly select one interview, the remaining interviews are imputed by one code per interview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' I fit a regression line to the upper 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' limits and extrapolate that to the x-axis for dataset II, presented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The extrapolated regression crosses the axis at approximately 70 interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The median is the statistic reported from the Kaplan-Meier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' For saturation, the interest is the probability estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The KM estimate starts at 1 (100%) because no interviews have occurred - the probability that saturation has not occurred is 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The K-M probability estimate declines to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='0%, at about 55 interviews, interpreted as probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='0% that saturation has not occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' I do not recommend nor adopt a hypothesis testing framework for interpreting saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The Kaplan Meier provides a confidence interval associated with the probability distribution to incorporate variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' As above, it may not be reasonable to assume individual interviews and elicited codes are not statistically independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Future research in this area may consider use of bootstrap resampling or other methods that account for the potential non-independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Results Figure 3 present a Kaplan-Meier estimate of the probability of saturation, based on anonymized interview data provided by Revicki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The x-axis is the interview in chronological sequence and y axis KM probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' An “event” is an interview with zero new codes otherwise censored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The upper confidence interval is extrapolated to approximately 55 using a linear fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The KM drops to 0 at approximately 48 Page 9 of 14 Figure 2 Anonymized Data: Probability of Saturation With Extrapolation The statistical methodology presented here, fits a probability distribution to the elicited code data from Guest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The example is simplified in part because the raw data is not available, and I extract the data from Figure 1 of Guest et all as appears in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Anonymized data: Probability of saturation with extrapolation 0 0 0 0 10 20 30 40 50 60 Interview numberPage 10 of 14 Figure 3 Guest Probability of Saturation Pragmatic Sample Size estimation There are well established methods for estimating sample size for survival “time to event” endpoint (Wu, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' When sample sizes (number of interviews) are on the order of 10 to as much as 50 interviews, then a simple pragmatic “trial and error” type method may be used estimating sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The Kaplan Meier can be estimated in excel or in an open source language such as R (R core team, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Note a reason for estimating sample size (number of interviews) could be for estimating a budget for an instrument development project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' In a pharmaceutical Phase III clinical trial sample size Guest: Probability of saturation 0 0 6 0 4 0 2 0 0 0 0 10 20 30 40 Interview numberPage 11 of 14 must be completely specified prior to enrolling the first patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' For instrument development the sample size (number of interviews) should be a “very good guess” and allow for the possibility to be revised during the interview process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' I present several sample size planning scenarios in Table 5, to illustrate a pragmatic approach to estimating the number of interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Use best available methods, such as expert judgement for an initial estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Then generate hypothetical sequences of interviews where each element is either 0 new codes, or 1 or more new codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Scenario I As an example, suppose hypothetically the initial expert judgement is that 10 interviews are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Below I denote “1”= interview with new codes , “0”: interview with zero (0) new codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Note ten interviews is completely hypothetical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Scenario1: All new codes elicited in early interviews : (1,1,1,1,1,0,0,0,0,0) Scenario2: All new codes elicited in the later interviews;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' (0,0,0,0,0,1,1,1,1,1) Scenario3: Interviews eliciting new codes uniformly distributed: (1,0,1,0,0,1,1,0,1,0) Table 5 Pragmatic Sample size (number of interviews) determination Assume base scenario of 10 interviews Scenario Interview Sequence 1=new code 0= zero new codes KM Probability of saturation (95% CI) Additional interviews with zero new codes required for KMprob=0 1: All new codes elicited in early interviews (1,1,1,1,1,0,0,0,0,0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='269 ,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='929) 3 2: all new codes elicited in the later interviews (0,0,0,0,0,1,1,1,1,1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='0 ( na, na) 0 3: Interviews eliciting new codes uniformly distributed: (1,0,1,0,0,1,1,0,1,0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='24 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='0479 ,1) 3 Na: not estimated Table 5, scenario 1, presents a scenario of ten interviews, where the KM probability of (not being saturated) at the last interview is larger than zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' By a pragmatic trial and error, adding three additional interviews with zero new code gives an estimate of probability of (not being ) saturated of zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Discussion and Conclusions This paper presents a probability foundation for the definition of saturation and provides a methodology for defining Sample size in the context of the FDA PRO guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Page 12 of 14 PRO experts planning and conducting interviews for development of PRO’s may use standard one sample statistical methods (Wu, 2016) for estimation of say, sample size required for estimation for a survival distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' There do not appear to be any publications that provide any summary of interview level data in the literature which could be used for sample size planning purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Probability Framework for interviews Guest (2020) argues that there may be no probability model for saturation and cites Galvin , Fugard &Potts, and others as examples of probability-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Limiting the discussion to qualitative research, I propose a probability framework for saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The probability assumption is based on a hypergeometric distribution arising in an estimation setting called “capture recapture” (CRC) (Tilling, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Capture-recapture estimates the total population (N) under the assumption there is a fixed and unknown number of “elements”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Common examples of capture-recapture are in estimation of total population (Wittes, 1974) , and completeness of a disease registry, cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Homelessness, mental health, drug use, and in software development number of undetected software errors ( “bugs”) (Chun 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' A key assumption of capture-recapture is that codes have an equal probability of elicitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Again this assumption may not be reasonable if successive interview questions change or evolve based on prior interview questions and answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' I used Capture-recapture (CRC) methods to estimate the total number of “codes” for an instrument for an “indirect estimate” of sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The CRC would give an estimate of the number of codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Consider an instrument such as the SF36 where there are 36 codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' An estimate at say, the second or third interview that the total number of codes is approximately 36 but at the hypothetical second interview only two 2 codes suggests that approximately 34 (36-2) codes remain to be elicited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The estimate of total number of codes is intended for use in arranging and budgeting for the remaining number of interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The estimate is not intended to be the definitive statistical estimate of number of codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' One key assumption for both Kaplan-Meier and capture-recapture is independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' This is unlikely to be true with items (codes) for a patient reported outcome where items are routinely assessed for their correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Preliminary assessments, using capture-recapture methods such as the Lincoln Peterson, or Chapman estimator were prepared but not presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The estimators were computed at each interview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Lincoln-Peterson and Chapman tended to underestimate the true number of codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' The Good-Turing (GT) is simple to compute and underestimated the true number for the initial 5- interviews, then overestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' For example, while the true known number of codes is 19, after 5 interviews the GT estimator was 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='5 and after 6 interviews was 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Hypothetically, If the estimator at 5 interviews was correct, then approximately 14 (19-5) or fewer interviews remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E3T4oBgHgl3EQf3AuS/content/2301.04760v1.pdf'} +page_content=' Again the discrepancies are likely due to the induced dependence among codes elicited at 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Leading methods based on normalizing +flows have tried to solve this problem with an invertible transfor- +mation instead of a deterministic mapping. However, the implicit +bijective mapping is not explored well. Inspired by a latent +observation that noise tends to appear in the high-frequency +part of the image, we propose a fully invertible denoising +method that injects the idea of disentangled learning into a +general invertible neural network to split noise from the high- +frequency part. More specifically, we decompose the noisy image +into clean low-frequency and hybrid high-frequency parts with +an invertible transformation and then disentangle case-specific +noise and high-frequency components in the latent space. In this +way, denoising is made tractable by inversely merging noiseless +low and high-frequency parts. Furthermore, we construct a +flexible hierarchical disentangling framework, which aims to +decompose most of the low-frequency image information while +disentangling noise from the high-frequency part in a coarse- +to-fine manner. Extensive experiments on real image denoising, +JPEG compressed artifact removal, and medical low-dose CT +image restoration have demonstrated that the proposed method +achieves competing performance on both quantitative metrics +and visual quality, with significantly less computational cost. +Index Terms—Invertible Image denoising, bijective transfor- +mation, disentangling learning, hierarchical representation. +I. INTRODUCTION +I +MAGE denoising, aiming to recover clean observation from +its noisy measurement, is a classic inverse problem. Based +on the Bayesian perspective, most traditional approaches view +it as a general maximum a posteriori (MAP) optimization +problem, with assumptions regarding both the image priors and +noise (usually Gaussian), which deviate from real cases and +lead to critical limitations in practical scenes. Deep learning- +based methods [1]–[4] have achieved superior denoising per- +formance in recent years. Most of them view it as a nonlinear +mapping between noisy and clean image pairs. However, real +image noise is generally accumulated from multiple degrading +sources, which results in non-injective mapping due to various +noise types and levels. +Bridging a bijective transformation between the noisy and +clean image pairs to solve such an ambiguous inverse problem +in low-level vision has been explored in recent years. Previous +methods [5], [6] have employed convolutional neural networks +(CNNs) to model wavelet transforms to solve image restora- +tions, in which wavelet decomposition and reconstruction +W. Du, H. Chen, and H. Yang are with the College of Computer Sci- +ence, Sichuan University, Chengdu 610065, China. Y. Zhang is with the +college of Cyber Science and Engineering, Sichuan University, Chengdu, +610065, China. Email: wenchaodu.scu@gmail.com; huchen@scu.edu.cn; +yzhang@scu.edu.cn; yanghongyu@scu.edu.cn. +17.55/0.09 +35.30/0.83 +35.56/0.84 +35.92/0.85 +18.47/0.14 +Noisy +31.62/0.71 +InvDn +31.91/0.72 +DANet +32.25/0.73 +Ours +Fig. 1. +Real image noise removal results on SIDD validation set. The two +representative methods are selected for comparisons, i.e. flow-based InvDN +[9] and GAN-based DANet [7], PSNR and SSIM values are also computed. +Zoomed in for better visualization. +processes are modeled separately, leading to an injective map- +ping procedure. Generative adversarial network-based methods +[7], [8] have also modeled the bijective transformation for +noisy image generation and restoration in supervised and +unsupervised manners. However, these methods need different +models to simulate these two processes separately, which leads +to complex training procedures. +Normalizing flow that allows for efficient and exact like- +lihood calculation and sampling by invertible transformation, +has been applied to solve ill-posed inverse problems in low- +level vision [9]–[12]. NoiseFlow [10] is a seminal flow-based +model for camera noise generation. Unlike general CNN- +based methods [13] that model camera imaging pipelines in +forward and reverse directions to simulate real noise gen- +eration, NoiseFlow learns the distribution of real noise and +generates diverse noisy images by latent variable sampling to +augment training data. However, it treats the clean image as +conditional prior and thus is not fully invertible between clean +and noisy image pairs. However, it treats the clean image as +conditional prior and thus is not fully invertible. Considering +that noise tends to appear in the high-frequency part of the +image, InvDn [9] discards the high-frequency content, and then +utilizes the invertible neural network (INN) [14] to capture +the distribution of the high-frequency part of the clean image, +image denoising is achieved by sampling a latent variable from +predefined distribution to approximate the lost high-frequency +information. Nevertheless, since the model is bijective, the +ill-posedness of the task is partly alleviated. However, InvDn +assumes that the low and high-frequency contents of the image +arXiv:2301.13358v1 [cs.CV] 31 Jan 2023 + +2 +are independent of each other and thus lacks the ability to +exploit their dependency for image denoising, as shown in +Fig. 1, which leads to the recovered image losing part of high- +frequency details, e.g., over-smoothed image edges. Further- +more, the implicit bijective mapping is not fully guaranteed +due to extra latent variable sampling. +Instead, in this paper, we propose a fully invertible model +for image denoising that injects the idea of disentangled learn- +ing into a general invertible architecture to explore feature- +level noise-high frequency signals splitting. Specifically, we +aim to remove noise from the hybrid high-frequency part of +the noisy image while reserving case-specific high-frequency +information as much as possible. To do so, we first decompose +the image into low and high-frequency representations by an +invertible transform during the forward propagation. Assum- +ing the decomposed low-frequency information is noiseless, +according to the information lossless characteristics of the +normalizing flow, the noise only appears in the high-frequency +part. Therefore, our goal is to split the noise component from +it, so that the noise-free high-frequency signal is preserved +well. +To this end, we introduce the concept of disentangled learn- +ing into the general invertible architecture, which transforms +the hybrid high-frequencies into the compact and independent +representations with internal characteristics, where we model +the high-frequency representation learning in the form of +distribution, e.g., the marginal distribution of the hybrid high- +frequency representation obeys a pre-specified distribution, +i.e. isotropic Gaussian. In this way, we directly disentangle +noise and clean high-frequency representations along specific +dimensions without extra latent variable sampling, which +leads to a more accurate and stable bijective transformation. +Image denoising is achieved by inversely merging noise-free +low and high-frequency representations only. Furthermore, we +hierarchically decompose the high-frequency representation +into low and high-frequency parts and disentangle noise from +fine-grained high-frequency parts, which results in a flexible +and efficient framework for generalized image denoising tasks. +In short, our contributions are summarized as follows: +1) We propose a fully invertible model for image denoising, +which introduces the idea of disentangled learning into a +general invertible architecture and achieves a more accu- +rate bijective mapping without latent variable sampling. +2) We construct a hierarchical high-frequency decompo- +sition framework, which could process diverse noise +with varying complexity, and achieve a better trade-off +between performance and computational costs. +3) Extensive experiments on natural image denoising, +JPEG compressed artifact removal and medical low- +dose CT image restoration demonstrate the proposed +method achieves competing performance in terms of +quantitative and qualitative evaluations. To the best of +our knowledge, this is the first full invertible method +capable of solving multiple real image denosing tasks. +The remainder of the paper is organized as follows: Section +II provides a brief review of related work. Section III presents +our approach in detail. In Section IV, extensive experiments +are conducted to evaluate the proposed method, and the +conclusion is presented in Section V. +II. RELATED WORK +A. Traditional Methods +Most traditional image denoising models usually construct a +MAP optimization problem with a data filed and an extra +regularization term. Along this direction, making assumptions +regarding the noise distribution is necessary for most methods +[15]–[17] to build the model (e.g., Mixture of Gaussian). +The regularization term is generally based on the natural +image prior. Classic total variation [18] uses the statistical +characteristics of images to remove noise. Sparse dictionary +learning and Field-of-Experts (FoE) also employ certain pri- +ors existing in image patches [19]–[21]. Non-local similarity +method [22] employ non-local similar patterns of image. In +addition, transform techniques are also explored, e.g., wavelet +domain methods [23] and block-matching and 3D filtering +(BM3D) [24]. +B. Deep Learning based Methods +Instead of preset image and noise distribution priors, DNNs +based methods directly learn a denoiser in a data-driven +manner. Previous method [25] first explored the multi-layer +perception (MLP). Chen et al. [4] further proposed a feed- +forward deep network called the trainable non-linear reaction +diffusion (TNRD) model. Furthermore, Mao et al. [3] utilized +a fully convolution encoder-decoder network with symmetric +skip connection to solve image restoration. Zhang et al. +[2] proposed the Gaussian denoising convolution network +(DnCNN) and achieved superior performance. Considering +non-local self-similarity characteristic of images, non-local +attention networks [26], [27] are also explored in image +restoration. +However, most of these works focus on the synthetic noisy +images with specific noise levels, spatially variant noise limit +the capacity of such models on real cases. To this end, some +methods employed self-adaptive noise level estimated from +inputs to serve as extra priors (e.g., FFDNet [28] and CBDNet +[29]). VDN [30] further integrated variational inference into +the noise estimation and image denoising with a unique +Bayesian framework. AINDNet [31] introduced the transfer +learning to mitigate the domain gap between the real and syn- +thetic noise distribution. In addition, simulating real noise with +a generative model was also explored in recent works [7], [32]. +Considering that DNNs-based image denoising is non-injective +in nature, a complex network architecture and powerful repre- +sentation ability is always required. Thus, Transformer-based +methods [33]–[35] have drawn more attention due to powerful +representation learning on global image features. +C. Normalizing Flow based Methods +Invertible neural networks (INNs) have drawn more attention +to solving ambiguous inverse problems [36]. Unlike DNNs, +INNs focus on learning the forward process and using ad- +ditional latent output variables to capture the information + +3 +that would otherwise be lost. Due to invertibility, a model +of the corresponding inverse process is learned implicitly. +NoiseFlow [10] modeled the distribution of real noise in +ISP imaging process to augment the training data. InvDn [9] +extended the idea of IRN [12] to real noise removal, which +discards hybrid high-frequency information containing noise +of the image and exploited extra latent variable sampling to +approximate the lost high-frequency information. However, a +key factor is ignored that noise is case-specific, which implies +the corresponded high-frequency information is also case- +specific. More recently, FDN [37] directly disentangled the +noise in the latent space with normalizing flows, which leads +to a more complex network. FINO [38] introduced the dual +flow models to bridge bijective mapping between the noisy +and clean image pairs. +Rather, we aim to solve image denoising by modeling +a reliable bijective mapping with single model only. We +empirically show that integrating the idea of disentangled +learning into the flow-based framework can result in fast and +stable training as well as good performance on generalized +image denoising tasks. +III. METHODOLOGY +A. Preliminaries +Typical INNs models a generative process with a known +distribution through a sequence of differentiable, invertible +mappings. Formally, let x0 ∈ RD be a random variable +with a known and tractable probability density function pX0 : +RD → R and let x1, . . . , xN be a sequence of random +variables such that xi = fi(xi−1) where fi : RD → RD +is a differentiable, bijective function. Then, if y = f(x0) = +fn ◦ fN−1 ◦ · · · ◦ f1(x0), the change of variables formula says +that the probability density function for y is +p(y) = pX0(g(y)) +N +� +j=1 +| det Jj(g(y))|−1 +(1) +where g = g1 ◦ · · · ◦ gN−1 ◦ gN is the inverse of f, and +Jj = ∂fj/∂xj−1 is the Jacobian of the jth transformation fj +with respect to its input xj−1 (i.e., the output of fj−1). +Due to such flexibility on accessing to the inverse mapping, +INN architecture could be used for variational inference [39], +[40], and representation learning without any information loss. +INN is composed of basic invertible blocks [41]. For the l-th +block, the input u is split into u1 and u2 along the channel axis, +and the typical additive affine transformation and corresponded +inverse transformation are formulated as: +� +v1 = u1 + φ(u2) +v2 = u2 + η(v1) ⇔ +� +u2 = v2 − η(v1) +u1 = v1 − φ(u2) +(2) +where φ and η are arbitrary neural networks. The output of a +single block is [v1, v2]. +To enhance the transformation ability of the identity branch, +Eq. (2) is always augmented as: +� +� +� +� +� +� +� +� +� +v1 = u1 ⊙ exp(ψ(u2)) + φ(u2) +v2 = u2 ⊙ exp(ρ(u1)) + η(u1) +u2 = (v2 − η(v1)) ⊙ exp(−ρ(v1)) +u1 = (v1 − φ(u2)) ⊙ exp(−ψ(u2)) +(3) +where ψ(·), φ(·) and η(·) denote the transformation functions, +which are arbitrary [41]. Function ρ(·) is further followed +by a centered sigmoid function and a scale term to prevent +numerical explosion due to the exp(·) function. +B. Problem Specification +We rethink the image denoising task from the perspective of +invertible transformation. Suppose the original noisy observa- +tion is y, its clean image is x and the noise is n. We have +p(y) = p(x, n) = p(x)p(n|x) +(4) +where the distribution of noisy image p(y) is a joint distri- +bution corresponded to x and n. Directly splitting the noise +component from y is difficult due to unknown p(n). In +addition, p(n|x) implies the noise is case-specific to image +content. A general observation that noise tends to appear +in the high-frequency component of the image. Therefore, +we employ a wavelet transformation to decompose the noisy +image y into low and high-frequency components, denoted as +yLF and yHF respectively. Then, Eq. (4) is reformulated by +p(y) = p(yLF, yHF) = p(yLF)p(yHF|yLF) +(5) +Ideally, wavelet decomposition is orthogonal, which could +yield dense representation without redundancies, so Eq. (5) +is rewritten as p(y) = p(yLF, yHF) = p(yLF)p(yHF). +Our goal is to disentangle noise representation from hy- +brid high-frequency component, which means that the low- +frequency part yLF is approximately noiseless, and noise is +only contained in p(yHF). Considering the unique characteris- +tics of information lossless in INNs, we could implement it +easily. Thus, we focus on how to split noise from p(yHF) such +that +p(yHF) = p(yhF, yn) = p(yhF)p(yn|yhF) +(6) +where yhF denotes noise-free high-frequency part, and yn is +case-specific noise. +Different from general flow based models, e.g., IRN [12] +and InvDn [9], a latent variable z is introduced to approximate +the case-specific yhF, and transformed z to be case-agnostic +with a specified distribution, where z is used to model the +high-frequency information lost in degrading. For real image +denoising, yn and yhF are always case-specific and tangled. +Therefore, it is difficult to model lost yhF accurately with a +case-agnostic variable without any conditional priors. Thus, +we attempt to solve this problem with a simple manner, i.e. +disentangling noise part from yHF in the latent space. In this +way, case-specific yhF and yn can be split well. +Directly disentangling hybrid yHF is difficult due to un- +known p(yn). To this end, we assume that yhF and yn are +independent, so that Eq. (6) is reformulated as +p(yHF) = p(yhF, yn) = p(yhF)p(yn) +(7) + +4 +Fig. 2. Illustration of invertible image denoising by disentangling representation. In the forward procedure, noisy image x is first decomposed into hybrid high +frequencies yHF and noisy low frequencies yLF with Discrete Wavelet Transform (DWT), which are feed into a parameterized invertible neural network fθ(·) +to transform into a clean low-frequency ˆyLF and a high-frequency yHF obeying specific distribution in the latent space, where case-specific high-frequency +yhF and noise yn would be disentangled. Image denoising is achieved through the inverse function f−1 +θ +and DWT operation with disentangled yhF and clean +ˆyLF. +To satisfy this assumption, we directly enforce p(yHF) to +obey pre-specific distribution, e.g., isotropic Gaussian, which +means p(yHF) is compact and explainable, a key characteristic +is that the feature vectors in yHF are independent of each other. +We disentangle the high-frequency representation yHF along +the specific dimension to split yhF and yn. To reconstruct clean +observation, we explicitly remove yn and only preserve specific +p(yhF), so that p(yHF) = p(yhF)p(yn) = p(yhF) · 1. Benefited +from the invertible characteristic of INNs, image denoising +could be achieved by inverse pass using Eq. (1). +C. Model Architecture +The sketch of disentangling framework is illustrated in Fig. 2, +which contains two processes: i.e. forward decomposition and +inverse reconstruction, and is easily injected into the general +invertible framework. +1) Forward Decomposition: The key to our approach is +to decompose the approximately noiseless low-frequency and +case-specific high-frequency representations during the for- +ward pass. To achieve this, we first exploit the Discrete +Wavelet Transform (DWT) to decompose the image into low +and high-frequency parts, where Haar Transformation [42] +is used to approximate it for simplicity. Haar Transform +explicitly decomposes the inputs, (i.e. images or a group of +feature maps) into an approximate low-pass representation +and high-frequency coefficients with three directions. More +concretely, it transforms the input with shape (H × W × C) +into a tensor of shape ( 1 +2H × 1 +2W ×4C). The first C slices of +the output tensor are effectively produced by average pooling, +which is approximately a low-pass representation equivalent +to the bilinear interpolation down-sampling. The remaining +three groups of C slices contain residual components in the +vertical, horizontal and diagonal directions, which are the +high-frequency information of the original input. By such a +transformation, the low and high-frequency information are +explicitly separated. +Then, an invertible network module is used to further +abstract the yLF and yHF, where we leverage the coupling layer +architecture in [41], [43], presented in i.e. Eqs. (2) and (3). Our +goal is to polish the low and high-frequency inputs to obtain +a suitable low-frequency representation and an independent +properly distributed high-frequency representation. Therefore, +we match yLF and yHF respectively to the split of u1, u2 in Eq. +(2). Furthermore, to increase the model capacity, we employ +the additive transformation (Eq. (2)) for the low-frequency part +u1, and the enhanced affine transformation (Eq. (3)) for the +high-frequency part u2. +After transformation, a noise-free low-frequency representa- +tion ˆyLF is expected. However, noise also appears in ˆyLF actu- +ally due to non-ideal decomposition. Inspired by the Nyquist- +Shannon sampling theorem that the lost information during +down-sampling a clean image amounts to high-frequency +contents, we utilize the bicubic method [44] to guide ˆyLF +decomposition. Let xguide be the down-sampled clean image x +corresponding to ˆyLF that is produced by the bicubic method. +To generate the clean low-frequency representation, we drive +the ˆyLF to resemble xguide: +Lguide := ℓX (xguide, ˆyLF) +(8) +where ℓX is a difference metric on X, i.e. the L2 loss. +Benefited from the characteristic of information lossless of +invertible architecture, our low-frequency guidance loss drives +noise appearing in yHF only. +Another goal during the forward pass is to disentangle noise +from high-frequency part yHF. To this end, we enforce the +yHF to obey specific distribution, i.e. pyHF ∼ N(0, IK) so that +it could be disentangled along the specific dimension. A KL +divergence loss is used as the distribution metric, where +DKL = − +� +p(z) log(p(z) +q(z))dz +(9) +which is referred to as our distribution guidance loss Ldist. +2) Inverse Reconstruction: Considering feature vectors in +transformed yHF is independent, we directly split it along the +channel axis into a specific high-frequency part yhF and noise +yn, i.e. yHF = [yhF, yn], and yhF and yn are case-specific. To +reconstruct the clean image x in the flow-based architecture, +we need to construct dimension-consistent noise-free high- +frequency representation ˆyHF. To do so, we simply replace +yn with 0 in reconstructed high-frequency representation ˆyHF, +i.e. ˆyHF = [yhF, 0], which lies in a subspace of yHF. Further, to + +DWT +InvBlock +InvBlock +m +Invertible Block5 +Fig. 3. Illustration of our hierarchical disentangling framework. Our approach exploits the three invertible modules to transform the noisy image y into clean +low frequencies and specific high frequencies with different scales, i.e. L = 3. Each invertible module is composed of a DWT layer and stacked InvBlocks. +guide noise component disentanglement, we minimize recon- +struction loss Lrecon with corresponded clean image x: +Lrecon := ℓX (x, iDWT(f −1 +θ +(ˆyLF, ˆyHF))) +(10) +where ℓX measures the difference between the clean image and +the reconstructed one, i.e. the L1 loss. iDWT denotes inverse +DWT, and f −1 +θ +denotes the inverse pass of parameterized +invertible neural networks. +In addition, implicit noisy image self-reconstruction could +also be achieved by injecting case-specific yn into yHF, it +doesn’t rely on any extra self-supervised constraints. This +implies there exists an implicit bijective mapping between the +noisy and clean image pairs. +D. Hierarchical Disentangled Representation +Furthermore, inspired by the latent observation that most +image information is located in the low-frequency part, and +high and low-frequencies are not decomposed exactly due +to non-ideal transform. Thus, exploiting the sufficient low- +frequency information while disentangling fine-grained high- +frequency signals is necessary. Based on this analysis, we +construct a multi-level decomposing framework in Fig. 3. +During the forward propagating, we stack multiple invertible +modules to decompose the high-frequency representation yHF +into low and high-frequency parts, this leads to multiscale +low-frequency outputs {ˆyl +LF}L +l=1. Therefore, the low-frequency +guidance loss Lguide is rewritten as +Lguide := +L +� +l=1 +ℓX (xl +guide, ˆyl +LF) +(11) +where L is the number of levels decomposed, ˆyl +LF represents +the decomposed low-frequency output in lth level, and corre- +sponded multiscale guided image xl +guide is generated by down- +sampling x to 1/2l scale with the bicubic method. +For the high-frequency part, we only minimize the KL +divergence loss for the last output yL +HF, and disentangle noise +component from it. In our experiments, we set L = 3 at most +for generalized image denoising tasks. +In inverse propagation, we reconstruct clean high frequen- +cies ˆyl +HF level by level, from level L to level 1. For lth +level, it leads to an implicit conditional flow model, i.e. +p(ˆyl−1 +HF |yl +HF, ˆyl +LF), which could be implemented by ˆyl−1 +HF += +(a) +(b) +Fig. 4. Online PSNR curves on SIDD validation set with 300k iterations. +TABLE I +THE ARCHITECTURE CONFIGURATIONS. +Metric +Single-level +Multi-level (Ours) +L=2 +L=3 +L=2 +L=3 +PSNR +39.358 +39.499 +39.364 +39.468 +SSIM +0.908 +0.910 +0.909 +0.910 +#Param(M) +4.419 +12.554 +4.021 +9.092 +f −1 +θ +(yl +HF, ˆyl +LF). Therefore, the case-specific high-frequency +signal is generated in a coarse-to-fine manner, it is efficient +and stable. +We optimize the whole architecture by minimizing the com- +pact loss Ltotal with the combination of reconstruction loss +Lrecon, low-frequency guidance loss Lguide and distribution +loss Ldist: +Ltotal := λ1Lrecon + λ2Lguide + λ3Ldist +(12) +where λ1, λ2 and λ3 are coefficients for balancing different +loss terms. +IV. EXPERIMENT +A. Experimental Setting +1) Datasets: To validate the effectiveness of our method, +two representative real image noise datasets, the Smartphone +Image Denoising Dataset (SIDD) [45] and Darmstadt Noise +Dataset (DND) [46], are utilized to verify our method’s +performance. The SIDD is taken by five smartphone cameras +with small apertures and sensor sizes from 10 scenes under +varied lighting conditions. Ground truth images are generated +through a systematic procedure. We use the medium version +of SIDD as the training set, which contains 320 clean-noisy + +三二39.50 +39.25 +39.00 +38.75 +PSNR +38.50 +L=1 +38.25 +L=2 +38.00 +37.75 +L=3 +37.50 +0 +10 +20 +30 +40 +50 +60 +Iterations (×5k)39.25 +39.00 +38.75 +PSNR +38.50 +38.25 +B=4 +38.00 +B=8 +37.75 +B=12 +37.50 +0 +10 +20 +30 +40 +50 +60 +Iterations (×5k)6 +(a) +(b) +(c) +(d) +(e) +(f) +Fig. 5. Visualization analysis for invertible bijective mapping. The top row denotes the results of invertible image denoising. (a) is input noisy image, (b), (c), +and (d) denote the decomposed low-frequency outputs from the level L = 1, 2, 3, separately. (e) is inverse denoised output, and (f) is the self-reconstructed +noisy image. The second row visualizes the high-frequency feature maps from L = 1 during forward propagating. The bottom row visualizes the disentangled +high-frequency feature maps from L = 1 during inverse propagating. +TABLE II +THE DISENTANGLED DIMENSION CONFIGURATIONS FOR MODEL WITH +B=8 AND L=2. +DIM(yn) +4/5 +3/5 +2/5 +1/5 +PSNR +39.348 +39.352 +39.403 +39.364 +SSIM +0.9086 +0.9085 +0.9086 +0.9086 +pairs for training and 1280 cropped patches from the other +40 pairs for validation. The reported test results are obtained +via an online submission system. The DND is captured by +four consumer-grade cameras of different sensor sizes. It +contains 50 pairs of real-world noisy and approximately noise- +free images. These images are cropped into 1000 patches of +size 512 × 512. Similarly, the performance is evaluated by +submitting the results to the online system. Considering DND +does not provide any training data, we employ a training +strategy by combining the training set of SIDD and Renoir +[47]. Results are submitted to the DND benchmark by utilizing +the same model that provides the best validation performance +on the SIDD benchmark. +2) Implementations: The proposed method cascades three +invertible modules at most, each module contains a Haar trans- +form layer and 8 basic invertible blocks, i.e. L = 3, B = 8. +For the single invBlock (as shown in Fig. 2), we implement the +non-linear functions φ, ρ, and η with the Densenet Block (DB) +[48]. All the models are trained with Adam as the optimizer, +with momentum of β1 = 0.9, β2 = 0.999. The batch size is set +as 16 with 144×144 size, and the initial learning rate is fixed +at 2 × 10−4, which decays by half for every 100k iterations. +The training is performed on a single Tesla P100 GPU. We +augment the training data with extra horizontal and vertical +flipping, as well as random rotations. For loss functions, we +set λ1 = 1, λ2 = 4 and λ3 = 1 separately for different loss +terms in all experiments. In addition, Peak-Signal-Noise-Ratio +(PNSR) and Structural Similarity (SSIM) are used to evaluate +the performance of methods in all experiments. +B. Ablation Study +We mainly explore three major determinants of our model: a). +Model capacity, which depends on the number of decomposed +levels, and the number of invertible blocks in single invertible +module; b). Effectiveness of architecture, including decom- +posing types of image and disentangling ways of framework; +and c). Disentangled representation. All the experiments are +performed in SIDD validation set with 300k iterations. +1) Model capability: We first study two key factors affect- +ing the model capability, i.e. multi-level decomposition and the +capacity of the single invertible module. As shown in Fig. 4- +(a) and (b), we observe that increasing the decomposed levels +leads to significant performance gains. In addition, fixing +decomposition levels (e.g., L = 2), stacking more invBlocks +in the single invertible module also further enhances the ability +of the model, but it also brings more parameters. Therefore, +we set the B = 8 and L = 3 at most in our architecture +configurations. +2) Architecture Designing: We also consider different ar- +chitectures, i.e. decompose low and high-frequency parts in +the last level only instead of each level, which is similar +to InvDn [9]. The quantitative results are illuminated in +Tab. I, our multilevel decomposition architecture with high- +frequencies disentangling achieves a better trade-off in terms +of the performance and complexity of the model. +3) Disentangled Representation: We further explore the +effects of disentangled dimensions in yHF. We split yn from yHF +with different dimensions along the channel axis. All models +are trained separately. Tab. II gives the detailed results, the + +7 +(17.56/0.1110) +(18.17/0.1205) +Noisy +(34.19/0.7965) +(35.08/0.8442) +DANet [7] +(34.00/0.7857) +(34.74/0.8365) +InvDn [9] +(34.80/0.8156) +(35.68/0.8564) +MAXIM [35] +(34.47/0.8067) +(35.14/0.8458) +Ours(L=2) +(PSNR/SSIM) +(PSNR/SSIM) +Reference +Noisy +(PSNR/SSIM) +CBDNet [29] +(31.40/0.8364) +VDN [30] +(34.08/0.9166) +DANet [7] +(33.66/0.9148) +InvDn [9] +(34.22/0.9216) +Ours(L=3) +(34.28/0.9224) +Fig. 6. Visualized denoising samples from SIDD and DND datasets. The first and second rows are from the SIDD, and bottom is from the DND. +best model is achieved by setting DIM(yn) = 2/5·DIM(yHF). +We use it in our final model configurations. +4) Analysis for Bijective Mapping: We further explore +the bijective relationship in our architecture. As shown in +Fig. 5, we first give the decomposed low-frequency outputs +from multi-level decomposition during forward propagation, +denoised output, and self-reconstructed noisy input by im- +plicitly inverse reconstruction. Our method only splits case- +specific noise from the hybrid high-frequency component in +the latent space to achieve image denoising, where it bridges +the bijective transformation between the noisy image genera- +tion and restoration. Moreover, we visualize the decomposed +hybrid high-frequency components and disentangled noise-free +high-frequencies components. It is obvious that our method +could remove case-specific noise signals in latent space while +retaining fine high-frequencies, which implies the effectiveness +of our method. +C. Real Image Noise Removal +We perform comprehensive comparisons, including model pa- +rameters, computational complexity (MACs), and quantitative +metrics, on two real denoising benchmarks, i.e. SIDD [45] +and DND [46]. Note that “MACs” is counted on a single +RGB image with 256 × 256 size. Moreover, we select two +kinds of representative methods for comparisons, one is the +deterministic mapping-based methods, including CNN-based +and Transformer methods; The other is bijective mapping- +based methods, e.g., Flow-based and GAN-based methods. +TABLE III +COMPREHENSIVE COMPARISONS WITH OTHER COMPETING METHODS. +Method +#Param +(M) +MACs +(G) +SIDD [45] +DND [46] +PSNR +SSIM +PSNR +SSIM +DnCNN [2] +0.67 +44.02 +37.73 +0.941 +37.90 +0.9430 +TNRD [4] +– +– +24.73 +0.643 +33.65 +0.8306 +BM3D [24] +– +– +25.65 +0.685 +33.51 +0.8507 +CBDNet [29] +4.34 +– +33.28 +0.868 +38.06 +0.9421 +RIDNet [49] +1.50 +98.26 +– +– +39.26 +0.8306 +GradNet [50] +1.60 +– +38.34 +0.946 +39.44 +0.9543 +AINDNet [31] +13.76 +– +39.08 +0.953 +39.53 +0.9561 +VDN [30] +7.82 +49.5 +39.28 +0.909 +39.38 +0.9518 +MIRNet [51] +31.78 +816.0 +39.72 +0.959 +39.88 +0.9563 +SADNet [52] +4.23 +18.97 +– +– +39.59 +0.9523 +MPRNet [53] +15.7 +1394.0 +39.71 +0.958 +39.80 +0.9540 +MAXIM [35] +22.20 +339.0 +39.96 +0.960 +39.84 +0.9540 +DANet [7] +63.01 +14.85 +39.25 +0.955 +39.47 +0.9548 +DANet++ [7] +63.01 +14.85 +39.43 +0.956 +39.58 +0.9545 +InvDn [9] +2.64 +47.96 +39.28 +0.955 +39.57 +0.9522 +FDN [37] +4.38 +77.64 +39.31 +0.955 +– +– +FINO [38] +– +– +39.40 +0.957 +– +– +Ours+RB+L2 +2.47 +46.54 +39.31 +0.956 +– +– +Ours+RB+L3 +5.26 +52.26 +39.40 +0.957 +– +– +Ours+DB+L2 +4.02 +74.02 +39.39 +0.956 +39.54 +0.9520 +Ours+DB+L3 +9.09 +84.40 +39.48 +0.957 +39.60 +0.9536 +In addition, in order to demonstrate the effectiveness of +our method, we further explore the lightweight architecture +configurations, e.g., using the more lightweight invBlocks with +Residual Block (RB) [54] and different decomposition levels. +Tab. III lists the detailed results, there exists a latent +tendency for the deterministic mapping-based methods that +the larger models bring better performance, e.g., MIRNet and + +8 +TABLE IV +PSNR| SSIM | PSNR-B VALUES COMPARISONS. THE BEST AND THE SECOND BEST RESULTS ARE BOLDFACED AND UNDERLINED. +Dataset +QF +SADCT [56] +LD [57] +PCA [58] +ARCNN [59] +TNRD [4] +DnCNN [2] +Classic5 +10 +28.88 +0.807 +28.16 +28.39 +0.780 +27.59 +29.32 +0.800 +29.08 +29.03 +0.793 +28.76 +29.28 +0.799 +29.04 +29.40 +0.803 +29.13 +20 +30.92 +0.866 +29.75 +30.30 +0.858 +29.37 +31.56 +0.858 +31.12 +31.15 +0.852 +30.59 +31.47 +0.858 +31.05 +31.63 +0.861 +31.19 +30 +32.14 +0.891 +30.83 +31.47 +0.833 +30.17 +32.86 +0.884 +32.31 +32.51 +0.881 +31.98 +32.78 +0.884 +32.24 +32.91 +0.886 +32.38 +LIVE1 +10 +28.65 +0.809 +28.01 +28.26 +0.805 +27.68 +29.01 +0.809 +28.83 +28.96 +0.808 +28.77 +29.14 +0.811 +28.88 +29.19 +0.812 +28.90 +20 +30.81 +0.878 +29.82 +30.19 +0.872 +29.64 +31.28 +0.875 +30.72 +31.29 +0.873 +30.79 +31.46 +0.877 +31.04 +31.59 +0.880 +31.07 +30 +32.08 +0.908 +30.92 +29.41 +0.896 +29.36 +32.62 +0.903 +32.18 +32.67 +0.904 +32.22 +32.84 +0.906 +32.28 +32.98 +0.909 +32.34 +BSD500 +10 +28.23 +0.778 +27.38 +28.03 +0.782 +27.29 +28.64 +0.779 +28.01 +28.56 +0.791 +27.87 +28.60 +0.793 +27.95 +28.84 +0.801 +28.44 +20 +30.09 +0.851 +28.61 +29.82 +0.851 +28.43 +30.73 +0.851 +29.42 +30.43 +0.859 +29.10 +30.51 +0.861 +29.34 +31.05 +0.874 +30.29 +30 +31.21 +0.884 +29.34 +30.87 +0.872 +29.15 +31.99 +0.884 +30.84 +31.52 +0.890 +29.92 +31.58 +0.890 +30.02 +32.36 +0.905 +31.43 +Twitter +27.61 +0.728 +27.53 +27.58 +0.727 +27.49 +27.71 +0.730 +27.66 +27.54 +0.730 +27.49 +27.60 +0.727 +27.52 +27.63 +0.729 +27.54 +WeChat +29.60 +0.780 +29.59 +29.48 +0.796 +29.47 +29.63 +0.799 +29.60 +29.30 +0.799 +29.29 +26.64 +0.791 +26.63 +29.57 +0.798 +29.57 +Dataset +QF +LIPIO [60] +M-Net [61] +DCSC [62] +RNAN [63] +Ours(L=2) +Ours(L=3) +Classic5 +10 +29.35 +0.802 +29.04 +29.69 +0.811 +29.31 +29.62 +0.827 +29.30 +29.87 +0.828 +29.42 +29.51 +0.816 +29.47 +29.53 +0.815 +29.50 +20 +31.58 +0.857 +31.12 +31.90 +0.866 +31.29 +31.81 +0.880 +31.34 +32.11 +0.869 +32.16 +31.57 +0.870 +31.52 +31.64 +0.871 +31.59 +30 +32.86 +0.884 +32.28 +32.97 +0.888 +32.49 +33.06 +0.903 +32.49 +33.38 +0.892 +32.35 +32.70 +0.894 +32.62 +32.81 +0.894 +32.72 +LIVE1 +10 +29.17 +0.812 +28.89 +29.45 +0.819 +29.04 +29.34 +0.832 +29.01 +29.63 +0.824 +29.13 +29.36 +0.824 +29.32 +29.37 +0.823 +29.33 +20 +31.52 +0.877 +31.07 +31.83 +0.885 +31.14 +31.70 +0.896 +31.18 +32.03 +0.888 +31.12 +31.64 +0.889 +31.57 +31.65 +0.889 +31.57 +30 +32.99 +0.907 +32.31 +33.07 +0.911 +32.47 +33.07 +0.922 +32.43 +33.45 +0.915 +32.22 +32.94 +0.915 +32.83 +32.95 +0.916 +32.83 +BSD500 +10 +28.81 +0.782 +28.39 +28.96 +0.804 +28.56 +28.95 +0.805 +28.55 +29.08 +0.805 +28.48 +28.97 +0.796 +28.92 +28.97 +0.795 +28.93 +20 +30.92 +0.855 +30.07 +31.05 +0.874 +30.36 +31.13 +0.876 +30.41 +31.25 +0.875 +30.27 +31.10 +0.868 +31.01 +31.11 +0.868 +31.02 +30 +32.31 +0.887 +31.27 +32.61 +0.907 +31.15 +32.42 +0.906 +31.52 +32.70 +0.907 +31.33 +32.34 +0.899 +32.22 +32.35 +0.899 +32.22 +Twitter +27.47 +0.733 +27.41 +27.98 +0.744 +27.87 +27.63 +0.731 +27.43 +27.43 +0.718 +27.42 +31.04 +0.794 +30.82 +31.09 +0.795 +30.90 +WeChat +28.90 +0.800 +28.90 +29.82 +0.807 +29.82 +29.58 +0.800 +29.58 +29.56 +0.800 +29.56 +32.32 +0.823 +32.07 +32.30 +0.823 +32.07 +MPRNet, but which also lead to expensive computational +costs. Moreover, MAXIM introduces the MLP-style Trans- +former achieve significant performance gains on the SIDD test +set, but it also brings great computations. Instead, bijective +mapping-based methods reverse this tendency, e.g., DANet +and InvDn. They achieve the better trade-off between the +performance and computational complexity. +Our method further balances the performance and com- +putational costs on the SIDD test set, where we only stack +two lightweight invertible modules (denote as “RB+L2”) and +achieves better results compared to general CNN-based meth- +ods and bijective mapping-based methods. Meanwhile, the +performance could be improved further by extending to 3- +level decomposition (denote as “RB+L3”). Compared with +DANet and InvDn, the PSNRs of our lightweight models +increase by 0.1 ∼ 0.2dB on the SIDD test set. Replacing +lightweight residual blocks with dense blocks (DB) in a single +InvBlock, our method (denote as “DB+L2” and “DB+L3”) +achieves consistent performance gains on two real denoising +benchmarks. In addition, our method is more flexible and +friendly to mobile applications, which achieves comparable +results with state-of-the-art methods, e.g., MIRNet, MPRNet, +and MAXIM, note that the performance of our approach +could be further improved with the more effective inverse +architecture, e.g., invertible attention networks [55], but it is +beyond the scope of this paper. +Visualized comparisons are shown in Fig. 6. It is clear +that other bijective mapping-based methods (e.g., DANet and +InvDn) could remove noise well but also bring over-smoothed +effects, e.g., blurred edges and local structures. Transformer- +based MAXIM could alleviate this problem by capturing +non-local high-frequency patterns with global self-attention. +Instead, our method only removes the noise appearing in the +high-frequency textures of degraded images, which doesn’t +depend on any nonlocal patterns or image priors, while with +finer details against others’. +D. JPEG Compression Artifact Removal +Beyond real image denoising, our method is further extended +to JPEG compression artifact reduction. Following the same +experimental setting as in [64], we first validate our approach +in synthetic datasets, where we use both the training and +testing sets from BSD500 [65] as training data. JPEG com- +pressed images were generated by the Matlab JPEG encoder. +To present the performance of our method on blind image +deblocking, the JPEG quality factors (QF) are randomly gen- +erated in [1, 40]. Note that we only train one single model (i.e. +blind image compressed artifact removal) to handle all the +JPEG compression factors, e.g., 10, 20 and 30. The whole +training process was conducted on the Y channel image of +YCrCb space. +In contrast to synthetic JPEG deblocking, real image com- +pressed artifact reduction generally contains two implicit tasks, +i.e., up-scaling and artifact reduction, where the original +images are usually compressed and rescaled for transmission +and storage. Two real datasets, which are collected from +popular social medias but with different compression rates, +are used to verify our method’s effectiveness, i.e., Twitter [59] +and WeChat [64]. Twitter contains 114 training images and +extra 10 images for validation. Each high-resolution image +(3264 × 2448) results in a compressed and rescaled version +with a fixed resolution of 600 × 450. For WeChat, which only +provides 300 testing images, each image (3000 × 4000 pixel) +has a corresponded compression version with 600 × 800. Our +approach is only trained on Twitter training set, and tested in +validation set and WeChat. +1) Comparisons on synthetic datasets: Tab. IV lists the +detailed results on the three widely used synthetic datasets, i.e., +5 images in Classic5 [66], 29 images in LIVE1 [67] and 100 + +9 +Example in “BSD500” +(PSNR-B/SSIM) +JPEG +(29.43/0.8595) +DnCNN [2] +(33.67/0.9223) +LD [57] +(31.34/0.8963) +LIPIO [60] +(32.80/0.9139) +PCA [58] +(32.90/0.9173) +DCSC [62] +(33.55/0.9314) +ARCNN [59] +(33.12/0.9148) +Ours(L=2) +(34.12/0.9327) +Fig. 7. Visualized comparisons on synthetic datasets with JPEG QF=10. Red rectangle denotes the zoomed ROI area. +Example in “Twitter” +Compressed +(25.57/0.7280) +DnCNN [2] +(26.35/0.7534) +PCA [58] +(25.60/0.7319) +LIPIO [60] +(25.71/0.7337) +LD [57] +(25.59/0.7322) +Ours(L=2) +(31.27/0.8375) +ARCNN [59] +(26.24/0.7488) +Reference +(PSNR-B/SSIM) +Fig. 8. Visualized comparisons on the real-world use case. Red rectangle denotes the zoomed ROI area. +images in the validation set of BSD500, where we select some +representative traditional methods [56]–[58] and deep learning +based methods [2], [4], [59], [61]–[63] for comparison. In +addition, we use the PSNR, SSIM and the PSNR-B [68] for +quantitative evaluations, where PSNR-B is more sensitive to +blocking artifacts than the PSNR. +As shown in Tab. IV, although the proposed method doesn’t +achieve the best PSNR and SSIM metrics, it has the best +PSNR-B values against other methods, which implies that the +proposed method is more effective on recovering the local +high-frequencies of the compressed image than other methods. +Further, we observe a latent tendency that our method achieves +approximate consistent results in terms of the PSNR and +PSNR-B metrics. In contrast, other methods all exhibit obvious +degradation for the single PSNR-B metric. +Visualized results are demonstrated in Fig. 7, the obvious +blocking effects couldn’t be reduced well in local high- +frequency areas with general CNN-based methods. Instead, our +approach can recover consistent textures and smoother edges +while reducing blocking artifacts significantly. +2) Comparisons on real cases: To avoid out-of-memory +caused by excessive image resolution during inference, we +first crop the image with a sliding window, which results in +local image blocks without overlap, and then perform artifact +reduction operation and measurement calculation. +Quantitative results are demonstrated in Tab. IV. Traditional +and CNN-based methods all appear heavy performance degra- +dation due to complex noise distribution and high-frequency +information loss. Instead, our approach still achieves obvious +performance gains on Twitter and WeChat, i.e. the PSNR +and PSNR-B increase by 2 ∼ 3 dB on average, which +implies that our method is more effective to process real + +10 +LDCT +(34.39/0.9318) +CTformer [69] +(37.80/0.9758) +RedCNN [70] +(38.84/0.9813) +Eformer [71] +(39.21/0.9799) +WGAN-VGG [72] +(35.98/0.9598) +Ours(L=2) +(40.22/0.9832) +InvDn [9] +(38.20/0.9791) +NDCT +(PSNR/SSIM) +Fig. 9. Visualized comparisons with state-of-the-art methods. The display window is [160, 240]HU. Red rectangles denote ROI areas, zoomed in for better +visualization. +TABLE V +QUANTITATIVE RESULTS OF ALL COMPETING METHODS ON MAYO. +Methods +#Param (M) +MACs (G) +PSNR +SSIM +LDCT +– +– +38.13 +0.961 +BM3D [24] +– +– +42.10 +0.983 +RedCNN [70] +1.85 +462.48 +43.36 +0.989 +WGAN-VGG [72] +34.07 +14.76 +40.08 +0.979 +InvDn [9] +2.07 +161.6 +42.44 +0.987 +Eformer [71] +1.11 +67.45 +43.12 +0.988 +CTformer [69] +1.45 +219.42 +42.76 +0.987 +Ours(L=1) +1.44 +189.34 +43.44 +0.989 +Ours(L=2) +2.10 +243.98 +43.57 +0.990 +compressed artifact removal. Furthermore, visualized results +are illuminated in Fig. 8, our method presents significant +advantages in removing real artifacts while preserving fine +high-frequency details. +E. Low-Dose CT Image Restoration +We further validate our method on the medical low-dose +CT image restoration. Mayo1, a real clinical dataset [73] +authorized by Mayo Clinic for the 2016 NIH-AAPM-Mayo +Clinic Low Dose CT Grand Challenge, is used to evaluate low- +dose CT (LDCT) reconstruction algorithms, i.e. recovering +norm-dose CT (NDCT) image from low-dose measurements. +It contains 5,936 slices with 512×512 sizes from 10 different +subjects, each LDCT slice is simulated by inserting real noise +into the NDCT to reach a noise level that corresponded to 25% +1http://www.aapm.org/GrandChallenge/LowDoseCT/ +of the full dose. In addition, due to extra metallic applicators +implanted in part patients, the heavy streak artifacts are +introduced into image domain, which leads to more complex +noise distribution. We shuffle the dataset and randomly select +4,000 slices as the training set, the rest are used as validation +and testing to evaluate the performance of our approach. +Representative methods, including BM3D, WGAN-VGG [72] +and RedCNN [70], are used for comparison. Further, to eval- +uate the performance of bijective mapping-based method, we +select the InvDn [9] for comparison, where we obey the best +hyperparameters setting and retrain it with 600k iterations. In +addition, we also select recent Transformer-based methods for +comprehensive comparisons, e.g., Eformer [71] and CTformer +[9]. +Quantitative results are illuminated in Tab. V. Compared +to the general CNN-based and Transformer-based methods, +our method presents powerful denoising ability with a single +decomposing module only, and significant gains are achieved +with a 2-level decomposition, i.e. L = 2. Our approach +surpasses RedCNN by an average margin of ∼ 0.2dB in PSNR +while with few computational costs. InvDn doesn’t present +obvious advantages with the bijective characteristic, where +complex high-frequency distribution in CT image domain is +hard to approximate with case-agnostic latent variable. +Visualized results are demonstrated in Fig. 9. A repre- +sentative LDCT slice with heavy streak artifacts is selected +for comparison. All the methods could remove noise better. +However, RedCNN could reduce the noise well, but the streak +artifacts are still preserved. WGAN-VGG generates visually + +11 +pleasing results with adversarial training, but it introduces +extra noise and artifacts into the results. InvDn oversmoothies +the local structures, meanwhile, the artifacts are also retained. +Eformer and CTformer could restore global structures better, +but noise and artifacts aren’t removed successfully. Instead, +our method has presented significant advantages in removing +noise and artifacts while recovering fine structure details. +V. 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Wang, “Low-dose ct image denoising using a generative +adversarial network with wasserstein distance and perceptual loss,” IEEE +Transactions on Medical Imaging, vol. 37, pp. 1348–1357, 2018. +[73] C. H. McCollough, A. Bartley, R. E. Carter, B. Chen, T. A. Drees, +P. Edwards, D. R. Holmes, A. E. Huang, F. Khan, S. Leng, K. McMillan, +G. Michalak, K. M. Nunez, L. Yu, and J. G. Fletcher, “Low-dose ct for +the detection and classification of metastatic liver lesions: Results of +the 2016 low dose ct grand challenge,” Medical Physics, vol. 44, p. +e339–e352, 2017. + diff --git a/etFQT4oBgHgl3EQfkjYE/content/tmp_files/load_file.txt b/etFQT4oBgHgl3EQfkjYE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5844a291f8421c19b9b6842424241e4c9a7e949f --- /dev/null +++ b/etFQT4oBgHgl3EQfkjYE/content/tmp_files/load_file.txt @@ -0,0 +1,1652 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf,len=1651 +page_content='1 Hierarchical Disentangled Representation for Invertible Image Denoising and Beyond Wenchao Du, Hu Chen, Yi Zhang Senior Member, IEEE and Hongyu Yang Abstract—Image denoising is a typical ill-posed problem due to complex degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Leading methods based on normalizing flows have tried to solve this problem with an invertible transfor- mation instead of a deterministic mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' However, the implicit bijective mapping is not explored well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Inspired by a latent observation that noise tends to appear in the high-frequency part of the image, we propose a fully invertible denoising method that injects the idea of disentangled learning into a general invertible neural network to split noise from the high- frequency part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' More specifically, we decompose the noisy image into clean low-frequency and hybrid high-frequency parts with an invertible transformation and then disentangle case-specific noise and high-frequency components in the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' In this way, denoising is made tractable by inversely merging noiseless low and high-frequency parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Furthermore, we construct a flexible hierarchical disentangling framework, which aims to decompose most of the low-frequency image information while disentangling noise from the high-frequency part in a coarse- to-fine manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Extensive experiments on real image denoising, JPEG compressed artifact removal, and medical low-dose CT image restoration have demonstrated that the proposed method achieves competing performance on both quantitative metrics and visual quality, with significantly less computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Index Terms—Invertible Image denoising, bijective transfor- mation, disentangling learning, hierarchical representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' INTRODUCTION I MAGE denoising, aiming to recover clean observation from its noisy measurement, is a classic inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Based on the Bayesian perspective, most traditional approaches view it as a general maximum a posteriori (MAP) optimization problem, with assumptions regarding both the image priors and noise (usually Gaussian), which deviate from real cases and lead to critical limitations in practical scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Deep learning- based methods [1]–[4] have achieved superior denoising per- formance in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Most of them view it as a nonlinear mapping between noisy and clean image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' However, real image noise is generally accumulated from multiple degrading sources, which results in non-injective mapping due to various noise types and levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Bridging a bijective transformation between the noisy and clean image pairs to solve such an ambiguous inverse problem in low-level vision has been explored in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Previous methods [5], [6] have employed convolutional neural networks (CNNs) to model wavelet transforms to solve image restora- tions, in which wavelet decomposition and reconstruction W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Du, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Chen, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Yang are with the College of Computer Sci- ence, Sichuan University, Chengdu 610065, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Zhang is with the college of Cyber Science and Engineering, Sichuan University, Chengdu, 610065, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Email: wenchaodu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='scu@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' huchen@scu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' yzhang@scu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' yanghongyu@scu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='55/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='09 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='30/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='83 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='56/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='84 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='92/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='85 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='47/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='14 Noisy 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='62/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='71 InvDn 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='91/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='72 DANet 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='25/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='73 Ours Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Real image noise removal results on SIDD validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' The two representative methods are selected for comparisons, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' flow-based InvDN [9] and GAN-based DANet [7], PSNR and SSIM values are also computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Zoomed in for better visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' processes are modeled separately, leading to an injective map- ping procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Generative adversarial network-based methods [7], [8] have also modeled the bijective transformation for noisy image generation and restoration in supervised and unsupervised manners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' However, these methods need different models to simulate these two processes separately, which leads to complex training procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Normalizing flow that allows for efficient and exact like- lihood calculation and sampling by invertible transformation, has been applied to solve ill-posed inverse problems in low- level vision [9]–[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' NoiseFlow [10] is a seminal flow-based model for camera noise generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Unlike general CNN- based methods [13] that model camera imaging pipelines in forward and reverse directions to simulate real noise gen- eration, NoiseFlow learns the distribution of real noise and generates diverse noisy images by latent variable sampling to augment training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' However, it treats the clean image as conditional prior and thus is not fully invertible between clean and noisy image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' However, it treats the clean image as conditional prior and thus is not fully invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Considering that noise tends to appear in the high-frequency part of the image, InvDn [9] discards the high-frequency content, and then utilizes the invertible neural network (INN) [14] to capture the distribution of the high-frequency part of the clean image, image denoising is achieved by sampling a latent variable from predefined distribution to approximate the lost high-frequency information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Nevertheless, since the model is bijective, the ill-posedness of the task is partly alleviated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' However, InvDn assumes that the low and high-frequency contents of the image arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='13358v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='CV] 31 Jan 2023 2 are independent of each other and thus lacks the ability to exploit their dependency for image denoising, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 1, which leads to the recovered image losing part of high- frequency details, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=', over-smoothed image edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Further- more, the implicit bijective mapping is not fully guaranteed due to extra latent variable sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Instead, in this paper, we propose a fully invertible model for image denoising that injects the idea of disentangled learn- ing into a general invertible architecture to explore feature- level noise-high frequency signals splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Specifically, we aim to remove noise from the hybrid high-frequency part of the noisy image while reserving case-specific high-frequency information as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' To do so, we first decompose the image into low and high-frequency representations by an invertible transform during the forward propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Assum- ing the decomposed low-frequency information is noiseless, according to the information lossless characteristics of the normalizing flow, the noise only appears in the high-frequency part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Therefore, our goal is to split the noise component from it, so that the noise-free high-frequency signal is preserved well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' To this end, we introduce the concept of disentangled learn- ing into the general invertible architecture, which transforms the hybrid high-frequencies into the compact and independent representations with internal characteristics, where we model the high-frequency representation learning in the form of distribution, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=', the marginal distribution of the hybrid high- frequency representation obeys a pre-specified distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' isotropic Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' In this way, we directly disentangle noise and clean high-frequency representations along specific dimensions without extra latent variable sampling, which leads to a more accurate and stable bijective transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Image denoising is achieved by inversely merging noise-free low and high-frequency representations only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Furthermore, we hierarchically decompose the high-frequency representation into low and high-frequency parts and disentangle noise from fine-grained high-frequency parts, which results in a flexible and efficient framework for generalized image denoising tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' In short, our contributions are summarized as follows: 1) We propose a fully invertible model for image denoising, which introduces the idea of disentangled learning into a general invertible architecture and achieves a more accu- rate bijective mapping without latent variable sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 2) We construct a hierarchical high-frequency decompo- sition framework, which could process diverse noise with varying complexity, and achieve a better trade-off between performance and computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 3) Extensive experiments on natural image denoising, JPEG compressed artifact removal and medical low- dose CT image restoration demonstrate the proposed method achieves competing performance in terms of quantitative and qualitative evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' To the best of our knowledge, this is the first full invertible method capable of solving multiple real image denosing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' The remainder of the paper is organized as follows: Section II provides a brief review of related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Section III presents our approach in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' In Section IV, extensive experiments are conducted to evaluate the proposed method, and the conclusion is presented in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Traditional Methods Most traditional image denoising models usually construct a MAP optimization problem with a data filed and an extra regularization term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Along this direction, making assumptions regarding the noise distribution is necessary for most methods [15]–[17] to build the model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=', Mixture of Gaussian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' The regularization term is generally based on the natural image prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Classic total variation [18] uses the statistical characteristics of images to remove noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Sparse dictionary learning and Field-of-Experts (FoE) also employ certain pri- ors existing in image patches [19]–[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Non-local similarity method [22] employ non-local similar patterns of image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' In addition, transform techniques are also explored, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=', wavelet domain methods [23] and block-matching and 3D filtering (BM3D) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Deep Learning based Methods Instead of preset image and noise distribution priors, DNNs based methods directly learn a denoiser in a data-driven manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Previous method [25] first explored the multi-layer perception (MLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' [4] further proposed a feed- forward deep network called the trainable non-linear reaction diffusion (TNRD) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Furthermore, Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' [3] utilized a fully convolution encoder-decoder network with symmetric skip connection to solve image restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' [2] proposed the Gaussian denoising convolution network (DnCNN) and achieved superior performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Considering non-local self-similarity characteristic of images, non-local attention networks [26], [27] are also explored in image restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' However, most of these works focus on the synthetic noisy images with specific noise levels, spatially variant noise limit the capacity of such models on real cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' To this end, some methods employed self-adaptive noise level estimated from inputs to serve as extra priors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=', FFDNet [28] and CBDNet [29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' VDN [30] further integrated variational inference into the noise estimation and image denoising with a unique Bayesian framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' AINDNet [31] introduced the transfer learning to mitigate the domain gap between the real and syn- thetic noise distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' In addition, simulating real noise with a generative model was also explored in recent works [7], [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Considering that DNNs-based image denoising is non-injective in nature, a complex network architecture and powerful repre- sentation ability is always required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Thus, Transformer-based methods [33]–[35] have drawn more attention due to powerful representation learning on global image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Normalizing Flow based Methods Invertible neural networks (INNs) have drawn more attention to solving ambiguous inverse problems [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Unlike DNNs, INNs focus on learning the forward process and using ad- ditional latent output variables to capture the information 3 that would otherwise be lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Due to invertibility, a model of the corresponding inverse process is learned implicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' NoiseFlow [10] modeled the distribution of real noise in ISP imaging process to augment the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' InvDn [9] extended the idea of IRN [12] to real noise removal, which discards hybrid high-frequency information containing noise of the image and exploited extra latent variable sampling to approximate the lost high-frequency information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' However, a key factor is ignored that noise is case-specific, which implies the corresponded high-frequency information is also case- specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' More recently, FDN [37] directly disentangled the noise in the latent space with normalizing flows, which leads to a more complex network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' FINO [38] introduced the dual flow models to bridge bijective mapping between the noisy and clean image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Rather, we aim to solve image denoising by modeling a reliable bijective mapping with single model only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' We empirically show that integrating the idea of disentangled learning into the flow-based framework can result in fast and stable training as well as good performance on generalized image denoising tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' METHODOLOGY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Preliminaries Typical INNs models a generative process with a known distribution through a sequence of differentiable, invertible mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Formally, let x0 ∈ RD be a random variable with a known and tractable probability density function pX0 : RD → R and let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' , xN be a sequence of random variables such that xi = fi(xi−1) where fi : RD → RD is a differentiable, bijective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Then, if y = f(x0) = fn ◦ fN−1 ◦ · · · ◦ f1(x0), the change of variables formula says that the probability density function for y is p(y) = pX0(g(y)) N � j=1 | det Jj(g(y))|−1 (1) where g = g1 ◦ · · · ◦ gN−1 ◦ gN is the inverse of f, and Jj = ∂fj/∂xj−1 is the Jacobian of the jth transformation fj with respect to its input xj−1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=', the output of fj−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Due to such flexibility on accessing to the inverse mapping, INN architecture could be used for variational inference [39], [40], and representation learning without any information loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' INN is composed of basic invertible blocks [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' For the l-th block, the input u is split into u1 and u2 along the channel axis, and the typical additive affine transformation and corresponded inverse transformation are formulated as: � v1 = u1 + φ(u2) v2 = u2 + η(v1) ⇔ � u2 = v2 − η(v1) u1 = v1 − φ(u2) (2) where φ and η are arbitrary neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' The output of a single block is [v1, v2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' To enhance the transformation ability of the identity branch, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' (2) is always augmented as: � � � � � � � � � v1 = u1 ⊙ exp(ψ(u2)) + φ(u2) v2 = u2 ⊙ exp(ρ(u1)) + η(u1) u2 = (v2 − η(v1)) ⊙ exp(−ρ(v1)) u1 = (v1 − φ(u2)) ⊙ exp(−ψ(u2)) (3) where ψ(·), φ(·) and η(·) denote the transformation functions, which are arbitrary [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Function ρ(·) is further followed by a centered sigmoid function and a scale term to prevent numerical explosion due to the exp(·) function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Problem Specification We rethink the image denoising task from the perspective of invertible transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Suppose the original noisy observa- tion is y, its clean image is x and the noise is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' We have p(y) = p(x, n) = p(x)p(n|x) (4) where the distribution of noisy image p(y) is a joint distri- bution corresponded to x and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Directly splitting the noise component from y is difficult due to unknown p(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' In addition, p(n|x) implies the noise is case-specific to image content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' A general observation that noise tends to appear in the high-frequency component of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Therefore, we employ a wavelet transformation to decompose the noisy image y into low and high-frequency components, denoted as yLF and yHF respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Then, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' (4) is reformulated by p(y) = p(yLF, yHF) = p(yLF)p(yHF|yLF) (5) Ideally, wavelet decomposition is orthogonal, which could yield dense representation without redundancies, so Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' (5) is rewritten as p(y) = p(yLF, yHF) = p(yLF)p(yHF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Our goal is to disentangle noise representation from hy- brid high-frequency component, which means that the low- frequency part yLF is approximately noiseless, and noise is only contained in p(yHF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Considering the unique characteris- tics of information lossless in INNs, we could implement it easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Thus, we focus on how to split noise from p(yHF) such that p(yHF) = p(yhF, yn) = p(yhF)p(yn|yhF) (6) where yhF denotes noise-free high-frequency part, and yn is case-specific noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Different from general flow based models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=', IRN [12] and InvDn [9], a latent variable z is introduced to approximate the case-specific yhF, and transformed z to be case-agnostic with a specified distribution, where z is used to model the high-frequency information lost in degrading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' For real image denoising, yn and yhF are always case-specific and tangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Therefore, it is difficult to model lost yhF accurately with a case-agnostic variable without any conditional priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Thus, we attempt to solve this problem with a simple manner, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' disentangling noise part from yHF in the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' In this way, case-specific yhF and yn can be split well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Directly disentangling hybrid yHF is difficult due to un- known p(yn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' To this end, we assume that yhF and yn are independent, so that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' (6) is reformulated as p(yHF) = p(yhF, yn) = p(yhF)p(yn) (7) 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Illustration of invertible image denoising by disentangling representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' In the forward procedure, noisy image x is first decomposed into hybrid high frequencies yHF and noisy low frequencies yLF with Discrete Wavelet Transform (DWT), which are feed into a parameterized invertible neural network fθ(·) to transform into a clean low-frequency ˆyLF and a high-frequency yHF obeying specific distribution in the latent space, where case-specific high-frequency yhF and noise yn would be disentangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Image denoising is achieved through the inverse function f−1 θ and DWT operation with disentangled yhF and clean ˆyLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' To satisfy this assumption, we directly enforce p(yHF) to obey pre-specific distribution, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=', isotropic Gaussian, which means p(yHF) is compact and explainable, a key characteristic is that the feature vectors in yHF are independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' We disentangle the high-frequency representation yHF along the specific dimension to split yhF and yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' To reconstruct clean observation, we explicitly remove yn and only preserve specific p(yhF), so that p(yHF) = p(yhF)p(yn) = p(yhF) · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Benefited from the invertible characteristic of INNs, image denoising could be achieved by inverse pass using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Model Architecture The sketch of disentangling framework is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 2, which contains two processes: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' forward decomposition and inverse reconstruction, and is easily injected into the general invertible framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 1) Forward Decomposition: The key to our approach is to decompose the approximately noiseless low-frequency and case-specific high-frequency representations during the for- ward pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' To achieve this, we first exploit the Discrete Wavelet Transform (DWT) to decompose the image into low and high-frequency parts, where Haar Transformation [42] is used to approximate it for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Haar Transform explicitly decomposes the inputs, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' images or a group of feature maps) into an approximate low-pass representation and high-frequency coefficients with three directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' More concretely, it transforms the input with shape (H × W × C) into a tensor of shape ( 1 2H × 1 2W ×4C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' The first C slices of the output tensor are effectively produced by average pooling, which is approximately a low-pass representation equivalent to the bilinear interpolation down-sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' The remaining three groups of C slices contain residual components in the vertical, horizontal and diagonal directions, which are the high-frequency information of the original input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' By such a transformation, the low and high-frequency information are explicitly separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Then, an invertible network module is used to further abstract the yLF and yHF, where we leverage the coupling layer architecture in [41], [43], presented in i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Our goal is to polish the low and high-frequency inputs to obtain a suitable low-frequency representation and an independent properly distributed high-frequency representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Therefore, we match yLF and yHF respectively to the split of u1, u2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Furthermore, to increase the model capacity, we employ the additive transformation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' (2)) for the low-frequency part u1, and the enhanced affine transformation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' (3)) for the high-frequency part u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' After transformation, a noise-free low-frequency representa- tion ˆyLF is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' However, noise also appears in ˆyLF actu- ally due to non-ideal decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Inspired by the Nyquist- Shannon sampling theorem that the lost information during down-sampling a clean image amounts to high-frequency contents, we utilize the bicubic method [44] to guide ˆyLF decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Let xguide be the down-sampled clean image x corresponding to ˆyLF that is produced by the bicubic method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' To generate the clean low-frequency representation, we drive the ˆyLF to resemble xguide: Lguide := ℓX (xguide, ˆyLF) (8) where ℓX is a difference metric on X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' the L2 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Benefited from the characteristic of information lossless of invertible architecture, our low-frequency guidance loss drives noise appearing in yHF only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Another goal during the forward pass is to disentangle noise from high-frequency part yHF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' To this end, we enforce the yHF to obey specific distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' pyHF ∼ N(0, IK) so that it could be disentangled along the specific dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' A KL divergence loss is used as the distribution metric, where DKL = − � p(z) log(p(z) q(z))dz (9) which is referred to as our distribution guidance loss Ldist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 2) Inverse Reconstruction: Considering feature vectors in transformed yHF is independent, we directly split it along the channel axis into a specific high-frequency part yhF and noise yn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' yHF = [yhF, yn], and yhF and yn are case-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' To reconstruct the clean image x in the flow-based architecture, we need to construct dimension-consistent noise-free high- frequency representation ˆyHF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' To do so, we simply replace yn with 0 in reconstructed high-frequency representation ˆyHF, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' ˆyHF = [yhF, 0], which lies in a subspace of yHF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Further, to DWT InvBlock InvBlock m Invertible Block5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Illustration of our hierarchical disentangling framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Our approach exploits the three invertible modules to transform the noisy image y into clean low frequencies and specific high frequencies with different scales, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' L = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Each invertible module is composed of a DWT layer and stacked InvBlocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' guide noise component disentanglement, we minimize recon- struction loss Lrecon with corresponded clean image x: Lrecon := ℓX (x, iDWT(f −1 θ (ˆyLF, ˆyHF))) (10) where ℓX measures the difference between the clean image and the reconstructed one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' the L1 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' iDWT denotes inverse DWT, and f −1 θ denotes the inverse pass of parameterized invertible neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' In addition, implicit noisy image self-reconstruction could also be achieved by injecting case-specific yn into yHF, it doesn’t rely on any extra self-supervised constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' This implies there exists an implicit bijective mapping between the noisy and clean image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Hierarchical Disentangled Representation Furthermore, inspired by the latent observation that most image information is located in the low-frequency part, and high and low-frequencies are not decomposed exactly due to non-ideal transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Thus, exploiting the sufficient low- frequency information while disentangling fine-grained high- frequency signals is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Based on this analysis, we construct a multi-level decomposing framework in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' During the forward propagating, we stack multiple invertible modules to decompose the high-frequency representation yHF into low and high-frequency parts, this leads to multiscale low-frequency outputs {ˆyl LF}L l=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Therefore, the low-frequency guidance loss Lguide is rewritten as Lguide := L � l=1 ℓX (xl guide, ˆyl LF) (11) where L is the number of levels decomposed, ˆyl LF represents the decomposed low-frequency output in lth level, and corre- sponded multiscale guided image xl guide is generated by down- sampling x to 1/2l scale with the bicubic method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' For the high-frequency part, we only minimize the KL divergence loss for the last output yL HF, and disentangle noise component from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' In our experiments, we set L = 3 at most for generalized image denoising tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' In inverse propagation, we reconstruct clean high frequen- cies ˆyl HF level by level, from level L to level 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' For lth level, it leads to an implicit conditional flow model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' p(ˆyl−1 HF |yl HF, ˆyl LF), which could be implemented by ˆyl−1 HF = (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Online PSNR curves on SIDD validation set with 300k iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' TABLE I THE ARCHITECTURE CONFIGURATIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Metric Single-level Multi-level (Ours) L=2 L=3 L=2 L=3 PSNR 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='358 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='499 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='364 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='468 SSIM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='908 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='910 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='909 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='910 #Param(M) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='419 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='554 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='021 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='092 f −1 θ (yl HF, ˆyl LF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Therefore, the case-specific high-frequency signal is generated in a coarse-to-fine manner, it is efficient and stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' We optimize the whole architecture by minimizing the com- pact loss Ltotal with the combination of reconstruction loss Lrecon, low-frequency guidance loss Lguide and distribution loss Ldist: Ltotal := λ1Lrecon + λ2Lguide + λ3Ldist (12) where λ1, λ2 and λ3 are coefficients for balancing different loss terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' EXPERIMENT A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Experimental Setting 1) Datasets: To validate the effectiveness of our method, two representative real image noise datasets, the Smartphone Image Denoising Dataset (SIDD) [45] and Darmstadt Noise Dataset (DND) [46], are utilized to verify our method’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' The SIDD is taken by five smartphone cameras with small apertures and sensor sizes from 10 scenes under varied lighting conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Ground truth images are generated through a systematic procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' We use the medium version of SIDD as the training set, which contains 320 clean-noisy 三二39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='50 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='25 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='00 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='75 PSNR 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='50 L=1 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='25 L=2 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='00 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='75 L=3 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='50 0 10 20 30 40 50 60 Iterations (×5k)39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='25 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='00 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='75 PSNR 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='50 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='25 B=4 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='00 B=8 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='75 B=12 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='50 0 10 20 30 40 50 60 Iterations (×5k)6 (a) (b) (c) (d) (e) (f) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Visualization analysis for invertible bijective mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' The top row denotes the results of invertible image denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' (a) is input noisy image, (b), (c), and (d) denote the decomposed low-frequency outputs from the level L = 1, 2, 3, separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' (e) is inverse denoised output, and (f) is the self-reconstructed noisy image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' The second row visualizes the high-frequency feature maps from L = 1 during forward propagating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' The bottom row visualizes the disentangled high-frequency feature maps from L = 1 during inverse propagating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' TABLE II THE DISENTANGLED DIMENSION CONFIGURATIONS FOR MODEL WITH B=8 AND L=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' DIM(yn) 4/5 3/5 2/5 1/5 PSNR 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='348 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='352 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='403 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='364 SSIM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9086 pairs for training and 1280 cropped patches from the other 40 pairs for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' The reported test results are obtained via an online submission system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' The DND is captured by four consumer-grade cameras of different sensor sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' It contains 50 pairs of real-world noisy and approximately noise- free images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' These images are cropped into 1000 patches of size 512 × 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Similarly, the performance is evaluated by submitting the results to the online system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Considering DND does not provide any training data, we employ a training strategy by combining the training set of SIDD and Renoir [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Results are submitted to the DND benchmark by utilizing the same model that provides the best validation performance on the SIDD benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 2) Implementations: The proposed method cascades three invertible modules at most, each module contains a Haar trans- form layer and 8 basic invertible blocks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' L = 3, B = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' For the single invBlock (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 2), we implement the non-linear functions φ, ρ, and η with the Densenet Block (DB) [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' All the models are trained with Adam as the optimizer, with momentum of β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9, β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' The batch size is set as 16 with 144×144 size, and the initial learning rate is fixed at 2 × 10−4, which decays by half for every 100k iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' The training is performed on a single Tesla P100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' We augment the training data with extra horizontal and vertical flipping, as well as random rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' For loss functions, we set λ1 = 1, λ2 = 4 and λ3 = 1 separately for different loss terms in all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' In addition, Peak-Signal-Noise-Ratio (PNSR) and Structural Similarity (SSIM) are used to evaluate the performance of methods in all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Ablation Study We mainly explore three major determinants of our model: a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Model capacity, which depends on the number of decomposed levels, and the number of invertible blocks in single invertible module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Effectiveness of architecture, including decom- posing types of image and disentangling ways of framework;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' and c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Disentangled representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' All the experiments are performed in SIDD validation set with 300k iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 1) Model capability: We first study two key factors affect- ing the model capability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' multi-level decomposition and the capacity of the single invertible module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 4- (a) and (b), we observe that increasing the decomposed levels leads to significant performance gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' In addition, fixing decomposition levels (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=', L = 2), stacking more invBlocks in the single invertible module also further enhances the ability of the model, but it also brings more parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Therefore, we set the B = 8 and L = 3 at most in our architecture configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 2) Architecture Designing: We also consider different ar- chitectures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' decompose low and high-frequency parts in the last level only instead of each level, which is similar to InvDn [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' The quantitative results are illuminated in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' I, our multilevel decomposition architecture with high- frequencies disentangling achieves a better trade-off in terms of the performance and complexity of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 3) Disentangled Representation: We further explore the effects of disentangled dimensions in yHF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' We split yn from yHF with different dimensions along the channel axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' All models are trained separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' II gives the detailed results, the 7 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='56/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='1110) (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='17/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='1205) Noisy (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='19/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='7965) (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='08/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='8442) DANet [7] (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='00/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='7857) (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='74/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='8365) InvDn [9] (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='80/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='8156) (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='68/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='8564) MAXIM [35] (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='47/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='8067) (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='14/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='8458) Ours(L=2) (PSNR/SSIM) (PSNR/SSIM) Reference Noisy (PSNR/SSIM) CBDNet [29] (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='40/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='8364) VDN [30] (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='08/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9166) DANet [7] (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='66/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9148) InvDn [9] (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='22/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9216) Ours(L=3) (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='28/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9224) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Visualized denoising samples from SIDD and DND datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' The first and second rows are from the SIDD, and bottom is from the DND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' best model is achieved by setting DIM(yn) = 2/5·DIM(yHF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' We use it in our final model configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 4) Analysis for Bijective Mapping: We further explore the bijective relationship in our architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 5, we first give the decomposed low-frequency outputs from multi-level decomposition during forward propagation, denoised output, and self-reconstructed noisy input by im- plicitly inverse reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Our method only splits case- specific noise from the hybrid high-frequency component in the latent space to achieve image denoising, where it bridges the bijective transformation between the noisy image genera- tion and restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Moreover, we visualize the decomposed hybrid high-frequency components and disentangled noise-free high-frequencies components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' It is obvious that our method could remove case-specific noise signals in latent space while retaining fine high-frequencies, which implies the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Real Image Noise Removal We perform comprehensive comparisons, including model pa- rameters, computational complexity (MACs), and quantitative metrics, on two real denoising benchmarks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' SIDD [45] and DND [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Note that “MACs” is counted on a single RGB image with 256 × 256 size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Moreover, we select two kinds of representative methods for comparisons, one is the deterministic mapping-based methods, including CNN-based and Transformer methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' The other is bijective mapping- based methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=', Flow-based and GAN-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' TABLE III COMPREHENSIVE COMPARISONS WITH OTHER COMPETING METHODS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Method #Param (M) MACs (G) SIDD [45] DND [46] PSNR SSIM PSNR SSIM DnCNN [2] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='67 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='02 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='941 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9430 TNRD [4] – – 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='643 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='8306 BM3D [24] – – 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='685 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='8507 CBDNet [29] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='34 – 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='868 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9421 RIDNet [49] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='50 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='26 – – 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='8306 GradNet [50] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='60 – 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='946 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9543 AINDNet [31] 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='76 – 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='953 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9561 VDN [30] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='82 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='5 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='909 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9518 MIRNet [51] 31.' metadata={'source': 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computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Moreover, MAXIM introduces the MLP-style Trans- former achieve significant performance gains on the SIDD test set, but it also brings great computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Instead, bijective mapping-based methods reverse this tendency, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=', DANet and InvDn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' They achieve the better trade-off between the performance and computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Our method further balances the performance and com- putational costs on the SIDD test set, where we only stack two lightweight invertible modules (denote as “RB+L2”) and achieves better results compared to general CNN-based meth- ods and bijective mapping-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Meanwhile, the performance could be improved further by extending to 3- level decomposition (denote as “RB+L3”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Compared with DANet and InvDn, the PSNRs of our lightweight models increase by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='1 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='2dB on the SIDD test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Replacing lightweight residual blocks with dense blocks (DB) in a single InvBlock, our method (denote as “DB+L2” and “DB+L3”) achieves consistent performance gains on two real denoising benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' In addition, our method is more flexible and friendly to mobile applications, which achieves comparable results with state-of-the-art methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=', MIRNet, MPRNet, and MAXIM, note that the performance of our approach could be further improved with the more effective inverse architecture, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=', invertible attention networks [55], but it is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Visualized comparisons are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' It is clear that other bijective mapping-based methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=', DANet and InvDn) could remove noise well but also bring over-smoothed effects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=', blurred edges and local structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Transformer- based MAXIM could alleviate this problem by capturing non-local high-frequency patterns with global self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Instead, our method only removes the noise appearing in the high-frequency textures of degraded images, which doesn’t depend on any nonlocal patterns or image priors, while with finer details against others’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' JPEG Compression Artifact Removal Beyond real image denoising, our method is further extended to JPEG compression artifact reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Following the same experimental setting as in [64], we first validate our approach in synthetic datasets, where we use both the training and testing sets from BSD500 [65] as training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' JPEG com- pressed images were generated by the Matlab JPEG encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' To present the performance of our method on blind image deblocking, the JPEG quality factors (QF) are randomly gen- erated in [1, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Note that we only train one single model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' blind image compressed artifact removal) to handle all the JPEG compression factors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=', 10, 20 and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' The whole training process was conducted on the Y channel image of YCrCb space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' In contrast to synthetic JPEG deblocking, real image com- pressed artifact reduction generally contains two implicit tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=', up-scaling and artifact reduction, where the original images are usually compressed and rescaled for transmission and storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Two real datasets, which are collected from popular social medias but with different compression rates, are used to verify our method’s effectiveness, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=', Twitter [59] and WeChat [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Twitter contains 114 training images and extra 10 images for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Each high-resolution image (3264 × 2448) results in a compressed and rescaled version with a fixed resolution of 600 × 450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' For WeChat, which only provides 300 testing images, each image (3000 × 4000 pixel) has a corresponded compression version with 600 × 800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Our approach is only trained on Twitter training set, and tested in validation set and WeChat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 1) Comparisons on synthetic datasets: Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' IV lists the detailed results on the three widely used synthetic datasets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=', 5 images in Classic5 [66], 29 images in LIVE1 [67] and 100 9 Example in “BSD500” (PSNR-B/SSIM) JPEG (29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='43/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='8595) DnCNN [2] (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='67/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9223) LD [57] (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='34/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='8963) LIPIO [60] (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='80/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9139) PCA [58] (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='90/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9173) DCSC [62] (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='55/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9314) ARCNN [59] (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='12/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9148) Ours(L=2) (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='12/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9327) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Visualized comparisons on synthetic datasets with JPEG QF=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Red rectangle denotes the zoomed ROI area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Example in “Twitter” Compressed (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='57/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='7280) DnCNN [2] (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='35/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='7534) PCA [58] (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='60/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='7319) LIPIO [60] (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='71/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='7337) LD [57] (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='59/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='7322) Ours(L=2) (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='27/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='8375) ARCNN [59] (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='24/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='7488) Reference (PSNR-B/SSIM) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Visualized comparisons on the real-world use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Red rectangle denotes the zoomed ROI area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' images in the validation set of BSD500, where we select some representative traditional methods [56]–[58] and deep learning based methods [2], [4], [59], [61]–[63] for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' In addition, we use the PSNR, SSIM and the PSNR-B [68] for quantitative evaluations, where PSNR-B is more sensitive to blocking artifacts than the PSNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' IV, although the proposed method doesn’t achieve the best PSNR and SSIM metrics, it has the best PSNR-B values against other methods, which implies that the proposed method is more effective on recovering the local high-frequencies of the compressed image than other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Further, we observe a latent tendency that our method achieves approximate consistent results in terms of the PSNR and PSNR-B metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' In contrast, other methods all exhibit obvious degradation for the single PSNR-B metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Visualized results are demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 7, the obvious blocking effects couldn’t be reduced well in local high- frequency areas with general CNN-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Instead, our approach can recover consistent textures and smoother edges while reducing blocking artifacts significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 2) Comparisons on real cases: To avoid out-of-memory caused by excessive image resolution during inference, we first crop the image with a sliding window, which results in local image blocks without overlap, and then perform artifact reduction operation and measurement calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Quantitative results are demonstrated in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Traditional and CNN-based methods all appear heavy performance degra- dation due to complex noise distribution and high-frequency information loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Instead, our approach still achieves obvious performance gains on Twitter and WeChat, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' the PSNR and PSNR-B increase by 2 ∼ 3 dB on average, which implies that our method is more effective to process real 10 LDCT (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='39/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9318) CTformer [69] (37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='80/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9758) RedCNN [70] (38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='84/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9813) Eformer [71] (39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='21/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9799) WGAN-VGG [72] (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='98/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9598) Ours(L=2) (40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='22/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9832) InvDn [9] (38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='20/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='9791) NDCT (PSNR/SSIM) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Visualized comparisons with state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' The display window is [160, 240]HU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Red rectangles denote ROI areas, zoomed in for better visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' TABLE V QUANTITATIVE RESULTS OF ALL COMPETING METHODS ON MAYO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Methods #Param (M) MACs (G) PSNR SSIM LDCT – – 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='961 BM3D [24] – – 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='983 RedCNN [70] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='85 462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='48 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='989 WGAN-VGG [72] 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='07 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='76 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='979 InvDn [9] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='07 161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='6 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='987 Eformer [71] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='11 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='45 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='988 CTformer [69] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='45 219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='42 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='987 Ours(L=1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='44 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='34 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='989 Ours(L=2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='10 243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='98 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='990 compressed artifact removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Furthermore, visualized results are illuminated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 8, our method presents significant advantages in removing real artifacts while preserving fine high-frequency details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Low-Dose CT Image Restoration We further validate our method on the medical low-dose CT image restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Mayo1, a real clinical dataset [73] authorized by Mayo Clinic for the 2016 NIH-AAPM-Mayo Clinic Low Dose CT Grand Challenge, is used to evaluate low- dose CT (LDCT) reconstruction algorithms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' recovering norm-dose CT (NDCT) image from low-dose measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' It contains 5,936 slices with 512×512 sizes from 10 different subjects, each LDCT slice is simulated by inserting real noise into the NDCT to reach a noise level that corresponded to 25% 1http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='aapm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='org/GrandChallenge/LowDoseCT/ of the full dose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' In addition, due to extra metallic applicators implanted in part patients, the heavy streak artifacts are introduced into image domain, which leads to more complex noise distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' We shuffle the dataset and randomly select 4,000 slices as the training set, the rest are used as validation and testing to evaluate the performance of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Representative methods, including BM3D, WGAN-VGG [72] and RedCNN [70], are used for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Further, to eval- uate the performance of bijective mapping-based method, we select the InvDn [9] for comparison, where we obey the best hyperparameters setting and retrain it with 600k iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' In addition, we also select recent Transformer-based methods for comprehensive comparisons, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=', Eformer [71] and CTformer [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Quantitative results are illuminated in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Compared to the general CNN-based and Transformer-based methods, our method presents powerful denoising ability with a single decomposing module only, and significant gains are achieved with a 2-level decomposition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' L = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Our approach surpasses RedCNN by an average margin of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content='2dB in PSNR while with few computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' InvDn doesn’t present obvious advantages with the bijective characteristic, where complex high-frequency distribution in CT image domain is hard to approximate with case-agnostic latent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Visualized results are demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' A repre- sentative LDCT slice with heavy streak artifacts is selected for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' All the methods could remove noise better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' However, RedCNN could reduce the noise well, but the streak artifacts are still preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' WGAN-VGG generates visually 11 pleasing results with adversarial training, but it introduces extra noise and artifacts into the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' InvDn oversmoothies the local structures, meanwhile, the artifacts are also retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Eformer and CTformer could restore global structures better, but noise and artifacts aren’t removed successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Instead, our method has presented significant advantages in removing noise and artifacts while recovering fine structure details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' CONCLUSION In this paper, we propose a flexible and efficient hierarchical disentangled representation architecture for invertible image denoising, which bridges a bijective transformation between noise image self-reconstruction and restoration, and largely mitigates the ill-posedness of the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Extensive experiments on real image denoising, JPEG compressed artifact removal, and medical low-dose CT image restoration have demonstrated that the proposed method achieves competing performance in terms of both quantitative and qualitative evaluations, while with varying complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Although many concrete implemen- tations of the advanced idea are possible, we show that a simple design already achieves excellent results on generalized image denoising tasks, which provides a new perspective for solving real image restoration tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' REFERENCES [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFQT4oBgHgl3EQfkjYE/content/2301.13358v1.pdf'} +page_content=' Jain and H.' metadata={'source': 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b/i9FAT4oBgHgl3EQf-B7W/content/tmp_files/2301.08761v1.pdf.txt @@ -0,0 +1,1277 @@ +Astronomy & Astrophysics manuscript no. main_AA +©ESO 2023 +January 24, 2023 +Letter to the Editor +Red Horizontal Branch stars: an asteroseismic perspective +Massimiliano Matteuzzi1, 2⋆, Josefina Montalbán1, 3, Andrea Miglio1, 2, 3, Mathieu Vrard4, Giada Casali1, 2, +Amalie Stokholm1, 5, Marco Tailo1, Warrick Ball3, Walter E. van Rossem3, 5, and Marica Valentini6 +1 Department of Physics & Astronomy "Augusto Righi", University of Bologna, via Gobetti 93/2, 40129 Bologna, Italy +2 INAF-Astrophysics and Space Science Observatory of Bologna, via Gobetti 93/3, 40129 Bologna, Italy +3 School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK +4 Department of Astronomy, The Ohio State University, Columbus, OH 43210, USA +5 Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, +Denmark +6 Leibniz-Institut für Astrophysik Potsdam, An der Sternwarte 16, Potsdam, 14482, Germany +January 24, 2023 +ABSTRACT +Robust age estimates of red giant stars are now possible thanks to the precise inference of their mass based on asteroseismic con- +straints. However, there are cases where such age estimates can be highly precise yet very inaccurate. An example is giants that have +undergone mass loss or mass transfer events that have significantly altered their mass. In this context, stars with “apparent” ages +significantly higher than the age of the Universe are candidates as stripped stars, or stars that have lost more mass than expected, most +likely via interaction with a companion star, or because of the poorly understood mass-loss mechanism along the red-giant branch. +In this work we identify examples of such objects among red giants observed by Kepler, both at low ([Fe/H] ≲ −0.5) and solar +metallicity. By modelling their structure and pulsation spectra, we find a consistent picture confirming that these are indeed low-mass +objects consisting of a He core of ≈ 0.5 M⊙ and an envelope of ≈ 0.1 − 0.2 M⊙. Moreover, we find that these stars are characterised +by a rather extreme coupling (q ≳ 0.4) between the pressure-mode and gravity-mode cavities, i.e. much higher than the typical value +for red clump stars, providing thus a direct seismic signature of their peculiar structure. +The complex pulsation spectra of these objects, if observed with sufficient frequency resolution, hold detailed information about the +structural properties of likely products of mass stripping, hence can potentially shed light on their formation mechanism. On the +other hand, our tests highlight the difficulties associated with measuring reliably the large frequency separation, especially in shorter +datasets, with impact on the reliability of the inferred masses and ages of low-mass Red Clump stars with e.g. K2 or TESS data. +Key words. asteroseismology – stars: evolution – stars: fundamental parameters – stars: horizontal branch – stars: interiors – stars: +mass-loss +1. Introduction +It is widely accepted that the large range in color shown by low +mass stars in the central He-burning phase, called Horizontal +Branch (HB), is mainly due to variations in the efficiency of the +H-burning shell, hence, to the mass of the H-envelope remaining +around a He-core of ≃ 0.5 M⊙ (e.g. Salaris & Cassisi 2006). In +a colour-magnitude diagram (CMD), low-mass core-He-burning +(CHeB) stars appear distributed in both bluer and redder colours +than the RR Lyrae instability strip (RRL−IS). Those located be- +tween the RR Lyrae and the red-clump (RC; e.g. Girardi 2016) +are called rHB (red Horizontal Branch) stars, and they have a +H-rich envelope of ≈ 0.1 − 0.2 M⊙ (e.g. Rood & Crocker 1989; +Valcarce & Catelan 2008; Girardi 2016; Tailo et al. 2020). This +HB component has been clearly observed in globular clusters of +different metallicity and age (e.g. Armandroff 1988; Stetson et al. +1989; Catelan 2009; Tailo et al. 2020), however, rHB objects also +exist in the field. While their identification is challenging, their +census has been considered extremely important for tracing old +stellar populations in the Milky Way (MW; e.g. Kaempf et al. +2005; Chen et al. 2010, 2011). Although mainly associated with +stars of low/intermediate metallicity, corresponding to the thick +disc and halo population, spectroscopic studies (e.g. Af¸sar et al. +⋆ E-mail: massimilia.matteuzz2@unibo.it +2012, 2018) have shown that rHB stars are also present in the +metal-rich component of the MW. This suggests that the pro- +genitors of these objects have followed a non-standard evolution +with significant mass loss, or envelope stripping due to binary +interactions. Signs of significant mass loss have been revealed +in red giants observed by the Kepler space telescope (Borucki +et al. 2010), in the field and in the open cluster NGC 6819 (e.g. +Handberg et al. 2017; Brogaard et al. 2021; Li et al. 2022). +Stellar evolution models predict different structures for rHB +and RC stars, with the latter having a similar He core, but a larger +H envelope. We thus expect their seismic properties to be dif- +ferent. The exquisite precision achieved after 4 years of Kepler +observations has revealed oscillation spectra of red giants with +an increasing level of complexity (see Chaplin & Miglio 2013, +for a review): frequency patterns in red giant branch (RGB) stars +similar to those found in main sequence stars (Universal pattern, +Mosser et al. 2011), spectra of RC stars with "forests" of dipole +modes around the nominal acoustic mode, but still with an evi- +dent regularity (i.e. Beck et al. 2011), and also "outlier" spectra +with a larger number of visible modes over the whole frequency +domain, which are hypothesised in this letter to belong to rHB +stars (see Figure 1). +In this work we identify a small sample of 11 rHB candidates +among the red giants in the Kepler field. Their global seismic +Article number, page 1 of 9 +arXiv:2301.08761v1 [astro-ph.SR] 20 Jan 2023 + +A&A proofs: manuscript no. main_AA +properties and atmospheric parameters suggest that they are low- +mass CHeB stars with low/intermediate and solar metallicity. +Combining numerical simulations of stellar structure and evo- +lution and stellar oscillations, we study the consistency between +the location of our rHB candidates in the Hertzsprung–Russell +diagram (HRD), their theoretically predicted internal structure +and their oscillation spectra. Our rHB sample is presented in +Section 2 and the theoretical models in Section 3. Section 4 dis- +cusses the properties of theoretical oscillation spectra of typical +rHB and RC stars, as well as the comparison with observations. +In Section 5 we summarise our findings. +2. Observational data +In addition to KIC4937011, a 0.71 M⊙ CHeB star belonging to +the open cluster NGC 6819 (see Handberg et al. 2017) which +has a turn-off mass of ∼ 1.6 M⊙, we found 11 red giants in the +Kepler database1 with peculiar power spectral density (PSD). +While their global seismic parameters (mean large frequency +separation ⟨∆ν⟩, frequency of maximum power νmax, and asymp- +totic period spacing of the dipole modes ∆Π1) are compatible +with low-mass CHeB stars, they have complex oscillation spec- +tra. They have, for instance, an unusually high number of observ- +able dipole g/p mixed modes without the amplitude modulation +around the p-like modes that is typically found in low-RGB and +RC stars. This fact suggests that all dipole modes have a signifi- +cant amplitude also in the outer region of the star, and hence, that +g and p resonant cavities in these objects are strongly coupled. +The ability to transfer the energy of the mode from one cav- +ity to the other, instead of remaining trapped mainly in one of +them, is quantified by the coupling factor q (e.g. Shibahashi +1979; Takata 2016). The analysis of Kepler light curves pro- +vides the seismic parameters mentioned above, as well as the +value of the coupling factor q (e.g. Vrard et al. 2016; Mosser +et al. 2017, 2018). Theoretically, the parameter q ranges from 0 +(uncoupled) to 1 (completely coupled). All the stars in our sam- +ple have q ≳ 0.4, while the median for RC stars is ≃ 0.25 − 0.3 +(Vrard et al. 2016; Mosser et al. 2017). +On the other hand, the values of radial mode-linewidths +(Γ0 > 0.2 µHz) are larger than the third quantile of the full sam- +ple of CHeB Kepler stars (median value Γ0 = 0.15 µHz, Vrard +et al. 2018). Both a high q and a high Γ0 contribute to increase +the complexity of the spectra. Moreover, given the dependence +of Γ0 on the effective temperature Teff (e.g. Chaplin et al. 2009), +the quadrupole modes are more difficult to detect in the hotter +metal-poor subsample than in the cooler metal-rich objects. +The seismic properties (νmax, ⟨∆ν⟩, ∆Π1 and q) for our +sample are reported in Table 1 and A.1, together with the at- +mospheric parameters (Teff and chemical composition) from +APOGEE-DR16/DR17 (Ahumada et al. 2020; Abdurro’uf et al. +2022). Around 25% (3 out of 12) of the sample are metal-rich +(0 ≤ [Fe/H] < 0.3) cool (4600 ≤ Teff/K ≤ 4800) stars, and the +rest are low/intermediate metallicity (−1.4 < [Fe/H] < −0.5) +stars with 5200 ≤ Teff/K ≤ 5600, that is, belonging to the "clas- +sical" rHB. +Tables 1 and A.1 contain also the stellar luminosity derived +using Gaia-DR3 astrometry data (see Appendix A for details), +and an estimate of their mass. The latter can be derived from +scaling relations involving atmospheric and global seismic pa- +1 https://archive.stsci.edu/missions-and-data/kepler +rameters (see e.g. Miglio et al. 2012). Here we use the one com- +bining L, Teff and νmax: +M +M⊙ += +�Teff,⊙ +Teff +�3.5 � νmax +νmax,⊙ +� � L +L⊙ +� +, +(2.1) +where the solar reference values are Teff,⊙ = 5777 K, νmax,⊙ = +3090 µHz (Huber et al. 2011). The mass uncertainties are calcu- +lated in quadrature by considering an uncertainty of at least 50 K +in Teff as estimated from an independent analysis of APOGEE +spectra (see Appendix A). In Appendix A.1 we also discuss the +stellar mass values from a model-based corrected scaling rela- +tion involving Teff, ⟨∆ν⟩ and νmax (Eq. A.1). +We notice that the mass of KIC 4937011 in Table A.1 is that +of Handberg et al. (2017), and its value is nevertheless compati- +ble with our results obtained with Eq. 2.1 or A.1. All the objects +in our sample are then very low-mass stars (M ≲ 0.8 M⊙) with a +high coupling2 between p-mode and g-mode cavities. +We select three stars (those in Table 1) as representative of +low-mass CHeB stars in different metallicity domains. Figure 2 +shows these stars in an HRD, together with the Kepler-APOGEE +red giant sample (Miglio et al. 2021, grey dots) and the red +edge of the RRL-IS (Marconi et al. 2015, dashed red line). The +two metal-poor stars (blue star symbols) are located between the +RRL-IS and the RC, as expected for rHB stars, while the metal- +rich CHeB star (orange star symbol) appears in the region of the +"ensemble" Kepler-RC. Its location is nevertheless redder than +the RC at solar-metallicity, and hence it is well a rHB metal-rich +star as suggested also by its mass (see also Handberg et al. 2017) +and oscillation spectra. As mentioned above, rHBs, especially +those metal-rich, must have followed non-standard evolution to +reach their current state within the age of the Universe. They are +probably the progeny of strongly interacting binary systems. It +has not been possible to confirm that hypothesis using the cur- +rently available Gaia-DR3 astrometry data (see Halbwachs et al. +2022, for the non-single star processing3), but we cannot exclude +that they were part of binary systems in the past. +3. Simulated data +The aim of this work is not to fit the available observational data, +but to analyse the relation between the structures of rHB stars, +according to stellar evolution theory, and their oscillation spec- +tra, and to compare the latter with those observed in our sample. +From a grid of models (see Appendix B) we selected two sets +of parameters that represent well the mass and chemical com- +position of the classical rHB (M = 0.65 M⊙, [α/Fe] = 0.2 and +[Fe/H] = −1.00) and metal-rich low-mass CHeB (M = 0.75 M⊙, +[α/Fe] = 0 and [Fe/H] = 0) stars. For comparison, we also con- +sider a typical RC star (M = 1.5 M⊙ with solar composition). +As it appears in Fig. 2, the parameters selected for our refer- +ence models provide indeed a good representation of the low- +intermediate metallicity and metal-rich rHBs in our sample. We +also note that without complementary information, such as that +provided by asteroseismology, a metal-rich rHB would be mis- +taken for a more massive star in RGB (see also Handberg et al. +2017). +2 We notice that stars in the CHeB stage could have multiple cavities +in the inner part due to semi-convection. This could lead to bias when +estimating q from the fit of observations with the asymptotic relation for +dipole modes (e.g. Pinçon & Takata 2022), thus it must be considered +in future. +3 We +also +checked +the +non-single +star +hypothesis +using +the +fidelity_v2 table. +Article number, page 2 of 9 + +Matteuzzi et al.: rHBs as viewed by asteroseismology +15 +20 +25 +30 +35 +40 +Frequency [µHz] +0 +50000 +100000 +150000 +200000 +250000 +300000 +350000 +400000 +PSD [ppm2 +µHz ] +(a) +KIC5271626 +[Fe/H] = 0.01 +Teff = 4770 K +Original +Smooth +25 +30 +35 +40 +45 +Frequency [µHz] +0 +10000 +20000 +30000 +40000 +50000 +60000 +70000 +80000 +PSD [ppm2 +µHz ] +(b) +KIC6032981 +[Fe/H] = −1.01 +Teff = 5300 K +Original +Smooth +25 +30 +35 +40 +45 +Frequency [µHz] +0 +20000 +40000 +60000 +80000 +PSD [ppm2 +µHz ] +(c) +KIC8694070 +[Fe/H] = −1.44 +Teff = 5300 K +Original +Smooth +26 +28 +30 +32 +34 +36 +38 +40 +42 +Frequency [µHz] +0 +20000 +40000 +60000 +80000 +100000 +120000 +140000 +160000 +PSD [ppm2 +µHz ] +(d) +KIC1161618 +RC star +[Fe/H] = −0.02 +Teff = 4740 K +Original +Smooth +25.0 +27.5 +30.0 +32.5 +35.0 +37.5 +40.0 +42.5 +45.0 +Frequency [µHz] +0 +100000 +200000 +300000 +400000 +PSD [ppm2 +µHz ] +(e) +KIC2436824 +RGB star +[Fe/H] = 0.34 +Teff = 4340 K +Original +Smooth +Fig. 1: PSD for five low-mass red giants (grey lines in the five panels) observed by Kepler. Panels (a), (b), and (c) show the three +low-mass CHeB stars KIC5271626, KIC6032981, and KIC8694070 (first three rows in Table 1 and colored stars in Fig. 2). Panels +(d) and (e) show the RC star KIC1161618 and the RGB star KIC2436824, for comparison. All the five panels contain a smoothed +PSD (red lines) computed with a box kernel of width 0.5 µHz in panels (a), (b), (c), (d), and of width 0.1 µHz in panel (e). +Table 1: Summary of the seismic and atmospheric properties for three rHB candidates of our sample (Sect. 2). +KIC +L [L⊙] +Teff [K] +[Fe/H] +[α/Fe] +⟨∆ν⟩ [µHz] +νmax [µHz] +q +∆Π1 [s] +M [M⊙] +5271626∗ +42 ± 4 +4769 ± 9 +0.03 +0.01 +3.91 ± 0.05 +25.1 ± 0.5 +0.61 +291.4 ± 1.7 +0.66 ± 0.07 +6032981+ +44 ± 4 +5300 ± 110 +-1.01 +0.37 +5.188 ± 0.017 +35.4 ± 0.6 +1.15 +321 ± 3 +0.68 ± 0.08 +8694070 +53 ± 5 +5300 ± 30 +-1.44 +0.25 +5.135 ± 0.018 +34.6 ± 0.6 +0.7 +332 ± 4 +0.81 ± 0.09 +Mock rHB +44 +5663 +-1.00 +0.2 +6.41 +42.5 +0.65 +324 +0.65 +Mock RC +59 +4891 +0.00 +0.0 +4.79 +44.1 +0.25 +313 +1.50 +Notes. For each Kepler ID (KIC) we report the effective temperature Teff, [Fe/H], [α/Fe] from APOGEE-DR17, or APOGEE-DR16 +(one star tagged with a + in apex); mean large frequency separation ⟨∆ν⟩, and frequency of maximum power νmax calculated by us using +the code in Davies & Miglio (2016), or Yu et al. (2018) data (one star tagged with a * in apex). The coupling factor q and asymptotic +period spacing of the dipole modes ∆Π1 as calculated from the stretched-period method (see e.g. Vrard et al. 2016). The current stellar +mass M is computed from Eq. 2.1. The last two rows show the properties of a simulated rHB and RC star (Sect. 3). +It is generally accepted that, except for the age, the properties +of a low-mass star with a He core of ≃ 0.5 M⊙ and an H-rich en- +velope of ∼ 0.1 − 0.2 M⊙ are largely independent of whether the +star was born with a small mass or whether it originates from a +more massive star (M ≲ 1.8 M⊙) that underwent significant mass +loss. Therefore, it is justified to use structure models calculated +without mass loss such as those in our grid. +In the following we concentrate on a metal-poor model since, +as described in Sect. 2, we expect metal-poor rHBs to present +more marked differences with respect to the spectra of typical +RC stars. We select structure models with a central He mass +Article number, page 3 of 9 + +A&A proofs: manuscript no. main_AA +3500 +4000 +4500 +5000 +5500 +6000 +Teff [K] +101 +102 +103 +L [L⊙] +Obs: metal-poor +Obs: metal-rich +rHB +RC +RR Lyrae red edge +M = 0.65 M⊙, [Fe/H] = −1.00, [α/Fe] = 0.2 +M = 1.50 M⊙, [Fe/H] = 0.00, [α/Fe] = 0.0 +M = 0.75 M⊙, [Fe/H] = 0.00, [α/Fe] = 0.0 +Fig. 2: HRD of a sample of red giants in the Kepler field. The +colored star symbols highlight the location of the first three rHB +candidates in Table 1 and the grey dots in the background cor- +respond to the Kepler-APOGEE sample in Miglio et al. (2021). +The blue and red lines represent the theoretical red giant evo- +lutionary tracks (from the RGB phase until the first thermal +pulse) of low-mass stars with two different chemical composi- +tion: M = 0.65 M⊙, [α/Fe] = 0.2, [Fe/H] = −1.00 (blue), and +for M = 0.75 M⊙, [α/Fe] = 0, [Fe/H] = 0 (red). The green +line is the evolutionary track for a 1.5 M⊙ with solar composi- +tion and the dashed red one is the red edge of the RRL-IS for the +composition of the blue track (see Marconi et al. 2015). Solid or- +ange and blue circles corresponds to our rHB and RC reference +models, with a central He mass fraction Yc ≃ 0.27. +fraction Yc ∼ 0.27 as representative of the CHeB phase. The +structures and oscillation spectra of these reference models will +be discussed in Sect. 4. +To simulate 4-yr long Kepler observations of such objects we +use the code AADG3 (AsteroFLAG Artificial Dataset Generator, +version 3.0.2; Ball et al. 2018, and references therein). Frequen- +cies and normalised inertiae Enorm (see the definition in, e.g., +Aerts et al. 2010) of radial (ℓ = 0) and non-radial (ℓ = 1 − 3) +adiabatic oscillation modes are computed using the code GYRE +(version 6.0.1, Townsend & Teitler 2013; Townsend et al. 2018; +Goldstein & Townsend 2020). AADG3 also requires information +on modes lifetimes, a quantity directly related to non-adiabatic +processes, and therefore not resulting from the GYRE computa- +tion. AADG3 uses a relation between Γ0, ν, νmax and Teff cali- +brated on a small sample of main sequence and RGB spectra. +Since the temperatures of our metal-poor rHBs are outside the +domain covered by the calibration sample, and since Γ0 also de- +pends on the evolutionary state (Vrard et al. 2018), we adopt as +values of Γ0 the ones obtained from peak-bagging radial modes +in the spectra of our CHeB sample (using the method described +in Davies & Miglio 2016). +4. Discussion +In this section we analyse the structures and oscillation spectra +of our reference models (rHB and RC), and we compare the sim- +ulated PSD with the observed ones (Section 4.3). +4.1. Propagation diagram +The propagation diagrams of dipole modes for our rHB and RC +reference models are shown in the upper panels of Fig. 3. In each +panel, we show the modified Brunt-Väisälä ( ˜N) and Lamb ( ˜S ) +frequencies (Takata 2006) as a function of the normalised radius +(x = r/Rphot, with Rphot the photospheric radius), as well as the +expected frequency domain of the solar-like oscillations. +The profiles of ˜N and ˜S define the inner limits of the g- and +p-cavities. For modes with frequency close to νmax these limits +are defined by the condition ˜S (x1) = νmax and ˜N(x2) = νmax, and +in the region between x1 and x2 the modes are evanescent. +The extent of the evanescent zone is one of the ingredi- +ents determining the coupling between resonant cavities (Takata +2016; Pinçon et al. 2020). From the zoom-in boxes in Fig. 3 it +appears that this region is smaller in the rHB model than in the +RC one and, therefore, we expect the coupling factor q to be +larger in the former than in the latter. Indeed, using the structure +of our reference models and the strong-coupling approximation4 +for the dipole modes (Takata 2016; Pinçon et al. 2020) we ob- +tain qrHB = 0.65 and qRC = 0.25 at ν = νmax. We note that these +values are consistent, given the typical uncertainties (σq ∼ 0.2), +with those measured from the observed PSDs (see Table 1, A.1, +and Vrard et al. 2016; Mosser et al. 2017, 2018). +We notice that the value of the coupling factor is also a func- +tion of the mode frequency (e.g. Pinçon et al. 2020; Jiang et al. +2020, and van Rossem in prep.). As shown in Fig. 3, the size of +the evanescent zone decreases (thus q increases) with increasing +frequency. The value of q varies from 0.56 to 0.74 in the solar- +like frequency domain for the rHB model, and from 0.22 to 0.24 +for the RC one. In Sect. 4.2 we discuss the effect of this variation +on the behavior of the period spacing. +4.2. Dipole mode properties +In this section we analyse the properties of the dipole mode spec- +tra computed for our reference models. The bottom panels of +Fig. 3 show Enorm and the period spacing ∆P (i.e. period differ- +ence between two consecutive modes of same angular degree) as +a function of the eigenfrequencies. +We remind that Enorm is an average of the mode energy and +its value indicates the main region probed by the mode. Modes +examining central, high-density regions have higher Enorm than +modes that are preferentially trapped in the outer regions. The +inertia of dipole modes of the RC model shows a significant vari- +ation between local minima and maxima (ratio up to ≈ 27 in the +observable region) corresponding to the p-like and g-like modes, +respectively. On the contrary, the inertia in the rHB is almost uni- +form, with a small contrast between maxima and minima (ratio +up to ≈ 3 in the observable region). This indicates that the dipole +modes in the rHB are not clearly trapped in any of the resonant +cavities, that is, they have an important mixed p/g character. This +behaviour is consistent with the coupling factor values derived in +the previous section. +Since the amplitude of the modes is inversely proportional +to the square root of the inertia (see e.g. Dupret et al. 2009), we +expect a modulation of the dipole mode amplitudes around the p- +like mode in the case of the RC, as observed in some Kepler red +giants, while many dipole modes with similar amplitudes may be +observed in the spectrum of the rHB. This implies an increasing +4 The weak-coupling approximation (see e.g. Shibahashi 1979; Unno +et al. 1989) does not hold for low-mass CHeB stars (see e.g. Vrard et al. +2016; Mosser et al. 2017, van Rossem in prep.). +Article number, page 4 of 9 + +Matteuzzi et al.: rHBs as viewed by asteroseismology +10−2 +10−1 +100 +r/Rphot +100 +101 +102 +103 +104 +105 +ν [µHz] +rHB +x1 x2 +30 +60 +˜N +˜S(ℓ = 1) +νmax +10−2 +10−1 +100 +r/Rphot +100 +101 +102 +103 +104 +105 +ν [µHz] +RC +x1 +x2 +30 +60 +˜N +˜S(ℓ = 1) +νmax +10−5 +10−3 +10−1 +Enorm +rHB +ℓ = 0 +ℓ = 1 +νmax +∆Pa +20 +30 +40 +50 +60 +70 +ν [µHz] +250 +300 +∆P [s] +10−4 +10−2 +100 +Enorm +RC +ℓ = 0 +ℓ = 1 +νmax +∆Pa +10 +20 +30 +40 +50 +60 +70 +ν [µHz] +200 +300 +∆P [s] +Fig. 3: Comparison between structure and seismic properties of rHB (left panels) and RC (right panels) reference models (see +Sect. 3). Upper panels: Propagation diagrams of the dipole modes, with blue and orange lines corresponding to the modified Brunt- +Väis¨lä and Lamb frequencies (Takata 2006) respectively. The grey bands represent the frequency domain of expected solar-like +oscillations, and at their centre the dashed cyan lines indicate the νmax values. The insets are zoom-in of the evanescent zones, +delimited at νmax by the red and black points. Their different extension translates in different coupling between g and p cavities (see +main text). Lower panels: Normalised inertia Enorm and period spacing of the dipole modes ∆P as functions of the eigenfrequencies, +with the red curve representing, for comparison, Enorm of radial modes. The dashed green line indicates the value of the period +spacing from the asymptotic theory of high-order g-modes (∆Pa, Tassoul 1980) and the grey band and the dashed cyan line have the +same meaning as in the upper panels. +complexity of the oscillation spectra, as shown by the stars in +our sample (see Fig. 1). +The high value of q also affects the behaviour of the period +spacing (see also Mosser et al. 2017). In the bottom part of the +lower panels of Fig. 3, we plot ∆P as a function of the eigenfre- +quencies as well as the constant value (green dashed line) pre- +dicted by the asymptotic g-mode approximation (∆Pa, Tassoul +1980). In the observable frequency domain we notice for the +rHB model a significant deviation of ∆P from the asymptotic +value even for modes with high inertia, as well as a decreasing +trend of ∆P with increasing frequency. To show that both effects +are a consequence of the high value of q and its frequency de- +pendence, we use the Ong & Basu (2020) formalism to separate +pure isolated p-modes (π-modes) from pure isolated g-modes (γ- +modes), that is, pure g-modes not affected by the coupling with +the acoustic cavity. In Fig. 4 we plot the period spacing of dipole +γ-modes, and, as we could expect, their average value is consis- +tent with that from the asymptotic approximation of pure high- +order g-modes. Therefore, the differences in the period spacing +of the RC and rHB models are explained by the high coupling +for the latter, which causes all dipole modes to have an important +acoustic component, thus decreasing the value of ∆P. +4.3. Power spectral density +Fig. 5 shows the simulated PSDs of our reference models to- +gether with the inertia of the ℓ = 0, 1, 2, 3 modes. The contribu- +tion of each degree to the PSD is shown in Appendix C. +Comparison between Fig. 5 and Fig. 1 shows many similar- +ities between the rHB-mock spectrum and the observed ones. +These spectra appear noisier than RC ones, with a large number +of peaks corresponding to non-radial modes. In particular, there +Article number, page 5 of 9 + +A&A proofs: manuscript no. main_AA +20 +30 +40 +50 +60 +70 +ν [µHz] +290 +300 +310 +320 +330 +340 +350 +360 +∆P [s] +ℓ = 1 +∆Pa +Fig. 4: Period spacing as a function of the eigenfrequencies of +the isolated dipole γ-modes (Ong & Basu 2020). Other symbols +and colors are the same as in Fig. 3. The high modulation in the +period spacing above the observable frequencies is connected to +structural glitches (see e.g. Bossini et al. 2015). +are observable dipole modes in the entire frequency range be- +tween two consecutive radial modes, unlike the behaviour in RC +and low-RGB stars, where only a few modes around the corre- +sponding p-like mode have observable amplitudes. +We see that the strong coupling also affects the quadrupole +modes. Several of them, with frequencies close to those of the +p-like modes, are expected to have similar contributions to the +PSD. Moreover, because of a higher inertia at the local minima +with respect to the RC model, quadrupole modes in rHB stars +would have lower amplitudes. All that makes more challenging +to detect and characterise ℓ = 2 modes in CHeB metal-poor stars. +Finally, ℓ = 3 modes have eigenfrequencies close to those of ra- +dial modes and their heights are similar to the background noise. +They tend to form a continuum that should be considered during +the background analysis (see Appendix C). +5. Conclusions +High-quality spectra obtained from the 4-year long Kepler obser- +vations of a large number of red giants allowed us to identify a +small number of red giants (12) whose oscillation spectra appear +to be very noisy or complex with respect to the typical behaviour +of oscillation spectra in Kepler red giants. Their global seismic +parameters are compatible with low-mass stars (M ≲ 0.8 M⊙) +in the central He-burning phase, and the fit of the asymptotic +relation for the dipole modes (e.g. Vrard et al. 2016) results in +coupling factor values q ≳ 0.4, i.e. much higher than the typical +value for stars classified as RC (q ∼ 0.25 − 0.30, e.g. Vrard et al. +2016; Mosser et al. 2017, 2018). In our sample we find stars with +low/intermediate metallicity (75%), and also with solar metallic- +ity. Their position in the HRD is compatible with the so-called +rHB stars, i.e. low-mass objects between the RRL-IS and the RC +at the corresponding metallicity. Stellar evolution theory predicts +for these stars a structure consisting of a He core of ∼ 0.5 M⊙ and +an envelope of ≈ 0.1 − 0.2 M⊙ (e.g. Rood & Crocker 1989; Val- +carce & Catelan 2008; Gratton et al. 2010; Girardi 2016; Tailo +et al. 2020). +In this work we have shown that the oscillation spectra we +expect for this type of stars are entirely consistent with those ob- +served in our sample. These spectra are clearly different from +those of the stars that, with a similar He core but a much larger +envelope, populate the RC. The main factor determining these +differences is the coupling between the inner and outer regions, +reflecting very different density profiles inside these stars. A +second factor increasing the complexity of these spectra is the +higher temperature of the less metallic stars, which decrease the +lifetime of the modes. In fact, solar-like oscillations in rHBs have +also been detected in the K2 (Howell et al. 2014) light curves +of the globular cluster M4 (e.g. Wallace et al. 2019), where +the complexity of the spectra and the reduced observation time +(80 days) have made it difficult to extract robust ⟨∆ν⟩ values (e.g. +Tailo et al. 2022; Howell et al. 2022). +rHB stars are well known and easily identified in globular +clusters. Here we have also shown the ability of asteroseismol- +ogy to identify these low-mass CHeB stars in the field and in +solar-metallicity environments where, even with high-precision +photometry, they would be hardly distinguishable from other +stars in RC or RGB phases. +It is clear that 0.7 M⊙ stars, especially those of solar metallic- +ity, must have followed a non-standard evolution during which +they have lost a large amount of mass (see also Li et al. 2022; +Bobrick et al. 2022). This work provides us with a solid frame- +work for the future study of these stars and of the processes that +have led them to their current mass. Knowing that is fundamen- +tal in order to derive their ages with accuracy, and to potentially +provide another piece of the puzzle in the sequence between RC +and subdwarf B stars or other stripped stars. +Acknowledgements. We are grateful to (in alphabetical order) Emma Willett, +Joel Ong, Joris De Ridder, Masao Takata and Saniya Khan for useful discussions. +We are also grateful to the anonymous referee for the constructive comments. +This work has made use of data from the European Space Agency (ESA) mission +Gaia (https://www.cosmos.esa.int/gaia) and from the Two Micron All Sky Sur- +vey (https://irsa.ipac.caltech.edu/Missions/2mass.html). This research made use +of Lightkurve, a Python package for Kepler and TESS data analysis (Lightkurve +Collaboration et al. 2018), and of dustmaps, a package for interstellar dust red- +dening and extinction (Green 2018). AM, AS, GC, JM, MM, MT acknowledge +support from the ERC Consolidator Grant funding scheme (project ASTER- +OCHRONOMETRY, https://www.asterochronometry.eu, G.A. n. 772293). MV +acknowledge support from NASA grant 80NSSC18K1582. Funding for the Stel- +lar Astrophysics Centre is provided by The Danish National Research Founda- +tion (Grant agreement No. DNRF106). +References +Abdurro’uf, Accetta, K., Aerts, C., et al. 2022, ApJS, 259, 35 +Aerts, C., Christensen-Dalsgaard, J., & Kurtz, D. W. 2010, Asteroseismology +Af¸sar, M., Bozkurt, Z., Böcek Topcu, G., et al. 2018, AJ, 155, 240 +Af¸sar, M., Sneden, C., & For, B. Q. 2012, AJ, 144, 20 +Ahumada, R., Prieto, C. A., Almeida, A., et al. 2020, ApJS, 249, 3 +Armandroff, T. 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R., et al. 2018, ApJS, 236, 42 +Article number, page 7 of 9 + +A&A proofs: manuscript no. main_AA +Appendix A: Physical properties of the full sample +In this appendix we give some details concerning the origin of +the physical quantities in Tables 1 and A.1. The latter comple- +ments the former providing the properties of the rest of our sam- +ple of rHB candidates (see also the HRD of the whole sample in +Fig. A.1). +The global seismic parameters νmax and ⟨∆ν⟩ of targets +tagged with an asterisk in Tables 1 and +A.1 are taken from +Yu et al. (2018), while those for the NGC 6819 cluster member +(KIC 4937011, tagged with R) are from Handberg et al. (2017). +For the rest of the sample, we employ the approach of Davies & +Miglio (2016) and the value of ⟨∆ν⟩ is computed using individ- +ual frequencies and the weighted fit of the asymptotic relation +for radial modes. As discussed in Handberg et al. (2017), this +method gives results in good agreement with the values of ⟨∆ν⟩ +derived by Yu et al. (2018) and allows a forward comparison +with model-based values. The asymptotic period spacing of the +dipole modes ∆Π1 and the coupling factor q are derived using +the stretched-period method (see e.g. Vrard et al. 2016). +The atmospheric parameters Teff and chemical composi- +tion come from APOGEE-DR17, except for four targets with +a STAR_BAD flag in that release. For them (’+’ apex in Table 1 +and A.1) we adopt the available values in APOGEE-DR16. To +check the reliability of these atmospheric parameters and of the +quoted uncertainties, we performed an independent analysis for +the three stars in Table 1. We used MOOG-synth5 with the as- +sumption of local thermodynamic equilibrium, the linelist of +APOGEE-DR17 (Shetrone et al. 2015; Smith et al. 2021) im- +plemented with lines from VALD database6 and MARCS model +atmospheres (Gustafsson et al. 2008). We get results in good +agreement with those in APOGEE-DR16/17 except for Teff un- +certainties. Even for the best situation in which log g is fixed to +the seismic values (e.g. Valentini et al. 2019), the uncertainty on +Teff is σTeff ∼ 50 K. Therefore, although in Tables 1 and A.1 we +keep the values from APOGEE, we assume a minimum value of +σTeff = 50 K in deriving stellar mass and its uncertainty. +Bolometric luminosities L are estimated by combining as- +trometry data from Gaia DR3 (Babusiaux et al. 2022) with +2MASS photometry (Skrutskie et al. 2006) in the Ks-band and +bolometric correction from Casagrande & VandenBerg (2014, +2018). We applied the Gaia-DR3 parallax zero-point correction +of Lindegren et al. (2021) and estimated reddening and extinc- +tion from the three-dimensional maps of Green et al. (2019). +The errors in L are calculated with a Markov chain Monte Carlo +(MCMC) method considering fixed the extinction and the value +of Mbol,⊙ (Mbol,⊙ = 4.75, Casagrande & VandenBerg 2014). +Stellar masses, as described in Sec. 2, have been estimated +using the scaling relation Eq. 2.1 and the values of L, Teff and +νmax just described. In the following paragraph we present the +results obtained with an alternative scaling relation. +Appendix A.1: Stellar mass from scaling relation involving +⟨∆ν⟩ and νmax +In order to test the mass estimations made with Eq. 2.1 of Sect. 2, +we employed the model-based corrected scaling relation (see +e.g. Kjeldsen & Bedding 1995; Gai et al. 2011) +M +M⊙ += f 4 +∆ν +� Teff +Teff,⊙ +�1.5 � νmax +νmax,⊙ +�3 �⟨∆ν⟩⊙ +⟨∆ν⟩ +�4 +(A.1) +5 https://www.as.utexas.edu/ chris/moog.html +6 http://vald.astro.uu.se +3500 +4000 +4500 +5000 +5500 +6000 +Teff [K] +101 +102 +103 +L [L⊙] +KIC8694070 +KIC3626807 +KIC12504765 +KIC11072164 +KIC6032981 +KIC9691704 +KIC2555126 +KIC3428926 +KIC9335415 +KIC4937011 +KIC5271626 +KIC11299941 +rHB +RC +RR Lyrae red edge +M = 0.65 M⊙, [Fe/H] = −1.00, [α/Fe] = 0.2 +M = 1.50 M⊙, [Fe/H] = 0.00, [α/Fe] = 0.0 +M = 0.75 M⊙, [Fe/H] = 0.00, [α/Fe] = 0.0 +Fig. A.1: The same as Fig. 2, but including all the CHeB stars in +our sample. These stars are colour-coded according to increasing +[Fe/H]. +for two metal-rich stars (KIC5271626 and KIC4937011) and for +two metal-poor stars (KIC6032981 and KIC11072164) of our +sample. Here we used the solar reference values of Sect. 2, and +⟨∆ν⟩⊙ = 135.1 µHz (Huber et al. 2011). The correction fac- +tor f∆ν on the ⟨∆ν⟩ scaling law (Ulrich 1986) is derived with +the procedure described in Rodrigues et al. (2017), i.e. by us- +ing the theoretical radial mode frequencies of stellar models to +compute ⟨∆ν⟩ from the weighted linear fit of the asymptotic +relation (see also Miglio et al. 2021; Tailo et al. 2022). We +based the iterative search for the correct f∆ν on evolutionary +tracks with the same metallicity (within the errors) as the four +stars aforementioned: solar composition for the metal-rich ones; +[Fe/H] = −1.00 with [α/Fe] = 0.2 and [α/Fe] = 0.4 for the +two metal-poor ones (see Appendix B for details on the models). +To correct the model-predicted ⟨∆ν⟩ from the surface effects, we +included ⟨∆ν⟩⊙ = 135.3 µHz of our solar-calibrated model to the +correction factor f∆ν (e.g. White et al. 2011). Finally, we com- +puted the theoretical radial oscillations with the tool GYRE. The +f∆ν we found are nearly equal to 1.03 and 1.01 for the metal-poor +and for the metal-rich stars respectively. In deriving the masses +with Eq. A.1, we considered a minimum error of 50 K in Teff +(as said previously in Appendix A), and an error of 0.01 on f∆ν +due to the impossibility of knowing the exact position, at fixed +νmax, of our observed stars along the evolutionary tracks. There- +fore, these masses are compatible within the errors with those +derived from Eq. 2.1. We want also to notice that it is difficult +to have a very precise ⟨∆ν⟩ estimate for these stars, because the +radial modes are located in crowded regions (see Appendix C). +This leads to systematic errors in the measurement of individual +radial modes that can be of the order of 4% by mass. +Article number, page 8 of 9 + +Matteuzzi et al.: rHBs as viewed by asteroseismology +Table A.1: Physical properties for the rest of our sample of rHB candidates. +KIC +L [L⊙] +Teff [K] +[Fe/H] +[α/Fe] +⟨∆ν⟩ [µHz] +νmax [µHz] +q +∆Π1 [s] +M [M⊙] +2555126 +41 ± 4 +5320 ± 20 +-0.72 +0.26 +5.66 ± 0.03 +36.4 ± 0.6 +0.93 +280 ± 20 +0.64 ± 0.06 +3428926+ +36 ± 3 +5560 ± 130 +-0.50 +0.27 +6.72 ± 0.02 +43.0 ± 0.6 +1.15 +270 ± 40 +0.58 ± 0.07 +3626807 +50 ± 6 +5310 ± 20 +-1.16 +0.26 +5.276 ± 0.011 +36.5 ± 0.6 +0.69 +308 ± 6 +0.79 ± 0.10 +9335415+ +46 ± 4 +5580 ± 120 +-0.50 +0.11 +5.808 ± 0.018 +34.9 ± 0.5 +0.53 +240 ± 40 +0.59 ± 0.07 +9691704 +55 ± 7 +5230 ± 20 +-0.88 +0.30 +4.802 ± 0.013 +32.6 ± 0.5 +0.23 +334 ± 5 +0.83 ± 0.11 +11072164 +43 ± 4 +5215 ± 18 +-1.01 +0.24 +4.761 ± 0.012 +32.8 ± 0.5 +1.11 +300 ± 50 +0.65 ± 0.06 +11299941∗ +32 ± 3 +4585 ± 7 +0.25 +0.05 +4.08 ± 0.09 +28.0 ± 0.8 +0.45 +300 ± 20 +0.64 ± 0.08 +12504765+ +51 ± 5 +5220 ± 130 +-1.15 +0.33 +4.817 ± 0.010 +32.4 ± 0.5 +0.65 +340 ± 20 +0.76 ± 0.10 +4937011R +37 ± 4 +4707 ± 8 +-0.02 +0.03 +4.08 ± 0.10 +28.3 ± 0.4 +0.53 +224.3 ± 1.4 +0.71 ± 0.08 +Notes. It includes also the properties of KIC4937011 (undermassive star in NGC 6819, tagged with a R in apex), for which we show +⟨∆ν⟩, νmax, and M from Handberg et al. (2017). See Table 1 for a description of the symbols +Appendix B: Grids of stellar models +As mentioned in Sect. 3, we chose three sets of stellar param- +eters to represent a rHB star, a metal-rich low-mass CHeB star, +and a RC star. The stellar models at the base of this work belong +to a grid of stellar evolutionary models computed with the code +MESA-r11532 (Modules for Experiments in Stellar Astrophysics +Paxton et al. 2011, 2013, 2015, 2016, 2018, 2019). In the compu- +tation we follow the evolution from the pre-main sequence phase +until the first thermal pulse in the asymptotic giant branch for +stellar masses from 0.6 M⊙ till 2.00 M⊙, with a step of 0.05 M⊙. +We consider 36 different chemical composition, with 12 values +of [Fe/H] (from -2.5 to 0.25) and three values of alpha-elements +enhancement: [α/Fe] = 0.0, 0.2 and 0.4. We adopt as refer- +ence solar mixture that from Asplund et al. (2009) and high- and +low-temperature radiative opacity tables have been computed for +these specific metal mixtures, solar and alpha-enhanced ones. +Envelope convection is described by the Mixing Length theory +Cox & Giuli (1968) and the corresponding αMLT parameter, the +same for all the grid, is derived from the solar calibration with the +same physics. We add below the convective envelope a diffusive +undershooting (Herwig 2000) with a size parameter f = 0.02 +(see Khan et al. 2018). Extra-mixing over the convective core +limit during the central-He burning phase is treated following +the formalism by Bossini et al. (2017). +Appendix C: Contribution of individual eigenmodes +to the PSDs of CHeB stars +In this section we break down the PSDs of our reference models +(Fig. 5) into the contributions from the modes of the different +angular degree. The smoothed PSDs for ℓ = 0, 1, 2, 3 are shown +in Fig. C.1. The smoothing is chosen just for showing purposes, +i.e. to resemble a Lorentzian fit of each eigenmode. The mod- +ulation around the p-like mode in the dipole modes of the RC +star and the higher number of observed mixed modes in the rHB +model are evident. Furthermore, the quadrupole modes of the +rHB model are less visible than those of the RC model, and its +octupole modes resemble a continuous background with small +peaks almost coinciding with the radial modes. Finally, we want +to notice that the presence, in rHB stars, of ℓ = 1, 2, 3 modes +very close to the radial ones (in some cases almost coinciding, +e.g. Fig. C.1) could introduce a non-negligible influence on the +analysis of the heights and the linewidths of the ℓ = 0 modes. +30 +35 +40 +45 +50 +55 +60 +Frequency [µHz] +0 +2000 +4000 +6000 +8000 +10000 +12000 +PSD [ppm2 +µHz ] +rHB +ℓ = 0 +ℓ = 1 +ℓ = 2 +ℓ = 3 +νmax +30 +35 +40 +45 +50 +55 +60 +Frequency [µHz] +0 +5000 +10000 +15000 +20000 +PSD [ppm2 +µHz ] +RC +ℓ = 0 +ℓ = 1 +ℓ = 2 +ℓ = 3 +νmax +Fig. C.1: Smoothed version of the PSDs presented in Section 4.3. +Here we show the individual degrees for the rHB (top) and RC +(bottom) simulated stars. The dashed cyan line is the correspond- +ing νmax. +Article number, page 9 of 9 + diff --git a/i9FAT4oBgHgl3EQf-B7W/content/tmp_files/load_file.txt b/i9FAT4oBgHgl3EQf-B7W/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..09e923499b3658759b53d3c8e2326b8f148f2060 --- /dev/null +++ b/i9FAT4oBgHgl3EQf-B7W/content/tmp_files/load_file.txt @@ -0,0 +1,1155 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf,len=1154 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' main_AA ©ESO 2023 January 24, 2023 Letter to the Editor Red Horizontal Branch stars: an asteroseismic perspective Massimiliano Matteuzzi1, 2⋆, Josefina Montalbán1, 3, Andrea Miglio1, 2, 3, Mathieu Vrard4, Giada Casali1, 2, Amalie Stokholm1, 5, Marco Tailo1, Warrick Ball3, Walter E.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Birmingham B15 2TT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' UK 4 Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Columbus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' OH 43210,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' USA 5 Stellar Astrophysics Centre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Aarhus University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Ny Munkegade 120,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' DK-8000 Aarhus C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Denmark 6 Leibniz-Institut für Astrophysik Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' An der Sternwarte 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 14482,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Germany January 24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2023 ABSTRACT Robust age estimates of red giant stars are now possible thanks to the precise inference of their mass based on asteroseismic con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' However, there are cases where such age estimates can be highly precise yet very inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' An example is giants that have undergone mass loss or mass transfer events that have significantly altered their mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' In this context, stars with “apparent” ages significantly higher than the age of the Universe are candidates as stripped stars, or stars that have lost more mass than expected, most likely via interaction with a companion star, or because of the poorly understood mass-loss mechanism along the red-giant branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' In this work we identify examples of such objects among red giants observed by Kepler, both at low ([Fe/H] ≲ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='5) and solar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' By modelling their structure and pulsation spectra, we find a consistent picture confirming that these are indeed low-mass objects consisting of a He core of ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='5 M⊙ and an envelope of ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='2 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Moreover, we find that these stars are characterised by a rather extreme coupling (q ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='4) between the pressure-mode and gravity-mode cavities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' much higher than the typical value for red clump stars, providing thus a direct seismic signature of their peculiar structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The complex pulsation spectra of these objects, if observed with sufficient frequency resolution, hold detailed information about the structural properties of likely products of mass stripping, hence can potentially shed light on their formation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' On the other hand, our tests highlight the difficulties associated with measuring reliably the large frequency separation, especially in shorter datasets, with impact on the reliability of the inferred masses and ages of low-mass Red Clump stars with e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' K2 or TESS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' asteroseismology – stars: evolution – stars: fundamental parameters – stars: horizontal branch – stars: interiors – stars: mass-loss 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Introduction It is widely accepted that the large range in color shown by low mass stars in the central He-burning phase, called Horizontal Branch (HB), is mainly due to variations in the efficiency of the H-burning shell, hence, to the mass of the H-envelope remaining around a He-core of ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='5 M⊙ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Salaris & Cassisi 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' In a colour-magnitude diagram (CMD), low-mass core-He-burning (CHeB) stars appear distributed in both bluer and redder colours than the RR Lyrae instability strip (RRL−IS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Those located be- tween the RR Lyrae and the red-clump (RC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Girardi 2016) are called rHB (red Horizontal Branch) stars, and they have a H-rich envelope of ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='2 M⊙ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Rood & Crocker 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Valcarce & Catelan 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Girardi 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Tailo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' This HB component has been clearly observed in globular clusters of different metallicity and age (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Armandroff 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Stetson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Catelan 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Tailo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2020), however, rHB objects also exist in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' While their identification is challenging, their census has been considered extremely important for tracing old stellar populations in the Milky Way (MW;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Kaempf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2010, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Although mainly associated with stars of low/intermediate metallicity, corresponding to the thick disc and halo population, spectroscopic studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Af¸sar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' ⋆ E-mail: massimilia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='matteuzz2@unibo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='it 2012, 2018) have shown that rHB stars are also present in the metal-rich component of the MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' This suggests that the pro- genitors of these objects have followed a non-standard evolution with significant mass loss, or envelope stripping due to binary interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Signs of significant mass loss have been revealed in red giants observed by the Kepler space telescope (Borucki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2010), in the field and in the open cluster NGC 6819 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Handberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Brogaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Stellar evolution models predict different structures for rHB and RC stars, with the latter having a similar He core, but a larger H envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' We thus expect their seismic properties to be dif- ferent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The exquisite precision achieved after 4 years of Kepler observations has revealed oscillation spectra of red giants with an increasing level of complexity (see Chaplin & Miglio 2013, for a review): frequency patterns in red giant branch (RGB) stars similar to those found in main sequence stars (Universal pattern, Mosser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2011), spectra of RC stars with "forests" of dipole modes around the nominal acoustic mode, but still with an evi- dent regularity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Beck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2011), and also "outlier" spectra with a larger number of visible modes over the whole frequency domain, which are hypothesised in this letter to belong to rHB stars (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' In this work we identify a small sample of 11 rHB candidates among the red giants in the Kepler field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Their global seismic Article number, page 1 of 9 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='08761v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='SR] 20 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' main_AA properties and atmospheric parameters suggest that they are low- mass CHeB stars with low/intermediate and solar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Combining numerical simulations of stellar structure and evo- lution and stellar oscillations, we study the consistency between the location of our rHB candidates in the Hertzsprung–Russell diagram (HRD), their theoretically predicted internal structure and their oscillation spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Our rHB sample is presented in Section 2 and the theoretical models in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Section 4 dis- cusses the properties of theoretical oscillation spectra of typical rHB and RC stars, as well as the comparison with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' In Section 5 we summarise our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Observational data In addition to KIC4937011, a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='71 M⊙ CHeB star belonging to the open cluster NGC 6819 (see Handberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2017) which has a turn-off mass of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='6 M⊙, we found 11 red giants in the Kepler database1 with peculiar power spectral density (PSD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' While their global seismic parameters (mean large frequency separation ⟨∆ν⟩, frequency of maximum power νmax, and asymp- totic period spacing of the dipole modes ∆Π1) are compatible with low-mass CHeB stars, they have complex oscillation spec- tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' They have, for instance, an unusually high number of observ- able dipole g/p mixed modes without the amplitude modulation around the p-like modes that is typically found in low-RGB and RC stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' This fact suggests that all dipole modes have a signifi- cant amplitude also in the outer region of the star, and hence, that g and p resonant cavities in these objects are strongly coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The ability to transfer the energy of the mode from one cav- ity to the other, instead of remaining trapped mainly in one of them, is quantified by the coupling factor q (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Shibahashi 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Takata 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The analysis of Kepler light curves pro- vides the seismic parameters mentioned above, as well as the value of the coupling factor q (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Vrard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Mosser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2017, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Theoretically, the parameter q ranges from 0 (uncoupled) to 1 (completely coupled).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' All the stars in our sam- ple have q ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='4, while the median for RC stars is ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='25 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='3 (Vrard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Mosser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' On the other hand, the values of radial mode-linewidths (Γ0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='2 µHz) are larger than the third quantile of the full sam- ple of CHeB Kepler stars (median value Γ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='15 µHz, Vrard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Both a high q and a high Γ0 contribute to increase the complexity of the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Moreover, given the dependence of Γ0 on the effective temperature Teff (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Chaplin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2009), the quadrupole modes are more difficult to detect in the hotter metal-poor subsample than in the cooler metal-rich objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The seismic properties (νmax, ⟨∆ν⟩, ∆Π1 and q) for our sample are reported in Table 1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1, together with the at- mospheric parameters (Teff and chemical composition) from APOGEE-DR16/DR17 (Ahumada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Abdurro’uf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Around 25% (3 out of 12) of the sample are metal-rich (0 ≤ [Fe/H] < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='3) cool (4600 ≤ Teff/K ≤ 4800) stars, and the rest are low/intermediate metallicity (−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='4 < [Fe/H] < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='5) stars with 5200 ≤ Teff/K ≤ 5600, that is, belonging to the "clas- sical" rHB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Tables 1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1 contain also the stellar luminosity derived using Gaia-DR3 astrometry data (see Appendix A for details), and an estimate of their mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The latter can be derived from scaling relations involving atmospheric and global seismic pa- 1 https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='edu/missions-and-data/kepler rameters (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Miglio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Here we use the one com- bining L, Teff and νmax: M M⊙ = �Teff,⊙ Teff �3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='5 � νmax νmax,⊙ � � L L⊙ � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1) where the solar reference values are Teff,⊙ = 5777 K, νmax,⊙ = 3090 µHz (Huber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The mass uncertainties are calcu- lated in quadrature by considering an uncertainty of at least 50 K in Teff as estimated from an independent analysis of APOGEE spectra (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' In Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1 we also discuss the stellar mass values from a model-based corrected scaling rela- tion involving Teff, ⟨∆ν⟩ and νmax (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' We notice that the mass of KIC 4937011 in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1 is that of Handberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' (2017), and its value is nevertheless compati- ble with our results obtained with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1 or A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' All the objects in our sample are then very low-mass stars (M ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='8 M⊙) with a high coupling2 between p-mode and g-mode cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' We select three stars (those in Table 1) as representative of low-mass CHeB stars in different metallicity domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Figure 2 shows these stars in an HRD, together with the Kepler-APOGEE red giant sample (Miglio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2021, grey dots) and the red edge of the RRL-IS (Marconi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2015, dashed red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The two metal-poor stars (blue star symbols) are located between the RRL-IS and the RC, as expected for rHB stars, while the metal- rich CHeB star (orange star symbol) appears in the region of the "ensemble" Kepler-RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Its location is nevertheless redder than the RC at solar-metallicity, and hence it is well a rHB metal-rich star as suggested also by its mass (see also Handberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2017) and oscillation spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' As mentioned above, rHBs, especially those metal-rich, must have followed non-standard evolution to reach their current state within the age of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' They are probably the progeny of strongly interacting binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' It has not been possible to confirm that hypothesis using the cur- rently available Gaia-DR3 astrometry data (see Halbwachs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2022, for the non-single star processing3), but we cannot exclude that they were part of binary systems in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Simulated data The aim of this work is not to fit the available observational data, but to analyse the relation between the structures of rHB stars, according to stellar evolution theory, and their oscillation spec- tra, and to compare the latter with those observed in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' From a grid of models (see Appendix B) we selected two sets of parameters that represent well the mass and chemical com- position of the classical rHB (M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='65 M⊙, [α/Fe] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='2 and [Fe/H] = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='00) and metal-rich low-mass CHeB (M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='75 M⊙, [α/Fe] = 0 and [Fe/H] = 0) stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' For comparison, we also con- sider a typical RC star (M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='5 M⊙ with solar composition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' As it appears in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2, the parameters selected for our refer- ence models provide indeed a good representation of the low- intermediate metallicity and metal-rich rHBs in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' We also note that without complementary information, such as that provided by asteroseismology, a metal-rich rHB would be mis- taken for a more massive star in RGB (see also Handberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2 We notice that stars in the CHeB stage could have multiple cavities in the inner part due to semi-convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' This could lead to bias when estimating q from the fit of observations with the asymptotic relation for dipole modes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Pinçon & Takata 2022), thus it must be considered in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 3 We also checked the non-single star hypothesis using the fidelity_v2 table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Article number, page 2 of 9 Matteuzzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' : rHBs as viewed by asteroseismology 15 20 25 30 35 40 Frequency [µHz] 0 50000 100000 150000 200000 250000 300000 350000 400000 PSD [ppm2 µHz ] (a) KIC5271626 [Fe/H] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='01 Teff = 4770 K Original Smooth 25 30 35 40 45 Frequency [µHz] 0 10000 20000 30000 40000 50000 60000 70000 80000 PSD [ppm2 µHz ] (b) KIC6032981 [Fe/H] = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='01 Teff = 5300 K Original Smooth 25 30 35 40 45 Frequency [µHz] 0 20000 40000 60000 80000 PSD [ppm2 µHz ] (c) KIC8694070 [Fe/H] = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='44 Teff = 5300 K Original Smooth 26 28 30 32 34 36 38 40 42 Frequency [µHz] 0 20000 40000 60000 80000 100000 120000 140000 160000 PSD [ppm2 µHz ] (d) KIC1161618 RC star [Fe/H] = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='02 Teff = 4740 K Original Smooth 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='0 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='0 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='5 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='5 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='5 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='0 Frequency [µHz] 0 100000 200000 300000 400000 PSD [ppm2 µHz ] (e) KIC2436824 RGB star [Fe/H] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='34 Teff = 4340 K Original Smooth Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 1: PSD for five low-mass red giants (grey lines in the five panels) observed by Kepler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Panels (a), (b), and (c) show the three low-mass CHeB stars KIC5271626, KIC6032981, and KIC8694070 (first three rows in Table 1 and colored stars in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Panels (d) and (e) show the RC star KIC1161618 and the RGB star KIC2436824, for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' All the five panels contain a smoothed PSD (red lines) computed with a box kernel of width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='5 µHz in panels (a), (b), (c), (d), and of width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1 µHz in panel (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Table 1: Summary of the seismic and atmospheric properties for three rHB candidates of our sample (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' KIC L [L⊙] Teff [K] [Fe/H] [α/Fe] ⟨∆ν⟩ [µHz] νmax [µHz] q ∆Π1 [s] M [M⊙] 5271626∗ 42 ± 4 4769 ± 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='05 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='61 291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='07 6032981+ 44 ± 4 5300 ± 110 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='37 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='188 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='017 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='15 321 ± 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='08 8694070 53 ± 5 5300 ± 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='135 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='018 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='7 332 ± 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='09 Mock rHB 44 5663 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='41 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='65 324 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='65 Mock RC 59 4891 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='79 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='25 313 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='50 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' For each Kepler ID (KIC) we report the effective temperature Teff, [Fe/H], [α/Fe] from APOGEE-DR17, or APOGEE-DR16 (one star tagged with a + in apex);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' mean large frequency separation ⟨∆ν⟩, and frequency of maximum power νmax calculated by us using the code in Davies & Miglio (2016), or Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' (2018) data (one star tagged with a * in apex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The coupling factor q and asymptotic period spacing of the dipole modes ∆Π1 as calculated from the stretched-period method (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Vrard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The current stellar mass M is computed from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The last two rows show the properties of a simulated rHB and RC star (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' It is generally accepted that, except for the age, the properties of a low-mass star with a He core of ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='5 M⊙ and an H-rich en- velope of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='2 M⊙ are largely independent of whether the star was born with a small mass or whether it originates from a more massive star (M ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='8 M⊙) that underwent significant mass loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Therefore, it is justified to use structure models calculated without mass loss such as those in our grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' In the following we concentrate on a metal-poor model since, as described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2, we expect metal-poor rHBs to present more marked differences with respect to the spectra of typical RC stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' We select structure models with a central He mass Article number, page 3 of 9 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' main_AA 3500 4000 4500 5000 5500 6000 Teff [K] 101 102 103 L [L⊙] Obs: metal-poor Obs: metal-rich rHB RC RR Lyrae red edge M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='65 M⊙, [Fe/H] = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='00, [α/Fe] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='2 M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='50 M⊙, [Fe/H] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='00, [α/Fe] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='0 M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='75 M⊙, [Fe/H] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='00, [α/Fe] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2: HRD of a sample of red giants in the Kepler field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The colored star symbols highlight the location of the first three rHB candidates in Table 1 and the grey dots in the background cor- respond to the Kepler-APOGEE sample in Miglio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The blue and red lines represent the theoretical red giant evo- lutionary tracks (from the RGB phase until the first thermal pulse) of low-mass stars with two different chemical composi- tion: M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='65 M⊙, [α/Fe] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='2, [Fe/H] = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='00 (blue), and for M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='75 M⊙, [α/Fe] = 0, [Fe/H] = 0 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The green line is the evolutionary track for a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='5 M⊙ with solar composi- tion and the dashed red one is the red edge of the RRL-IS for the composition of the blue track (see Marconi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Solid or- ange and blue circles corresponds to our rHB and RC reference models, with a central He mass fraction Yc ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' fraction Yc ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='27 as representative of the CHeB phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The structures and oscillation spectra of these reference models will be discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' To simulate 4-yr long Kepler observations of such objects we use the code AADG3 (AsteroFLAG Artificial Dataset Generator, version 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Ball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2018, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Frequen- cies and normalised inertiae Enorm (see the definition in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=', Aerts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2010) of radial (ℓ = 0) and non-radial (ℓ = 1 − 3) adiabatic oscillation modes are computed using the code GYRE (version 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1, Townsend & Teitler 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Townsend et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Goldstein & Townsend 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' AADG3 also requires information on modes lifetimes, a quantity directly related to non-adiabatic processes, and therefore not resulting from the GYRE computa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' AADG3 uses a relation between Γ0, ν, νmax and Teff cali- brated on a small sample of main sequence and RGB spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Since the temperatures of our metal-poor rHBs are outside the domain covered by the calibration sample, and since Γ0 also de- pends on the evolutionary state (Vrard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2018), we adopt as values of Γ0 the ones obtained from peak-bagging radial modes in the spectra of our CHeB sample (using the method described in Davies & Miglio 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Discussion In this section we analyse the structures and oscillation spectra of our reference models (rHB and RC), and we compare the sim- ulated PSD with the observed ones (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Propagation diagram The propagation diagrams of dipole modes for our rHB and RC reference models are shown in the upper panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' In each panel, we show the modified Brunt-Väisälä ( ˜N) and Lamb ( ˜S ) frequencies (Takata 2006) as a function of the normalised radius (x = r/Rphot, with Rphot the photospheric radius), as well as the expected frequency domain of the solar-like oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The profiles of ˜N and ˜S define the inner limits of the g- and p-cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' For modes with frequency close to νmax these limits are defined by the condition ˜S (x1) = νmax and ˜N(x2) = νmax, and in the region between x1 and x2 the modes are evanescent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The extent of the evanescent zone is one of the ingredi- ents determining the coupling between resonant cavities (Takata 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Pinçon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' From the zoom-in boxes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 3 it appears that this region is smaller in the rHB model than in the RC one and, therefore, we expect the coupling factor q to be larger in the former than in the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Indeed, using the structure of our reference models and the strong-coupling approximation4 for the dipole modes (Takata 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Pinçon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2020) we ob- tain qrHB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='65 and qRC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='25 at ν = νmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' We note that these values are consistent, given the typical uncertainties (σq ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='2), with those measured from the observed PSDs (see Table 1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1, and Vrard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Mosser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2017, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' We notice that the value of the coupling factor is also a func- tion of the mode frequency (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Pinçon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2020, and van Rossem in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 3, the size of the evanescent zone decreases (thus q increases) with increasing frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The value of q varies from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='56 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='74 in the solar- like frequency domain for the rHB model, and from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='22 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='24 for the RC one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='2 we discuss the effect of this variation on the behavior of the period spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Dipole mode properties In this section we analyse the properties of the dipole mode spec- tra computed for our reference models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The bottom panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 3 show Enorm and the period spacing ∆P (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' period differ- ence between two consecutive modes of same angular degree) as a function of the eigenfrequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' We remind that Enorm is an average of the mode energy and its value indicates the main region probed by the mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Modes examining central, high-density regions have higher Enorm than modes that are preferentially trapped in the outer regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The inertia of dipole modes of the RC model shows a significant vari- ation between local minima and maxima (ratio up to ≈ 27 in the observable region) corresponding to the p-like and g-like modes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' On the contrary, the inertia in the rHB is almost uni- form, with a small contrast between maxima and minima (ratio up to ≈ 3 in the observable region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' This indicates that the dipole modes in the rHB are not clearly trapped in any of the resonant cavities, that is, they have an important mixed p/g character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' This behaviour is consistent with the coupling factor values derived in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Since the amplitude of the modes is inversely proportional to the square root of the inertia (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Dupret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2009), we expect a modulation of the dipole mode amplitudes around the p- like mode in the case of the RC, as observed in some Kepler red giants, while many dipole modes with similar amplitudes may be observed in the spectrum of the rHB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' This implies an increasing 4 The weak-coupling approximation (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Shibahashi 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Unno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 1989) does not hold for low-mass CHeB stars (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Vrard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Mosser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2017, van Rossem in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Article number, page 4 of 9 Matteuzzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' : rHBs as viewed by asteroseismology 10−2 10−1 100 r/Rphot 100 101 102 103 104 105 ν [µHz] rHB x1 x2 30 60 ˜N ˜S(ℓ = 1) νmax 10−2 10−1 100 r/Rphot 100 101 102 103 104 105 ν [µHz] RC x1 x2 30 60 ˜N ˜S(ℓ = 1) νmax 10−5 10−3 10−1 Enorm rHB ℓ = 0 ℓ = 1 νmax ∆Pa 20 30 40 50 60 70 ν [µHz] 250 300 ∆P [s] 10−4 10−2 100 Enorm RC ℓ = 0 ℓ = 1 νmax ∆Pa 10 20 30 40 50 60 70 ν [µHz] 200 300 ∆P [s] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 3: Comparison between structure and seismic properties of rHB (left panels) and RC (right panels) reference models (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Upper panels: Propagation diagrams of the dipole modes, with blue and orange lines corresponding to the modified Brunt- Väis¨lä and Lamb frequencies (Takata 2006) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The grey bands represent the frequency domain of expected solar-like oscillations, and at their centre the dashed cyan lines indicate the νmax values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The insets are zoom-in of the evanescent zones, delimited at νmax by the red and black points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Their different extension translates in different coupling between g and p cavities (see main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Lower panels: Normalised inertia Enorm and period spacing of the dipole modes ∆P as functions of the eigenfrequencies, with the red curve representing, for comparison, Enorm of radial modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The dashed green line indicates the value of the period spacing from the asymptotic theory of high-order g-modes (∆Pa, Tassoul 1980) and the grey band and the dashed cyan line have the same meaning as in the upper panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' complexity of the oscillation spectra, as shown by the stars in our sample (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The high value of q also affects the behaviour of the period spacing (see also Mosser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' In the bottom part of the lower panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 3, we plot ∆P as a function of the eigenfre- quencies as well as the constant value (green dashed line) pre- dicted by the asymptotic g-mode approximation (∆Pa, Tassoul 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' In the observable frequency domain we notice for the rHB model a significant deviation of ∆P from the asymptotic value even for modes with high inertia, as well as a decreasing trend of ∆P with increasing frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' To show that both effects are a consequence of the high value of q and its frequency de- pendence, we use the Ong & Basu (2020) formalism to separate pure isolated p-modes (π-modes) from pure isolated g-modes (γ- modes), that is, pure g-modes not affected by the coupling with the acoustic cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 4 we plot the period spacing of dipole γ-modes, and, as we could expect, their average value is consis- tent with that from the asymptotic approximation of pure high- order g-modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Therefore, the differences in the period spacing of the RC and rHB models are explained by the high coupling for the latter, which causes all dipole modes to have an important acoustic component, thus decreasing the value of ∆P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Power spectral density Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 5 shows the simulated PSDs of our reference models to- gether with the inertia of the ℓ = 0, 1, 2, 3 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The contribu- tion of each degree to the PSD is shown in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Comparison between Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 1 shows many similar- ities between the rHB-mock spectrum and the observed ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' These spectra appear noisier than RC ones, with a large number of peaks corresponding to non-radial modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' In particular, there Article number, page 5 of 9 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' main_AA 20 30 40 50 60 70 ν [µHz] 290 300 310 320 330 340 350 360 ∆P [s] ℓ = 1 ∆Pa Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 4: Period spacing as a function of the eigenfrequencies of the isolated dipole γ-modes (Ong & Basu 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Other symbols and colors are the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The high modulation in the period spacing above the observable frequencies is connected to structural glitches (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Bossini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' are observable dipole modes in the entire frequency range be- tween two consecutive radial modes, unlike the behaviour in RC and low-RGB stars, where only a few modes around the corre- sponding p-like mode have observable amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' We see that the strong coupling also affects the quadrupole modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Several of them, with frequencies close to those of the p-like modes, are expected to have similar contributions to the PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Moreover, because of a higher inertia at the local minima with respect to the RC model, quadrupole modes in rHB stars would have lower amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' All that makes more challenging to detect and characterise ℓ = 2 modes in CHeB metal-poor stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Finally, ℓ = 3 modes have eigenfrequencies close to those of ra- dial modes and their heights are similar to the background noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' They tend to form a continuum that should be considered during the background analysis (see Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Conclusions High-quality spectra obtained from the 4-year long Kepler obser- vations of a large number of red giants allowed us to identify a small number of red giants (12) whose oscillation spectra appear to be very noisy or complex with respect to the typical behaviour of oscillation spectra in Kepler red giants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Their global seismic parameters are compatible with low-mass stars (M ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='8 M⊙) in the central He-burning phase, and the fit of the asymptotic relation for the dipole modes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Vrard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2016) results in coupling factor values q ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' much higher than the typical value for stars classified as RC (q ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='25 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='30, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Vrard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Mosser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2017, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' In our sample we find stars with low/intermediate metallicity (75%), and also with solar metallic- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Their position in the HRD is compatible with the so-called rHB stars, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' low-mass objects between the RRL-IS and the RC at the corresponding metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Stellar evolution theory predicts for these stars a structure consisting of a He core of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='5 M⊙ and an envelope of ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='2 M⊙ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Rood & Crocker 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Val- carce & Catelan 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Gratton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Girardi 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Tailo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' In this work we have shown that the oscillation spectra we expect for this type of stars are entirely consistent with those ob- served in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' These spectra are clearly different from those of the stars that, with a similar He core but a much larger envelope, populate the RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The main factor determining these differences is the coupling between the inner and outer regions, reflecting very different density profiles inside these stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' A second factor increasing the complexity of these spectra is the higher temperature of the less metallic stars, which decrease the lifetime of the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' In fact, solar-like oscillations in rHBs have also been detected in the K2 (Howell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2014) light curves of the globular cluster M4 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Wallace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2019), where the complexity of the spectra and the reduced observation time (80 days) have made it difficult to extract robust ⟨∆ν⟩ values (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Tailo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Howell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' rHB stars are well known and easily identified in globular clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Here we have also shown the ability of asteroseismol- ogy to identify these low-mass CHeB stars in the field and in solar-metallicity environments where, even with high-precision photometry, they would be hardly distinguishable from other stars in RC or RGB phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' It is clear that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='7 M⊙ stars, especially those of solar metallic- ity, must have followed a non-standard evolution during which they have lost a large amount of mass (see also Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Bobrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' This work provides us with a solid frame- work for the future study of these stars and of the processes that have led them to their current mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Knowing that is fundamen- tal in order to derive their ages with accuracy, and to potentially provide another piece of the puzzle in the sequence between RC and subdwarf B stars or other stripped stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' We are grateful to (in alphabetical order) Emma Willett, Joel Ong, Joris De Ridder, Masao Takata and Saniya Khan for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' We are also grateful to the anonymous referee for the constructive comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='int/gaia) and from the Two Micron All Sky Sur- vey (https://irsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='ipac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='edu/Missions/2mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='html).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' This research made use of Lightkurve, a Python package for Kepler and TESS data analysis (Lightkurve Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2018), and of dustmaps, a package for interstellar dust red- dening and extinction (Green 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' AM, AS, GC, JM, MM, MT acknowledge support from the ERC Consolidator Grant funding scheme (project ASTER- OCHRONOMETRY, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='asterochronometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='eu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 772293).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' MV acknowledge support from NASA grant 80NSSC18K1582.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Funding for the Stel- lar Astrophysics Centre is provided by The Danish National Research Founda- tion (Grant agreement No.' 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A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2018, ApJ, 859, 156 Kjeldsen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' & Bedding, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 1995, A&A, 293, 87 Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=', Bedding, T.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2021, A&A, 649, A4 Marconi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=', Coppola, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=', Bono, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2015, ApJ, 808, 50 Miglio, A.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=', Stello, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2011, ApJ, 743, 161 Yu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=', Huber, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=', Bedding, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2018, ApJS, 236, 42 Article number, page 7 of 9 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' main_AA Appendix A: Physical properties of the full sample In this appendix we give some details concerning the origin of the physical quantities in Tables 1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The latter comple- ments the former providing the properties of the rest of our sam- ple of rHB candidates (see also the HRD of the whole sample in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The global seismic parameters νmax and ⟨∆ν⟩ of targets tagged with an asterisk in Tables 1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1 are taken from Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' (2018), while those for the NGC 6819 cluster member (KIC 4937011, tagged with R) are from Handberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' For the rest of the sample, we employ the approach of Davies & Miglio (2016) and the value of ⟨∆ν⟩ is computed using individ- ual frequencies and the weighted fit of the asymptotic relation for radial modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' As discussed in Handberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' (2017), this method gives results in good agreement with the values of ⟨∆ν⟩ derived by Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' (2018) and allows a forward comparison with model-based values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The asymptotic period spacing of the dipole modes ∆Π1 and the coupling factor q are derived using the stretched-period method (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Vrard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The atmospheric parameters Teff and chemical composi- tion come from APOGEE-DR17, except for four targets with a STAR_BAD flag in that release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' For them (’+’ apex in Table 1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1) we adopt the available values in APOGEE-DR16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' To check the reliability of these atmospheric parameters and of the quoted uncertainties, we performed an independent analysis for the three stars in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' We used MOOG-synth5 with the as- sumption of local thermodynamic equilibrium, the linelist of APOGEE-DR17 (Shetrone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2021) im- plemented with lines from VALD database6 and MARCS model atmospheres (Gustafsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' We get results in good agreement with those in APOGEE-DR16/17 except for Teff un- certainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Even for the best situation in which log g is fixed to the seismic values (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Valentini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2019), the uncertainty on Teff is σTeff ∼ 50 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Therefore, although in Tables 1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1 we keep the values from APOGEE, we assume a minimum value of σTeff = 50 K in deriving stellar mass and its uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Bolometric luminosities L are estimated by combining as- trometry data from Gaia DR3 (Babusiaux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2022) with 2MASS photometry (Skrutskie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2006) in the Ks-band and bolometric correction from Casagrande & VandenBerg (2014, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' We applied the Gaia-DR3 parallax zero-point correction of Lindegren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' (2021) and estimated reddening and extinc- tion from the three-dimensional maps of Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The errors in L are calculated with a Markov chain Monte Carlo (MCMC) method considering fixed the extinction and the value of Mbol,⊙ (Mbol,⊙ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='75, Casagrande & VandenBerg 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Stellar masses, as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2, have been estimated using the scaling relation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1 and the values of L, Teff and νmax just described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' In the following paragraph we present the results obtained with an alternative scaling relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1: Stellar mass from scaling relation involving ⟨∆ν⟩ and νmax In order to test the mass estimations made with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1 of Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2, we employed the model-based corrected scaling relation (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Kjeldsen & Bedding 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Gai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2011) M M⊙ = f 4 ∆ν � Teff Teff,⊙ �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='5 � νmax νmax,⊙ �3 �⟨∆ν⟩⊙ ⟨∆ν⟩ �4 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1) 5 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='edu/ chris/moog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='html 6 http://vald.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='uu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='se 3500 4000 4500 5000 5500 6000 Teff [K] 101 102 103 L [L⊙] KIC8694070 KIC3626807 KIC12504765 KIC11072164 KIC6032981 KIC9691704 KIC2555126 KIC3428926 KIC9335415 KIC4937011 KIC5271626 KIC11299941 rHB RC RR Lyrae red edge M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='65 M⊙, [Fe/H] = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='00, [α/Fe] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='2 M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='50 M⊙, [Fe/H] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='00, [α/Fe] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='0 M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='75 M⊙, [Fe/H] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='00, [α/Fe] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1: The same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2, but including all the CHeB stars in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' These stars are colour-coded according to increasing [Fe/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' for two metal-rich stars (KIC5271626 and KIC4937011) and for two metal-poor stars (KIC6032981 and KIC11072164) of our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Here we used the solar reference values of Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2, and ⟨∆ν⟩⊙ = 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1 µHz (Huber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The correction fac- tor f∆ν on the ⟨∆ν⟩ scaling law (Ulrich 1986) is derived with the procedure described in Rodrigues et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' (2017), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' by us- ing the theoretical radial mode frequencies of stellar models to compute ⟨∆ν⟩ from the weighted linear fit of the asymptotic relation (see also Miglio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Tailo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' We based the iterative search for the correct f∆ν on evolutionary tracks with the same metallicity (within the errors) as the four stars aforementioned: solar composition for the metal-rich ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' [Fe/H] = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='00 with [α/Fe] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='2 and [α/Fe] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='4 for the two metal-poor ones (see Appendix B for details on the models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' To correct the model-predicted ⟨∆ν⟩ from the surface effects, we included ⟨∆ν⟩⊙ = 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='3 µHz of our solar-calibrated model to the correction factor f∆ν (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Finally, we com- puted the theoretical radial oscillations with the tool GYRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The f∆ν we found are nearly equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='03 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='01 for the metal-poor and for the metal-rich stars respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' In deriving the masses with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1, we considered a minimum error of 50 K in Teff (as said previously in Appendix A), and an error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='01 on f∆ν due to the impossibility of knowing the exact position, at fixed νmax, of our observed stars along the evolutionary tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' There- fore, these masses are compatible within the errors with those derived from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' We want also to notice that it is difficult to have a very precise ⟨∆ν⟩ estimate for these stars, because the radial modes are located in crowded regions (see Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' This leads to systematic errors in the measurement of individual radial modes that can be of the order of 4% by mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Article number, page 8 of 9 Matteuzzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' : rHBs as viewed by asteroseismology Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1: Physical properties for the rest of our sample of rHB candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' KIC L [L⊙] Teff [K] [Fe/H] [α/Fe] ⟨∆ν⟩ [µHz] νmax [µHz] q ∆Π1 [s] M [M⊙] 2555126 41 ± 4 5320 ± 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='26 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='03 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='93 280 ± 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='06 3428926+ 36 ± 3 5560 ± 130 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='27 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='02 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='15 270 ± 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='07 3626807 50 ± 6 5310 ± 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='26 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='276 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='011 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='69 308 ± 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='10 9335415+ 46 ± 4 5580 ± 120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='808 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='018 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='9 ± 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='013 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='23 334 ± 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='11 11072164 43 ± 4 5215 ± 18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='761 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='012 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='11 300 ± 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='06 11299941∗ 32 ± 3 4585 ± 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='09 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='0 ± 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='010 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='65 340 ± 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='10 4937011R 37 ± 4 4707 ± 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='03 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='10 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='53 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='08 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' It includes also the properties of KIC4937011 (undermassive star in NGC 6819, tagged with a R in apex), for which we show ⟨∆ν⟩, νmax, and M from Handberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' See Table 1 for a description of the symbols Appendix B: Grids of stellar models As mentioned in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 3, we chose three sets of stellar param- eters to represent a rHB star, a metal-rich low-mass CHeB star, and a RC star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The stellar models at the base of this work belong to a grid of stellar evolutionary models computed with the code MESA-r11532 (Modules for Experiments in Stellar Astrophysics Paxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2011, 2013, 2015, 2016, 2018, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' In the compu- tation we follow the evolution from the pre-main sequence phase until the first thermal pulse in the asymptotic giant branch for stellar masses from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='6 M⊙ till 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='00 M⊙, with a step of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='05 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' We consider 36 different chemical composition, with 12 values of [Fe/H] (from -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='5 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='25) and three values of alpha-elements enhancement: [α/Fe] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' We adopt as refer- ence solar mixture that from Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' (2009) and high- and low-temperature radiative opacity tables have been computed for these specific metal mixtures, solar and alpha-enhanced ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Envelope convection is described by the Mixing Length theory Cox & Giuli (1968) and the corresponding αMLT parameter, the same for all the grid, is derived from the solar calibration with the same physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' We add below the convective envelope a diffusive undershooting (Herwig 2000) with a size parameter f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='02 (see Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Extra-mixing over the convective core limit during the central-He burning phase is treated following the formalism by Bossini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Appendix C: Contribution of individual eigenmodes to the PSDs of CHeB stars In this section we break down the PSDs of our reference models (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 5) into the contributions from the modes of the different angular degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The smoothed PSDs for ℓ = 0, 1, 2, 3 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The smoothing is chosen just for showing purposes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' to resemble a Lorentzian fit of each eigenmode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The mod- ulation around the p-like mode in the dipole modes of the RC star and the higher number of observed mixed modes in the rHB model are evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Furthermore, the quadrupole modes of the rHB model are less visible than those of the RC model, and its octupole modes resemble a continuous background with small peaks almost coinciding with the radial modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Finally, we want to notice that the presence, in rHB stars, of ℓ = 1, 2, 3 modes very close to the radial ones (in some cases almost coinciding, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1) could introduce a non-negligible influence on the analysis of the heights and the linewidths of the ℓ = 0 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' 30 35 40 45 50 55 60 Frequency [µHz] 0 2000 4000 6000 8000 10000 12000 PSD [ppm2 µHz ] rHB ℓ = 0 ℓ = 1 ℓ = 2 ℓ = 3 νmax 30 35 40 45 50 55 60 Frequency [µHz] 0 5000 10000 15000 20000 PSD [ppm2 µHz ] RC ℓ = 0 ℓ = 1 ℓ = 2 ℓ = 3 νmax Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='1: Smoothed version of the PSDs presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Here we show the individual degrees for the rHB (top) and RC (bottom) simulated stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' The dashed cyan line is the correspond- ing νmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} +page_content=' Article number, page 9 of 9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FAT4oBgHgl3EQf-B7W/content/2301.08761v1.pdf'} diff --git a/iNE0T4oBgHgl3EQfpgG_/vector_store/index.faiss b/iNE0T4oBgHgl3EQfpgG_/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..843b98f6feeb45e372acdcad9f14c51324e527a0 --- /dev/null +++ b/iNE0T4oBgHgl3EQfpgG_/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f4c94f951e59a42be7a8ae94602a61cc104e4dd222dd5b543908fc6ad96418c7 +size 2818093 diff --git a/iNE4T4oBgHgl3EQfrw2W/content/tmp_files/2301.05211v1.pdf.txt b/iNE4T4oBgHgl3EQfrw2W/content/tmp_files/2301.05211v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..df36a5ec7a736cf7768151fd9067b963018c8e88 --- /dev/null +++ b/iNE4T4oBgHgl3EQfrw2W/content/tmp_files/2301.05211v1.pdf.txt @@ -0,0 +1,1106 @@ +Accidental Light Probes +Hong-Xing Yu1 +Samir Agarwala1 +Charles Herrmann2 +Richard Szeliski2 +Noah Snavely2 +Jiajun Wu1 +Deqing Sun2 +1Stanford University +2Google Research +Abstract +Recovering lighting in a scene from a single image is +a fundamental problem in computer vision. While a mir- +ror ball light probe can capture omnidirectional lighting, +light probes are generally unavailable in everyday images. +In this work, we study recovering lighting from accidental +light probes (ALPs)—common, shiny objects like Coke cans, +which often accidentally appear in daily scenes. We propose +a physically-based approach to model ALPs and estimate +lighting from their appearances in single images. The main +idea is to model the appearance of ALPs by photogram- +metrically principled shading and to invert this process via +differentiable rendering to recover incidental illumination. +We demonstrate that we can put an ALP into a scene to allow +high-fidelity lighting estimation. Our model can also recover +lighting for existing images that happen to contain an ALP*. +I’d rather be Shiny. — Tamatoa from Moana, 2016 +1. Introduction +Traditionally, scene lighting has been captured through +the use of light probes, typically a chromium mirror ball; +their shape (perfect sphere) and material (perfect mirror) +allow for a perfect measurement of all light that intersects +the probe. Unfortunately, perfect light probes rarely appear +in everyday photos, and it is unusual for people to carry +them around to place in scenes. Fortunately, many everyday +objects share the desired properties of light probes: Coke +cans, rings, and thermos bottles are shiny (high reflectance) +and curved (have a variety of surface normals). These ob- +jects can reveal a significant amount of information about +the scene lighting, and can be seen as imperfect “accidental” +light probes (e.g., the Diet Pepsi in Figure 1). Unlike perfect +light probes, they can easily be found in casual photos or +acquired and placed in a scene. In this paper, we explore us- +ing such everyday, shiny, curved objects as Accidental Light +Probes (ALPs) to estimate lighting from a single image. +*Project website: https://kovenyu.com/ALP +Figure 1. (Left) From an image that has an accidental light probe +(a Diet Pepsi can), we insert a virtual object (a Diet Coke can) +with estimated lighting using the accidental light probe (Middle), +and using estimated lighting from a recent state-of-the-art lighting +estimation method [49] (Right). Note how our method better re- +lights the inserted can to produce an appearance consistent with the +environment. +In general, recovering scene illumination from a single +view is fundamental for many computer vision applications +such as virtual object insertion [9], relighting [46], and pho- +torealistic data augmentation [51]. Yet, it remains an open +problem primarily due to its highly ill-posed nature. Images +are formed through a complex interaction between geometry, +material, and lighting [21], and without precise prior knowl- +edge of a scene’s geometry or materials, lighting estimation +is extremely under-constrained. For example, scenes that +consist primarily of matte materials reveal little information +about lighting, since diffuse surfaces behave like low-pass +filters on lighting during the shading process [38], eliminat- +ing high-frequency lighting information. To compensate for +the missing information, the computer vision community has +explored using deep learning to extract data-driven priors for +lighting estimation [14,44]. However, these methods gener- +ally do not leverage physical measurements to address these +ambiguities, yet physical measurements can offer substantial +benefits in such an ill-posed setting. +For images with ALPs, we propose a physically-based +1 +arXiv:2301.05211v1 [cs.CV] 12 Jan 2023 + +modeling approach for lighting estimation. The main idea is +to model the ALP appearance using physically-based shad- +ing and to invert this process to estimate lighting. This +inversion process involves taking an input image, estimating +the ALP’s 6D pose and scale, and then using the object’s +surface geometry and material to infer lighting. Compared to +purely data-driven learning approaches that rely on diverse, +high-quality lighting datasets, which are hard to acquire, our +physically-based approach generalizes to different indoor +and outdoor scenes. +To evaluate this technique, we collect a test set of real +images, where we put ALPs in daily scenes and show that +our approach can estimate high-fidelity lighting. We also +demonstrate lighting estimation and object insertion based +on existing images (Figure 1). +In summary, we make the following three contributions: +• We propose the concept of accidental light probes +(ALPs), which can provide strong lighting cues in ev- +eryday scenes and casual photos. +• We develop a physically-based approach for lighting +estimation for images with an ALP and show improved +visual performance compared to existing light estima- +tion techniques. +• We collect a dataset of ALPs and a dataset of images +with ALPs and light probes in both indoor and out- +door scenes. We demonstrate that our physically-based +model outperforms existing methods on these datasets. +2. Related Work +Lighting estimation. A traditional light probe (i.e., a metal +mirror ball) can capture omnidirectional lighting [9]. How- +ever, light probes are usually not present in existing im- +ages and people do not often carry mirror balls around +or place them in scenes when taking pictures. Hence, re- +searchers have attempted to use everyday objects like human +faces [6, 23, 27, 46, 56] and eyes [33] to estimate lighting. +These methods focus on predicting lighting in portrait images +where human faces are the main subject; in contrast, we tar- +get images with everyday, high-reflectance objects. Another +line of research focuses on lighting estimation from known, +non-reflective objects. Both Weber et al. [53] and Park et +al. [34] learn to regress illumination directly from images of +homogeneous-material objects. Wei et al. [54] extend object- +based illumination regression to spatially-varying materials. +They train object-specific deep networks that require large, +diverse lighting data to generalize to novel scenes. Other +approaches use a “scan” of the objects in a scene, relying +on RGBD video. Park et al. [35] estimate scene lighting +for novel view synthesis from an RGBD video of a shiny +object like a bag of chips; Richter-Trummer et al. [39] use +an RGBD video of an arbitrarily-shaped object. We instead +focus on lighting estimation from a single RGB image. +In addition to object-based lighting estimation, another +popular line of work focuses on learning lighting estima- +tion directly from images of scenes [12–14,43,44,58,59]. +Many of these methods rely heavily on supervised training +on synthetic data. As a result, they are sensitive to domain +shifts between training and test data and, in particular, suffer +from a synthetic-to-real domain gap. In contrast, our ap- +proach is based on physically principled modeling and is not +vulnerable to this issue. +Inverse rendering. Our approach is closely related to in- +verse rendering methods that aim to jointly recover geom- +etry, material, and lighting from images. Recent work in +this area uses multi-view observations of an object with +known camera poses to recover scene lighting and object +properties [18,32]. These methods jointly optimize geom- +etry, material, and lighting and generalize to diverse scene +settings. However, in single-view settings, the optimization +problem for inverse rendering is highly ill-posed, and these +methods often produce degenerate solutions. +Learning-based inverse rendering techniques have also +gained popularity in material and geometry estimation tasks +[30,42,52,57,61]. These methods include differential render- +ing as part of their training pipeline and can learn priors to +model geometry and materials of scenes and objects. How- +ever, they are limited in their ability to generalize to a diverse +set of scenes. +Material reconstruction. Material modeling and recon- +struction have a long history in computer vision and graphics. +Early papers [4, 15, 47] developed early analytical models +of material reflection based on general experimental obser- +vations. More recent works [8, 20, 28] have attempted to +directly solve for a general bidirectional reflectance distribu- +tion function (BRDF), which analytically defines how light +is reflected at a given point on an object’s surface; however, +many of these techniques fail for highly specular or curved +objects. For example, traditional BRDF acquisition [10,31] +requires a gonioreflectometer, which tries to precisely mea- +sure reflectance at different angles. This machine typically +runs on flat objects and struggles on curved objects like Diet +Coke cans. Modern approaches [62] use RGBD sensors +and joint optimization on differentiably rendered objects and +multi-view images [18, 32, 60]; our approach builds upon +differentiable rendering to optimize material reconstruction +and adapts them for ALPs. +3. Approach +Accidental Light Probes (ALPs) are daily metallic shiny +objects, such as a soda can, a thermoflask, or a ring. Given a +single image containing an ALP, we aim to recover the inci- +dental illumination by inverting physically-based rendering, +as shown in Fig. 2. Our main idea is that we can first acquire +the shape and reconstruct the spatially-varying BRDF of +2 + +Multiview capture +Geometry +Lighting +B) Single-image lighting estimation +Rendered +Differentiable +rendering +A) Offline ALP reconstruction (material estimation) +- +Geometry +Material +Lighting +- +Rendered +Single ALP image +Material +6D Pose + Scale +Optimizable +Fixed (ALP Model) +Differentiable +rendering +Optimizable +Fixed (Measured) +6D Pose + Scale +Figure 2. Our physically-based approach to lighting estimation consists of two stages, namely (A) offline Accidental Light Probe (ALP) +reconstruction and (B) inference-time single-image lighting estimation. For (A), we use a capture-optimization hybrid method to reconstruct +the ALP model with high fidelity. For (B), we formulate lighting estimation as a joint optimization of 7D pose (scale + 6D pose) and +environment lighting. +the ALP offline (Fig. 2 top), and then optimize incidental +lighting as well as the 6D pose of the ALP (Fig. 2 bottom). +3.1. Formulation +Our goal is to estimate high-fidelity lighting from the +appearance of an ALP in a single image. We approach this +goal through the perspective of inverse rendering, where the +forward process is described by the rendering equation [21]: +L(ωo) = +� +H +Li(ωi)f(ωi, ωo)(n · ωi)dωi, +(1) +where L(ωo) is the outgoing radiance to direction ωo (cor- +responding to pixel intensity), Li(ωi) is incidental radiance +from direction ωi (lighting), f is the bidirectional reflectance +distribution function (BRDF) at the surface location (ma- +terial), n is the normal direction (geometry), and H is the +upper hemisphere along the normal. Recovering lighting by +inverting Eqn 1 is a highly ill-posed problem, as infinitely +many combinations of geometry, material, and lighting can +generate the same appearance in the image. Fortunately, for +ALPs, we can pre-acquire prior physical knowledge of their +shapes and materials as they are everyday objects. Thus, +we can reduce the full inverse rendering problem to a joint +estimation of 6D ALP pose and lighting, which is relatively +more constrained and tractable: +min +π,Li L +� +Irender(π, Li|f, S), Iref +� +, +(2) +where Irender is generated by a differentiable renderer that +takes the shape S (represented by a mesh), the 6D pose of the +ALP π, the spatially-varying BRDF f, and the environment +lighting Li as inputs. Iref denotes the observed single image. +L denotes an image-space loss that we define in Section 3.3. +Our physically-based formulation entails the high-fidelity +acquisition of shape and spatially-varying material of the +ALP, as well as a robust single-view joint optimization al- +gorithm. We show an overview in Fig. 2 and elaborate the +components in Section 3.2 and Section 3.3, respectively. +Shading model. +We adopt physically-based rendering +(PBR) [36] due to its principled photogrammetry and ra- +diometry. Specifically, we consider metallic materials as +they have little diffuse reflection. Diffuse reflection is unde- +sirable as it behaves like a low-pass filter of lighting in the +shading process [38], eliminating the physically recoverable +lighting information. To model metallic material, we use a +microfacet model [47] with a GGX distribution [48]: +f(ωi, ωo) = +D · F · G +4(n · ωi)(n · ωo), +(3) +where D is the GGX normal distribution [48], F is the +Fresnel reflection, and G is the geometric attenuation. We +adopt Disney’s parameterization [5], where the metallic +material is modeled by its specular albedo A and rough- +ness r. +Specifically, the specular albedo A is used to +model Fresnel reflection by Schlick’s approximation [40] +F = A + (1 − A)(1 − |h · ωo|)5, where h = +ωi+ωo +|ωi+ωo| de- +notes the half vector. The roughness r controls the shape +of the specular reflection lobe via the micro-normal distri- +bution D = +r4 +π(|n·h|(r4−1)+1)2 and the geometric attenuation +G = +2|n·ωi| +|n·ωi|+√ +r4+(1−r4)|n·ωi|2 · +2|n·ωo| +|n·ωo|+√ +r4+(1−r4)|n·ωo|2 . +Lighting model. To recover lighting for arbitrary conditions, +we use an environment map to represent omnidirectional +lighting and adopt image-based lighting for shading each +3 + +ke +ALORIES +12FLOZ +(355ml)RECYCLE ME +ORQILOLDiet +ke +0 +TES +12FLOZ +(355ml)RECYCLEMEok +NOCALORIES +NO SUGARoK +NOCRLOAIESY +7 +X1ZFLOZ +DietCo +RECYCLE ME +O2021TH COCACOIACOMPANY +AFFEINECONTENT:46mg/121ol +PHENYLKETONURICS CONTAINSPHENYIAL +Opixel. For efficiency, we only consider direct lighting and +use a differentiable rasterizer with deferred shading [24] to +render Irender. This is inaccurate for concave objects with +self-occlusion and self-reflections. To mitigate this without +expensive global illumination, we include a soft visibility +term to Eqn 1 to approximate it such that the shading output +is modulated as vL(ωo), where v denotes the soft visibility +that is optimized and treated as a surface texture. +3.2. Reconstructing ALPs +Recovering high-fidelity lighting by physically-based in- +verse rendering requires high-quality geometry and mate- +rial reconstruction of the ALPs. While existing state-of- +the-art inverse rendering methods can jointly optimize for +geometry, material, and lighting from dense multi-view im- +ages [18, 32, 60], they still struggle for real metallic ob- +jects under arbitrary lighting due to high specularity (Fig. 4). +Moreover, several challenges exist when the goal is not view +synthesis but photogrammetrically correct reconstruction. +For highly specular objects such as metallic ones, the re- +flected lights from the near field can lead to environment +baking, as it breaks the distant light assumption (we show an +example of the environment-baked material reconstruction +in the supplementary material). The color ambiguity of mate- +rial albedo and lighting is also not resolved. In addition, the +geometry reconstruction quality heavily relies on the quality +of object silhouettes in multi-view images. +To overcome these challenges, we reconstruct ALPs by a +hybrid method. First, we use a light box with a turntable to +control environment lighting for multi-view capture, and us- +ing a thin supporting stand to alleviate near-field reflections +(setup shown in Fig. 3) and environment baking. Second, in- +stead of optimizing the incidental lighting to the ALP under +capture, we record it by a calibrated light probe to remove +the color ambiguity between material and lighting. And +third, we provide a high-quality shape using a range scan- +ner [1] to reduce the geometry reconstruction down to 6D +pose and size fitting. Thus, as demonstrated in the top row of +Figure 2, our ALP reconstruction is cast as an optimization +for its spatially-varying material and shape fitting: +min +π,α,f +� +{Icapture} +L +� +Irender(π, α, f|Li, S), Icapture +� +, +(4) +where π and α are the 6D pose and size to fit the +shape S to multi-view camera coordinate frame solved by +COLMAP [41], and f is the material parameterized by +spatially-varying albedo A and roughness r. We show the +reconstruction of a Coke can in Fig. 4. We include the opti- +mization and loss details in the supplementary material. +3.3. Single-view physically-based light estimation +Given an image containing an ALP, we first extract an +object segmentation mask for the ALP by manually crop- +Figure 3. (Left) We use a light box with controllable lighting for +our capture. To mitigate near-field reflections, we leverage a thin +stand to support the object. (Right) To minimize environmental +changes due to camera and photographer movement, we cover the +lightbox with a cloth and use a turntable for multi-view capture. +ping the image and then using an off-the-shelf foreground +segmentation tool [2]; however, this could alternately be +obtained by object detection [7] with salient object segmen- +tation [37] or semantic segmentation [45]. We then retrieve +the appropriate ALP model, containing its reflectance and +geometric information. Yet, even given the ALP’s 3D model +and 2D segmentation in the input image, accurately aligning +these two elements is still challenging. Traditional feature +point detection and Perspective-n-Point methods do not work +on textureless objects such as rings and thermoflasks. Addi- +tionally, modern learning-based single-view pose estimation +methods [29,50,55] require diverse, realistic lighting to syn- +thesize training data and do not generalize well outside the +training distribution. +Therefore, we formulate the lighting estimation and pose +estimation as a joint estimation problem in Eqn 2, and we +solve it via a differentiable rendering-based optimization +which is generalizable to arbitrary scenes for both textured +and textureless objects (see the bottom of Figure 2). Here +we need a joint estimation as the appearance of a specular +object (and thus the differentiable rendering gradient sig- +nals) is highly dependent on both the object pose and the +environment lighting. We use Monte Carlo ray tracing with +Visible Normal Distribution Function (VNDF) importance +sampling [19] for unbiased shading. +Losses and regularizations. +Our loss function used in +Eqn 2 is given by: +L = LRGB + Lmask +λ1 Lpose-reg +λ2 Llight-reg, +(5) +where LRGB denotes a L1 loss on RGB images, Lmask de- +notes a combination of a L1 loss and a Chamfer loss on +masks [3], where the mask is given by the differentiable +rasterizer [24]. Lpose-reg and Llight-reg denote a pose regular- +ization and a lighting regularization with their weights λ1 +and λ2, respectively. +Without multi-view constraints, the joint optimization +problem has multiple local minima for the 6D pose; thus, we +introduce a pose regularization and a lighting regularization. +4 + +pepsMENU +C3 +F10四±0.0150100 +1/10Nvdiffrec +Nvdiffrecmc +Ours +Albedo Roughness Normal +Albedo Roughness Normal +Albedo +Roughness Normal +Figure 4. Visual comparison of ALP reconstruction from state- +of-the-art optimization-based inverse rendering methods [18,32] +versus our hybrid method. Recent inverse rendering methods strug- +gle on real textured metallic objects. +The pose regularization is given by: +Lpose-reg = ∥B(Mrender) − B(Mref)∥2 +2 + ∥q − qref∥2 +2, +(6) +where Mrender is the rendered mask, B(Mrender) denotes the +pixel-space barycenter of the mask, q denotes the quaternion +representation of the ALP orientation, and qref denotes a com- +mon orientation (we use a front-facing canonical orientation +obtained by aligning principal axes). The barycenter term +prevents vanished gradient due to non-overlapping pose ini- +tialization, and the orientation term prevents hard-to-escape +local minima like upside-down cans. We decay the weight +of the pose regularization to zero through optimization. +To accurately estimate omnidirectional lighting by invert- +ing Eqn 1, we need to evaluate the Monte Carlo integral +interval densely over light rays coming from all directions. +However, from a single view, an ALP often only covers a +limited subset of normal directions compared to a perfect +sphere. Thus, light rays coming from a certain subset of +directions contribute little to the appearance of the ALP. +These directions are then under-sampled, and the lighting +estimation for them is less informed and unconfident. +To mitigate this, we introduce a lighting smoothness reg- +ularization which “fills in” the less confident regions in the +environment map by propagating the confident information +from nearby directions. The regularization is given by: +Llight-reg = ∥Li(ω) − Li(ω + ∆ω)∥1, +(7) +where ∆ω denotes a small deviation of a solid angle sampled +from a normal distribution, and ω is sampled uniformly in +all solid angles. Note that in addition to propagating confi- +dent lighting estimates, the lighting regularization also helps +improve pose estimation, since many pose estimation errors +come from trying to fix mistakes in high-frequency [17] +lighting changes, which light regularization alleviates. +4. Experiments +4.1. Setup +Accidental Light Probes Dataset. We acquire 5 common +accidental light probes that have different shapes or sptially- +varying BRDFs, including 3 soda cans (diet Coke, diet Pepsi, +Diet Coke +Cleaner +Sprite +Thermos cap +Diet Pepsi +Figure 5. Close up of our ALP dataset. +and Sprite), a thermos cap, and a solder tip cleaner. We show +example images in Figure 5. +Evaluation Dataset. +We collect a dataset of 10 indoor +scenes and 13 outdoor scenes. The indoor and outdoor +scenes are taken at different points of time, such as day +and night, at different locations. We show examples in Fig- +ure 6. We place each of our ALPs in the scenes and capture +HDR images of the ALPs. We also capture ground-truth +lighting by placing a chromium ball (a perfect light probe) +in the scene. +Baselines. We compare our method to several state-of-the- +art lighting estimation methods [12, 14, 49]. Unlike our +method, all of these techniques utilize deep learning. Since +[12, 14] do not have publicly available models, we asked +their authors to run inference on our dataset. +4.2. Comparison to Baseline Methods +Qualitative Results. For all object insertion comparisons, +we compute an environment map either through an ALP +with our proposed method or by running the other baselines +on a single image of the scene. Note that for the baselines, +we use the image with the perfect light probe as input; this +should provide a slight advantage to these techniques since +the image with the perfect light probe contains the most +information regarding scene lighting. +In Figure 7, we insert various objects into the scene and +relight them using the computed environment maps; we then +qualitatively compare the results. We demonstrate that our +computed environment map produces significantly more ac- +curate and compelling results than other single-image light- +ing estimation methods. In particular, note that our method +is the only approach that can recover the overall tone of the +lighting: other methods are either too yellow or gray. +In Figure 8, we show relighting results on perfect spheres +of various finishes from all methods and ALPs on both indoor +and outdoor scenes. Only our technique produces results +similar to the ground truth for mirror finishes. We note that +for all the three soda cans, the relighting on mirror spheres +are slightly blurry, since their materials are much rougher +than perfect mirror, which behaves as low-pass filters of +lighting in the shading process [38]. We also note that for +Sprite and diet Coke, there is some texture color baking in +the recovered lighting due to imperfectly aligned 6D poses, +which lead to high-frequency lighting artifacts to compensate +5 + +loke +D3Soke +DietDiet +oke +NO SUGAR +12FL0Z +NO CALORIES +(355ml)OLEMON-LIME +pritpepsiDiet +keFigure 6. Examples of our collected dataset for evaluating lighting estimation under different illumination conditions, including indoor and +outdoor scenes at daytime and nighttime. +Groundtruth lighting +DeepParam +Ours +StyleLight +Garon et.al. +Input image +Figure 7. Object insertion results on our test scenes for both indoor (first two rows) and outdoor (last two rows). We compare to Garon et +al. [14], Deep parametric lighting [12], and StyleLight [49]. We center-crop the result images for better visualization. +the pixel-space misalignment. Our lighting regularization +mitigates this type of artifacts, yet a highly robust algorithm +remains as future work. +Quantitative Results. In Table 1, we report quantitative +results on relighting perfect spheres with various representa- +tive materials (mirror, shiny, diffuse). Similar to [26,49], we +compute angular error [11] and scale-invariant RMSE [16] +to compare the relighted spheres from each technique to the +ground truth relighting. +Quantitatively, for the relighting task, our method, applied +to any of the ALPs, significantly outperforms the baselines. +In particular, w.r.t. angular error, the Thermos cap provides +a 3 to 4 times improvement over the best baseline. +4.3. Analysis +Capture Setup. We also analyze the quality of our recon- +struction compared to two recent multi-view inverse render- +ing methods, Nvdiffrec [32] and Nvdiffrecmc [18] using our +lightbox capture setups. Table 2 shows the results of using +our lighting estimation pipeline with various reconstructions +of a Diet Coke can. Our reconstruction performs the best +and leads to a decrease of angular error by 20% or more. +Both Nvdiffrec and Nvdiffrecmc are normally applied to +multiview casual images, so for completeness, we also com- +pute reconstructions and quantitative results for this setting +(included in the supplementary materials). These reconstruc- +tions perform strictly worse than those computed from the +lightbox setup. We also show a qualitative comparison of the +geometry and materials in Figure 4, of each technique in its +default setting, where ours are clearly better than the alter- +native methods. Table 2 shows that our ALP reconstruction +pipelines give us better results than using current state-of-art +inverse rendering methods to get our ALP models. +Ablation for 6D Pose + Scale Estimation. As mentioned +in Sec. 3.3, the pose estimation problem for aligning a 3D +model of an ALP and its appearance in a real image is chal- +lenging. The appearance of the object in the real image +6 + +lok +ant,oKokloklokMethod +Indoor +Outdoor +Angular Error↓ +Scale-invariant RMSE↓ +Angular Error↓ +Scale-invariant RMSE↓ +Mirror +Shiny +Diffuse +Mirror +Shiny +Diffuse +Mirror +Shiny +Diffuse +Mirror +Shiny +Diffuse +StyleLight [49] +12.572 +7.700 +5.949 +3.087 +0.837 +0.264 +15.088 +9.830 +8.539 +1.867 +0.918 +0.294 +Deep Param. [12] +7.204 +6.252 +6.166 +3.137 +0.958 +0.287 +8.803 +7.228 +6.525 +1.963 +1.056 +0.305 +Garon et al. [14] +9.403 +8.215 +6.626 +3.030 +0.754 +0.207 +8.062 +6.873 +6.118 +1.706 +0.766 +0.237 +Cleaner +5.682 +4.550 +3.965 +2.204 +0.252 +0.073 +6.395 +4.920 +5.155 +1.081 +0.245 +0.101 +Diet Coke +4.733 +3.405 +3.067 +2.901 +0.550 +0.101 +6.011 +3.877 +2.587 +1.460 +0.501 +0.136 +Diet Pepsi +3.972 +2.712 +2.190 +2.726 +0.408 +0.064 +4.890 +2.830 +1.472 +1.352 +0.396 +0.108 +Sprite +5.952 +4.445 +3.767 +2.913 +0.556 +0.112 +7.023 +4.923 +3.892 +1.468 +0.513 +0.154 +Thermos cap +3.744 +2.080 +1.622 +2.555 +0.288 +0.057 +3.965 +2.159 +1.516 +1.092 +0.217 +0.053 +Table 1. Comparison to state-of-the-art single image lighting estimation methods: StyleLight [49], Deep Parametric [12] and Garon et +al [14]. We evaluate them using relighting on different materials. +StyleLight +Deep +Parametric +Garon +et al +GT +Diet +Pepsi +Diet +Coke +Sprite +Cap +Cleaner +Input +Mirror +Shiny +Diffuse +Input +Mirror +Shiny +Diffuse +Input +Mirror +Shiny +Diffuse +Figure 8. Qualitative comparison of relighting results in outdoor (left and center) and indoor (right) scenes. We compare our approach to +StyleLight [49], Deep Parametric [12] and Garon et al [14] on relighting mirror, shiny and diffuse spheres. +depends on both its pose and lighting; trying to jointly opti- +mize these can introduce potential failure cases. In Sec 3.3, +we describe several design choices w.r.t. the optimization and +loss which address some of these failures cases. In Table 3, +we perform an ablation study on each of these decisions and +report quantitative results. We show that all design decisions +(Silhouette loss, Chamfer loss [3], joint optimization, pose, +and light regularization) contribute to the final overall perfor- +mance. We also show representative examples in Figure 9. +They demonstrate how each design choice helps the pose +estimation, which in return helps lighting estimation. +Visualizing confident regions for ALPs. As briefly dis- +cussed in Sec. 3.3, an ALP has a subset of surface normals +compared to a perfect sphere light probe, which leads to +under-sampled lighting directions. For example, cylindri- +cal objects (Diet Coke can, ring, etc.) tend to sample well +light rays perpendicular to the can while significantly under- +sampling light rays above and below the can. Since we +7 + +okeOkekeMO-LIME +priteOurs +Ours w/o +Joint +optimization +Groundtruth +Ours +Ours w/o +Light +regularization +Groundtruth +Ours +Ours w/o +Pose +regularization +Groundtruth +Rendering +Environment Map +Rendering +Environment Map +Rendering +Environment Map +Figure 9. Qualitative ablation of the losses we use in our method. Each of our design choices contributes to improvements in pose and +lighting optimization which can be observed qualitatively. +Method +Mirror +Shiny +Diffuse +Nvdiffrec [32] +6.99 +5.06 +3.59 +Nvdiffrecmc [18] +6.55 +4.60 +3.84 +ALP (Ours) +5.46 +3.67 +2.80 +Table 2. Evaluation on our ALP model acquisition for a Diet Coke +can using our lightbox setup. We compare our acquisition method +to Nvdiffrec [32] and Nvdiffrecmc [18]. We use the same lighting +estimation approach for compared methods and report average +angular error across all test scenes. +Method +Mirror +Shiny +Diffuse +Silhouette loss [3] +6.812 +4.976 +3.919 +Ours w/o joint optimization +5.401 +3.726 +3.044 +Ours w/o pose regularization +5.962 +4.180 +3.338 +Ours w/o light regularization +6.032 +3.647 +2.954 +Ours +5.291 +3.610 +2.923 +Table 3. Ablation study on our joint pose-lighting optimization. +We compare to a baseline that uses a silhouette loss and a Chamfer +loss [3], and variants of our approach. We show angular errors +averaged on all test scenes. +ALP image +Normal +map +Sphere +normal map +Observed +normals +Confident regions in the +environment map +Figure 10. Visualization of sampling directions for a diet Coke can. +See the text in 4.3 for a full description of these visualizations. +use VNDF importance sampling which aligns well with our +BRDF’s density lobe, we visualize a “confidence map” as +normalized sampling frequency. We show this confidence +map in Figure 10 for a representative ALP (i.e., Diet Coke). +This demonstrates that the visible surface of a Coke can from +a single view only under-samples lighting directions from +the top and the bottom. +Discussion. Our method shows strong promise for recover- +ing scene lighting from a single image containing an ALP. +One exciting potential application is improved image editing +for in-the-wild images; however, to enable this for any im- +age, we would either need to increase the number of ALPs +or explore methods that enable us to dynamically edit one of +the collected measurements (geometry or material). Another +limitation is that we assume our input is an HDR image, +generally not available for in-the-wild images. However, we +note that recent work has sought to convert LDR images to +HDR [22,25] and HDR has also become more available due +to support on recent phones. +5. Conclusion +In this paper, we introduce the use of accidental light +probes to estimate environmental lighting from single im- +ages. We do this by first scanning common 3D objects and +establishing their reflective properties using a controlled +lighting environment (a simple turntable and light box). We +then use differentiable rendering combined with a physically- +based rendering model to recover the unknown object pose +and environment lighting when the object is placed (or nat- +urally occurs) in an image. We create a new dataset of +materials and geometry for several common, shiny, curved +objects along with image showing these in a variety of in- +door and outdoor environments. 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State of the art on 3d reconstruction with rgb-d cameras. +In Computer graphics forum, volume 37, pages 625–652. +Wiley Online Library, 2018. 2 +10 + diff --git a/iNE4T4oBgHgl3EQfrw2W/content/tmp_files/load_file.txt b/iNE4T4oBgHgl3EQfrw2W/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4fbb7d7a711dff738750df580fe5577d4381b020 --- /dev/null +++ b/iNE4T4oBgHgl3EQfrw2W/content/tmp_files/load_file.txt @@ -0,0 +1,621 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf,len=620 +page_content='Accidental Light Probes Hong-Xing Yu1 Samir Agarwala1 Charles Herrmann2 Richard Szeliski2 Noah Snavely2 Jiajun Wu1 Deqing Sun2 1Stanford University 2Google Research Abstract Recovering lighting in a scene from a single image is a fundamental problem in computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' While a mir- ror ball light probe can capture omnidirectional lighting, light probes are generally unavailable in everyday images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' In this work, we study recovering lighting from accidental light probes (ALPs)—common, shiny objects like Coke cans, which often accidentally appear in daily scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We propose a physically-based approach to model ALPs and estimate lighting from their appearances in single images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' The main idea is to model the appearance of ALPs by photogram- metrically principled shading and to invert this process via differentiable rendering to recover incidental illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We demonstrate that we can put an ALP into a scene to allow high-fidelity lighting estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Our model can also recover lighting for existing images that happen to contain an ALP*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' I’d rather be Shiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' — Tamatoa from Moana, 2016 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Introduction Traditionally, scene lighting has been captured through the use of light probes, typically a chromium mirror ball;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' their shape (perfect sphere) and material (perfect mirror) allow for a perfect measurement of all light that intersects the probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Unfortunately, perfect light probes rarely appear in everyday photos, and it is unusual for people to carry them around to place in scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Fortunately, many everyday objects share the desired properties of light probes: Coke cans, rings, and thermos bottles are shiny (high reflectance) and curved (have a variety of surface normals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' These ob- jects can reveal a significant amount of information about the scene lighting, and can be seen as imperfect “accidental” light probes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=', the Diet Pepsi in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Unlike perfect light probes, they can easily be found in casual photos or acquired and placed in a scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' In this paper, we explore us- ing such everyday, shiny, curved objects as Accidental Light Probes (ALPs) to estimate lighting from a single image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Project website: https://kovenyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='com/ALP Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' (Left) From an image that has an accidental light probe (a Diet Pepsi can), we insert a virtual object (a Diet Coke can) with estimated lighting using the accidental light probe (Middle), and using estimated lighting from a recent state-of-the-art lighting estimation method [49] (Right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Note how our method better re- lights the inserted can to produce an appearance consistent with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' In general, recovering scene illumination from a single view is fundamental for many computer vision applications such as virtual object insertion [9], relighting [46], and pho- torealistic data augmentation [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Yet, it remains an open problem primarily due to its highly ill-posed nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Images are formed through a complex interaction between geometry, material, and lighting [21], and without precise prior knowl- edge of a scene’s geometry or materials, lighting estimation is extremely under-constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' For example, scenes that consist primarily of matte materials reveal little information about lighting, since diffuse surfaces behave like low-pass filters on lighting during the shading process [38], eliminat- ing high-frequency lighting information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' To compensate for the missing information, the computer vision community has explored using deep learning to extract data-driven priors for lighting estimation [14,44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' However, these methods gener- ally do not leverage physical measurements to address these ambiguities, yet physical measurements can offer substantial benefits in such an ill-posed setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' For images with ALPs, we propose a physically-based 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='05211v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='CV] 12 Jan 2023 modeling approach for lighting estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' The main idea is to model the ALP appearance using physically-based shad- ing and to invert this process to estimate lighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' This inversion process involves taking an input image, estimating the ALP’s 6D pose and scale, and then using the object’s surface geometry and material to infer lighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Compared to purely data-driven learning approaches that rely on diverse, high-quality lighting datasets, which are hard to acquire, our physically-based approach generalizes to different indoor and outdoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' To evaluate this technique, we collect a test set of real images, where we put ALPs in daily scenes and show that our approach can estimate high-fidelity lighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We also demonstrate lighting estimation and object insertion based on existing images (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' In summary, we make the following three contributions: We propose the concept of accidental light probes (ALPs), which can provide strong lighting cues in ev- eryday scenes and casual photos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We develop a physically-based approach for lighting estimation for images with an ALP and show improved visual performance compared to existing light estima- tion techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We collect a dataset of ALPs and a dataset of images with ALPs and light probes in both indoor and out- door scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We demonstrate that our physically-based model outperforms existing methods on these datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Related Work Lighting estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' A traditional light probe (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=', a metal mirror ball) can capture omnidirectional lighting [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' How- ever, light probes are usually not present in existing im- ages and people do not often carry mirror balls around or place them in scenes when taking pictures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Hence, re- searchers have attempted to use everyday objects like human faces [6, 23, 27, 46, 56] and eyes [33] to estimate lighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' These methods focus on predicting lighting in portrait images where human faces are the main subject;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' in contrast, we tar- get images with everyday, high-reflectance objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Another line of research focuses on lighting estimation from known, non-reflective objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Both Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' [53] and Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' [34] learn to regress illumination directly from images of homogeneous-material objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' [54] extend object- based illumination regression to spatially-varying materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' They train object-specific deep networks that require large, diverse lighting data to generalize to novel scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Other approaches use a “scan” of the objects in a scene, relying on RGBD video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' [35] estimate scene lighting for novel view synthesis from an RGBD video of a shiny object like a bag of chips;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Richter-Trummer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' [39] use an RGBD video of an arbitrarily-shaped object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We instead focus on lighting estimation from a single RGB image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' In addition to object-based lighting estimation, another popular line of work focuses on learning lighting estima- tion directly from images of scenes [12–14,43,44,58,59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Many of these methods rely heavily on supervised training on synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' As a result, they are sensitive to domain shifts between training and test data and, in particular, suffer from a synthetic-to-real domain gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' In contrast, our ap- proach is based on physically principled modeling and is not vulnerable to this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Inverse rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Our approach is closely related to in- verse rendering methods that aim to jointly recover geom- etry, material, and lighting from images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Recent work in this area uses multi-view observations of an object with known camera poses to recover scene lighting and object properties [18,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' These methods jointly optimize geom- etry, material, and lighting and generalize to diverse scene settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' However, in single-view settings, the optimization problem for inverse rendering is highly ill-posed, and these methods often produce degenerate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Learning-based inverse rendering techniques have also gained popularity in material and geometry estimation tasks [30,42,52,57,61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' These methods include differential render- ing as part of their training pipeline and can learn priors to model geometry and materials of scenes and objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' How- ever, they are limited in their ability to generalize to a diverse set of scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Material reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Material modeling and recon- struction have a long history in computer vision and graphics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Early papers [4, 15, 47] developed early analytical models of material reflection based on general experimental obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' More recent works [8, 20, 28] have attempted to directly solve for a general bidirectional reflectance distribu- tion function (BRDF), which analytically defines how light is reflected at a given point on an object’s surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' however, many of these techniques fail for highly specular or curved objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' For example, traditional BRDF acquisition [10,31] requires a gonioreflectometer, which tries to precisely mea- sure reflectance at different angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' This machine typically runs on flat objects and struggles on curved objects like Diet Coke cans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Modern approaches [62] use RGBD sensors and joint optimization on differentiably rendered objects and multi-view images [18, 32, 60];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' our approach builds upon differentiable rendering to optimize material reconstruction and adapts them for ALPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Approach Accidental Light Probes (ALPs) are daily metallic shiny objects, such as a soda can, a thermoflask, or a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Given a single image containing an ALP, we aim to recover the inci- dental illumination by inverting physically-based rendering, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Our main idea is that we can first acquire the shape and reconstruct the spatially-varying BRDF of 2 Multiview capture Geometry Lighting B) Single-image lighting estimation Rendered Differentiable rendering A) Offline ALP reconstruction (material estimation) Geometry Material Lighting Rendered Single ALP image Material 6D Pose + Scale Optimizable Fixed (ALP Model) Differentiable rendering Optimizable Fixed (Measured) 6D Pose + Scale Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Our physically-based approach to lighting estimation consists of two stages, namely (A) offline Accidental Light Probe (ALP) reconstruction and (B) inference-time single-image lighting estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' For (A), we use a capture-optimization hybrid method to reconstruct the ALP model with high fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' For (B), we formulate lighting estimation as a joint optimization of 7D pose (scale + 6D pose) and environment lighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' the ALP offline (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' 2 top), and then optimize incidental lighting as well as the 6D pose of the ALP (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' 2 bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Formulation Our goal is to estimate high-fidelity lighting from the appearance of an ALP in a single image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We approach this goal through the perspective of inverse rendering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' where the forward process is described by the rendering equation [21]: L(ωo) = � H Li(ωi)f(ωi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' ωo)(n · ωi)dωi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' (1) where L(ωo) is the outgoing radiance to direction ωo (cor- responding to pixel intensity),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Li(ωi) is incidental radiance from direction ωi (lighting),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' f is the bidirectional reflectance distribution function (BRDF) at the surface location (ma- terial),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' n is the normal direction (geometry),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' and H is the upper hemisphere along the normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Recovering lighting by inverting Eqn 1 is a highly ill-posed problem, as infinitely many combinations of geometry, material, and lighting can generate the same appearance in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Fortunately, for ALPs, we can pre-acquire prior physical knowledge of their shapes and materials as they are everyday objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Thus, we can reduce the full inverse rendering problem to a joint estimation of 6D ALP pose and lighting, which is relatively more constrained and tractable: min π,Li L � Irender(π, Li|f, S), Iref � , (2) where Irender is generated by a differentiable renderer that takes the shape S (represented by a mesh), the 6D pose of the ALP π, the spatially-varying BRDF f, and the environment lighting Li as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Iref denotes the observed single image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' L denotes an image-space loss that we define in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Our physically-based formulation entails the high-fidelity acquisition of shape and spatially-varying material of the ALP, as well as a robust single-view joint optimization al- gorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We show an overview in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' 2 and elaborate the components in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='2 and Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Shading model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We adopt physically-based rendering (PBR) [36] due to its principled photogrammetry and ra- diometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Specifically, we consider metallic materials as they have little diffuse reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Diffuse reflection is unde- sirable as it behaves like a low-pass filter of lighting in the shading process [38], eliminating the physically recoverable lighting information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' To model metallic material, we use a microfacet model [47] with a GGX distribution [48]: f(ωi, ωo) = D · F · G 4(n · ωi)(n · ωo), (3) where D is the GGX normal distribution [48], F is the Fresnel reflection, and G is the geometric attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We adopt Disney’s parameterization [5], where the metallic material is modeled by its specular albedo A and rough- ness r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Specifically, the specular albedo A is used to model Fresnel reflection by Schlick’s approximation [40] F = A + (1 − A)(1 − |h · ωo|)5, where h = ωi+ωo |ωi+ωo| de- notes the half vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' The roughness r controls the shape of the specular reflection lobe via the micro-normal distri- bution D = r4 π(|n·h|(r4−1)+1)2 and the geometric attenuation G = 2|n·ωi| |n·ωi|+√ r4+(1−r4)|n·ωi|2 · 2|n·ωo| |n·ωo|+√ r4+(1−r4)|n·ωo|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Lighting model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' To recover lighting for arbitrary conditions, we use an environment map to represent omnidirectional lighting and adopt image-based lighting for shading each 3 ke ALORIES 12FLOZ (355ml)RECYCLE ME ORQILOLDiet ke 0 TES 12FLOZ (355ml)RECYCLEMEok NOCALORIES NO SUGARoK NOCRLOAIESY 7 X1ZFLOZ DietCo RECYCLE ME O2021TH COCACOIACOMPANY AFFEINECONTENT:46mg/121ol PHENYLKETONURICS CONTAINSPHENYIAL Opixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' For efficiency, we only consider direct lighting and use a differentiable rasterizer with deferred shading [24] to render Irender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' This is inaccurate for concave objects with self-occlusion and self-reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' To mitigate this without expensive global illumination, we include a soft visibility term to Eqn 1 to approximate it such that the shading output is modulated as vL(ωo), where v denotes the soft visibility that is optimized and treated as a surface texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Reconstructing ALPs Recovering high-fidelity lighting by physically-based in- verse rendering requires high-quality geometry and mate- rial reconstruction of the ALPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' While existing state-of- the-art inverse rendering methods can jointly optimize for geometry, material, and lighting from dense multi-view im- ages [18, 32, 60], they still struggle for real metallic ob- jects under arbitrary lighting due to high specularity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Moreover, several challenges exist when the goal is not view synthesis but photogrammetrically correct reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' For highly specular objects such as metallic ones, the re- flected lights from the near field can lead to environment baking, as it breaks the distant light assumption (we show an example of the environment-baked material reconstruction in the supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' The color ambiguity of mate- rial albedo and lighting is also not resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' In addition, the geometry reconstruction quality heavily relies on the quality of object silhouettes in multi-view images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' To overcome these challenges, we reconstruct ALPs by a hybrid method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' First, we use a light box with a turntable to control environment lighting for multi-view capture, and us- ing a thin supporting stand to alleviate near-field reflections (setup shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' 3) and environment baking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Second, in- stead of optimizing the incidental lighting to the ALP under capture, we record it by a calibrated light probe to remove the color ambiguity between material and lighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' And third, we provide a high-quality shape using a range scan- ner [1] to reduce the geometry reconstruction down to 6D pose and size fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Thus, as demonstrated in the top row of Figure 2, our ALP reconstruction is cast as an optimization for its spatially-varying material and shape fitting: min π,α,f � {Icapture} L � Irender(π, α, f|Li, S), Icapture � , (4) where π and α are the 6D pose and size to fit the shape S to multi-view camera coordinate frame solved by COLMAP [41], and f is the material parameterized by spatially-varying albedo A and roughness r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We show the reconstruction of a Coke can in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We include the opti- mization and loss details in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Single-view physically-based light estimation Given an image containing an ALP, we first extract an object segmentation mask for the ALP by manually crop- Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' (Left) We use a light box with controllable lighting for our capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' To mitigate near-field reflections, we leverage a thin stand to support the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' (Right) To minimize environmental changes due to camera and photographer movement, we cover the lightbox with a cloth and use a turntable for multi-view capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' ping the image and then using an off-the-shelf foreground segmentation tool [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' however, this could alternately be obtained by object detection [7] with salient object segmen- tation [37] or semantic segmentation [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We then retrieve the appropriate ALP model, containing its reflectance and geometric information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Yet, even given the ALP’s 3D model and 2D segmentation in the input image, accurately aligning these two elements is still challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Traditional feature point detection and Perspective-n-Point methods do not work on textureless objects such as rings and thermoflasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Addi- tionally, modern learning-based single-view pose estimation methods [29,50,55] require diverse, realistic lighting to syn- thesize training data and do not generalize well outside the training distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Therefore, we formulate the lighting estimation and pose estimation as a joint estimation problem in Eqn 2, and we solve it via a differentiable rendering-based optimization which is generalizable to arbitrary scenes for both textured and textureless objects (see the bottom of Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Here we need a joint estimation as the appearance of a specular object (and thus the differentiable rendering gradient sig- nals) is highly dependent on both the object pose and the environment lighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We use Monte Carlo ray tracing with Visible Normal Distribution Function (VNDF) importance sampling [19] for unbiased shading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Losses and regularizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Our loss function used in Eqn 2 is given by: L = LRGB + Lmask +λ1 Lpose-reg +λ2 Llight-reg, (5) where LRGB denotes a L1 loss on RGB images, Lmask de- notes a combination of a L1 loss and a Chamfer loss on masks [3], where the mask is given by the differentiable rasterizer [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Lpose-reg and Llight-reg denote a pose regular- ization and a lighting regularization with their weights λ1 and λ2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Without multi-view constraints, the joint optimization problem has multiple local minima for the 6D pose;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' thus, we introduce a pose regularization and a lighting regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' 4 pepsMENU C3 F10四±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='0150100 1/10Nvdiffrec Nvdiffrecmc Ours Albedo Roughness Normal Albedo Roughness Normal Albedo Roughness Normal Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Visual comparison of ALP reconstruction from state- of-the-art optimization-based inverse rendering methods [18,32] versus our hybrid method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Recent inverse rendering methods strug- gle on real textured metallic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' The pose regularization is given by: Lpose-reg = ∥B(Mrender) − B(Mref)∥2 2 + ∥q − qref∥2 2, (6) where Mrender is the rendered mask, B(Mrender) denotes the pixel-space barycenter of the mask, q denotes the quaternion representation of the ALP orientation, and qref denotes a com- mon orientation (we use a front-facing canonical orientation obtained by aligning principal axes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' The barycenter term prevents vanished gradient due to non-overlapping pose ini- tialization, and the orientation term prevents hard-to-escape local minima like upside-down cans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We decay the weight of the pose regularization to zero through optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' To accurately estimate omnidirectional lighting by invert- ing Eqn 1, we need to evaluate the Monte Carlo integral interval densely over light rays coming from all directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' However, from a single view, an ALP often only covers a limited subset of normal directions compared to a perfect sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Thus, light rays coming from a certain subset of directions contribute little to the appearance of the ALP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' These directions are then under-sampled, and the lighting estimation for them is less informed and unconfident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' To mitigate this, we introduce a lighting smoothness reg- ularization which “fills in” the less confident regions in the environment map by propagating the confident information from nearby directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' The regularization is given by: Llight-reg = ∥Li(ω) − Li(ω + ∆ω)∥1, (7) where ∆ω denotes a small deviation of a solid angle sampled from a normal distribution, and ω is sampled uniformly in all solid angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Note that in addition to propagating confi- dent lighting estimates, the lighting regularization also helps improve pose estimation, since many pose estimation errors come from trying to fix mistakes in high-frequency [17] lighting changes, which light regularization alleviates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Setup Accidental Light Probes Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We acquire 5 common accidental light probes that have different shapes or sptially- varying BRDFs, including 3 soda cans (diet Coke, diet Pepsi, Diet Coke Cleaner Sprite Thermos cap Diet Pepsi Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Close up of our ALP dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' and Sprite), a thermos cap, and a solder tip cleaner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We show example images in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Evaluation Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We collect a dataset of 10 indoor scenes and 13 outdoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' The indoor and outdoor scenes are taken at different points of time, such as day and night, at different locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We show examples in Fig- ure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We place each of our ALPs in the scenes and capture HDR images of the ALPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We also capture ground-truth lighting by placing a chromium ball (a perfect light probe) in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We compare our method to several state-of-the- art lighting estimation methods [12, 14, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Unlike our method, all of these techniques utilize deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Since [12, 14] do not have publicly available models, we asked their authors to run inference on our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Comparison to Baseline Methods Qualitative Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' For all object insertion comparisons, we compute an environment map either through an ALP with our proposed method or by running the other baselines on a single image of the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Note that for the baselines, we use the image with the perfect light probe as input;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' this should provide a slight advantage to these techniques since the image with the perfect light probe contains the most information regarding scene lighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' In Figure 7, we insert various objects into the scene and relight them using the computed environment maps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' we then qualitatively compare the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We demonstrate that our computed environment map produces significantly more ac- curate and compelling results than other single-image light- ing estimation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' In particular, note that our method is the only approach that can recover the overall tone of the lighting: other methods are either too yellow or gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' In Figure 8, we show relighting results on perfect spheres of various finishes from all methods and ALPs on both indoor and outdoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Only our technique produces results similar to the ground truth for mirror finishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We note that for all the three soda cans, the relighting on mirror spheres are slightly blurry, since their materials are much rougher than perfect mirror, which behaves as low-pass filters of lighting in the shading process [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We also note that for Sprite and diet Coke, there is some texture color baking in the recovered lighting due to imperfectly aligned 6D poses, which lead to high-frequency lighting artifacts to compensate 5 loke D3Soke DietDiet oke NO SUGAR 12FL0Z NO CALORIES (355ml)OLEMON-LIME pritpepsiDiet keFigure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Examples of our collected dataset for evaluating lighting estimation under different illumination conditions, including indoor and outdoor scenes at daytime and nighttime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Groundtruth lighting DeepParam Ours StyleLight Garon et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Input image Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Object insertion results on our test scenes for both indoor (first two rows) and outdoor (last two rows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We compare to Garon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' [14], Deep parametric lighting [12], and StyleLight [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We center-crop the result images for better visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' the pixel-space misalignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Our lighting regularization mitigates this type of artifacts, yet a highly robust algorithm remains as future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Quantitative Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' In Table 1, we report quantitative results on relighting perfect spheres with various representa- tive materials (mirror, shiny, diffuse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Similar to [26,49], we compute angular error [11] and scale-invariant RMSE [16] to compare the relighted spheres from each technique to the ground truth relighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Quantitatively, for the relighting task, our method, applied to any of the ALPs, significantly outperforms the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' In particular, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' angular error, the Thermos cap provides a 3 to 4 times improvement over the best baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Analysis Capture Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We also analyze the quality of our recon- struction compared to two recent multi-view inverse render- ing methods, Nvdiffrec [32] and Nvdiffrecmc [18] using our lightbox capture setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Table 2 shows the results of using our lighting estimation pipeline with various reconstructions of a Diet Coke can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Our reconstruction performs the best and leads to a decrease of angular error by 20% or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Both Nvdiffrec and Nvdiffrecmc are normally applied to multiview casual images, so for completeness, we also com- pute reconstructions and quantitative results for this setting (included in the supplementary materials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' These reconstruc- tions perform strictly worse than those computed from the lightbox setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We also show a qualitative comparison of the geometry and materials in Figure 4, of each technique in its default setting, where ours are clearly better than the alter- native methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Table 2 shows that our ALP reconstruction pipelines give us better results than using current state-of-art inverse rendering methods to get our ALP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Ablation for 6D Pose + Scale Estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' As mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='3, the pose estimation problem for aligning a 3D model of an ALP and its appearance in a real image is chal- lenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' The appearance of the object in the real image 6 lok ant,oKokloklokMethod Indoor Outdoor Angular Error↓ Scale-invariant RMSE↓ Angular Error↓ Scale-invariant RMSE↓ Mirror Shiny Diffuse Mirror Shiny Diffuse Mirror Shiny Diffuse Mirror Shiny Diffuse StyleLight [49] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='572 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='700 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='949 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='087 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='287 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='803 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='228 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='525 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='963 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='305 Garon et al.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='217 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='053 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Comparison to state-of-the-art single image lighting estimation methods: StyleLight [49], Deep Parametric [12] and Garon et al [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We evaluate them using relighting on different materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' StyleLight Deep Parametric Garon et al GT Diet Pepsi Diet Coke Sprite Cap Cleaner Input Mirror Shiny Diffuse Input Mirror Shiny Diffuse Input Mirror Shiny Diffuse Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Qualitative comparison of relighting results in outdoor (left and center) and indoor (right) scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We compare our approach to StyleLight [49], Deep Parametric [12] and Garon et al [14] on relighting mirror, shiny and diffuse spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' depends on both its pose and lighting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' trying to jointly opti- mize these can introduce potential failure cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' In Sec 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='3, we describe several design choices w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' the optimization and loss which address some of these failures cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' In Table 3, we perform an ablation study on each of these decisions and report quantitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We show that all design decisions (Silhouette loss, Chamfer loss [3], joint optimization, pose, and light regularization) contribute to the final overall perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We also show representative examples in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' They demonstrate how each design choice helps the pose estimation, which in return helps lighting estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Visualizing confident regions for ALPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' As briefly dis- cussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='3, an ALP has a subset of surface normals compared to a perfect sphere light probe, which leads to under-sampled lighting directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' For example, cylindri- cal objects (Diet Coke can, ring, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=') tend to sample well light rays perpendicular to the can while significantly under- sampling light rays above and below the can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Since we 7 okeOkekeMO-LIME priteOurs Ours w/o Joint optimization Groundtruth Ours Ours w/o Light regularization Groundtruth Ours Ours w/o Pose regularization Groundtruth Rendering Environment Map Rendering Environment Map Rendering Environment Map Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Qualitative ablation of the losses we use in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Each of our design choices contributes to improvements in pose and lighting optimization which can be observed qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Method Mirror Shiny Diffuse Nvdiffrec [32] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='99 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='06 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='59 Nvdiffrecmc [18] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='55 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='60 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='84 ALP (Ours) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='46 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='67 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='80 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Evaluation on our ALP model acquisition for a Diet Coke can using our lightbox setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We compare our acquisition method to Nvdiffrec [32] and Nvdiffrecmc [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We use the same lighting estimation approach for compared methods and report average angular error across all test scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Method Mirror Shiny Diffuse Silhouette loss [3] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='812 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='976 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='919 Ours w/o joint optimization 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='401 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='726 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='044 Ours w/o pose regularization 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='962 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='180 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='338 Ours w/o light regularization 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='032 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='647 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='954 Ours 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='291 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='610 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='923 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Ablation study on our joint pose-lighting optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We compare to a baseline that uses a silhouette loss and a Chamfer loss [3], and variants of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We show angular errors averaged on all test scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' ALP image Normal map Sphere normal map Observed normals Confident regions in the environment map Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Visualization of sampling directions for a diet Coke can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' See the text in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='3 for a full description of these visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' use VNDF importance sampling which aligns well with our BRDF’s density lobe, we visualize a “confidence map” as normalized sampling frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We show this confidence map in Figure 10 for a representative ALP (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=', Diet Coke).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' This demonstrates that the visible surface of a Coke can from a single view only under-samples lighting directions from the top and the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Our method shows strong promise for recover- ing scene lighting from a single image containing an ALP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' One exciting potential application is improved image editing for in-the-wild images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' however, to enable this for any im- age, we would either need to increase the number of ALPs or explore methods that enable us to dynamically edit one of the collected measurements (geometry or material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Another limitation is that we assume our input is an HDR image, generally not available for in-the-wild images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' However, we note that recent work has sought to convert LDR images to HDR [22,25] and HDR has also become more available due to support on recent phones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Conclusion In this paper, we introduce the use of accidental light probes to estimate environmental lighting from single im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We do this by first scanning common 3D objects and establishing their reflective properties using a controlled lighting environment (a simple turntable and light box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We then use differentiable rendering combined with a physically- based rendering model to recover the unknown object pose and environment lighting when the object is placed (or nat- urally occurs) in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We create a new dataset of materials and geometry for several common, shiny, curved objects along with image showing these in a variety of in- door and outdoor environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Finally, we compare our approach to previous single-image lighting estimation alter- natives and demonstrate that it strongly outperforms previous approaches in terms of realism and fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' We would like to thank William T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfrw2W/content/2301.05211v1.pdf'} +page_content=' Freeman for the invaluable discussion and for the photo credit, Varun Jampani for helping us with data collection, and Henrique Weber and Jean-Franc¸ois Lalonde for running their methods as comparisons for us.' metadata={'source': 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a/iNE_T4oBgHgl3EQf4Byz/content/tmp_files/2301.08350v1.pdf.txt b/iNE_T4oBgHgl3EQf4Byz/content/tmp_files/2301.08350v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..516251938bed81aa52883286cf336eea9718b51f --- /dev/null +++ b/iNE_T4oBgHgl3EQf4Byz/content/tmp_files/2301.08350v1.pdf.txt @@ -0,0 +1,1868 @@ + +1 + +Abstract-- This paper presents a novel 2-stage microgrid unit +commitment (Microgrid-UC) algorithm considering cold-load +pickup (CLPU) effects, three-phase load balancing requirements, +and feasible reconfiguration options. Microgrid-UC schedules the +operation of switches, generators, battery energy storage systems, +and demand response resources to supply 3-phase unbalanced +loads in an islanded microgrid for multiple days. A performance- +based CLPU model is developed to estimate additional energy +needs of CLPU so that CLPU can be formulated into the +traditional 2-stage UC scheduling process. A per-phase demand +response budget term is added to the 1st stage UC objective +function to meet 3-phase load unbalance limits. To reduce +computational complexity in the 1st stage UC, we replace the +spanning tree method with a feasible reconfiguration topology list +method. The proposed algorithm is developed on a modified IEEE +123-bus system and tested on the real-time simulation testbed +using actual load and PV data. Simulation results show that +Microgrid-UC successfully accounts for CLPU, phase imbalance, +and feeder reconfiguration requirements. + +Index Terms—cold load pickup, demand response, feeder +reconfiguration, microgrid energy management, resiliency, +restoration, unbalance load management, unit commitment. +I. INTRODUCTION +ICROGRIDS powered by distributed energy resources +(DERs), primarily renewable generation resources and +grid-forming battery energy storage systems (BESS), have +attracted great interests in recent years as an effective operation +mechanism to provide grid services and enhance distribution +system resiliency [1]. +Unit commitment (UC) is the key algorithm of the energy +management system (EMSs) for scheduling generation +resources in the bulk power system (BPS). However, directly +applying BPS UC for microgrid EMS is oftentimes infeasible, +especially for microgrids at the feeder-level. First, on a +distribution feeder, intermittency of distributed renewables is +compounded with uncertainty in loads, making combined +forecasting errors much higher than those in BPSs [2]. Second, +unlike large, synchronous generators in the BPS, DERs are +constrained by both power and energy limits [3]. Particularly, +the installed power and energy capacity of grid-forming BESSs +or distributed generators are oftentimes insufficient to supply +the microgrid load at all times in a prolonged outage that lasts + +This research is supported by the U.S. Department of Energy's Office of +Energy Efficiency and Renewable Energy (EERE) under the Solar Energy +Technologies Office Award Number DE-EE0008770. +for days. Therefore, demand response (DR) and feeder +reconfiguration have to be frequently used to shed loads for +meeting power and energy limits. Third, in the BPS, cold load +pick-up (CLPU) [4] is seldom considered in UC. However, in +an islanded microgrid, due to interruptions mainly caused by +the intermittency of DERs and feeder reconfigurations, cutting +off a load and resupplying it is often time required during +outages, making CLPU occur more frequently. Those +additional CLPU energy needs so far cannot yet be predicted by +load forecasting algorithms. Note that in a distribution grid, +CLPU is mainly caused by the recovery of Heating, Ventilation, +and Air Conditioning (HVAC) systems, the electricity +consumption of which accounts for approximately 50% of +energy use in residential and office buildings [5]. Fourth, in +BPS EMSs, loads are mostly 3-phase balanced. However, in a +distribution grid, even under normally operation conditions, +loads are normally unbalanced. Load imbalance can also be +exacerbated by CLPU, feeder reconfiguration, or DR events. +Because highly unbalanced loads can cause power quality +issues, violate voltage regulation requirements, lower the +sensitivity of the protection systems [6], it is critical in +microgrid operation to maintain the phase imbalance within a +given limit. +Thus, in this paper, we propose a novel 2-stage microgrid +unit commitment (Microgrid-UC) algorithm that accounts for +CLPU, using DR for three-phase load balancing, and the +feasibility of reconfiguration options. Microgrid-UC manages +the islanded operation of a 3-phase unbalanced distribution +feeder for multiple days. The controllable resources include +breakers/switches, DERs (e.g., PV farms, diesel generators, +BESSs), and DR resources. The four unique considerations of +Microgrid-UC are explained as follows. +CLPU modeling: In the literature, there are two approaches +for modeling CLPU: model-based and data-driven methods. +The model-based approach predicts CLPU effects by physics- +based models. Using system on/off status and ambient +temperatures as inputs, the electricity consumptions of HVACs +are simulated to predict CLPU needs. Either exponential [7] or +linearized models [8] can be used for modeling the +thermodynamics process that causes CLPU. The drawback of +this method is that predetermined HVAC model parameters +cannot +produce +simulation +results +matching +field +The authors are with the Department of Electrical and Computer +Engineering, North Carolina State University, Raleigh, NC 27695 USA +(emails: {rhu5, ashirsa, vmuthuk2, szhang56, yli257, lsong4, bxu8, vdaldeg, +nlu2, baran, wtang8}@ncsu.edu). +Rongxing Hu, Ashwin Shirsat, Valliappan Muthukaruppan, Si Zhang, Yiyan Li, Lidong Song, Bei Xu, Victor +Paduani, Ning Lu, Fellow, IEEE, Mesut Baran, Fellow, IEEE, Wenyuan Tang, Member, IEEE +A Novel Feeder-level Microgrid Unit +Commitment Algorithm Considering Cold-load +Pickup, Phase Balancing, and Reconfiguration +M + + +2 +measurements. In contrast, the data-driven approach estimates +the CLPU curve parameters using historical data [9-11]. The +drawback of this approach is the lack of CLPU event data. Thus, +in this paper, we develop a hybrid CLPU modeling method. +Instead of directly estimating the CLPU curve parameters, we +use smart meter data to derive the HVAC model parameters +[12]. The HVAC models can then be used to model CLPU +effects for different ambient temperature and interruption +durations, the results of which can be used to derive the CLPU +curve parameters. +Formulating CLPU constraints into EMS: Conventional +distribution system EMS problem formulations only account +for CLPU in restoration algorithms. For example, in [13], a +Mixed-Integer Linear Programming (MILP) service restoration +algorithm accounts for CLPU using a linearized, delayed +exponential CLPU curve. In [14], a two-block representation +(one for normal loads and the other for CLPU increments) is +used to eliminate the CLPU nonlinear characteristic. In [15], to +capture the CLPU power after short outages, a time dependent +CLPU model based on operating state evolution of +thermostatically controlled loads (TCLs) is proposed. In [16], +the uncertainty of CLPU, captured by the probability density +functions (PDFs) of CLPU peak and duration, is included in the +restoration service. However, the drawback of such +formulations is the use of a predefined set of CLPU parameters, +through which variations of outdoor temperature and +interruption duration [17] cannot be considered. Thus, in this +paper, we proposed an adaptive CLPU estimation method that +can account for accumulated CLPU effects when picking up +load groups (LGs) with different “off” durations under different +ambient temperatures. +Load unbalance: In microgrid operation, maintaining 3- +phase load balance is essential for maintaining voltage balance +[18] and assuring protection relays to take correct actions [6]. +In [19], the authors propose two methods for controlling +distributed generations to balance 3-phase loads: adding a +penalty term representing the current unbalance to the objective +function and using phase power for a conservative linear +approximation of the current unbalance. In [20], voltage +unbalance is considered. However, using DR for mitigating +unbalance in UC is an uncharted area. Thus, we formulate a DR +budget term into the UC problem for meeting 3-phase load +balancing requirements. +Topology Scheduling: In the literature, spanning tree (ST) +[21-23] is a typical approach for distribution feeder +reconfiguration. In [24], the authors analyze other radiality +constraints including single-commodity flow (SCF) and the +combined ST and SCF constraints. However, if we need to +formulate feasibility into topology options, the complexity and +runtime of the algorithm will increase drastically. In practice, +because of protection settings and circuit operational limits, not +all topology options are feasible under different operation +conditions. Thus, we propose a feasible topology candidate +method to ensure the feasibility of the selected topology while +shortening the runtime by over 50%. +To summarize, the novelties of Microgrid-UC are three-fold. +First, we develop an adaptive CLPU model so that CLPU can +be formulated into both 24-hour ahead and intra-hour +optimization problem formulations. Second, a DR budget term +is formulated into the 24-hour ahead UC problem to balance 3- +phase system loads. Third, we use a feasible topology candidate +method to guarantee operational feasibility and reduce runtime. +The rest of this paper is organized as follows. Section II +presents the proposed microgrid-UC method. Results are +presented in Section III and Section IV concludes the paper. +II. METHODOLOGY +In this section, we first introduce the layout of a typical +feeder-level microgrid. Then, the assumptions, the overall +framework, the problem formulation and the operational +constraints of the 2-stage microgrid-UC are presented. +A. Typical Layout of a Feeder-level Microgrid +In this paper, our focus is to develop a 2-stage microgrid-UC +algorithm for managing a feeder-level microgrid by accounting +for cold-load pickup, three-phase load balancing, and feeder +reconfiguration. As shown in Fig. 1, a typical feeder-level +microgrid is powered by a hybrid system (e.g., the MW-level +PV plant collocated with a grid-forming BESS at bus 7) and +supplied multiple LGs, which can be prioritized into “critical” +(the red triangles) and “non-critical”. The microgrid controller +controls five switches (S1-S5) remotely to switch on/off LGs. + +Fig. 1. An illustration of the typical layout of a feeder-level microgrid. +B. Assumptions +We make the following assumptions regarding microgrid +operation, data availability, and device controllability. First, the +microgrid controller has access to smart meter data. The +controller controls the switches and DR resources in each LG +remotely via a fully functional communication network. Second, +critical loads have their own backup generators so the goal of +the microgrid controller is to reduce the use of their backup +generators by weighting the critical load with higher supply +priorities than the non-critical load. Only non-critical loads +participate DR. Third, no circuit loop is allowed and there is +only one grid-forming resource in the microgrid. In this paper, +the BESS at bus 7 is the grid-forming resource. +C. Scheduling Horizons and Intervals +Microgrid-UC is designed for multi-day, off-grid operations. +As shown in Fig. 2, a 24-hour ahead rolling forecaster and a 30- + +32姜 +29 +O250 +S5 +Q.350 +30 +33 +251 +110 +112 +113 +114 +28 +50 +3001 +31 +49 +25 +47 +109 +107 +LG5 +48 +46 +26 +45 +108 +270 +104 +451 +64 +106 +44 +43 +65 +103 +023 +C +450 +1050 +102 +635 +100 +24 +420 +41 +660 +101 +LG1 +40Q +39 +93 +71 +22 +39. +620 +S4 +70 +38. +36. +69 +19 +35 +20 +S2 +68 +75 +Hybrid +37 +LG3 +67 +60 +74 +LG4 +57 +S3 +73 +plant +11 +14 +58. +59 +72 +79 +85 +P +文610 +V +10 +萧 +9 +53 +54 +S1 +52 +7778 +2# +55 +56 +13 +76 +8 +800 +94 +84 +149 +12 +34# +96 +90# +88* +810 +17 +15°. +O +O +91 +87 +86 +83 +95 +89 +82 +T +3 +6 +16# +195 +PV farm +BESS +CriticalLoad +Switch + Load Group +No load node +3-ph load node +000 +A/B/Cloadnode +3 +minute ahead forecaster are used to provide Microgrid-UC with +PV, load, and weather forecasts. Figure 3 shows the scheduling +horizons and intervals. +In the first stage, a rolling 24-hour ahead unit commitment +is conducted every 30-minute using 24-hour-ahead PV, load, +weather forecast as inputs. The outputs are the operation +schedules for the BESS, DR resources, and switches. Note that +switches are switched on/off to supply/disconnect which LG for +48 30-minute scheduling intervals considering CLPU needs, +phase-balancing needs, and reconfiguration. +In the second stage, a 30-minute ahead power dispatch is +conducted using 30-minute ahead PV and load forecast as +inputs, while the weather input remains the same. The outputs +are the operation schedules for the BESS and DR resources for +six 5-minute intervals. + +Fig. 2. The flowchart of the two-stage Microgrid-UC. + + +Fig. 3. Scheduling horizons and intervals of the 2-stage Microgrid-UC. + +D. Problem formulation for the 1st Stage +Let the total weighted served load be 𝑓� +����, the total PV +curtailment penalty be 𝑓� +�� , and the total cold load pickup +(CLPU) penalty be 𝑓� +����. The 1st stage objective function can +be formulated as +max 𝑓� +���� � 𝑘��𝑓� +�� � 𝑘����𝑓� +���� +�1� + +𝑓� +���� � � � +� +𝑈�,� +� 𝑤� +�����𝑃�,�,� +����� � 𝑤�,� +����𝑃�,�,� +���� �∆𝑡 +�∈��,�,�� +�� +��� +�� +��� +�𝑘�� � +� +𝑤� +����𝑃�,� +��𝛥𝑡 +�∈��,�,�� +�� +��� + +�2� + + +𝑓� +�� � 3 � 𝑃��,� +����𝛥𝑡 +�� +��� +�3� + +𝑃��,� +���� � 𝑃��,� +���� � 𝑃��,� +�4� + +where 𝑘�� and 𝑘���� are coefficients of the PV curtailment +and the CLPU penalty; 𝑁� is the number of scheduling intervals +(𝑁� �48); ∆𝑡 is the scheduling interval (∆𝑡 � 30 minutes); 𝑚 +is the group index, 𝑁� represents the total number of LGs, 𝑝 is +the phase index; 𝑈�,� +� denotes the status of the mth LG (1: “on” +and 0: “off”); 𝑃�,�,� +����� and 𝑃�,�,� +����� are the forecasted non-critical +load and critical load in the mth LG on phase 𝑝 at time 𝑡 without +considering the CLPU effect; 𝑤�,� +���� is the priority weighting of +the critical load; 𝑤� +���� is the weighting of the customer +preferred supply period[25] at time 𝑡; 𝑃�,� +�� is the DR budget at +time 𝑡 on phase 𝑝 (note that only non-critical loads will provide +DR); 𝑃��,� +����, 𝑃��,� +���� and 𝑃��,� are the PV curtailment, prediction +and scheduled PV output on each phase. +For PV and BESS operational constraints, microgrid reserve +constraints, and polygon-based linearization of active power +and reactive power constraints of the inverters and the switches, +please refer to [25]. + +E. Minimum Service Duration Constraints +To avoid frequently switching on/off LGs, minimum service +duration (MSD) constraints are introduced in [25], If an LG can +be served, it is expected to be “on” for at least 𝐷� +��� +consecutive scheduling intervals. In this paper, we modify the +formulation to allow rolling scheduling in the 1st stage +considering initial service duration at the first step. As shown +in Fig. 4, 𝐷��,� is the remaining service duration need, which is +determined by + +𝐷��,� � max �min�𝐷� +��� � 𝐷�,��� +���������, 𝑁� � 𝑡 � 1�, 0� +�5� +� 𝑈�,��� +� +���,� +��� +� 𝐷��,��𝑈�,� +� +� 𝑈�,��� +� +�, 𝑡 � 1 +�6� + +where 𝐷�,� +��������� is the service duration already fulfilled in the +latest 𝐷� +��� steps, so 𝐷�,� +��������� � 𝐷� +���. Note that MSD is +treated as a soft constraint. Microgrid-UC can shut down an LG +before MSD is fulfilled when there is insufficient energy supply +for subsequent hours (reduce the default MSD, see Fig. 2). For +example, the actual load is significantly higher or the PV is +significantly lower than predicted values. + +Fig. 4. The MSD requirements. + +F. CLPU Constraints +In a microgrid powered by intermittent renewable generation +resources (e.g., PV and wind) and BESSs, CLPU may occur +frequently. Not all LGs can be served for the entire scheduling +period due to the uncertainty in generation and limitations in the +BESS energy and power capacities. Thus, oftentimes, shutting +down LGs is inevitable. + +Outage & +PV & load +weather +forecast +1st stage, energy scheduling: +information +(interval: 30min, horizon: 24h) +Resolution +LG status, topology, storage budget, +30min +DR budget, cold load pickup estimation +Reduce the MSD +Obtain solution ? +No +Yes +Switching operation post-process +optimization (if needed) +2nd stage, power dispatch: +Resolution +..5min.. +(interval: 5min, horizon: 30Omin) +DR resource dispatch, BESS dispatch, +voltage regulation +LG status +Real-time Implementation: +BESS SOC +HVAC load simulation, HIL/OpenDSSFirst-stage scheduling at 00: 00 +t=1 +t=2 +Nr = 48 +Secondstage +time +00:00 +00:30 +01:00 +1:30 +24:00 +dispatch +First-stage scheduling at 00: 30 +t=1 +t=2 +Nπ = 48 +time +1:30 +24:00 +00:30 +dispatchDMSD +m +DMSD +MSDserved +m +m,ini +m.t +m,ini +m, +Past : +Past Grid outage ends +4 +However, resupplying a previously “off” LG requires +additional energy and power capacity to be allocated than +supplying a previously “on” LG. The additional energy required +in CLPU is mainly consumed by HVAC loads, which can +account for approximately 50% of the total building load [5]. +After an LG is turned off for a prolonged period (e.g., 60- +minute), room temperature inside a building will coast out of +the thermostat deadband. Thus, once the LG is switched on, all +thermostatically-controlled HVAC loads will be turned on +simultaneously, causing a synchronized load peak. This process +can last from tens of minutes to hours depending on the “off” +duration, the ambient temperature, and the thermostat setting. +To account for CLPU in Microgrid-UC, we formulate +additional energy budgets required for CLPU in the 1st stage +scheduling using a novel hybrid CLPU modeling approach, +which is a major contribution to the state-of-the-art UC problem +formulation. + + 1) Develop the hybrid CLPU model +To predict the CLPU effect under different ambient +temperatures for different outage durations, we first need to +know the HVAC model parameters. As a first step, we derive +the HVAC parameters for each load profile in the Pecan Street +dataset [26]. Thus, once the load profile of a load node on the +feeder is selected from the Pecan Street dataset [25], the HVAC +parameters for the node are known. Note that in practice, if sub- +meter HVAC load profiles are not available, load +disaggregation algorithms [27-28] are needed to disaggregate +HVAC loads from smart meter data, after which the HVAC +model parameters can be derived. +In the 123-bus test system, there are 1100 HVACs in total. +Using weather forecast and LG on/off status as inputs, we can +then predict the CLPU effects by modeling the HVAC +consumptions for different outdoor temperatures and for +different outage durations. +As shown in Figs. 5 and 6, a significant amount of additional +energy beyond the “normal” consumption is needed when +picking up LGs that are previously “off”. After a prolonged +outage, the CLPU peak is the synchronized peak of all HVAC +loads, 𝑃� +�������. Note that if the outage occurs in a mild day, +the CLPU peak may be lower than the synchronized peak (see +Fig. 6, the 26 ºC case). However, if a scheduling interval is 30- +minute or longer, we can simplify the computation by assuming +that the CLPU peak equals to 𝑃� +������� regardless of how +many scheduling intervals the LG has been “off”. +The simulation results can be used to generate Figs. 7 and 8. +As shown in Fig. 7, at a given outdoor temperature, 𝑇� +���, the +CLPU peak duration, 𝑑�,� +����, is a function of outage duration, +𝑑�,� +��� . The longer the outage lasts, the longer the CLPU peak +duration will be. To simplify the calculation, we can linearize +𝑑�,� +���� versus 𝑑�,� +��� curves so that for a given temperature, an +incremental peak duration can be calculated from the slope of +the curve. Note that if the maximum outage duration, 𝑑�,� +�������, +is reached, 𝑑�,� +���� is capped at 𝑑� +��������. +As shown in Fig. 8, the CLPU power decay rate, 𝛾�,� +����, is a +function of outdoor temperature, 𝑇� +���. The higher 𝑇� +��� is, the +slower the CLPU peak decays from 𝑃� +������� to the steady- +state HVAC consumption level, 𝑃�,� +������ . To simplify the +calculation, we ignore the impact of the interruption duration +on the decay rate so that an equivalent power decay rate curve +can be computed with respect to 𝑇� +���. +From those results, a performance-based CLPU model (See +Fig. 9) having the following parameters can be derived: the +synchronized HVAC peak load of the LG (𝑃� +�������in Fig. 6), +the CLPU peak duration rate and saturation ( 𝜏�,� +���� and +𝑑� +�������� in Fig. 7) to get the CLPU peak duration (𝑑�,� +���� in +Fig. 9), the CLPU decay rate (𝛾�,� +���� in Fig. 8), and the CLPU +steady-state load (𝑃�,� +������ in Fig. 6) estimated from the outdoor +temperature range in steady-state operation. +Note that we select 𝑃� +������� to be the power base to make +the normalized CLPU peak as 1.0 p.u. Thus, 𝑘�,� is the power +of the HVAC load in the mth LG at time 𝑡 in per unit values with +steady state value as 𝑘�,� +������, as shown in Fig. 9. + +Fig. 5. CLPU effects for different interruption durations (𝑇��� �36 °C). + +Fig. 6. CLPU effects for different outdoor temperatures (2-hour outage). + +Fig. 7. CLPU effects for different outage durations and under different outdoor +temperatures (𝜏�,� +���� � ∆𝑑�,� +����/∆𝑡, where ∆𝑡 � 30 minutes). + +3000 +m.t +m.t +h +2500 +PMaxCLPU +m +7h +2000 +pSteady +1500 +m,t +1000 +500 +0 +10 +12 +14 +16 +18 +20 +22 +24 +26 +Time (hour)3000 +26°C +28°C +2500 +PMaxCLPU +30°C +m +32°C +Power(kw) +2000 +34°C +36°C +values +1500 +38'C +m,t +40°C +at different +1000 +Tout +500 +0 +10 +12 +14 +16 +18 +20 +Time (hour)00 +Tout= 38℃ +m,t +80 +(minute) +Tout=34 +60 +m,t +Tout +Tout=32 +40 +u +Tout= 28℃ +20 +0 +1 +2 +4 +5 +6 +7 +8 +9 +10 +At +doff +(hour) +m.t +5 + +Fig. 8. CLPU decay rates with respect to outdoor temperature with dots +representing simulated decay rates for different outage durations 𝑑�,� +��� (Note +that the curve is normalized to the synchronized CLPU peak, 𝑃�,� +�������). + +Fig. 9. Model structure of the proposed performance-based CLPU model. + + 2) Estimate the CLPU peak duration +To estimate the accumulated CLPU peak duration, we have +𝑈�,� +���� � 1 � 𝑈�,� +� +�7� +0 � 𝑑�,� � 𝑀𝑈�,� +���� +�8� +𝑑�,� � 𝑀𝑈�,� +� +� 𝑑�,��� � 𝜏�,� +����𝑈�,� +����Δ𝑡 +�9� +𝑈� +���� � 𝑈�,� +��� +�10� +𝑑�,� +���� � 𝐷�,� +����𝑈� +���� +�11� +𝑑�,� +���� � 𝑑�,� � 𝑀𝑈� +���� +�12� +𝑑�,� +�� � 𝑑�,��� +�� +� 𝑑�,��� +���� � 𝑈�,��� +� +Δ𝑡 � 𝑀𝑈�,� +���� +�13� +0 � 𝑑�,� +�� � 𝑀𝑈�,� +� +�14� +where 𝑑�,� is the estimated accumulated CLPU peak duration +during interruptions without considering saturation; 𝑈�,� +���� is the +interruption status; 𝜏�,� +���� is incremental CLPU duration for +scheduling interval; 𝐷�,� +���� is the saturated value at time 𝑡 +derived (Fig. 7); 𝑀 is a large number greater than 24 � 60 +minutes (in our case, 𝑀 �1500); 𝑈� +���� is a binary variable +indicating peak duration saturation status; 𝑑�,� +���� and 𝑑�,� +�� are +the estimated CLPU peak duration and the remaining peak +duration, respectively. +Equation (7) determines whether the LG is “off”; (8) and (10) +ensure when the LG is served, CLPU peak and CLPU peak +saturation status could be 0. For each consecutive “off” interval, +a resultant CLPU peak duration increment is added to the +previous CLPU peak duration using (9). Note that in (9), we do +not consider the saturation effect. +If the CLPU peak duration is saturated, (11) calculates the +saturated CLPU peak duration and saturation status; if not, (12) +ensures the accumulated CLPU peak duration by the end of the +interruption duration. Thus, (12) is disabled when the CLPU +peak duration is saturated. Note that the maximum CLPU peak +duration is capped according to the temperature of the step (see +Fig. 7). +When LGs are served intermittently, unfulfilled CLPU +needs may be carried over to the next “on” cycle. To account +for it, remaining CLPU peak durations can be estimated by (13) +and (14). Note that minimizing the CLPU peak duration also +leads to the minimization of additional energy needs for CLPU. +In the results section, we will demonstrate that as a result of +such considerations, Microgrid-UC tends to supply loads in +consecutive intervals instead of turning them on/off frequently +to minimize the total energy needed for CLPU. + + 3) Set CLPU decay status +To determine the CLPU decay status for the mth LG at time +𝑡, 𝑈�,� +�����, and ensure that the CLPU decay will start only when +the CLPU peak duration elapses, we add the following +constraints +𝑈�,� +����� � 𝑈�,� +� +�15� +𝑀�1 � 𝑈�,� +������ � 𝑑�,� +�� +�16� +�𝑀𝑈�,� +����� � 𝑀�𝑈�,� +� +� 𝑑�,� +�� +�17� +where 𝑀� is a small constant (in our case, 𝑀� � 0.001). + + 4) Calculate CLPU power +After 𝑃�,� +������ is estimated from Fig. 6 based on 𝑇� +���, the +steady state value, 𝑘�,� +������, can be obtained by (18). The HVAC +load factor 𝑘�,� is within peak value (1.0 p.u.) by (19). +𝑘�,� +������ � +𝑃�,� +������ +𝑃�,� +������� +�18� +𝑘�,� +������𝑈�,� +� +� 𝑘�,� � 𝑈�,� +� +�19� + +If the LG is “on” at interval 𝑡 and the peak decay status +(𝑈�,� +�����) is still 1, the CLPU factor, 𝑘�,� +����, can be calculated by +(20-21). We assume when a LG is turned on all HVAC loads in +the LG will be turned on such that the CLPU peak is 𝑃�,� +�������, +(1 p.u.), this is ensured by (20). +𝑘�,� � 𝑘�,��� � �𝑈�,� +� +� 𝑈�,��� +� +� � 𝛾�,�𝑈�,� +����� +�20� +𝑘�,� +���� � 𝑘�,� � 𝑘�,� +������𝑈�,� +� +�21� +Finally, the CLPU power in kW value, 𝑃�,� +����, is calculated +as +𝑃�,� +���� � 𝑘�,� +����𝑃�,� +������� +�22� + + 5) Formulate CLPU into the 1st stage Microgrid-UC +To mitigate the CLPU effect, a CLPU penalty term, 𝑓� +����, +consisting of three penalty factors, 𝑘� +���� , 𝑘� +����, and 𝑘� +���� is +added to the 1st stage Microgrid-UC problem formulation as: +𝑓� +���� � � ��𝑘� +����𝑃�,� +����Δ𝑡 � 𝑘� +����𝑑�,� +���� � 𝑘� +����𝑑�,� +�� � +�� +��� +�� +��� + + �23� +𝑃�,�,� � 𝑃�,�,� +����� � 𝑃�,�,� +���� +�24� + + +� +𝑃��,�,� +�∈�� +���� +� 𝑈�,� +� 𝑃�,�,� � 𝑃�,�,� +���� � � 𝑃��,�,� +�∈�� +�� +�25� + +Tout +0.8 +0.6 +0.4 +0.2 +0 +26 +28 +30 +32 +34 +36 +38 +40 +(℃)km,t (p.u.) +m,t +1.0 +,Steady +m,t +decay +m,t +10 +Josn +=1 +m,t +6 + +where 𝑃�,�,� is the baseload (i.e., the non-HVAC portion of the +forecasted load) on phase 𝑝 in the mth LG; 𝑃��,�,� is the active +power at switch 𝑚𝑛 flowing from LG 𝑚 to LG 𝑛; Ω� +���� and +Ω� +�� represent the “from” LG set and “to” LG set of the mth LG, +, respectively. Power balance within a non-source group is +ensured by (25). Note that for the LG the hybrid PV plant is +located at, (25) will need to consider the outputs of PV and +BESS. Also, reactive power has similar. +G. 3-Phase Load Balancing Requirements +Microgrid-UC is designed to serve 3-phase unbalanced +loads. To reflect the Point of Common Coupling (PCC) power +unbalance requirements, an unbalance factor 𝑘��� is defined as +[19]. +𝑘��� � max��𝑃���,�,� � 𝑃���,� +��� �� +𝑃���,� +��� +, ∀𝑝 ∈ �𝑎, 𝑏, 𝑐� +�26� +𝑃���,� +��� � 1 +3 +� +𝑃���,�,� +�∈��,�,�� +�27� +In a feeder-level microgrid, the load can be highly +unbalanced. The current control mechanism does not allow 3- +phase inverters to deliver highly unbalanced 3-phase power to +the loads. Thus, it is crucial that 1-phase DR resources can be +scheduled to reduce the unbalance among the 3-phase loads +actively. Thus, a DR budget term is formulated in the 1st stage +Microgrid-UC for balancing 3-phase loads. +𝑃���,�,� � ��𝑈�,� +� 𝑃�,�,� � 𝑃�,�,� +����� +�� +��� +� 𝑃�,� +�� +�28� +0 � 𝑃�,� +�� � 𝑘�� ��𝑈�,� +� 𝑃�,�,� � 𝑃�,�,� +����� +�� +��� +�29� + +�𝑘��� +���𝑃��� +��� � 𝑃���,�,� � 𝑃���,� +��� � 𝑘��� +���𝑃��� +���, ∀𝑝 ∈ �𝑎, 𝑏, 𝑐��30� + +where 𝑃�,� +�� denotes the DR budget of phase 𝑝 ; 𝑃���,�,� and +𝑃���, � +��� represent PCC phase power and the average power; 𝑘�� +is the DR budget limit factor for phase balancing; 𝑘��� +��� denotes +the unbalance limit. +Equation (28) is the PCC power balance equation +considering the per-phase DR budget (𝑃�,� +��) for each scheduling +interval 𝑡; (29) limits the amount of DR resource used for phase +balancing; (30) represents the 3-phase power balancing +constraints at the PCC. Budgeting DR in UC for meeting 3- +phase power balancing requirements is a novel consideration in +UC problem formulation so we consider it as one of the main +contributions of the paper. +H. Microgrid Reconfiguration +Although Spanning tree (ST) is widely used for formulating +radial topology constraints in feeder reconfiguration problems +[24], there are a few drawbacks. First, because in the 1st stage +Microgrid-UC, we won’t consider voltage regulation and +losses. Thus, an underlying assumption is that all topology +options supplying the same LGs have the same performance. +Second, as shown in Fig. 10, ST could assign three different +topologies at time 𝑡, 𝑡 � 1, and 𝑡 � 2 to serve the five LGs. +However, this leads to many unnecessary switching operations. +Thus, a post-process is needed to minimize the total number of +switching by choosing only one topology for the three time +periods (see Fig. 2). Third, additional constraints are needed to +be added for excluding infeasible options (e.g., limited by +protection settings, low voltage, or line faults). If there are many +infeasible topology options, the computational complexity +increases quickly. Lastly, ST does not allow multiple grid- +forming resources to coordinate in a microgrid directly [25]. + +Fig. 10. Multiple feeder reconfiguration options when supplying all 5-LGs. +Thus, we propose to replace ST with searching through a list +of predefined topology candidates. All candidates on the list +satisfy protection settings and meet voltage regulation +requirements. Thus, no post-process is needed. +The topology candidates set, 𝛺� +����� , contains 𝑁����� +feasible topology candidates at time 𝑡 . Let 𝑈�,�, +����� be the +selection status of the 𝑥�� topology option with 1 as being +selected and 0 not selected. +Because only one topology candidate can be selected at each +interval, we have +� 𝑈�,� +����� +������ +��� +� 1 +�31� +Then, the LG on/off status is determined by +�𝑈�,�, +� 𝑈�,�, +� … 𝑈��,�, +� +� +� + � ℳ� +� �𝑈�,�, +�����𝑈�,�, +����� … 𝑈������,�, +����� +� +� +�32� +where ℳ� +� is the 𝑁� � 𝑁����� LG mapping matrix. +Finally, the switch status is determined by +�𝑈�,�, +��𝑈�,�, +�� … 𝑈���,�, +�� +� +� +� ℳ� +�� �𝑈�,�, +�����𝑈�,�, +����� … 𝑈������,�, +����� +� +� +�33� +where ℳ� +��is the 𝑁�� � 𝑁����� switch status mapping matrix. +I. The 2nd Stage Microgrid-UC Problem Formulation +The objective function of the 2nd stage Microgrid-UC +minimizes the amount of DR usage, 𝑓� +��, the PV curtailment, +𝑓� +��, and the BESS energy deviation from its budget, 𝑓� +����. + +min 𝑓� +�� � 𝑘� +����𝑓� +���� � 𝑘� +��𝑓� +�� +�34� + +𝑓� +�� � � � � +� +𝑈�,�,�,�� +�� +𝑃�,�,�,��∆𝑡� +�∈��,�,�� +�� +���� +��� +�� +��� +�� +���� +�35� +𝑃�,�,�,�� � 𝑈�,�,�,�� +�� +�𝑃�,�,�,�� � 𝑘�,�� +����𝑃�,�,�,�� +�������� +�36� +𝑓� +���� � ∆𝐸� +� +�37� +∆𝐸� +� � ∆𝐸� +� � 𝐸�,�� � 𝐸�,� +�38� +𝑓� +�� � 3 � 𝑃��, ��∆𝑡� +�� +���� +�39� +where 𝑘� +���� and 𝑘� +�� are the weight coefficients of the BESS +energy storage deviation and the PV curtailment; ∆𝑡� is the +scheduling interval (5 minutes); 𝑇� is the scheduling horizon + +t + 1 +t + 2 +LGwith/withoutDER(served +Switch (on) +Switch(off) +7 +(30 minutes) of the 2nd stage; 𝑁� +���� is the number of nodes in +LG 𝑚, 𝑖 is the node index; 𝑈�,�,�,�� +�� + is the demand response +actions (1: turn off the load); 𝑘�,�� +���� is the CLPU factor +calculated by interpolating the first stage’s estimated CLPU +factor 𝑘�,� +���� linearly; 𝐸�,� is the BESS energy setpoint for the +2nd stage (it is the BESS energy by the end of step 𝑡 from the +first stage optimization); 𝐸�,�� is the BESS energy by the end of +the last step 𝑇� at the 2nd stage; ∆𝐸� +� and ∆𝐸� +� are the positive +and negative energy storage deviations (both are non-negative). +The CLPU effect is included by (36) in the 2nd stage, (37) +only penalizes the negative BESS deviation and (38) calculates +the BESS energy deviations. +In the 2nd stage, we perform linear power flow [29] with +voltage regulation constraints, the hybrid PV plant voltage is set +at 1.03 p.u., for other nodes, the voltage should be regulated +within [0.95,1.05] p.u.[25]. The DR is dispatched to meet power +balance, and 3-phase power balancing requirements and system +reserve limits. The PV and BESS operational constraints, +microgrid reserve constraints, and polygon-based linearization +of active power and reactive power constraints of the inverters +and the switches are similar to those in the 1st stage, the +imbalance limit (30) is also included. +III. SIMULATION RESULTS +In this paper, the performance of the proposed algorithm is +demonstrated on a modified IEEE-123 bus system, as shown in +Fig. 1. If the main grid power is lost, the feeder will be supplied +by a hybrid PV plant (on node 7) consisting of a PV farm (rated +at 4.2 MW) and a BESS (rated at 3 MW/6 MWh). +The PV forecast data are obtained based on actual +measurements of a 5MW PV farm. As shown in Fig. 11, there +are significant PV forecast errors on the second day. The +charging/discharging efficiency of the BESS is 95%. The +minimum and maximum SOC of the BESS are set at 20% and +90%, respectively, and the initial SOC is 90%. As the only grid- +forming source, the BESS regulates the system voltage at 1.03 +p.u.. +Nodes 47 and 65 have critical loads with the load priority +weighting (𝑤�,� +����) set as 2. The outage length is 48 hours. There +are two customer preferred service periods: 7:00-9:00 a.m. and +18:00-20:00 p.m.. The preferred time weighting (𝑤� +����) is 1.5. +The load profiles at each load node in the 123-bus system +are populated using the Pecan Street dataset [26] using methods +introduced in [25]. As shown in Figs. 12 and 13, the 3-phase +loads in an LG can be very unbalanced (e.g. LG1). The non- +HVAC load (i.e., the base load) is forecasted using the method +introduced in [30]. By combining the baseload and the modeled +HVAC loads, we can calculate the total house level loads. +CLPU parameters are obtained by the method presented in +Section II.F, we assume each LG has the same CLPU +parameters except 𝑃�,� +������. The corresponding weather data are +downloaded from NOAA. +The proposed Microgrid-UC algorithm has been formulated +as an MILP problem, it is solved by the CPLEX 12.10 +integrated with MATLAB 2019b on a desktop with I9-9900 +CPU and 64G RAM. The EMS-OpenDSS simulation is +conducted using the Matlab COM interface. + +Fig. 11. Total load and per-phase PV profiles (forecasted and real) on the 123- +bus feeder. Forecast-NR denotes the non-rolling day-ahead forecast. + +Fig. 12. Unbalance factor of the total feeder load. + +Fig. 13. Per-phase load forecast (5-min) for the five LGs. +A. Cold Load Pickup +Two cases (with and without CLPU estimation) are set up to +quantify the impact of CLPU on microgrid EMS. To show the +CLPU effect more clearly, the phase imbalance limits are +ignored in the 2 stages and the DR budget is not considered in +the 1st stage UC. The reconfiguration is constrained by ST. We +run the EMS-OpenDSS co-simulation using baseload forecast +and the simulated HVAC load, both of which are 1-minute data. +The results are presented in Figs. 14-19. In the figures, “RT” +denotes real-time simulation results and “EMS” denotes the 2nd +stage Microgrid-UC dispatching results. From the results, we +have made the following observations: + +As shown in Fig. 14 (a), if we don’t consider CLPU, +MSD will be frequently violated and there will be many +short interruptions (e.g., nine 30-minute interruptions). +This is caused by frequent overspending of the given +energy budget when picking up previously “off” LGs. +In contrast, when CLPU is considered (see Fig. 15 (a)), +MSD is satisfied most of the time. The minimum +service duration is 1 hour, but it occurred only once. + +As shown in Fig. 14 (b), without the CLPU estimation, +the actually served load can be over two times of the +Microgrid-UC dispatch values, showing that significant +amount of additional energy is needed to meet CLPU +when picking up those “off” groups. This leads to a +deficiency in energy budgeted for subsequent hours. + +TCS500C3TC4TC204p85054583035343Q38↑08T0005000300006EOLGcg2! +(30-JJJJ- OL6GC92f-MK +30-JJJEOLGC92! +O-JJJBGST +(-JIIUOH20000℃10'3MOS +30-JJJ1OQ85054583035343038048O-JUAOLS +J26O91O1500rC500B +8 +Thus, LGs are frequent turned off for compensating the +deficiency. As shown in Fig. 15 (b), by taking CLPU +into consideration, the served load matches the actual +load closely. This shows that Microgrid-UC optimizes +the supply sequence of LGs to minimize CLPU effects. + +Fig. 14. Without CLPU estimation: (a) LG status with “yellow” as served, (b) +served load profiles. + +Fig. 15. With CLPU estimation: (a) LG status with “yellow” as served, (b) +served load profiles. + +Fig. 16. (a) BESS and DR power dispatched by Microgrid-UC, (b) BESS +deviation between the real-time simulation results and the dispatch results. + +Fig. 17. With CLPU estimation: estimated and simulated (a) HVAC load, (b) +CLPU effect. + +As shown in Fig. 16 (b), without considering CLPU, the +actual BESS storage frequently drops below the +dispatched value, causing suboptimal solutions, +violation of MSD constraints, and more interruptions. +When considering CLPU (see Fig. 17), even if CLPU +lasts for hours, Microgrid-UC can still capture the +CLPU magnitude and duration very well. + +We do observe that the DR dispatching (see Fig. 16 (a)), +mainly caused by the forecast error, may cause more +CLPU estimation errors because the load shedding +action can also lead to CLPU (see Fig. 17). The DR +caused CLPU has not yet been account for in the +proposed algorithm. + +Figures 18 and 19 show that phase unbalance and +voltage fluctuations are more severe when CLPU is not +considered. This is because the unbalance can be +exacerbated by the additional CLPU energy needs. + +As shown in Table I, considering CLPU in the +Microgrid-UC can serve more loads and provide better +service to critical loads while meeting MSD +requirements and mitigating CLPU effects. + + +Fig. 18. Phase unbalance in real-time implementation: (a) without CLPU +estimation, (b) with CLPU estimation. + +Fig. 19. Load node voltages in real-time implementation: (a) without CLPU +estimation, (b) with CLPU estimation. + +TABLE I +MICROGRID PERFORMANCE COMPARISON WITH AND WITHOUT CLPU +CLPU +estimation +Served +energy +(kWh) +Served energy +during preferred +periods +(kWh) +Critical load served +time (h) +[Node 47 + Node 65] +Served HVAC +energy +(kWh) +CLPU +(kWh) +without +52913 +12284 +22.5+26.5 +30682 +5277 +with +53271 +10420 +25+26.5 +30104 +3711 +B. DR Budget +In this section, the benefit of budgeting DR for balancing the +3-phase load in the first stage UC is quantified. As shown in +Table II, we set up six cases with different DR budget limits +and 3-phase load imbalance requirements. Note that we only + +432 +LG +(a) +4000 +Utilized PV (RT) +Served Load (EMS) +3000 +Served normal Load (EMS) +(kW) +Served Load (RT) +Power +2000 +1000 +0 +4 +6 +(b)432 +LG +(a) +4000 +Utilized PV (RT) +Served Load (EMS) +3000 +Served Load (RT) +Power (kW) +2000 +1000 +0 +4 +6 +10 +(b)6000 +Storage (w/o CLPU estimation) +Storage (w/ CLPU estimation) +150 +DR (w/o CLPU estimation) +4000 +DR (kWh) +DR (w/ CLPU estimation) +100 +2000 +50 +(a) +RT-EMS (w/o CLPU estimation) +200 +RT-EMS (w/ CLPU estimation) +0 +200 +400 +600 +0 +10 +141618202224262830 +34.3638.4042.444648 +(b)2500 +40 +EMS (1st) +EMS (2nd) +HVAC Load (k W) +2000 +RT +30 +Tempereture (°C) +Temperature +1500 +20 +1000 +10 +500 +(a) +40 +EMS (1st) +EMS (2nd) +1000 +.RT +30 +CLPU (kW) +Temperature +20 +500 +10 +0 +0 +2 +6 +P +(b)0.6 +0.4 +0.2 +0 +(a) +0.6 +balance +0.4 +0.2 +0 +0 +(b)1.05 +Range +Averag +0.93 +(a) +1.05 +Range +Average +0.95 +0 +4 +32.34.36.384042444648 +(b) +9 +run the Microgrid-UC in standalone and CLPU is considered +for all 6 cases. +TABLE II +MICROGRID PERFORMANCE COMPARISON CONSIDERING DR BUDGETS +Case Unbalance +limits +1st stage +2nd stage +DR +budget +（% +served +load） +Served +energy +(kWh) +DR +budget +(kWh) +Served +energy +(preferred +period) +(kWh) +Critical +load +served +time +(h) +Served +energy +(kWh) +Demand +Response +(kWh) +PV +Curtailment +(kWh) +1 +- +0 +56264 + 0 +9966 +26.5+26.5 54020 +1963 +127 +2 +- +10% 55752 1503 +10045 +26+29 +54191 +2767 +34 +3 +0.2 +0 +21677 +0 +0 +8.5+8.5 +21505 +80 +29234 +4 +0.2 +10% 55811 2101 +9237 +25.5+26 +54164 +3433 +19 +5 +0.2 +20% 54999 3923 +10226 +28+28 +53789 +4806 +420 +6 +0.2 +30% 55177 6754 +11013 +28+28 +54119 +7509 +100 + +The following observations are made: + +If phase imbalance is not considered (Cases 1 and 2), +the DR budget mainly prolongs the service time of the +critical load, increases the served load during preferred +periods, and reduces PV curtailments. + +If phase imbalance is considered in both stages (Case 3) +without DR budget considerations in the 1st stage, the +microgrid serves the least load, curtails the most PV, +results in the least service time for critical loads, and +serves the smallest amount load in preferred periods. + +If phase imbalance is considered in both stages, with the +DR budget in 1st stage at 10% (case 4), there are more +load severed with the least PV curtailment. However, +further increasing the DR budget limit from 10% to +30% (cases 5 and 6) will not lead to system performance +improvement. This indicates that recruiting 10% load +for DR for each scheduling interval is the optimal +operation setting for maintaining phase balance, serving +critical loads, and reducing PV curtailments. + +Note that the flexibility of microgrid service is topology +specific. In our case, the limiting factor is the +imbalance load in LG1 because the hybrid PV plant is +located in LG1. To compensate for the imbalance in +LG1, Microgrid-UC may give priority to serve LGs that +can balance the LG1 loads. By scheduling DR to +remove the unbalance in each LG, we not only enable +the microgrid to provide service in more hours (see Fig. +20), but also improves the fairness and flexibility when +serving LGs. + +Figure 21 shows how budgeting DR facilitates the +unbalance control in case 4. Note that there are a few +intervals when all three phase DR resources are used for +achieving other benefits (like the improvements of case +2 compared to case 1). +We conduct EMS-OpenDSS co-simulation for case 4 to +verify the performance of the unbalance control. As shown in +Fig. 22, the unbalance level is significantly lower than that of +the case without unbalance limit (see Fig. 18). Note that the +unbalance factor can be slightly over 0.2 due to forecast errors. + +Fig. 20. Service status of each LG with different DR budget settings (𝑘��). + + +Fig. 21. PCC power scheduling (1st stage) of case 4. (a) PCC load forecast (no +DR), (b) Scheduled DR actions, (c) unbalance factors before DR, and (d) +unbalance factors after DR. + + +Fig. 22. Phase unbalance in real-time implementation (case 4). + +C. Topology Scheduling +The 5-LG system has 16 topology options (see Fig. 23). To +compare the topology candidate method with the ST method, +we set up four cases: ST (spanning tree), Candi-all (all 16 +options), Candi-R4 (omit options 13, 14, 16), and Candi-R5 +(omit options 8, 13, 14, 16). Note that options 13, 14, and 16 +are omitted because, when all 5 LGs are served, option 15 is the +feeder’s default topology (better protection settings). Option 8 +is not selected because it leads to voltage regulation issues due +to heavy loading. The DR budget limit is 10% in stage 1 and +the unbalance limit is 0.2. CLPU is considered in all cases. +The service performances of the four cases are the same. The +topology scheduling mainly impacts the runtime of the 1st stage +UC (the 2nd stage only needs less than 5 seconds). Removing +infeasible/problematic topology candidates from the searching +list has two benefits: simplifying the protection settings and +shortening the Microgrid-UC runtime. +As shown in Fig. 24, the maximum runtime of Candi-all is +the highest, followed by ST. Candi-R4 and Candi-R5 reduce +calculation time significantly, making the runtime for each +scheduling within 80s. In practice, to serve the same LGs, there +may be only 1 path that is feasible due to protection settings or +Unbalance + +5 +DR +4 +G +3 +L +2 +5 +0.1 +4 +G +3 +L +2 +5 +4 +G +3 +L +2 +5 +G +4 +L +3 +2 +0 +2 +4 +61000 +B +kw +500 +0 +(a) +100 +kw +50 +(b) +lance +Unbal +0.2 +(c) +lance +0.4 +Jnbal +0.2 +0 +0 +2 +4 +6 +(d)1JO485054583035343Q3840454480°4OS0.0 +10 +for meeting service requirements. Therefore, using a list of +candidates will greatly simplify the UC computing time while +avoiding infeasible solutions. + + +Fig. 23. Sixteen microgrid topologies. Each circle represents an LG, a grey +filled circle denotes a served LG. Each line represents a switchable branch. Red +solid lines as “on” branches and black dotted lines are “off”. + + +Fig. 24. 1st stage UC runtime comparison. +IV. CONCLUSIONS +This paper proposed a 2-stage Microgrid-UC method for +supplying 3-phase unbalanced loads in an islanded microgrid +for multiple days. Microgrid-UC uses an adaptive CLPU +estimation method to schedule for CLPU, which can account +for temperature variations when interruption durations vary. +Microgrid-UC balances the 3-phase loads to improve microgrid +service flexibility by budgeting DR for load balancing services +in the first stage. Lastly, Microgrid-UC used topology candidate +based reconfiguration method to improve the computational +efficiency. The superior performance of Microgrid-UC +compared with the conventional UC algorithms are verified by +the test cases. In our follow-up paper, we plan to extend the +Microgrid-UC algorithm for managing microgrids with +multiple grid-forming resources and across multiple feeders. +V. REFERENCES +[1] +A. Hussain, V.-H. Bui, and H.-M. Kim, “Microgrids as a resilience +resource and strategies used by microgrids for enhancing resilience,” +Appl. Energy, vol. 240, pp. 56–72, Apr. 2019. +[2] +H. Jiang, Y. Zhang, E. Muljadi, J. J. Zhang, and D. W. Gao, “A short-term +and high-resolution distribution system load forecasting approach using +support vector regression with hybrid parameters optimization,” IEEE +Trans. Smart Grid, vol. 9, no. 4, pp. 3341–3350, Jul. 2018. +[3] +H. A. Rahman, et al., “Operation and control strategies of Integrated +Distributed Energy Resources: A Review,” Renewable and Sustain. +Energy Rev., vol. 51, pp. 1412–1420, 2015. +[4] +E. Agneholm and J. Daalder, “Cold load pick-up of residential load,” +Proc. Inst. Elect. Eng., Gen. Transm. 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Butler-Purry, “Linear power flow +formulations and optimal operation of three-phase autonomous droop- +controlled microgrids,” Elec. Power Syst. Res., vol. 196, p. 107231, Apr. +2021. +[30] Y. Li, S. Zhang, R. Hu, and N. Lu, “A meta-learning based distribution +system load forecasting model selection framework,” Appl. Energy, vol. +294, Jul. 2021. + +3 +1 +5 +6 +8 +5 +5 +10 +11 +12 +X +4 +5 +5 +13 +14 +15 +16 +1 +3 +3 +4 +3 +4250 +200 +150 +100 +50 +0 +ST +Candi-all +Candi-R4 +Candi-R5 \ No newline at end of file diff --git a/iNE_T4oBgHgl3EQf4Byz/content/tmp_files/load_file.txt b/iNE_T4oBgHgl3EQf4Byz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..38f8a52ab9fa14f1597b549e159085e2429a443e --- /dev/null +++ b/iNE_T4oBgHgl3EQf4Byz/content/tmp_files/load_file.txt @@ -0,0 +1,930 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf,len=929 +page_content='1 \uf020 Abstract-- This paper presents a novel 2-stage microgrid unit commitment (Microgrid-UC) algorithm considering cold-load pickup (CLPU) effects, three-phase load balancing requirements, and feasible reconfiguration options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Microgrid-UC schedules the operation of switches, generators, battery energy storage systems, and demand response resources to supply 3-phase unbalanced loads in an islanded microgrid for multiple days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' A performance- based CLPU model is developed to estimate additional energy needs of CLPU so that CLPU can be formulated into the traditional 2-stage UC scheduling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' A per-phase demand response budget term is added to the 1st stage UC objective function to meet 3-phase load unbalance limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' To reduce computational complexity in the 1st stage UC, we replace the spanning tree method with a feasible reconfiguration topology list method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The proposed algorithm is developed on a modified IEEE 123-bus system and tested on the real-time simulation testbed using actual load and PV data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Simulation results show that Microgrid-UC successfully accounts for CLPU, phase imbalance, and feeder reconfiguration requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Index Terms—cold load pickup, demand response, feeder reconfiguration, microgrid energy management, resiliency, restoration, unbalance load management, unit commitment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' INTRODUCTION ICROGRIDS powered by distributed energy resources (DERs), primarily renewable generation resources and grid-forming battery energy storage systems (BESS), have attracted great interests in recent years as an effective operation mechanism to provide grid services and enhance distribution system resiliency [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Unit commitment (UC) is the key algorithm of the energy management system (EMSs) for scheduling generation resources in the bulk power system (BPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' However, directly applying BPS UC for microgrid EMS is oftentimes infeasible, especially for microgrids at the feeder-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' First, on a distribution feeder, intermittency of distributed renewables is compounded with uncertainty in loads, making combined forecasting errors much higher than those in BPSs [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Second, unlike large, synchronous generators in the BPS, DERs are constrained by both power and energy limits [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Particularly, the installed power and energy capacity of grid-forming BESSs or distributed generators are oftentimes insufficient to supply the microgrid load at all times in a prolonged outage that lasts This research is supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=" Department of Energy's Office of Energy Efficiency and Renewable Energy (EERE) under the Solar Energy Technologies Office Award Number DE-EE0008770." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' for days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Therefore, demand response (DR) and feeder reconfiguration have to be frequently used to shed loads for meeting power and energy limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Third, in the BPS, cold load pick-up (CLPU) [4] is seldom considered in UC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' However, in an islanded microgrid, due to interruptions mainly caused by the intermittency of DERs and feeder reconfigurations, cutting off a load and resupplying it is often time required during outages, making CLPU occur more frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Those additional CLPU energy needs so far cannot yet be predicted by load forecasting algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Note that in a distribution grid, CLPU is mainly caused by the recovery of Heating, Ventilation, and Air Conditioning (HVAC) systems, the electricity consumption of which accounts for approximately 50% of energy use in residential and office buildings [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Fourth, in BPS EMSs, loads are mostly 3-phase balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' However, in a distribution grid, even under normally operation conditions, loads are normally unbalanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Load imbalance can also be exacerbated by CLPU, feeder reconfiguration, or DR events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Because highly unbalanced loads can cause power quality issues, violate voltage regulation requirements, lower the sensitivity of the protection systems [6], it is critical in microgrid operation to maintain the phase imbalance within a given limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Thus, in this paper, we propose a novel 2-stage microgrid unit commitment (Microgrid-UC) algorithm that accounts for CLPU, using DR for three-phase load balancing, and the feasibility of reconfiguration options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Microgrid-UC manages the islanded operation of a 3-phase unbalanced distribution feeder for multiple days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The controllable resources include breakers/switches, DERs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=', PV farms, diesel generators, BESSs), and DR resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The four unique considerations of Microgrid-UC are explained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' CLPU modeling: In the literature, there are two approaches for modeling CLPU: model-based and data-driven methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The model-based approach predicts CLPU effects by physics- based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Using system on/off status and ambient temperatures as inputs, the electricity consumptions of HVACs are simulated to predict CLPU needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Either exponential [7] or linearized models [8] can be used for modeling the thermodynamics process that causes CLPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The drawback of this method is that predetermined HVAC model parameters cannot produce simulation results matching field The authors are with the Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC 27695 USA (emails: {rhu5, ashirsa, vmuthuk2, szhang56, yli257, lsong4, bxu8, vdaldeg, nlu2, baran, wtang8}@ncsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Rongxing Hu, Ashwin Shirsat, Valliappan Muthukaruppan, Si Zhang, Yiyan Li, Lidong Song, Bei Xu, Victor Paduani, Ning Lu, Fellow, IEEE, Mesut Baran, Fellow, IEEE, Wenyuan Tang, Member, IEEE A Novel Feeder-level Microgrid Unit Commitment Algorithm Considering Cold-load Pickup, Phase Balancing, and Reconfiguration M 2 measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' In contrast, the data-driven approach estimates the CLPU curve parameters using historical data [9-11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The drawback of this approach is the lack of CLPU event data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Thus, in this paper, we develop a hybrid CLPU modeling method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Instead of directly estimating the CLPU curve parameters, we use smart meter data to derive the HVAC model parameters [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The HVAC models can then be used to model CLPU effects for different ambient temperature and interruption durations, the results of which can be used to derive the CLPU curve parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Formulating CLPU constraints into EMS: Conventional distribution system EMS problem formulations only account for CLPU in restoration algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' For example, in [13], a Mixed-Integer Linear Programming (MILP) service restoration algorithm accounts for CLPU using a linearized, delayed exponential CLPU curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' In [14], a two-block representation (one for normal loads and the other for CLPU increments) is used to eliminate the CLPU nonlinear characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' In [15], to capture the CLPU power after short outages, a time dependent CLPU model based on operating state evolution of thermostatically controlled loads (TCLs) is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' In [16], the uncertainty of CLPU, captured by the probability density functions (PDFs) of CLPU peak and duration, is included in the restoration service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' However, the drawback of such formulations is the use of a predefined set of CLPU parameters, through which variations of outdoor temperature and interruption duration [17] cannot be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Thus, in this paper, we proposed an adaptive CLPU estimation method that can account for accumulated CLPU effects when picking up load groups (LGs) with different “off” durations under different ambient temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Load unbalance: In microgrid operation, maintaining 3- phase load balance is essential for maintaining voltage balance [18] and assuring protection relays to take correct actions [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' In [19], the authors propose two methods for controlling distributed generations to balance 3-phase loads: adding a penalty term representing the current unbalance to the objective function and using phase power for a conservative linear approximation of the current unbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' In [20], voltage unbalance is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' However, using DR for mitigating unbalance in UC is an uncharted area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Thus, we formulate a DR budget term into the UC problem for meeting 3-phase load balancing requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Topology Scheduling: In the literature, spanning tree (ST) [21-23] is a typical approach for distribution feeder reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' In [24], the authors analyze other radiality constraints including single-commodity flow (SCF) and the combined ST and SCF constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' However, if we need to formulate feasibility into topology options, the complexity and runtime of the algorithm will increase drastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' In practice, because of protection settings and circuit operational limits, not all topology options are feasible under different operation conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Thus, we propose a feasible topology candidate method to ensure the feasibility of the selected topology while shortening the runtime by over 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' To summarize, the novelties of Microgrid-UC are three-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' First, we develop an adaptive CLPU model so that CLPU can be formulated into both 24-hour ahead and intra-hour optimization problem formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Second, a DR budget term is formulated into the 24-hour ahead UC problem to balance 3- phase system loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Third, we use a feasible topology candidate method to guarantee operational feasibility and reduce runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Section II presents the proposed microgrid-UC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Results are presented in Section III and Section IV concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' METHODOLOGY In this section, we first introduce the layout of a typical feeder-level microgrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Then, the assumptions, the overall framework, the problem formulation and the operational constraints of the 2-stage microgrid-UC are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Typical Layout of a Feeder-level Microgrid In this paper, our focus is to develop a 2-stage microgrid-UC algorithm for managing a feeder-level microgrid by accounting for cold-load pickup, three-phase load balancing, and feeder reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 1, a typical feeder-level microgrid is powered by a hybrid system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=', the MW-level PV plant collocated with a grid-forming BESS at bus 7) and supplied multiple LGs, which can be prioritized into “critical” (the red triangles) and “non-critical”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The microgrid controller controls five switches (S1-S5) remotely to switch on/off LGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' An illustration of the typical layout of a feeder-level microgrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Assumptions We make the following assumptions regarding microgrid operation, data availability, and device controllability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' First, the microgrid controller has access to smart meter data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The controller controls the switches and DR resources in each LG remotely via a fully functional communication network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Second, critical loads have their own backup generators so the goal of the microgrid controller is to reduce the use of their backup generators by weighting the critical load with higher supply priorities than the non-critical load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Only non-critical loads participate DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Third, no circuit loop is allowed and there is only one grid-forming resource in the microgrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' In this paper, the BESS at bus 7 is the grid-forming resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Scheduling Horizons and Intervals Microgrid-UC is designed for multi-day, off-grid operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 2, a 24-hour ahead rolling forecaster and a 30- 32姜 29 O250 S5 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='350 30 33 251 110 112 113 114 28 50 3001 31 49 25 47 109 107 LG5 48 46 26 45 108 270 104 451 64 106 44 43 65 103 023 C 450 1050 102 635 100 24 420 41 660 101 LG1 40Q 39 93 71 22 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 620 S4 70 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 69 19 35 20 S2 68 75 Hybrid 37 LG3 67 60 74 LG4 57 S3 73 plant 11 14 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 59 72 79 85 P 文610 V 10 萧 9 53 54 S1 52 7778 2# 55 56 13 76 8 800 94 84 149 12 34# 96 90# 88* 810 17 15°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' O O 91 87 86 83 95 89 82 T 3 6 16# 195 PV farm BESS CriticalLoad Switch Load Group No load node 3-ph load node 000 A/B/Cloadnode 3 minute ahead forecaster are used to provide Microgrid-UC with PV, load, and weather forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Figure 3 shows the scheduling horizons and intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' In the first stage, a rolling 24-hour ahead unit commitment is conducted every 30-minute using 24-hour-ahead PV, load, weather forecast as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The outputs are the operation schedules for the BESS, DR resources, and switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Note that switches are switched on/off to supply/disconnect which LG for 48 30-minute scheduling intervals considering CLPU needs, phase-balancing needs, and reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' In the second stage, a 30-minute ahead power dispatch is conducted using 30-minute ahead PV and load forecast as inputs, while the weather input remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The outputs are the operation schedules for the BESS and DR resources for six 5-minute intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The flowchart of the two-stage Microgrid-UC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Scheduling horizons and intervals of the 2-stage Microgrid-UC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Problem formulation for the 1st Stage Let the total weighted served load be 𝑓� ����, the total PV curtailment penalty be 𝑓� �� , and the total cold load pickup (CLPU) penalty be 𝑓� ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The 1st stage objective function can be formulated as max 𝑓� ���� � 𝑘��𝑓� �� � 𝑘����𝑓� ���� �1� 𝑓� ���� � � � � 𝑈�,� � 𝑤� �����𝑃�,�,� ����� � 𝑤�,� ����𝑃�,�,� ���� �∆𝑡 �∈��,�,�� �� ��� �� ��� �𝑘�� � � 𝑤� ����𝑃�,� ��𝛥𝑡 �∈��,�,�� �� ��� �2� 𝑓� �� � 3 � 𝑃��,� ����𝛥𝑡 �� ��� �3� 𝑃��,� ���� � 𝑃��,� ���� � 𝑃��,� �4� where 𝑘�� and 𝑘���� are coefficients of the PV curtailment and the CLPU penalty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑁� is the number of scheduling intervals (𝑁� �48);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' ∆𝑡 is the scheduling interval (∆𝑡 � 30 minutes);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑚 is the group index, 𝑁� represents the total number of LGs, 𝑝 is the phase index;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑈�,� � denotes the status of the mth LG (1: “on” and 0: “off”);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑃�,�,� ����� and 𝑃�,�,� ����� are the forecasted non-critical load and critical load in the mth LG on phase 𝑝 at time 𝑡 without considering the CLPU effect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑤�,� ���� is the priority weighting of the critical load;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑤� ���� is the weighting of the customer preferred supply period[25] at time 𝑡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑃�,� �� is the DR budget at time 𝑡 on phase 𝑝 (note that only non-critical loads will provide DR);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑃��,� ����, 𝑃��,� ���� and 𝑃��,� are the PV curtailment, prediction and scheduled PV output on each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' For PV and BESS operational constraints, microgrid reserve constraints, and polygon-based linearization of active power and reactive power constraints of the inverters and the switches, please refer to [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Minimum Service Duration Constraints To avoid frequently switching on/off LGs, minimum service duration (MSD) constraints are introduced in [25], If an LG can be served, it is expected to be “on” for at least 𝐷� ��� consecutive scheduling intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' In this paper, we modify the formulation to allow rolling scheduling in the 1st stage considering initial service duration at the first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 4, 𝐷��,� is the remaining service duration need, which is determined by 𝐷��,� � max �min�𝐷� ��� � 𝐷�,��� ���������, 𝑁� � 𝑡 � 1�, 0� �5� � 𝑈�,��� � ���,� ��� � 𝐷��,��𝑈�,� � � 𝑈�,��� � �, 𝑡 � 1 �6� where 𝐷�,� ��������� is the service duration already fulfilled in the latest 𝐷� ��� steps, so 𝐷�,� ��������� � 𝐷� ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Note that MSD is treated as a soft constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Microgrid-UC can shut down an LG before MSD is fulfilled when there is insufficient energy supply for subsequent hours (reduce the default MSD, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' For example, the actual load is significantly higher or the PV is significantly lower than predicted values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The MSD requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' CLPU Constraints In a microgrid powered by intermittent renewable generation resources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=', PV and wind) and BESSs, CLPU may occur frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Not all LGs can be served for the entire scheduling period due to the uncertainty in generation and limitations in the BESS energy and power capacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Thus, oftentimes, shutting down LGs is inevitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Outage & PV & load weather forecast 1st stage, energy scheduling: information (interval: 30min, horizon: 24h) Resolution LG status, topology, storage budget, 30min DR budget, cold load pickup estimation Reduce the MSD Obtain solution ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' No Yes Switching operation post-process optimization (if needed) 2nd stage, power dispatch: Resolution .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='.5min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='. (interval: 5min, horizon: 30Omin) DR resource dispatch, BESS dispatch, voltage regulation LG status Real-time Implementation: BESS SOC HVAC load simulation, HIL/OpenDSSFirst-stage scheduling at 00: 00 t=1 t=2 Nr = 48 Secondstage time 00:00 00:30 01:00 1:30 24:00 dispatch First-stage scheduling at 00: 30 t=1 t=2 Nπ = 48 time 1:30 24:00 00:30 dispatchDMSD m DMSD MSDserved m m,ini m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='t m,ini m, Past : Past Grid outage ends 4 However, resupplying a previously “off” LG requires additional energy and power capacity to be allocated than supplying a previously “on” LG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The additional energy required in CLPU is mainly consumed by HVAC loads, which can account for approximately 50% of the total building load [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' After an LG is turned off for a prolonged period (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=', 60- minute), room temperature inside a building will coast out of the thermostat deadband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Thus, once the LG is switched on, all thermostatically-controlled HVAC loads will be turned on simultaneously, causing a synchronized load peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' This process can last from tens of minutes to hours depending on the “off” duration, the ambient temperature, and the thermostat setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' To account for CLPU in Microgrid-UC, we formulate additional energy budgets required for CLPU in the 1st stage scheduling using a novel hybrid CLPU modeling approach, which is a major contribution to the state-of-the-art UC problem formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 1) Develop the hybrid CLPU model To predict the CLPU effect under different ambient temperatures for different outage durations, we first need to know the HVAC model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' As a first step, we derive the HVAC parameters for each load profile in the Pecan Street dataset [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Thus, once the load profile of a load node on the feeder is selected from the Pecan Street dataset [25], the HVAC parameters for the node are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Note that in practice, if sub- meter HVAC load profiles are not available, load disaggregation algorithms [27-28] are needed to disaggregate HVAC loads from smart meter data, after which the HVAC model parameters can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' In the 123-bus test system, there are 1100 HVACs in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Using weather forecast and LG on/off status as inputs, we can then predict the CLPU effects by modeling the HVAC consumptions for different outdoor temperatures and for different outage durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' As shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 5 and 6, a significant amount of additional energy beyond the “normal” consumption is needed when picking up LGs that are previously “off”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' After a prolonged outage, the CLPU peak is the synchronized peak of all HVAC loads, 𝑃� �������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Note that if the outage occurs in a mild day, the CLPU peak may be lower than the synchronized peak (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 6, the 26 ºC case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' However, if a scheduling interval is 30- minute or longer, we can simplify the computation by assuming that the CLPU peak equals to 𝑃� ������� regardless of how many scheduling intervals the LG has been “off”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The simulation results can be used to generate Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 7, at a given outdoor temperature, 𝑇� ���, the CLPU peak duration, 𝑑�,� ����, is a function of outage duration, 𝑑�,� ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The longer the outage lasts, the longer the CLPU peak duration will be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' To simplify the calculation, we can linearize 𝑑�,� ���� versus 𝑑�,� ��� curves so that for a given temperature, an incremental peak duration can be calculated from the slope of the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Note that if the maximum outage duration, 𝑑�,� �������, is reached, 𝑑�,� ���� is capped at 𝑑� ��������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 8, the CLPU power decay rate, 𝛾�,� ����, is a function of outdoor temperature, 𝑇� ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The higher 𝑇� ��� is, the slower the CLPU peak decays from 𝑃� ������� to the steady- state HVAC consumption level, 𝑃�,� ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' To simplify the calculation, we ignore the impact of the interruption duration on the decay rate so that an equivalent power decay rate curve can be computed with respect to 𝑇� ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' From those results, a performance-based CLPU model (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 9) having the following parameters can be derived: the synchronized HVAC peak load of the LG (𝑃� �������in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 6), the CLPU peak duration rate and saturation ( 𝜏�,� ���� and 𝑑� �������� in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 7) to get the CLPU peak duration (𝑑�,� ���� in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 9), the CLPU decay rate (𝛾�,� ���� in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 8), and the CLPU steady-state load (𝑃�,� ������ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 6) estimated from the outdoor temperature range in steady-state operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Note that we select 𝑃� ������� to be the power base to make the normalized CLPU peak as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='0 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Thus, 𝑘�,� is the power of the HVAC load in the mth LG at time 𝑡 in per unit values with steady state value as 𝑘�,� ������, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' CLPU effects for different interruption durations (𝑇��� �36 °C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' CLPU effects for different outdoor temperatures (2-hour outage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' CLPU effects for different outage durations and under different outdoor temperatures (𝜏�,� ���� � ∆𝑑�,� ����/∆𝑡, where ∆𝑡 � 30 minutes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 3000 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='t m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content="t h 2500 PMaxCLPU m 7h 2000 pSteady 1500 m,t 1000 500 0 10 12 14 16 18 20 22 24 26 Time (hour)3000 26°C 28°C 2500 PMaxCLPU 30°C m 32°C Power(kw) 2000 34°C 36°C values 1500 38'C m,t 40°C at different 1000 Tout 500 0 10 12 14 16 18 20 Time (hour)00 Tout= 38℃ m,t 80 (minute) Tout=34 60 m,t Tout Tout=32 40 u Tout= 28℃ 20 0 1 2 4 5 6 7 8 9 10 At doff (hour) m." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='t 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' CLPU decay rates with respect to outdoor temperature with dots representing simulated decay rates for different outage durations 𝑑�,� ��� (Note that the curve is normalized to the synchronized CLPU peak, 𝑃�,� �������).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Model structure of the proposed performance-based CLPU model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 2) Estimate the CLPU peak duration To estimate the accumulated CLPU peak duration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' we have 𝑈�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� ���� � 1 � 𝑈�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� � �7� 0 � 𝑑�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� � 𝑀𝑈�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� ���� �8� 𝑑�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� � 𝑀𝑈�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� � � 𝑑�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='��� � 𝜏�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� ����𝑈�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� ����Δ𝑡 �9� 𝑈� ���� � 𝑈�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� ��� �10� 𝑑�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� ���� � 𝐷�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� ����𝑈� ���� �11� 𝑑�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� ���� � 𝑑�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� � 𝑀𝑈� ���� �12� 𝑑�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� �� � 𝑑�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='��� �� � 𝑑�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='��� ���� � 𝑈�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='��� � Δ𝑡 � 𝑀𝑈�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� ���� �13� 0 � 𝑑�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� �� � 𝑀𝑈�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� � �14� where 𝑑�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� is the estimated accumulated CLPU peak duration during interruptions without considering saturation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑈�,� ���� is the interruption status;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝜏�,� ���� is incremental CLPU duration for scheduling interval;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝐷�,� ���� is the saturated value at time 𝑡 derived (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑀 is a large number greater than 24 � 60 minutes (in our case, 𝑀 �1500);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑈� ���� is a binary variable indicating peak duration saturation status;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑑�,� ���� and 𝑑�,� �� are the estimated CLPU peak duration and the remaining peak duration, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Equation (7) determines whether the LG is “off”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' (8) and (10) ensure when the LG is served, CLPU peak and CLPU peak saturation status could be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' For each consecutive “off” interval, a resultant CLPU peak duration increment is added to the previous CLPU peak duration using (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Note that in (9), we do not consider the saturation effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' If the CLPU peak duration is saturated, (11) calculates the saturated CLPU peak duration and saturation status;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' if not, (12) ensures the accumulated CLPU peak duration by the end of the interruption duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Thus, (12) is disabled when the CLPU peak duration is saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Note that the maximum CLPU peak duration is capped according to the temperature of the step (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' When LGs are served intermittently, unfulfilled CLPU needs may be carried over to the next “on” cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' To account for it, remaining CLPU peak durations can be estimated by (13) and (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Note that minimizing the CLPU peak duration also leads to the minimization of additional energy needs for CLPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' In the results section, we will demonstrate that as a result of such considerations, Microgrid-UC tends to supply loads in consecutive intervals instead of turning them on/off frequently to minimize the total energy needed for CLPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 3) Set CLPU decay status To determine the CLPU decay status for the mth LG at time 𝑡, 𝑈�,� �����, and ensure that the CLPU decay will start only when the CLPU peak duration elapses, we add the following constraints 𝑈�,� ����� � 𝑈�,� � �15� 𝑀�1 � 𝑈�,� ������ � 𝑑�,� �� �16� �𝑀𝑈�,� ����� � 𝑀�𝑈�,� � � 𝑑�,� �� �17� where 𝑀� is a small constant (in our case, 𝑀� � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 4) Calculate CLPU power After 𝑃�,� ������ is estimated from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 6 based on 𝑇� ���, the steady state value, 𝑘�,� ������, can be obtained by (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The HVAC load factor 𝑘�,� is within peak value (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='0 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=') by (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑘�,� ������ � 𝑃�,� ������ 𝑃�,� ������� �18� 𝑘�,� ������𝑈�,� � � 𝑘�,� � 𝑈�,� � �19� If the LG is “on” at interval 𝑡 and the peak decay status (𝑈�,� �����) is still 1, the CLPU factor, 𝑘�,� ����, can be calculated by (20-21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' We assume when a LG is turned on all HVAC loads in the LG will be turned on such that the CLPU peak is 𝑃�,� �������, (1 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' ), this is ensured by (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑘�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� � 𝑘�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='��� � �𝑈�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� � � 𝑈�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='��� � � � 𝛾�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='�𝑈�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� ����� �20� 𝑘�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� ���� � 𝑘�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� � 𝑘�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� ������𝑈�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� � �21� Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' the CLPU power in kW value,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑃�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� ����,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' is calculated as 𝑃�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� ���� � 𝑘�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� ����𝑃�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� ������� �22� 5) Formulate CLPU into the 1st stage Microgrid-UC To mitigate the CLPU effect,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' a CLPU penalty term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑓� ����,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' consisting of three penalty factors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑘� ���� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑘� ����,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' and 𝑘� ���� is added to the 1st stage Microgrid-UC problem formulation as: 𝑓� ���� � � ��𝑘� ����𝑃�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� ����Δ𝑡 � 𝑘� ����𝑑�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� ���� � 𝑘� ����𝑑�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� �� � �� ��� �� ��� �23� 𝑃�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� � 𝑃�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� ����� � 𝑃�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� ���� �24� � 𝑃��,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� �∈�� ���� � 𝑈�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� � 𝑃�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� � 𝑃�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� ���� � � 𝑃��,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='� �∈�� �� �25� Tout 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='2 0 26 28 30 32 34 36 38 40 (℃)km,t (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=') m,t 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='0 ,Steady m,t decay m,t 10 Josn =1 m,t 6 where 𝑃�,�,� is the baseload (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=', the non-HVAC portion of the forecasted load) on phase 𝑝 in the mth LG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑃��,�,� is the active power at switch 𝑚𝑛 flowing from LG 𝑚 to LG 𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Ω� ���� and Ω� �� represent the “from” LG set and “to” LG set of the mth LG, , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Power balance within a non-source group is ensured by (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Note that for the LG the hybrid PV plant is located at, (25) will need to consider the outputs of PV and BESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Also, reactive power has similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 3-Phase Load Balancing Requirements Microgrid-UC is designed to serve 3-phase unbalanced loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' To reflect the Point of Common Coupling (PCC) power unbalance requirements, an unbalance factor 𝑘��� is defined as [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑘��� � max��𝑃���,�,� � 𝑃���,� ��� �� 𝑃���,� ��� , ∀𝑝 ∈ �𝑎, 𝑏, 𝑐� �26� 𝑃���,� ��� � 1 3 � 𝑃���,�,� �∈��,�,�� �27� In a feeder-level microgrid, the load can be highly unbalanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The current control mechanism does not allow 3- phase inverters to deliver highly unbalanced 3-phase power to the loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Thus, it is crucial that 1-phase DR resources can be scheduled to reduce the unbalance among the 3-phase loads actively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Thus, a DR budget term is formulated in the 1st stage Microgrid-UC for balancing 3-phase loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑃���,�,� � ��𝑈�,� � 𝑃�,�,� � 𝑃�,�,� ����� �� ��� � 𝑃�,� �� �28� 0 � 𝑃�,� �� � 𝑘�� ��𝑈�,� � 𝑃�,�,� � 𝑃�,�,� ����� �� ��� �29� �𝑘��� ���𝑃��� ��� � 𝑃���,�,� � 𝑃���,� ��� � 𝑘��� ���𝑃��� ���, ∀𝑝 ∈ �𝑎, 𝑏, 𝑐��30� where 𝑃�,� �� denotes the DR budget of phase 𝑝 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑃���,�,� and 𝑃���, � ��� represent PCC phase power and the average power;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑘�� is the DR budget limit factor for phase balancing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑘��� ��� denotes the unbalance limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Equation (28) is the PCC power balance equation considering the per-phase DR budget (𝑃�,� ��) for each scheduling interval 𝑡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' (29) limits the amount of DR resource used for phase balancing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' (30) represents the 3-phase power balancing constraints at the PCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Budgeting DR in UC for meeting 3- phase power balancing requirements is a novel consideration in UC problem formulation so we consider it as one of the main contributions of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Microgrid Reconfiguration Although Spanning tree (ST) is widely used for formulating radial topology constraints in feeder reconfiguration problems [24], there are a few drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' First, because in the 1st stage Microgrid-UC, we won’t consider voltage regulation and losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Thus, an underlying assumption is that all topology options supplying the same LGs have the same performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Second, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 10, ST could assign three different topologies at time 𝑡, 𝑡 � 1, and 𝑡 � 2 to serve the five LGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' However, this leads to many unnecessary switching operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Thus, a post-process is needed to minimize the total number of switching by choosing only one topology for the three time periods (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Third, additional constraints are needed to be added for excluding infeasible options (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=', limited by protection settings, low voltage, or line faults).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' If there are many infeasible topology options, the computational complexity increases quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Lastly, ST does not allow multiple grid- forming resources to coordinate in a microgrid directly [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Multiple feeder reconfiguration options when supplying all 5-LGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Thus, we propose to replace ST with searching through a list of predefined topology candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' All candidates on the list satisfy protection settings and meet voltage regulation requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Thus, no post-process is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The topology candidates set, 𝛺� ����� , contains 𝑁����� feasible topology candidates at time 𝑡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Let 𝑈�,�, ����� be the selection status of the 𝑥�� topology option with 1 as being selected and 0 not selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Because only one topology candidate can be selected at each interval, we have � 𝑈�,� ����� ������ ��� � 1 �31� Then, the LG on/off status is determined by �𝑈�,�, � 𝑈�,�, � … 𝑈��,�, � � � � ℳ� � �𝑈�,�, �����𝑈�,�, ����� … 𝑈������,�, ����� � � �32� where ℳ� � is the 𝑁� � 𝑁����� LG mapping matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Finally, the switch status is determined by �𝑈�,�, ��𝑈�,�, �� … 𝑈���,�, �� � � � ℳ� �� �𝑈�,�, �����𝑈�,�, ����� … 𝑈������,�, ����� � � �33� where ℳ� ��is the 𝑁�� � 𝑁����� switch status mapping matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The 2nd Stage Microgrid-UC Problem Formulation The objective function of the 2nd stage Microgrid-UC minimizes the amount of DR usage, 𝑓� ��, the PV curtailment, 𝑓� ��, and the BESS energy deviation from its budget, 𝑓� ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' min 𝑓� �� � 𝑘� ����𝑓� ���� � 𝑘� ��𝑓� �� �34� 𝑓� �� � � � � � 𝑈�,�,�,�� �� 𝑃�,�,�,��∆𝑡� �∈��,�,�� �� ���� ��� �� ��� �� ���� �35� 𝑃�,�,�,�� � 𝑈�,�,�,�� �� �𝑃�,�,�,�� � 𝑘�,�� ����𝑃�,�,�,�� �������� �36� 𝑓� ���� � ∆𝐸� � �37� ∆𝐸� � � ∆𝐸� � � 𝐸�,�� � 𝐸�,� �38� 𝑓� �� � 3 � 𝑃��, ��∆𝑡� �� ���� �39� where 𝑘� ���� and 𝑘� �� are the weight coefficients of the BESS energy storage deviation and the PV curtailment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' ∆𝑡� is the scheduling interval (5 minutes);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑇� is the scheduling horizon t + 1 t + 2 LGwith/withoutDER(served Switch (on) Switch(off) 7 (30 minutes) of the 2nd stage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑁� ���� is the number of nodes in LG 𝑚, 𝑖 is the node index;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑈�,�,�,�� �� is the demand response actions (1: turn off the load);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝑘�,�� ���� is the CLPU factor calculated by interpolating the first stage’s estimated CLPU factor 𝑘�,� ���� linearly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝐸�,� is the BESS energy setpoint for the 2nd stage (it is the BESS energy by the end of step 𝑡 from the first stage optimization);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 𝐸�,�� is the BESS energy by the end of the last step 𝑇� at the 2nd stage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' ∆𝐸� � and ∆𝐸� � are the positive and negative energy storage deviations (both are non-negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The CLPU effect is included by (36) in the 2nd stage, (37) only penalizes the negative BESS deviation and (38) calculates the BESS energy deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' In the 2nd stage, we perform linear power flow [29] with voltage regulation constraints, the hybrid PV plant voltage is set at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='03 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=', for other nodes, the voltage should be regulated within [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='95,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='05] p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='[25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The DR is dispatched to meet power balance, and 3-phase power balancing requirements and system reserve limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The PV and BESS operational constraints, microgrid reserve constraints, and polygon-based linearization of active power and reactive power constraints of the inverters and the switches are similar to those in the 1st stage, the imbalance limit (30) is also included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' SIMULATION RESULTS In this paper, the performance of the proposed algorithm is demonstrated on a modified IEEE-123 bus system, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' If the main grid power is lost, the feeder will be supplied by a hybrid PV plant (on node 7) consisting of a PV farm (rated at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='2 MW) and a BESS (rated at 3 MW/6 MWh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The PV forecast data are obtained based on actual measurements of a 5MW PV farm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 11, there are significant PV forecast errors on the second day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The charging/discharging efficiency of the BESS is 95%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The minimum and maximum SOC of the BESS are set at 20% and 90%, respectively, and the initial SOC is 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' As the only grid- forming source, the BESS regulates the system voltage at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='03 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='. Nodes 47 and 65 have critical loads with the load priority weighting (𝑤�,� ����) set as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The outage length is 48 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' There are two customer preferred service periods: 7:00-9:00 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' and 18:00-20:00 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='. The preferred time weighting (𝑤� ����) is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The load profiles at each load node in the 123-bus system are populated using the Pecan Street dataset [26] using methods introduced in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' As shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 12 and 13, the 3-phase loads in an LG can be very unbalanced (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' LG1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The non- HVAC load (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=', the base load) is forecasted using the method introduced in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' By combining the baseload and the modeled HVAC loads, we can calculate the total house level loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' CLPU parameters are obtained by the method presented in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='F, we assume each LG has the same CLPU parameters except 𝑃�,� ������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The corresponding weather data are downloaded from NOAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The proposed Microgrid-UC algorithm has been formulated as an MILP problem, it is solved by the CPLEX 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='10 integrated with MATLAB 2019b on a desktop with I9-9900 CPU and 64G RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The EMS-OpenDSS simulation is conducted using the Matlab COM interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Total load and per-phase PV profiles (forecasted and real) on the 123- bus feeder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Forecast-NR denotes the non-rolling day-ahead forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Unbalance factor of the total feeder load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Per-phase load forecast (5-min) for the five LGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Cold Load Pickup Two cases (with and without CLPU estimation) are set up to quantify the impact of CLPU on microgrid EMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' To show the CLPU effect more clearly, the phase imbalance limits are ignored in the 2 stages and the DR budget is not considered in the 1st stage UC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The reconfiguration is constrained by ST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' We run the EMS-OpenDSS co-simulation using baseload forecast and the simulated HVAC load, both of which are 1-minute data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The results are presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 14-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' In the figures, “RT” denotes real-time simulation results and “EMS” denotes the 2nd stage Microgrid-UC dispatching results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' From the results, we have made the following observations: As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 14 (a), if we don’t consider CLPU, MSD will be frequently violated and there will be many short interruptions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=', nine 30-minute interruptions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' This is caused by frequent overspending of the given energy budget when picking up previously “off” LGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' In contrast, when CLPU is considered (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 15 (a)), MSD is satisfied most of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The minimum service duration is 1 hour, but it occurred only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 14 (b), without the CLPU estimation, the actually served load can be over two times of the Microgrid-UC dispatch values, showing that significant amount of additional energy is needed to meet CLPU when picking up those “off” groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' This leads to a deficiency in energy budgeted for subsequent hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' TCS500C3TC4TC204p85054583035343Q38↑08T0005000300006EOLGcg2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' (30-JJJJ- OL6GC92f-MK 30-JJJEOLGC92!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=" O-JJJBGST (-JIIUOH20000℃10'3MOS 30-JJJ1OQ85054583035343038048O-JUAOLS J26O91O1500rC500B 8 Thus, LGs are frequent turned off for compensating the deficiency." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 15 (b), by taking CLPU into consideration, the served load matches the actual load closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' This shows that Microgrid-UC optimizes the supply sequence of LGs to minimize CLPU effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Without CLPU estimation: (a) LG status with “yellow” as served, (b) served load profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' With CLPU estimation: (a) LG status with “yellow” as served, (b) served load profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' (a) BESS and DR power dispatched by Microgrid-UC, (b) BESS deviation between the real-time simulation results and the dispatch results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' With CLPU estimation: estimated and simulated (a) HVAC load, (b) CLPU effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 16 (b), without considering CLPU, the actual BESS storage frequently drops below the dispatched value, causing suboptimal solutions, violation of MSD constraints, and more interruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' When considering CLPU (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 17), even if CLPU lasts for hours, Microgrid-UC can still capture the CLPU magnitude and duration very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' We do observe that the DR dispatching (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 16 (a)), mainly caused by the forecast error, may cause more CLPU estimation errors because the load shedding action can also lead to CLPU (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The DR caused CLPU has not yet been account for in the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Figures 18 and 19 show that phase unbalance and voltage fluctuations are more severe when CLPU is not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' This is because the unbalance can be exacerbated by the additional CLPU energy needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' As shown in Table I, considering CLPU in the Microgrid-UC can serve more loads and provide better service to critical loads while meeting MSD requirements and mitigating CLPU effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Phase unbalance in real-time implementation: (a) without CLPU estimation, (b) with CLPU estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Load node voltages in real-time implementation: (a) without CLPU estimation, (b) with CLPU estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' TABLE I MICROGRID PERFORMANCE COMPARISON WITH AND WITHOUT CLPU CLPU estimation Served energy (kWh) Served energy during preferred periods (kWh) Critical load served time (h) [Node 47 + Node 65] Served HVAC energy (kWh) CLPU (kWh) without 52913 12284 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='5+26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='5 30682 5277 with 53271 10420 25+26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='5 30104 3711 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' DR Budget In this section, the benefit of budgeting DR for balancing the 3-phase load in the first stage UC is quantified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' As shown in Table II, we set up six cases with different DR budget limits and 3-phase load imbalance requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Note that we only ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='432 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='LG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='Utilized PV (RT) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='Served Load (EMS) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='Served normal Load (EMS) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='(kW) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='Served Load (RT) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='Power ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='(b)432 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='LG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='Utilized PV (RT) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='Served Load (EMS) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='Served Load (RT) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='Power (kW) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='(b)6000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='Storage (w/o CLPU estimation) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='Storage (w/ CLPU estimation) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='DR (w/o CLPU estimation) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='DR (kWh) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='DR (w/ CLPU estimation) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='RT-EMS (w/o CLPU estimation) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='RT-EMS (w/ CLPU estimation) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='141618202224262830 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='3638.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='4042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='444648 (b)2500 40 EMS (1st) EMS (2nd) HVAC Load (k W) 2000 RT 30 Tempereture (°C) Temperature 1500 20 1000 10 500 (a) 40 EMS (1st) EMS (2nd) 1000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='RT 30 CLPU (kW) Temperature 20 500 10 0 0 2 6 P (b)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='2 0 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='6 balance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='2 0 0 (b)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='05 Range Averag 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='93 (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='05 Range Average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='95 0 4 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='384042444648 (b) 9 run the Microgrid-UC in standalone and CLPU is considered for all 6 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' TABLE II MICROGRID PERFORMANCE COMPARISON CONSIDERING DR BUDGETS Case Unbalance limits 1st stage 2nd stage DR budget （% served load） Served energy (kWh) DR budget (kWh) Served energy (preferred period) (kWh) Critical load served time (h) Served energy (kWh) Demand Response (kWh) PV Curtailment (kWh) 1 0 56264 0 9966 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='5+26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='5 54020 1963 127 2 10% 55752 1503 10045 26+29 54191 2767 34 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='2 0 21677 0 0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='5+8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='5 21505 80 29234 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='2 10% 55811 2101 9237 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='5+26 54164 3433 19 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='2 20% 54999 3923 10226 28+28 53789 4806 420 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='2 30% 55177 6754 11013 28+28 54119 7509 100 The following observations are made: If phase imbalance is not considered (Cases 1 and 2), the DR budget mainly prolongs the service time of the critical load, increases the served load during preferred periods, and reduces PV curtailments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' If phase imbalance is considered in both stages (Case 3) without DR budget considerations in the 1st stage, the microgrid serves the least load, curtails the most PV, results in the least service time for critical loads, and serves the smallest amount load in preferred periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' If phase imbalance is considered in both stages, with the DR budget in 1st stage at 10% (case 4), there are more load severed with the least PV curtailment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' However, further increasing the DR budget limit from 10% to 30% (cases 5 and 6) will not lead to system performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' This indicates that recruiting 10% load for DR for each scheduling interval is the optimal operation setting for maintaining phase balance, serving critical loads, and reducing PV curtailments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Note that the flexibility of microgrid service is topology specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' In our case, the limiting factor is the imbalance load in LG1 because the hybrid PV plant is located in LG1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' To compensate for the imbalance in LG1, Microgrid-UC may give priority to serve LGs that can balance the LG1 loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' By scheduling DR to remove the unbalance in each LG, we not only enable the microgrid to provide service in more hours (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 20), but also improves the fairness and flexibility when serving LGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Figure 21 shows how budgeting DR facilitates the unbalance control in case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Note that there are a few intervals when all three phase DR resources are used for achieving other benefits (like the improvements of case 2 compared to case 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' We conduct EMS-OpenDSS co-simulation for case 4 to verify the performance of the unbalance control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 22, the unbalance level is significantly lower than that of the case without unbalance limit (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Note that the unbalance factor can be slightly over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='2 due to forecast errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Service status of each LG with different DR budget settings (𝑘��).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' PCC power scheduling (1st stage) of case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' (a) PCC load forecast (no DR), (b) Scheduled DR actions, (c) unbalance factors before DR, and (d) unbalance factors after DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Phase unbalance in real-time implementation (case 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Topology Scheduling The 5-LG system has 16 topology options (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' To compare the topology candidate method with the ST method, we set up four cases: ST (spanning tree), Candi-all (all 16 options), Candi-R4 (omit options 13, 14, 16), and Candi-R5 (omit options 8, 13, 14, 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Note that options 13, 14, and 16 are omitted because, when all 5 LGs are served, option 15 is the feeder’s default topology (better protection settings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Option 8 is not selected because it leads to voltage regulation issues due to heavy loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The DR budget limit is 10% in stage 1 and the unbalance limit is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' CLPU is considered in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The service performances of the four cases are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The topology scheduling mainly impacts the runtime of the 1st stage UC (the 2nd stage only needs less than 5 seconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Removing infeasible/problematic topology candidates from the searching list has two benefits: simplifying the protection settings and shortening the Microgrid-UC runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 24, the maximum runtime of Candi-all is the highest, followed by ST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Candi-R4 and Candi-R5 reduce calculation time significantly, making the runtime for each scheduling within 80s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' In practice, to serve the same LGs, there may be only 1 path that is feasible due to protection settings or Unbalance 5 DR 4 G 3 L 2 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='1 4 G 3 L 2 5 4 G 3 L 2 5 G 4 L 3 2 0 2 4 61000 B kw 500 0 (a) 100 kw 50 (b) lance Unbal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='2 (c) lance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='4 Jnbal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='2 0 0 2 4 6 (d)1JO485054583035343Q3840454480°4OS0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='0 10 for meeting service requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Therefore, using a list of candidates will greatly simplify the UC computing time while avoiding infeasible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Sixteen microgrid topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Each circle represents an LG, a grey filled circle denotes a served LG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Each line represents a switchable branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Red solid lines as “on” branches and black dotted lines are “off”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 1st stage UC runtime comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' CONCLUSIONS This paper proposed a 2-stage Microgrid-UC method for supplying 3-phase unbalanced loads in an islanded microgrid for multiple days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Microgrid-UC uses an adaptive CLPU estimation method to schedule for CLPU, which can account for temperature variations when interruption durations vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Microgrid-UC balances the 3-phase loads to improve microgrid service flexibility by budgeting DR for load balancing services in the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Lastly, Microgrid-UC used topology candidate based reconfiguration method to improve the computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' The superior performance of Microgrid-UC compared with the conventional UC algorithms are verified by the test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' In our follow-up paper, we plan to extend the Microgrid-UC algorithm for managing microgrids with multiple grid-forming resources and across multiple feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Hussain, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Bui, and H.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Elect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=', Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Transm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Distrib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 147, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 44-50, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' [5] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Energy Information Administration (2013, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Hambrick, “Modeling and testing of unbalanced loading and voltage regulation: Final report,” Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Renew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Energy Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=', U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE_T4oBgHgl3EQf4Byz/content/2301.08350v1.pdf'} +page_content=' Dept.' metadata={'source': 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However, domain discrepancies in low-level image statistics and high-level contexts compromise the segmentation +performance over the target domain. A key idea to tackle this problem is to perform both image-level and feature-level adaptation +jointly. Unfortunately, there is a lack of such unified approaches for UDA tasks in the existing literature. This paper proposes a novel +UDA pipeline for semantic segmentation that unifies image-level and feature-level adaptation. Concretely, for image-level domain shifts, +we propose a global photometric alignment module and a global texture alignment module that align images in the source and target +domains in terms of image-level properties. For feature-level domain shifts, we perform global manifold alignment by projecting pixel +features from both domains onto the feature manifold of the source domain; and we further regularize category centers in the source +domain through a category-oriented triplet loss, and perform target domain consistency regularization over augmented target domain +images. Experimental results demonstrate that our pipeline significantly outperforms previous methods. In the commonly tested +GTA5→Cityscapes task, our proposed method using Deeplab V3+ as the backbone surpasses previous SOTA by 8%, achieving 58.2% +in mIoU. +Index Terms—Semantic Segmentation, Unsupervised Domain Adaptation, Photometric Alignment, Texture Alignment, Manifold +Modelling, Category Triplet Loss, Consistency Regularization. +! +1 +INTRODUCTION +S +EMANTIC segmentation, a classical and fundamental research +task in computer vision, aims to assign category labels to indi- +vidual pixels in an image. It has been extensively investigated and +has inspired many downstream applications including autonomous +driving [1], [2] and medical image analysis [3], [4], [5]. Although +the performance of existing semantic segmentation models have +enjoyed a significant improvement in the wave of deep neural net- +works [6], [7], [8], training a semantic segmentation model usually +requires a large number of images with pixel-level annotations, +the collection process of which is laborious and time-consuming. +Unsupervised Domain Adaptation (UDA) for semantic segmen- +tation is an alternative to avoid the data annotation problem: +it aims at learning a well-performing model from an unlabeled +target dataset by jointly exploiting labeled images from a different +source dataset (the label spaces of the two datasets must be +compatible). However, domain shifts/discrepancies exist between +different datasets. The most obvious differences are low-level +image statistics related to colors, textures, or even illumination +conditions. These differences can be partly alleviated by image- +level adaptation. However, there are also object-level differences, +such as object poses and spatial distributions, between different +datasets, which give rise to different feature distributions. All these +domain shifts have a detrimental impact on the final performance +of the semantic segmentation model. Therefore, it is crucial to +learn a feature representation capable of overcoming both image- +level and feature-level domain shifts for unsupervised domain +• +This work was supported in part by Hong Kong Research Grants Council +through Research Impact Fund (Grant R-5001-18). H. Ma was supported +by the Hong Kong PhD Fellowship. (Corresponding author: Yizhou Yu). +• +H. Ma, X. Lin and Y. Yu are with the Department of Computer Science, +the University of Hong Kong, Pokfulam Road, Hong Kong. E-mail: +mahaoyu@connect.hku.hk; xrlin2@cs.hku.hk; yizhouy@acm.org. +• +§ H. Ma and X. Lin have equal contribution. +adaptive semantic segmentation. +The causes of domain shifts/discrepancies have been exten- +sively studied in previous works. In general, the primary causes +can be categorized into image-level domain shifts and feature-level +domain shifts. Image-level domain shifts refer to the differences +in imaging conditions, such as lighting and settings in the camera +imaging pipeline. They affect the overall appearance of an image +and have a subtle influence on feature-level distributions. Existing +work addressing image-level domain shifts is in general based on +image-level style transfer, which makes use of deep models such +as generative models or image-to-image translation models [9], +[10], or Fourier Transformation [11]. We term these methods +image-level adaptation methods. These methods have proven that +transferring image styles or aligning feature distributions can bring +the two domains closer. However, generative methods usually +require a computationally expensive training process, whose in- +stability is notorious. Generative models also suffer from mode +collapse, which makes the range of the generated features unusu- +ally small (more explanation in Related Work). On the other hand, +the Fourier Transformation based method [11] produces inferior +style-transferred images, as shown in Figure 8. +We have observed that previous work in domain adaptive +semantic segmentation focusing on image-level domain align- +ment [10], [12] usually has inferior final segmentation perfor- +mance in comparison to recent work that adopts a more complete +pipeline [13], [14]. Such recent work further demonstrates that +replacing the original source domain images with image-level +domain aligned images can further improve the final performance +of feature-level adaptation techniques. This indicates that the +domain gap can only be partially alleviated with aforementioned +image-level adaptation methods, and feature-level alignment can +still benefit from an extra image translation module. Therefore, +feature-level adaptation is still necessary after image-level adapta- +arXiv:2301.01149v1 [cs.CV] 3 Jan 2023 + +IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE +2 +tion. For feature-level adaptation, a common practice in previous +studies employs an adversarial method [14], [15], which considers +features from the source and target domains aligned if they +cannot be distinguished by a trained discriminator. But adversarial +methods tend to generate a narrow range of feature distributions +to fool the discriminator. When different images share similar +feature distributions, trained models would have poor generaliza- +tion performance. On the other hand, to perform category-level +feature adaptation, some existing methods use category anchors +computed in the source domain to align the two domains [16], +[17], which can be regarded as imposing hard constraints on +category-level feature distributions. This method ignores feature +distances across different categories, and categories with similar +feature distributions in the source domain may still have similar +ones in the target domain, resulting in erroneous pseudo-labels +when no supervision signals are available in the target domain. +Our experiments demonstrate that imposing soft regularization +on category-level feature distributions by adjusting the relative +magnitude of inter-category and intra-category feature distances +can improve model capacity. +According to the above analysis, performing either image- +level adaptation or feature-level adaptation alone could not address +domain shifts adequately. Moreover, existing work on UDA for se- +mantic segmentation lacks a unified approach to minimize domain +shifts. Therefore, we approach the problem from both perspectives +and propose a novel and efficient pipeline that unifies image- +level and feature-level adaptation. For image-level domain shifts, +we propose two novel and training-free image-level operation, +called global photometric alignment and global texture alignment, +to adapt images from the source domain to the target domain. +However, image-level adaptation alone does not guarantee domain +alignment in the feature space. Therefore, we devise a global man- +ifold alignment module for feature-level adaptation. This module +represents the source domain feature manifold with a set of atoms, +and any pixel feature from the source domain or the target domain +can be projected onto this manifold. By minimizing the projection +errors between the input features and the manifold, all source +and target domain features are aligned to the same manifold. +To perform category-level feature adaptation, we also introduce +two category-level feature distribution regularization methods: a +category-oriented triplet loss is proposed in the source domain +to softly regularize category centers by enlarging the margin +between inter-category and intra-category feature distances. It is +only adopted in the source domain because the measurement +of inter-category and intra-category distances require reliable +annotations that only exist in the source domain. The category- +level feature adaptation method applied to the target domain is +the self-supervised consistency regularization. This regularization +makes the prediction on an augmented target image consistent +with the pseudo-label of the corresponding non-augmented image, +thus forcing the class labels of similar semantic contents to be +consistent in the target domain. By addressing domain shifts from +all perspectives simultaneously, experimental results demonstrate +that our proposed method is capable of achieving significant +performance improvements. +Domain adaptive semantic segmentation methods can be ap- +plied to either synthetic source images or real source images as +long as there exist significant domain gaps. For the application +to synthetic source images, we follow the common practice [11], +[13], [14], [15], [16], [18] and use the GTA5→Cityscapes and +SYNTHIA→Cityscapes benchmarks to evaluate our proposed +domain adaptation algorithm. In addition to synthetic source data, +we also construct a new task on two open-source real-world +endoscopic image datasets, Hyper-Kvasir [19] and Piccolo [20]. +This task can serve as a new medical image benchmark for future +studies in domain adaptive semantic segmentation. Experiment +results on all three benchmarks demonstrate that our proposed +method is capable of achieving significant performance improve- +ments over existing state-of-the-art algorithms. +To this end, this paper is an extension of +[18] and the +contributions of [18] can be summarized as follows, +• A novel image-to-feature domain adaptive semantic segmen- +tation pipeline is proposed to seamlessly combine coarse +image-level adaptation with category-level feature distribu- +tion regularization. +• Two novel and effective category-level regularization meth- +ods are proposed to deal with the source and target domain +shifts, respectively. The first one is category-oriented triplet +loss which regularizes category centers in the source domain, +and the second one performs target domain consistency +regularization over augmented target domain images. +• The +proposed +method +in +[18] +outperforms +all +previ- +ous methods, achieving state-of-the-art performances on +both GTA5→Cityscapes and SYNTHIA→Cityscapes bench- +marks. +Compared to the conference version [18], this paper gives a +more complete introduction and analysis of the proposed non- +adversarial image-to-feature domain adaptive semantic segmen- +tation pipeline. We provide more insights and discussions about +the modules proposed in [18]. More importantly, we extend our +work in +[18] by introducing global manifold alignment in the +high-level feature space. This manifold alignment algorithm serves +as a feature-level adaptation strategy complementary to global +photometric alignment proposed in [18]. An auxiliary data aug- +mentation scheme for global texture alignment is also proposed to +reduce the domain gap caused by texture variations. Experimental +results demonstrate that our proposed global manifold alignment +and global texture alignment modules make our proposed method +more robust and achieve new state-of-the-art performance. +To sum up, this paper has the following new contributions: +• A manifold alignment algorithm is proposed to represent the +high-level feature space via dimension reduction and clus- +tering algorithms. To the best of our knowledge, this is the +first piece of work that tackles unsupervised domain adaptive +semantic segmentation with explicit manifold modeling. All +related ablation studies have been conducted for this new +module. +• Global texture alignment is proposed as a data augmenta- +tion scheme for domain adaptive semantic segmentation. It +reduces the sensitivity of the trained model with respect to +domain-specific textures. +• For synthetic source data, our updated method outperforms +all previous UDA methods by a large margin, achieving new +state-of-the-art performance on both GTA5→Cityscapes and +SYNTHIA→Cityscapes benchmarks. +• We further construct a new medical image domain adaptive +semantic segmentation task on the basis of two open-source +real-world endoscopic image datasets. Our proposed method +also achieves state-of-the-art performance on this new task. + +IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE +3 +2 +RELATED WORK +Photometric Alignment. Previous works [21], [22], [23] in +unsupervised domain adaptation for image classification do not +pay attention to image-to-image translation. However, it has been +proven that a model trained with source images transferred into the +target domain style can significantly improve the final performance +in semantic segmentation tasks [14], [16]. This is perhaps because +deep features for semantic segmentation are relatively more sensi- +tive to local information compared to image classification. +In order to achieve image-level photometric alignment, ad- +versarial methods have been widely used in previous work on +domain adaptation +[12], [14], [17], [24], [25], [26], [27], [28], +such as GAN [9], [29] and CycleGAN [10]. These GAN-based +methods can transfer the styles of the images in the target domain +to that of the source domain and thus significantly reduce image- +level photometric differences [12], [14], [17]. Then, these style +transferred source domain images are used to train a segmentation +model. Because they are photometrically aligned with the target +domain images, the models trained with these style transferred +source domain images usually yield better performance compared +to the model trained with the source domain images [17] only. +However, it is also noted that adversarial models are unstable +during training. Previous work has shown that image-level ad- +versarial methods generally convert the source domain image- +level distribution to the one in the target domain to improve the +performance of the domain adapted model [14], [17]. But it is still +an open question whether the style transferred images distribution +roughly covers the whole target domain image-level distribution or +just a small part of it. The non-adversarial photometric alignment +methods for unsupervised semantic segmentation are rare. One +latest line of research is the Fourier Domain Adaptation proposed +in +[11]. The motivation is that the low-frequency component +of an image consists of the major photometric information, and +replacing the low-frequency component in a source domain image +with its reference image counterpart in the target domain could +align the photometric information between different domains. +However, the decomposition of frequency components is very +sensitive to the image’s content, and simply replacing the low- +frequency information of an image with that of another image +often introduces extra noises and leaves unsatisfactory visual +artifacts. According to their experiments, the model’s performance +trained on the frequency-aligned samples also relies heavily on +a multi-band ensemble with multiple models [11]. Unlike the +Fourier Domain Adaptation, our proposed method is directly +applied to color channels without the frequency decomposition, +which provides us with comparable (superior) performance and +image quality to its generative (Fourier Transformation-based) +counterpart. Moreover, our proposed method only consists of +several image-level operations which do not require standalone +training and can be used with arbitrary source-target image pairs. +Adversarial Methods for Domain Adaptation. There are tra- +ditional manifold learning methods that model high-dimensional +feature spaces before the deep learning era [30], [31], [32], [33], +but they are usually computationally costly when transplanted to +deep learning applications. Previous work on handling feature +spaces in UDA typically adopts adversarial methods [14], [15], +which do not directly model the feature manifold, but consider +features from the source and target domains aligned if they are +indistinguishable by a trained discriminator. However, generators +trained by adversarial methods are inclined to produce outputs +with similar feature distributions [34]. They can surely reduce +cross-domain feature distribution discrepancies and make image +features agnostic to the input domain. However, it also reduces +the diversity of image-level feature distributions from the same +domain. It is difficult to visualize high-dimensional feature dis- +tributions resulting from adversarial methods, but we can take +style-transferred RGB images generated by adversarial methods +as an example. As shown in Figure 8c, all images generated by +GAN are dark and smooth regardless of diverse image-level color +distributions in the target domain. This phenomenon is called the +mode collapse problem and is detrimental to the generalization +capability of the domain adapted model in the target domain. Most +recent algorithms [11], [15], [16] choose to remove adversarial +methods from their last stages due to this mode collapse problem. +Our approach differs from adversarial methods in that we model +the feature manifold directly by learning a feature manifold from +the source domain denoted by a set of representative feature +vectors. Then, we propose a pixel feature projection loss that +learns to project pixel features from both domains to the source do- +main feature manifold using these representative feature vectors. +Therefore, minimizing the projection errors from both domains +benefits domain alignment from a feature-level perspective. +Category-Based Methods. The distribution of different cat- +egory proportions can be very different between the source +domain and the target domain. Existing work [15], [24], [27], +[35] typically utilizes category labels/predictions to enforce global +semantic constraints on category distribution of predicted labels +in the target domain. Similar to their counterparts in image +classification [21], [22], [23], some previous works in semantic +segmentation (e.g. [16] and [17]) take one step forward to utilize +category information: the penultimate image features which are +used for generating pseudo-labels in the output layer in the target +domain are mapped to their corresponding counterparts in the +source domain. Another concurrent work [36] proposes to learn +category prototypes online and correct pseudo labels according +to the distance measurements between pixels features and those +learned category prototypes, which is an improved version of [16]. +However, category feature centroids used in [16], [36], or instance +features used in +[17] only serve as anchors for category-based +feature adaptation. The margins between different categories are +not explicitly enlarged. This is a problematic alignment strategy +because category centroids close to each other in the source +domain are still difficult to separate in the target domain. There- +fore, we propose a method that differs from theirs in two major +perspectives: first, a category-oriented triplet loss is proposed for +the source domain to impose a soft constraint that regularizes the +category centers for different categories. This approach actively +makes inter-category distances between different categories in the +high-level feature space larger than intra-category distances of a +certain category by a specified margin; secondly, we enforce the +predictions on augmented target domain images to be consistent +with the pseudo-labels generated by the segmentation model of +the corresponding non-augmented images. This is essentially a +self-supervision based consistency regularization method and the +design philosophy is based on the fact that the supervision signal +in the target domain is weak due to the lack of confident pseudo- +labels. + +IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE +4 +3 +METHOD +3.1 +Algorithm Pipeline +The underlying philosophy of our proposed pipeline is straight- +forward: first, we exploit the photometric differences in the two +domains to coarsely adapt the source domain images with the +target domain images to minimize the image-level domain shifts, +and the high-frequency distribution from the target domain is +also randomly transfered into the source domain image.; then, we +perform feature-level adaptation by aligning pixel features from +both domains with the feature manifold generated by the coarsely +adapted model regardless of its categories; finally, we impose +soft constraints on inter-class center distances and intra-class +feature variations to regularize category-level feature distributions. +Overview of the pipeline is presented in Figure 1 and is illustrated +as follows. +Settings. Suppose the labeled source domain dataset be +Ds = {(Is +m, Ys +m)}N s +I +m=1 where Is +m is a source image, Ys +m is +the pixel-level annotation of Is +m, and N s +I is the number of source +images in the source domain dataset. The target domain dataset +Du contains a large number of unlabeled images Du = {Iu +n}N u +I +n=1. +We assume the shape of all images is h × w × 3, and the +number of target classes to be segmented is Mc. Hence, we have +Ys +m ∈ {1, 2, · · · , Mc}h×w. The purpose is to learn a semantic +segmentation model for the target domain. +Step 0: Image-level Adaptation. Given a source domain +image Is +m in the training batch and a randomly selected target +domain reference image Iu +n, Is +m and Iu +n are converted into Lab +color space as (Ls +m, as +m, bs +m) and (Lu +n, au +n, bu +n) by our pro- +posed GPA module respectively. The histogram mapping function +fmatch(·) is then used to process both as +m and bs +m channels, and +gamma correction function fgamma(·) is applied to Ls +m to form +(fgamma(Ls +m), fmatch(as +m), fmatch(bs +m)). After the mappings, +the image is then converted back to RGB space to generate +the aligned image �Is +m. All these randomly generated adapted +images are used to construct the adapted source domain training +set �Ds = {( �Is +m, Ys +m)}N s +m=1 for each training epoch. Then, a +stochastic function τ1(·) is applied to the source domain training +set �Ds to produce an augmented version. A segmentation model +F0 is then trained based on the augmented style-transferred source +domain images τ1(�Ds) with the cross-entropy loss Lseg. +Step +1: +Feature-level +Adaptation. The aforementioned +image-level adaptation only diminishes the image-level domain +shifts between the source and target domains. But image-level +adaptation operations do not guarantee the adaptation of high- +level features because image components such as textures are not +altered by image-level photometric operations, and still impact the +high-level features. Therefore, we further modify a random subset +of the photometrically aligned images, and make their texture- +related high frequency components follow the corresponding +distributions in the target domain. Let �Is +m be a photometrically +aligned image, whose texture components are further updated. +The resulting image I +s +m is the actual input to the segmentation +model in this step. We also introduce a global manifold alignment +module to tackle the feature-level domain shifts. Before training a +new segmentation model, we learn a representation of the feature +manifold in the source domain offline. We first apply the initial +model F0 to all source domain images to obtain their feature +maps and prediction probability maps. Correctly classified feature +vectors from these feature maps are randomly sampled to form +matrix X . Then both PCA and K-Means clustering are applied +to X to learn a feature manifold represented with a set of cluster +centers z in a dimension reduced feature subspace. When a new +segmentation model is trained, features from both the source and +target domains are projected onto this manifold, and the projection +error, Lmfd, is minimized. +In addition to cross-entropy loss Lseg and manifold projec- +tion error Lmfd, two loss functions for category-level feature +distribution regularization are also adopted in the training process. +Category center fc for every category c is calculated as the L2 +normalized mean of all pixel features from category c in the source +domain. One of the two loss functions is a category-oriented +triplet loss Ltriplet defined over the image style transferred +source domain dataset �Ds to enlarge the inter-category distances +and minimize intra-category variances. In the target domain, the +pseudo-label at a certain pixel location and its associated confi- +dence are defined according to the prediction probability maps +produced from the initial segmentation model F0. Pseudo-labels +with confidence higher than an adaptive threshold are considered +valid samples, and are used to define a target domain consistency +loss Lcst to regularize category-level feature distributions in the +target domain. The remaining pixels are left out during back- +propagation. +We fine-tune the segmentation model F0 for U iterations by +minimizing Lseg + Lmfd + Ltriplet + Lcst to produce a new +segmentation model F1 for the current step. +Step 2 to K: Iterative Self-Supervised Training. Model F1 +trained in Step 1 can be further improved with iterative steps sim- +ilar to Step 1. Such an iterative approach is called self-supervised +training and is widely adopted in the area of unsupervised domain +adaptive semantic segmentation [11], [14], [16], [17]. The same +Step 1 is performed, but the pre-trained model F0 is replaced with +Fi−1. And model Fi−1 is also used to update manifold atoms z, +pseudo-labels and category centers fc. This process is repeated +for K − 1 times. Refined pseudo-labels generated by models +from each stage further improve the segmentation performance +and reduce the domain gap (Figure 2). But erroneous pseudo- +labels also accumulate false supervision signals and limit the +magnitude of performance improvement. Our proposed image-to- +feature pipeline is shown in Figure 1. +3.2 +Global Photometric Alignment +Since global domain shifts are mostly related to low-level image +attributes, global photometric alignment is proposed in our work to +transfer low-level image attributes of the target domain to source +domain images. It is observed that the spatial lightness distribution +of an image can be very complicated in different scenarios. It is +also important to note that directly operating on RGB channels +would cause severe artifacts and fake colors. In contrast, the +spatial color distribution of the a and b color channels always have +similar bell-shaped histograms. Therefore, we approach lightness +and color with different treatments: we perform classic histogram +matching [37] between the source domain image and the target +domain reference image only on color channels a and b to +avoid introducing artifacts commonly seen in histogram matching +results. +Lightness Gamma Correction. On the other hand, the L +channel is much more sophisticated under different circumstances. +This is because light interacts with the 3D structure of a scene +in a complicated manner. Simple histogram matching function +results in large areas of overexposure and fake structures. Thus, + +IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE +5 +Source Image +Aligned +Target Image +Pseudo-labels +𝓛𝒔𝒆𝒈 +𝓛𝒄𝒔𝒕 +𝓛𝒕𝒓𝒊𝒑𝒍𝒆𝒕 +Reference Target Image +: Shared weights +: Feat. encoder +: Classifier +(c) Feature-level Adaptation +Source Label +(a) Pipeline Overview +Source Image +Aligned +ℒ𝑠𝑒𝑔 +Reference target Image +(b) Image-level Adaptation Source Label +𝝉𝟏 +𝝉𝟐 +𝝉𝟐 +𝓛𝒎𝒇𝒅 +… +𝒛 +𝒇𝒄 +Fig. 1. (a) The pipeline consists of 1 image-level adaptation stage and K feature-level adaptation stages. (b) At first image-level adaptation is +implemented using the global photometric alignment operation. (c) Then the obtained model Fi is used to compute pseudo-labels, manifold atoms +z, category centers fc, category thresholds tc, as well as initialize the segmentation model for the subsequent feature-level adaptation stages in +an iterative self-supervised manner. +instead of using histogram matching for every histogram bin to +prescribe strict mapping, we choose to constrain the mean value +of the lightness channel in the source domain image and make +it equal to the mean value of the target domain reference image. +Because mean-variance policy might make the pixel value smaller +than 0 or larger than 1, we choose the power-law function, which is +also widely used in gamma correction. But the difference between +our proposed method and the classic gamma correction is that +our coefficients for the power-law function are not pre-defined by +users. They are automatically calculated with given source-target +image pairs. Specifically, the power-law function can be written +as fgamma(L) = Lγ, where L is the normalized lightness value +from 0 to 1 at each pixel location. γ = 1 when it is an identical +transformation. The mean value constraint can then be written as +� +L +fgamma(L)hs +m(L) = +� +L +Lγhs +m(L) = +� +L +Lhu +n(L), +(1) +where hs +m is the lightness histogram of a source image Is +m, and +hu +n is the lightness histogram of a target reference image Iu +n. +In practice, we introduce a regularization term β to prevent γ +from deviating too much away from 1. Thus, γ can be solved +numerically in the following nonlinear optimization, +γ∗ = arg min +γ +�� +L +Lγhs +m(L) − +� +L +Lhu +n(L) +�2 ++ β(γ − 1)2. +(2) +This optimization problem is a simple convex optimization with +only one variable γ, and can be easily solved with few steps of gra- + +GPAGPAStep 1-K:Feature- +level AdaptationUpdate +Tifc,tc,zStep 0:Image-level +AdaptationGTEXAIEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE +6 +20 +22 +24 +26 +28 +30 +32 +34 +0 +1 +2 +3 +4 +5 +6 +Domain Gap +Alternative Iteration Number +45 +47 +49 +51 +53 +55 +57 +59 +0 +1 +2 +3 +4 +5 +6 +mIoU +Alternative Iteration Number +Fig. 2. Iterative self-supervised training further improves the segmenta- +tion performance. +dient descent. Source-target image pairs are generated randomly +on the fly during training epochs because our proposed GPA is +highly efficient and does not require training in comparison to +GAN-based methods [14], [17]. The process of the proposed GPA +module is illustrated in Figure 3. +𝐿𝑠𝑟𝑐 → 𝐿𝑡𝑔𝑡 +𝑎𝑠𝑟𝑐 → 𝑎𝑡𝑔𝑡 +𝑏𝑠𝑟𝑐 → 𝑏𝑡𝑔𝑡 +(a) +(b) +(d) +(c) +Fig. 3. (a)Input source domain image and (b) a randomly chosen target +domain image is aligned in (c) Lab channels to generate (d) aligned +image. +3.3 +Global Texture Alignment +As discussed in previous work [38], CNN-based models are +sensitive to high-frequency information. We observe that synthetic +images have different and often stronger high-frequency informa- +tion in comparison to real-world images, which jeopardizes the +generalization performance of our model in the target domain. +Although the proposed GPA module maintains the diversity of +the source domain dataset, it modifies the photometric properties +of an image instead of the high-frequency texture. To alleviate +this problem, a global texture alignment module is proposed as an +auxiliary data augmentation scheme. The idea is straightforward: +we modify the high frequency components of a random subset +of the source domain images to make their distribution in each +image more consistent with that of the corresponding reference +image, which is sampled from the target domain. The process is +illustrated in Figure 1. This data augmentation scheme teaches the +segmentation model to ignore texture information and focus on +structural information. +To be specific, a bilateral filter fbilateral(·)|d,σc,σs is applied +to a source domain image �Is to generate the filtered image +I +s = fbilateral( �Is)|d,σc,σs. We use the bilateral filter to preserve +image structures and modify the texture component only. In order +to determine the parameters (d, σc and σs) of the bilateral filter, we +quantify the distribution of high-frequency image components, and +ensure that I +s and its target domain reference image Iu have sim- +ilar distributions of high-frequency components. We convert both +I +s and Iu to grayscale images and apply the Laplacian operator +fLap to obtain their high-frequency components Hs = fLap(I) +and Hu = fLap(Iu), respectively. Let h(Hs) and h(Hu) be +the respective histogram of Hs and Hu, and represent their +distribution of high-frequency components. To align h(Hs) and +h(Hu), the parameters of the bilateral filter are determined by +solving the following optimization problem, +d∗, σc∗, σs∗ = arg min +d,σc,σs KL( +� +s +h(Hs), +� +u +h(Hu))). +(3) +By applying the bilateral filter with optimized parameters, the KL +divergence between the distributions of the high-frequency com- +ponents of �Is and Iu can be significantly reduced. Note that d, σc +and σs are fixed once optimized. To introduce stochasticity, each +source domain image has 50% chance to be bilaterally filtered +before being fed to the segmentation model. We find that adding +this data augmentation scheme in the image-level adaptation step +would damage the final performance, and therefore, only use it as +an additional source domain data augmentation scheme during the +feature-level adaptation steps. +3.4 +Training Loss +The only training loss during image-level adaptation step is the +segmentation cross-entropy loss. The overall loss function we use +during feature-level adaptation steps consists of four parts: the +cross-entropy segmentation loss, the global manifold alignment +loss, the category-oriented triplet loss, and the target domain +consistency regularization loss. +𝒙𝑗 ∈ ℝ1×𝐷𝑐 +… +𝔃 ∈ ℝ𝑁𝑧×𝐷𝑐′ +𝑅(𝒙𝑗) ∈ ℝ1×𝐷𝑐′ +𝑊1 ∈ ℝ𝐷𝑐′×𝑁ℎ +𝑊2 ∈ ℝ𝐷𝑐′×𝑁ℎ +∗ +∗ +ෝ𝒙′𝑗 ∈ ℝ1×𝐷𝑐′ +𝑅−1(ො𝑥′𝑗) ∈ ℝ1×𝐷𝑐 +𝒘 ∈ ℝ𝑁𝑧×𝐷𝑐′ +𝓛𝒎𝒇𝒅 +softmax(⋅) +− +Fig. 4. By minimizing the projection error of source/target domain +features onto the manifold, our proposed manifold loss mitigates the +discrepancies between source domain feature distribution and target +domain feature distribution. + +1IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE +7 +Global Manifold Alignment. Methods such as Locally Linear +Embedding (LLE) and Isomap are commonly used to depict +manifolds, but they are too computationally costly for gradient +backpropagation based training. Here we use the K-means algo- +rithm to simplify the computation. As LLE uses a piecewise linear +model to approximate a high dimensional feature manifold, K- +means can be considered as a piecewise constant approximation +of the manifold. Every centroid obtained by K-means is a constant +approximation of a local region. By approximating the manifold +with a set of representative feature vectors, we can further align +features from the source and target domains. +In order to acquire feature representations, first we need to +apply the segmentation model obtained in the previous step Fi−1 +to each source image Is +m to compute the feature map of the second +last layer Xs +m and the final prediction probability map P s +m. The +feature vector and prediction probability at a given pixel location +j is denoted as xs +j and ps +j, respectively. The true category label at +location j is denoted as ys +j. Next, the predicted probabilities (ps +j) +are compared with true category labels (ys +j), and the correctly +classified feature vectors are randomly sampled to form the source +domain sample matrix X ∈ RNp×Dc, where Np is the total +number of sampled feature vectors and Dc is the dimensionality +of each feature vector. +Then, principal component analysis (PCA) is applied to X +to keep around 90% of the total explained ratio of energy and +obtain the dimension reduced version R(X) ∈ RNp×Dc′ , where +Dc′ << Dc. Afterwards, the classic K-Means clustering algo- +rithm is applied to R(X) to find the representative locations on the +feature manifold. These locations are denoted as z ∈ RNz×Dc′ , +which are essentially the atom vectors of the source domain feature +manifold. Any pixel feature from the source domain (xs +j) or the +target domain (xu +j ) can be projected to the subspace spanned by +the atoms in z, and the projection is represented as ˆx′ +j = wT z +(we omit the superscript for simplicity). The projection mapping +w is applied here for two reasons: first, although .Let R−1 +be the reconstruction operator of PCA. The projection error +||R−1(ˆx′ +j) − xj||2 is considered as the deviation from the source +domain manifold and is part of the projection error loss Lmfd. +The motivation of our proposed global manifold alignment is +straightforward: minimizing the source domain projection error +makes the feature manifold smoother, and minimizing the target +domain projection error decreases the distance (i.e. improves +the alignment) between feature distributions of the source and +target domains respectively. Specifically, we adopt an attention +mechanism to calculate the linear coefficients of atom vectors. The +manifold projection error Lmfd and reconstructed feature vector +ˆx′ +j can be computed using the following equations, +Lmfd = +� +j +||R−1(ˆx′ +j) − xj||2 +ˆx′ +j = wT z +wT = softmax +�(R(xj)W T +1 )(W2zT ) +√Nz +� +, +(4) +where R(xj) is the j-th row of R(X), both W1 ∈ RNh×Dc′ +and W2 ∈ RNh×Dc′ are trainable linear matrices respectively. +They are introduced to further lower the memory overhead of +the attention mechanism. They also project the manifold and all +features to a lower dimensional space, and two distinct projection +matrices enable better alignment between the projected manifold +and features. Nh is a hyperparameter representing the number of +hidden neurons, w ∈ RNz is the vector of atom coefficients. The +details to calculate the manifold projection loss is illustrated in +Figure 4. Although the global manifold loss is defined for the +global alignment of features, it cannot be adopted in the image +adaptation stage because it relies on a pre-trained model to provide +manifold atoms z. +Category-oriented Triplet Loss. Although the aforemen- +tioned GPA and GMA modules could learn domain-invariant +features to some extent, the losses used in previous training do not +explicitly control the category-wise feature distribution, and some +category-sensitive domain shifts are overlooked. Pixel features +from different categories are naturally distributed unevenly, and +some category centers are close to each other. To tackle this issue, +we propose a category-oriented triplet loss that aims to push the +category-wise features further closer to the corresponding category +centers the pixel belongs to and further away from other category +centers. Note that category centers are intentionally introduced to +make the calculation of category-oriented triplet loss practical. If +we use the traditional triplet loss without category centers, we need +to store pairwise distances among all pixels with a tremendous +GPU memory overload. Therefore, the category center fc of +category c is calculated as follows, +fc = G( 1 +Nc +� +s +� +j +1 +�ys +j = c +� xs +j), +(5) +where xs +j refers to the pixel-wise features in the penultimate +feature map, and ys +j be the ground truth pixel-wise labels of a +source domain image at pixel location (j). Nc refers to the total +number of pixels in category c, s refers to the source domain +image index, and G(·) is an L2 normalization function. Note that +it is crucial to use the L2 normalization G(·) to keep the category +centers on the unit sphere and avoid scaling issues among stages. +The category centers are updated after the training, allowing the +centers to become further and further from each other on the +sphere surface. +Our category-oriented triplet loss is formulated as follows, +Ltriplet = 1 +Ns +� +s +� +j +max +c,c̸=ys +j +max( +���G(xs +j) − fys +j +��� +− +��G(xs +j) − fc +�� + α, 0), +(6) +where Ns is the total number of pixels in all images, and α is a +prescribed margin. The loss will be zero if every feature xs +j is at +least α closer to its corresponding category center ys +j than other +category centers. Because triplet loss is focused on hard samples +and only reliable category labels for hard samples in the source +domain, the proposed category-oriented triplet loss is only applied +to the source domain images. +������������������������������������������������ +������������������������������������������������ +: Tgt. feats of cat. Green and Blue +: Feat centers of cat. Green and Blue +: Src. feats of cat Green and Blue +: Aug. feats of cat. Green and Blue +Fig. 5. Our proposed category-oriented triplet loss exploits hard samples +and further enlarge category margins. dpos and dneg represent the +distance of positive and negative pairs respectively. + +IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE +8 +The working principles of our proposed category-oriented +triplet loss are illustrated in Figure 5. In cooperation with the +proposed global photometric alignment and data augmentation +in the source domain, our proposed triplet loss can exploit hard +samples in the source domain that have been coarsely aligned to +the target domain and further improve the generalization capability +of the trained model. The proposed category triplet loss can +be considered complementary to the cross-entropy loss and the +manifold projection loss. +Target Domain Consistency Regularization. The category- +wise features are regularized by our proposed category-oriented +triplet loss in the source domain, where the annotated ground +truth labels are available. However, the supervision signal is weak +in the target domain where there is no labeled data provided. +Consistency regularization is an important component of many +recent state-of-the-art self-supervised learning algorithms, which +utilizes unlabeled data by relying on the assumption that the model +should output similar predictions when fed perturbed versions of +the same image [39], [40]. Motivated by this, we propose a target +domain consistency regularization method shown in Figure 1 to +perform category-level feature distribution regularization in the +target domain. +In the target domain, the pseudo-label at a certain pixel +location is defined as the category corresponding to the largest +component of the probability vector produced from the segmen- +tation model Fi−1 trained in the previous step, and the largest +component of the probability vector itself defines the confidence +of the pseudo label. We further pre-define a pair of probability +threshold Ph and percentage threshold p for all categories. p +is a constant value but leads to a category-specific probability +threshold Ps,c, meaning p% pixels in the category c have confi- +dence above threshold Ps,c. Thus the final confidence threshold +for category c is tc = min(Ph, Ps,c), and any pseudo-labels with +a confidence higher than tc in category c are considered valid +samples. +Our proposed target domain consistency regularization is +straightforward: given a target domain image Iu +n, with the trained +segmentation model Fi−1, we extract the pseudo-label ˆyu +n by +forwarding Iu +n to Fi−1 followed by applying the arg max(.) +function to its output; and the corresponding pixel prediction +is converted to a hard label vector 1[c=ˆyu +n,j]; then, a stochastic +function τ2(·) is applied to Iu +n to obtain a perturbed version �Iu +n; +after that, we forward �Iu +n to Fi to obtain the prediction � +P u +n +on the perturbed image; finally, the prediction � +P u +n is forced to +be consistent with ˆyu +n by using a cross entropy loss function +at pixel locations whose largest class probability is above the +previously defined category-level confidence threshold tc. Note +that the perturbed target domain image generated via the stochastic +function τ2(·) makes prediction harder. Thus more samples in +the target domain can be converted into hard samples, but the +generation of pseudo-labels is unaffected. In this way, category- +level feature distributions in the target domain are regularized +under the supervision of valid pseudo-labels. The overall formula +is defined as follows, +Lcst = +� +j +1(max(Fi−1(Iu +n)|j) ≥ tc) +CELoss(1[c=ˆyu +n,j], � +P u +n,j), +ˆyu +n,j = arg max(Fi−1(Iu +n)|j), +� +P u +n,j =Fi( �Iu +n)|j. +(7) +It is essential to use the trained model Fi−1 rather than model +Fi to generate pseudo-labels. This is because Fi is still being +trained and unstable. Fluctuating pseudo-labels generated by Fi +would be catastrophic to the training process. Experimental results +illustrate that this consistency regularization method is simple yet +efficient. It strengthens the supervision signal in the target domain +and improves the final performance. +4 +EXPERIMENTS +4.1 +Datasets and Implementation Details +For commonly used synthetic datasets, we follow the same evalu- +ation settings as used in [16]. Our proposed method are evaluated +with datasets GTA5 [42], Synthia [43], and Cityscapes [44]. The +Cityscapes dataset is the target domain dataset with 2, 957 of +size 2048 × 1024 training images and 500 validation images of +the same resolution. Cityscapes has 19 categories of objects in +total. The GTA5 and Synthia are two source domain datasets of +computer generated synthetic images, which contain 24, 966 of +size 1914 × 1052 training images and 9400 of size 1280 × 760 +training images respectively. The GTA5 dataset shares 19 common +categories with the Cityscapes dataset, and all the irrelevant +categories are ignored during training. The Synthia dataset shares +16 common categories with the Cityscapes dataset. Some previous +work [11], [14] only train and test on a 13-category subset of the +Synthia dataset, or train two models on both subset and the whole +set for better performance [16]. Here we follow the practice in +[13], [15] to train a model only on the whole set and test it on +both settings. +In order to evaluate the performance of our proposed method +on real-world source images, we construct a new domain adaptive +semantic segmentation task Kvasir→Piccolo on the basis of two +open-source datasets Hyper-Kvasir [19] and Piccolo [20]. The +Hyper-Kvasir dataset is the source dataset and consists of 1000 +wide-band (WL) gastrointestinal images. The image resolution of +the Hyper-Kvasir dataset is not fixed and is roughly 625 × 530. +The Piccolo dataset is the target dataset. Among the 1302 narrow- +band (NBI) colonoscopy images of size 854×480, 1161 are used +as training images and 141 as validation images. We follow the +rule in [41] to construct this new task specifically designed for +medical images: the images were collected with different modes +(NBI vs. WL), different locations (GI tract vs. colonoscopy) and +different devices to create a significant domain gap between the +source and target domains. +According to Figure 1, the photometrically adapted source +domain images are used first to train the initial segmentation +model F0 in the image-level adaptation step. Then, the model +is trained in an iterative self-supervision manner with K = 6 +and U = 20k. We compare to previous work in domain adapted +semantic segmentation [16], [17] based on self-supervision and +set the total number of training iterations to 140k. As reported +in [14], the best performance is achieved when Ph = 0.9 and +p = 10 for pseudo-labels, and we follow this setting in our +experiment. The regularization term β used by GPA module in (7) +is set to 0.01. For the proposed global texture alignment (GTEXA) +module, d = 5, σc = 75 and σs = 25 are the optimized +parameters of the bilateral filter for the GTA5→Cityscapes and +Synthia→Cityscapes tasks, the KL divergence is reduced roughly +from 0.16 to 0.10 and from 0.43 to 0.07, respectively. Because +images from both Hyper-Kvasir and Piccolo are real images, they +have quite similar distributions of high-frequency components. We + +IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE +9 +TABLE 1 +Performance comparison with state-of-the-art methods on the GTA5→Cityscapes task. Results with image-level adaptation only and our whole +image-to-feature pipeline are also presented. The best performing result is marked in bold +road +sidewalk +building +wall +fence +pole +light +sign +vege +. +terrace +sky +person +rider +car +truck +bus +train +motor +bike +mIoU +DeeplabV2 +BDL [14] +91.0 +44.7 +84.2 +34.6 +27.6 +30.2 +36.0 +36.0 +85.0 +43.6 +83.0 +58.6 +31.6 +83.3 +35.3 +49.7 +3.3 +28.8 +35.6 +48.5 +IDA [13] +90.6 +36.1 +82.6 +29.5 +21.3 +27.6 +31.4 +23.1 +85.2 +39.3 +80.2 +59.3 +29.4 +86.4 +33.6 +53.9 +0.0 +32.7 +37.6 +46.3 +DTST [17] +90.6 +44.7 +84.8 +34.3 +28.7 +31.6 +35.0 +37.6 +84.7 +43.3 +85.3 +57.0 +31.5 +83.8 +42.6 +48.5 +1.9 +30.4 +39.0 +49.2 +FGGAN [15] +91.0 +50.6 +86.0 +43.4 +29.8 +36.8 +43.4 +25.0 +86.8 +38.3 +87.4 +64.0 +38.0 +85.2 +31.6 +46.1 +6.5 +25.4 +37.1 +50.1 +FDA [11] +92.5 +53.3 +82.3 +26.5 +27.6 +36.4 +40.5 +38.8 +82.2 +39.8 +78.0 +62.6 +34.4 +84.9 +34.1 +53.1 +16.8 +27.7 +46.4 +50.4 +image adap. (ours) +84.6 +37.4 +81.0 +25.6 +12.9 +35.7 +33.8 +16.5 +83.5 +31.2 +82.7 +64.8 +35.7 +85.3 +30.0 +31.9 +8.0 +25.7 +32.2 +44.1 +I2F [18] +89.8 +46.0 +85.8 +32.5 +22.3 +41.0 +43.9 +28.9 +86.4 +31.0 +89.4 +65.6 +36.9 +87.9 +42.4 +54.4 +6.5 +38.9 +56.2 +51.9 +I2F all (ours) +90.8 +48.7 +85.2 +30.6 +28.0 +33.3 +46.4 +40.0 +85.6 +39.1 +88.1 +61.8 +35.0 +86.7 +46.3 +55.6 +11.6 +44.7 +54.3 +53.3 +DeeplabV3/+ +CAG [16] +90.4 +51.6 +83.8 +34.2 +27.8 +38.4 +25.3 +48.4 +85.4 +38.2 +78.1 +58.6 +34.6 +84.7 +21.9 +42.7 +41.1 +29.3 +37.2 +50.2 +WCBT [41] +89.4 +50.1 +83.9 +35.9 +27.0 +32.4 +38.6 +37.5 +84.5 +39.6 +85.7 +61.6 +33.7 +82.2 +36.0 +50.4 +0.3 +33.6 +32.1 +49.2 +image adapt. (ours) +83.9 +37.5 +82.7 +28.7 +18.9 +35.3 +41.3 +31.1 +85.2 +29.5 +86.6 +62.8 +30.9 +82.4 +23.0 +39.3 +33.0 +26.0 +39.7 +47.3 +I2F [18] +92.5 +58.3 +86.5 +27.4 +28.8 +38.1 +46.7 +42.5 +85.4 +38.4 +91.8 +66.4 +37.0 +87.8 +40.7 +52.4 +44.6 +41.7 +59.0 +56.1 +I2F all (ours) +92.6 +59.1 +87.0 +33.3 +32.2 +37.2 +46.3 +47.5 +86.2 +40.2 +90.7 +67.6 +41.1 +88.1 +48.7 +53.7 +47.0 +47.4 +60.6 +58.2 +TABLE 2 +Performance comparison with state-of-the-art methods on the Synthia→Cityscapes task (mIoU: 16-class; mIoU*: 13-class). The best performing +result is marked in bold +road +sidewalk +building +wall +fence +pole +light +sign +vege +. +sky +person +rider +car +bus +motor +bike +mIoU +mIoU* +DeeplabV2 +BDL [14] +86.0 +46.7 +80.3 +- +- +- +14.1 +11.6 +79.2 +81.3 +54.1 +27.9 +73.7 +42.2 +25.7 +45.3 +- +51.4 +IDA [13] +84.3 +37.7 +79.5 +5.3 +0.4 +24.9 +9.2 +8.4 +80.0 +84.1 +57.2 +23.0 +78.0 +38.1 +20.3 +36.5 +41.7 +48.9 +DTST [17] +83.0 +44.0 +80.3 +- +- +- +17.1 +15.8 +80.5 +81.8 +59.9 +33.1 +70.2 +37.3 +28.5 +45.8 +- +52.1 +FGGAN [15] +84.5 +40.1 +83.1 +4.8 +0.0 +34.3 +20.1 +27.2 +84.8 +84.0 +53.5 +22.6 +85.4 +43.7 +26.8 +27.8 +45.2 +52.5 +FDA [11] +79.3 +35.0 +73.2 +- +- +- +19.9 +24.0 +61.7 +82.6 +61.4 +31.1 +83.9 +40.8 +38.4 +51.1 +- +52.5 +image adapt. (ours) +76.4 +28.8 +71.6 +7.7 +0.5 +31.0 +13.8 +27.8 +69.3 +70.0 +59.7 +26.4 +75.7 +29.9 +22.1 +25.2 +39.7 +45.9 +I2F [18] +81.9 +33.7 +78.5 +11.0 +1.9 +36.7 +32.6 +33.4 +79.6 +78.2 +67.3 +33.6 +84.0 +33.5 +25.9 +47.6 +47.5 +54.6 +I2F all (ours) +84.9 +44.7 +82.2 +9.1 +1.9 +36.2 +42.1 +40.2 +83.8 +84.2 +68.9 +35.3 +83.0 +49.8 +30.1 +52.4 +51.8 +60.1 +DeeplabV3/+ +CAG (13 classes) [16] +84.8 +41.7 +85.5 +- +- +- +13.7 +23.0 +86.5 +78.1 +66.3 +28.1 +81.8 +21.8 +22.9 +49.0 +- +52.6 +CAG (16 classes) [16] +84.7 +40.8 +81.7 +7.8 +0.0 +35.1 +13.3 +22.7 +84.5 +77.6 +64.2 +27.8 +80.9 +19.7 +22.7 +48.3 +44.5 +- +WCBT [41] +81.7 +43.8 +80.1 +22.3 +0.5 +29.4 +28.6 +21.2 +83.4 +82.3 +63.1 +26.3 +83.7 +34.9 +26.3 +48.4 +47.2 +54.1 +image adapt. (ours) +64.0 +25.7 +73.9 +9.6 +0.8 +33.3 +12.3 +25.9 +81.6 +85.5 +62.4 +26.2 +80.6 +30.9 +26.8 +23.8 +41.5 +47.7 +I2F [18] +75.7 +30.0 +81.9 +11.5 +2.5 +35.3 +18.0 +32.7 +86.2 +90.1 +65.1 +33.2 +83.3 +36.5 +35.3 +54.3 +48.2 +55.5 +I2F all (ours) +79.9 +34.2 +83.6 +11.4 +4.8 +35.3 +28.0 +34.8 +86.9 +91.1 +64.9 +32.5 +86.2 +50.1 +42.6 +52.9 +51.2 +59.1 +use a gentle bilateral filter with d = 5, σc = 10 and σs = 25. +The KL divergence stays at 0.10 before and after the bilateral +filter is applied. For the proposed GMA module, Dc′ is set to +keep roughly 90% explained ratio of the energy of xj, which +is Dc′ = 32 for DeeplabV3+ and Dc′ = 256 for DeeplabV2, +compared to Dc = 256 for DeeplabV3+ model and Dc = 2048 +for DeeplabV2 respectively. The K-Means is used because it is +the simplest clustering algorithm to validate our motivation. The +number of clusters centers are set to Nz = 64 with hidden neuron +dimension Nh = 32. Note that a larger Nz or a more advanced +clustering method like KSVD might improve the performance. +Still, it is not pragmatic because of the memory consumption +or the computational complexity. We adopt the standard color- +jittering as the stochastic function τ1(·) in both source, and target +domains as in [15] in the image-level adaptation stages. We utilize +standard color-jittering, elastic deformation [45], and standard +random blurring in the feature-level adaptation stages. Elastic +deformation is used to mimic the differences between shapes in +different domains, and random blurring is used to simulate the +resolution differences. We conduct different settings for τ1(·) +and τ2(·) because we observed that both elastic deformation +and random blurring are strong data augmentations, and using +them in the image adaptation stage will distort the distribution +of the training data, which undermines the final segmentation +performance. +We follow the same experiment settings in [18]. In addition +to the DeeplabV3+(ResNet101) [6] discussed in +[18], we also +compare our proposed model with another commonly adopted +segmentation model DeeplabV2(ResNet101) used by other state- +of-the-art studies [14], [15], [17]. We implement our proposed +method with PyTorch [46], and deploy our experiments on 4 +NVIDIA GeForce 2080Ti GPUs with 1 source domain image and +1 target domain image randomly selected and stored on each GPU +for each backpropagation step. The stochastic gradient descent is +used during the image-level adaptation with a momentum of 0.9 +and weight decay of 1e − 4. The learning rate is initially set +to 5e − 4 and is decreased using the polynomial learning rate +policy with a power of 0.9 during training. For the feature-level +adaptation steps, we halve the learning rate to 2.5e − 4 based +on the learning rate from the image-level adaptation to fine-tune +previously trained models. +4.2 +Comparisons with State-of-the-Art Methods +In this section, we compare our method against all the existing +state-of-the-art methods [11], [13], [14], [15], [16], [17], on the +GTA5→Cityscapes, Synthia→Cityscapes and Kvasir→Piccolo +tasks. As shown in Table 1, the performance improvement +achieved +by +our +proposed +method +outperforms +all +previ- +ous methods with all different segmentation models. On the +GTA5→Cityscapes task, our model achieves a new state-of-the- +art mIoU (58.2%) on DeeplabV3+ and (53.3%) on DeeplabV2, +which are 8.0% and 2.9% higher than the previous best re- +sult using the same backbone (Table 1), respectively. On the +Synthia→Cityscapes task, the performances of our proposed +method are 60.1% and 59.1%, which are 7.6% and 5.0% higher +than that of the previous method, respectively. Note that we do +not include the comparison between our work and a concurrent +work [36] because three major differences make the comparison +unfair, which are presented as follows: (1) much stronger baseline +models. [36] adopts the domain adapted model from [24] as + +IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE +10 +(a) +(b) +(c) +(d) +Fig. 6. Qualitative examples of the comparisons between our method and CAG [16] on the GTA5→Cityscapes task. Specifically, (a) Input images, +(b) CAG [16], (c)Ours, (d) Labels +the baseline model, which generates much stronger performance +than ours. For example, our baseline model (37.6%) is 5.8% +lower than theirs (43.4%) on the GTA5→Cityscapes task; (2) +much stronger segmentation networks. [36] uses a more advanced +segmentation network proposed in +[47], [48] instead of the +standard DeeplabV2 that we used; (3) much stronger augmen- +tation strategies. RandAug [49] and Cutout [50] are utilized as +data augmentations in [36], which are much superior than our +augmentations. On the Kvasir→Piccolo task, the segmentation +performance of our proposed method is 81.5% and 84.2% on +DeeplabV2 and DeeplabV3+ respectively, which are 3.7% and +6.0% higher than the previous best results, as shown in Table 3. +We re-implemented three general-purpose state-of-the-art meth- +ods [11], [15], [16] for this comparison. Our proposed method +also significantly outperforms the domain adaptive segmentation +algorithm in [41], which was specifically designed for endoscopic +images. +Our method achieves the best performance in many important +categories, including ‘road’, ‘sidewalk’, ‘building’, ‘fence’, ‘veg- +etation’, ‘terrace’, ‘person’, ‘car’, ‘rider’, ‘truck’, ‘train’, ‘bus’, +‘motor’, and ‘bike’. In particular, our model performs the best +when classifying over ‘road’, ‘sidewalk’, ‘motor’, and ‘bike’, +even if some of these categories have very similar local appear- +ances. This is because our proposed category-oriented triplet loss +maximizes the inter-category distances and minimizes the intra- +category distances by exploiting the most difficult samples in + +IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE +11 +(a) +(b) +(c) +(d) +Fig. 7. Qualitative examples of the comparisons between our method and CAG [16] on the Kvasir→Piccolo task. Specifically, (a) Input images, (b) +CAG [16], (c)Ours, (d) Labels +the source domain, which improves the generalization capability +across different domains. Moreover, our proposed global mani- +fold/texture alignment brings an extra 2.1% improvements in the +final performance compared to the experiments in [18]. This is +because global photometric alignment is generally conducted in +the image-level inputs, and the features/textures between different +domains are still not explicitly aligned. Finally, our proposed +target consistency regularization also strengthens the relatively +weak supervision signal in the target domain. It improves the +segmentation accuracy of categories with large intra-category +variances, such as ‘building’ and ‘sky’, by regularizing its feature +distribution. According to our qualitative analysis, the previously +over-exposed white buildings are easily misclassified as skies but +can be corrected by our proposed target consistency regularization. +One interesting fact is that our proposed method has larger +overall performance improvements in the GTA5→Cityscapes +task compared to the Synthia→Cityscapes task. This is because +DeeplabV3+ adopts high-resolution feature maps and lower fea- +ture dimensions, which improves the segmentation performance. +The Synthia dataset is mostly constituted by large objects, limiting +the improvements that a DeeplabV3+ segmentation model can +bring. +Another interesting fact is that although our proposed +GMA has achieved clear performance improvements in both +DeeplabV3+ and DeeplabV2, yet the performance improvement +with DeeplabV2 is higher as showin in Table 1. This is because +the feature dimension for DeeplabV2 is much larger than the +feature dimension for DeeplabV3+(2048 vs. 256). This results in +a more complicated feature manifold which is difficult to align +with simple image-level photometric alignment. By modeling +the manifold and minimize the projection error, our proposed +GMA can effectively align high-dimensional features. Although +DeeplabV3+ is powerful, by cooperating with our proposed +GMA module, the performance of our proposed model on the +Synthia→Cityscapes task with DeeplabV2 is even higher than the +one with DeeplabV3+. +We further show some of the segmentation examples in Fig- +ure 7 and Figure 6 to qualitatively demonstrate the superiority +of our method. Our proposed method generates finer edges and +makes fewer mistakes. We also compare the style-transferred +images generated by our proposed GPA model with other state- +TABLE 3 +Performance comparison with state-of-the-art methods on the +Kvasir→Piccolo task. Best results are marked in bold. +polyps +background +mIoU +DeeplabV2 +FGGAN [15] +62.2 +88.5 +75.4 +FDA [11] +66.2 +89.5 +77.8 +image adapt. [18] +54.7 +84.9 +69.8 +I2F all (ours) +72.0 +90.9 +81.5 +DeeplabV3/+ +WCBT [41] +56.9 +86.5 +76.3 +CAG [16] +67.0 +89.4 +78.2 +image adapt. [18] +55.8 +82.0 +68.9 +I2F all (ours) +76.2 +92.2 +84.2 +of-the-art style-transfer techniques used by the domain adaptation +methods in Figure 8, and our proposed method has better quality +and a higher level of diversity. +4.3 +Ablation Studies +Component Analysis. In most previous work, a source-only +model trained on the source domain training set only is often +required to serve as initial pseudo label producer. Although we +do not use the source-only model during training, we train one to +provide a baseline so that it is convenient to verify the primary +performance gains from our proposed pipeline. Then, following +the experiment settings in [16], we perform extensive ablative +experiments using DeeplabV3+ on the GTA5→Cityscapes task +to verify the effectiveness of each of our proposed component. +As shown in Table 4, the source-only baseline using Deeplab +v3+ has a performance of 37.6% for the GTA5→Cityscapes task, +and our proposed overall pipeline improves the baseline perfor- +mance by 20.6%. Following the same settings in previous state- +of-the-art methods [13], [15], [16], we further evaluate the impact +of each proposed component according to the final performance +of our model on the GTA5→Cityscapes task by removing one +component at a time. The results are shown in Table 4. According +to our results, the final performance of the segmentation model +has the most deterioration when the global photometric alignment +module is removed. This is because the GPA module is critical +to the image-level adaptation. Removing it literally removes the +first image-level adaptation stage, and thus, the resulting erroneous + +IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE +12 +(a) +(b) +(c) +(d) +(e) +Fig. 8. Qualitative analysis on the global photometric alignment (GPA) module. (a) Input images, (b) Reference images, (c) BDL-GAN [14], (d) +Fourier Adaptation [11], (e) Global Photometric Alignment. +pseudo-labels are harmful to later stages. Although our proposed +global manifold alignment module can align the feature distri- +butions from different domains, the error from unaligned models +still accumulates across steps, which is detrimental to the final +performance of the model. This also validates the necessity of +a image-level adaptation step and the importance of an accurate +initial model. +Even though our proposed GPA module serves as an image- +level adaptation between domains, the feature-level domain shifts +are still not aligned completely. Our proposed GMA module can +further improve the feature-level adaptation between the source +domain and the target domain, decreasing the model perfor- +mance by 1.4% when removing the GMA module. In addition, +the GTEXA module is designed to modify the high-frequency +components of an image with a certain probability. This action +makes trained models robust to texture variations and further im- +proves the performance by roughly 0.7%. Furthermore, despite its +simplicity, our proposed target domain consistency regularization +has been proved to be very effective. The main reason for this +phenomenon is that there are fewer valid training samples in the +target domain than the source domain, and our proposed TCR +essentially serves as a data augmentation technique that increases +the number of valid training samples in the target domain. It +also introduces more hard samples without damaging the pseudo- +labels. Therefore, it gives rise to a significant performance gain, +decreasing the model performance by 4.8% when removing the +TCR module. Our proposed category-oriented triplet loss applied +on the source domain also boosts the performance by 3.1% +as it exploits hard samples in the source domain and improves +generalization capability across different domains. +Photometric Alignment. There are currently other methods, +which can achieve the goal of the image-level adaptation, such as +the GAN-based method in [14] and the frequency-based method +in [11]. We substitute our proposed global photometric alignment +and global texture alignment with these two methods and retrain +our whole pipeline. The result is shown in Table 5. We also visu- + +IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE +13 +TABLE 4 +Ablation study of the proposed components on the GTA5→Cityscapes +task. GPA: global photometric alignment, GTEXA: global texture +alignment, CTL: category-oriented triplet loss, TCR: target domain +consistency regularization. +GPA +GTEXA +GMA +CTL +TCR +mIoU +Source only +37.6 +Image adapt. +√ +47.3 +w/o Alignments +√ +√ +47.5 +w/o GPA +√ +√ +√ +√ +51.0 +w/o GMA +√ +√ +√ +√ +56.6 +w/o CTL +√ +√ +√ +√ +55.1 +w/o TCR +√ +√ +√ +√ +53.4 +w/o GTEXA +√ +√ +√ +√ +57.5 +all +√ +√ +√ +√ +√ +58.2 +alize some representative aligned images produced with different +methods in Figure 8. Our proposed GPA can generate the aligned +image according to a randomly chosen target domain reference +image. Simultaneously, the GAN-based model [14] performs de- +terministically and generates aligned images with a similar style, +only covering part of the actual target domain image span. This +explains why our proposed model works even better than the +pre-trained deep adversarial model. Although the frequency-based +method proposed in [11] can generate style-transferred images +randomly, the concatenation of frequencies usually introduces +significant noises during training, which largely limits its final +performance. +Based on our observation, gamma correction on Lab channels +does not have sufficient adaptation capability, while histogram +matching on all three channels results in image artifacts. We use +the simple mean-variance of RGB channels as the benchmark, and +run a comparison for the image-level adaptation stage. The result +in Table 5 shows our hybrid scheme performs the best. +56.50 +56.70 +56.90 +57.10 +57.30 +57.50 +57.70 +57.90 +58.10 +58.30 +58.50 +1 +2 +3 +4 +mIoU +𝛼=0.0 +𝛼=0.2 +𝛼=0.6 +𝛼=0.4 +Fig. 9. Quantitative analysis on the selection of category margin α. Best +performance is achived with α = 0.2. +Category Triplet Loss. We only apply the category-oriented +triplet loss to the source domain category labels in our proposed +method but not the pseudo-labels in the target domain. Although +the target domain images with pseudo-labels can be used as +supplementary samples when the pseudo-labels are of high con- +fidence, our proposed triplet loss aims to deal with hard samples +and pseudo-labels of hard samples in the target domain are not +reliable. To verify this, we include pseudo-labels in our category- +oriented triplet loss, and the result is shown in Table 5. We follow +the default settings as in [51] and set α = 0.2. But we also tested +our proposed category triplet loss with other settings as shown +in Figure 9, which shows that the best result is achieved when +α = 0.2. +Manifold Alignment. In our proposed model, we directly use +the PCA+K-Means to model the feature manifold. It shares similar +56.50 +56.70 +56.90 +57.10 +57.30 +57.50 +57.70 +57.90 +58.10 +58.30 +58.50 +1 +2 +3 +mIoU +𝑁ℎ = 16 +𝑁ℎ = 32 +𝑁ℎ = 64 +(b) +56.50 +56.70 +56.90 +57.10 +57.30 +57.50 +57.70 +57.90 +58.10 +58.30 +58.50 +1 +2 +3 +mIoU +𝑁𝑧 = 16 +𝑁𝑧 = 32 +𝑁𝑧 = 64 +(a) +Fig. 10. Quantitative analysis on the hyper-parameters of global manifold +alignment (GMA) module. (a) higher Nz leads to better performance, (b) +mIoU is generally not very sensitive to the choice of Nh. +functionality with the adversarial methods used in previous work. +But our proposed method suffers little from mode collapse. The +mode collapse is easily observed in the style-translated images +as in Figure 8. Still, it also exists in the high-level features and +undermines the diversity of the training set. To make fair compar- +isons, we substitute our proposed manifold alignment module with +a traditional global discriminator as in [17]. According to Tabel 5, +the result illustrates that it performs even worse than the version +without a global discriminator, manifesting the superiority of our +GMA module. We also conduct extensive experiments to verify +the values of the hyperparameters Nh and Nz. The experiment +results are presented in Figure 10. In general, the performance +of the model is insensitive to the choice of Nh, and larger Nz +leads to better performance, the best performance is achieved with +settings Nh = 32 and Nz = 64. Note that we can not increase +atom number Nz to more than 64 because the calculation of atom +weights and the reconstruction of pixel feature xj require a large +amount of GPU memory. +TABLE 5 +Ablation studies of the image adaptation strategy, photometric +alignment scheme, and using pseudo-labels for the category-oriented +triplet loss on the GTA5→Cityscapes task. +Modules +Methods +mIoU +Image Aapt. +Frequency Align [11]. +54.0 +BDL-GAN [14] +55.4 +Photometric+Texture +58.2 +GPA Scheme +RGB Mean-Variance +42.3 +Lab Gamma Correction +44.5 +Lab Histogram Match +43.3 +Hybrid +47.3 +Pseudo-labels +Triplet loss with pseudo-labels +56.1 +Triplet loss w/o pseudo-labels +58.2 +Manifold Align. +Adversarial Method +55.8 +Manifold Alignment +58.2 +5 +CONCLUSIONS +In this paper, we have explored non-adversarial methods in both +image-level and feature-level domain adaptation, and proposed a +novel unified image-to-feature adaptation pipeline for unsuper- +vised domain adaptive semantic segmentation. During this study, +we have found out that for this specific problem, adversarial meth- +ods could damage the diversity of feature distributions, and a sim- +ple photometric alignment module can achieve better performance. +We have also found out that a simple self-supervised consistency +loss is capable of regularizing category-level feature distributions + +IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE +14 +in the target domain. The proposed pipeline effectively integrates +global image-level and feature-level adaptation and category-level +feature distribution regularization. The global texture alignment +module also serves as an auxiliary data augmentation scheme for +the proposed pipeline. +In particular, we have introduced a novel and efficient global +photometric alignment module to adapt source domain images to +the target domain. A global texture alignment module has been +designed to modify the high-frequency components of images +from the source domain and make the trained model robust to +domain gaps caused by domain-specific textures. We have also +proposed a global manifold alignment module to directly model +the distribution of the pixel features from the source domain and +align the feature distributions from both domains. To our best +knowledge, this is the first piece of work that models the feature +manifold directly in unsupervised domain adaptation for semantic +segmentation. A category-oriented triplet loss has been devised for +the source domain to regularize source domain category centers. +A target domain consistency regularization method has also been +introduced for the target domain to regularize category-level +feature distributions. Extensive experiments have shown that each +of our proposed techniques improves the generalization capability +of our model significantly. The proposed three modules form a +complete adaptation strategy to tackle domain shifts. Integrating +them gives rise to a significant improvement over existing state- +of-the-art unsupervised domain adaptive semantic segmentation +methods, demonstrating that minimizing global and category-level +domain shifts simultaneously deserves more attention. +Limitations. Our work still has a few limitations. First of +all, the photometric alignment module is isotropic, which means +texture information is not altered by our proposed module. Al- +though we have proposed a global texture alignment scheme, it +is activated only when the source domain images have stronger +or similar high-frequency components in comparison to the target +domain images. It deserves more attention to develop a method +that can better close the domain gap without hurting the feature +diversity of source domain samples. In addition, our scheme for +global feature manifold alignment is the first attempt to model +the feature manifold directly. However, when we design our +scheme, the priority is making manifold alignment compatible +with gradient back-propagation based training, but not achieving +optimal alignment performance. Nonetheless, it demonstrates the +potential of direct feature manifold modeling in domain adaptation +tasks. At last, some of our proposed components are only designed +for the close-set setting because they are based on the assumption +that deep features from the same category should be similar, +which is not well suited for open-set tasks where different unseen +categories are all labeled as “unknown”. Thus how to extend our +algorithm to the open-set scenario remains an open problem. +REFERENCES +[1] +M. Treml, J. Arjona-Medina, T. Unterthiner, R. Durgesh, F. Friedmann, +P. Schuberth, A. Mayr, M. Heusel, M. Hofmarcher, M. 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Bengio, “Large scale online +learning of image similarity through ranking,” 2010. + diff --git a/kdAzT4oBgHgl3EQfNftb/content/tmp_files/load_file.txt b/kdAzT4oBgHgl3EQfNftb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c10d7d1dcab35d89178864662bb190f19e5c175d --- /dev/null +++ b/kdAzT4oBgHgl3EQfNftb/content/tmp_files/load_file.txt @@ -0,0 +1,1559 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf,len=1558 +page_content='IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 1 I2F: A Unified Image-to-Feature Approach for Domain Adaptive Semantic Segmentation Haoyu Ma§, Xiangru Lin§ and Yizhou Yu, Fellow, IEEE Abstract—Unsupervised domain adaptation (UDA) for semantic segmentation is a promising task freeing people from heavy annotation work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' However, domain discrepancies in low-level image statistics and high-level contexts compromise the segmentation performance over the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' A key idea to tackle this problem is to perform both image-level and feature-level adaptation jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Unfortunately, there is a lack of such unified approaches for UDA tasks in the existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This paper proposes a novel UDA pipeline for semantic segmentation that unifies image-level and feature-level adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Concretely, for image-level domain shifts, we propose a global photometric alignment module and a global texture alignment module that align images in the source and target domains in terms of image-level properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' For feature-level domain shifts, we perform global manifold alignment by projecting pixel features from both domains onto the feature manifold of the source domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' and we further regularize category centers in the source domain through a category-oriented triplet loss, and perform target domain consistency regularization over augmented target domain images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Experimental results demonstrate that our pipeline significantly outperforms previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' In the commonly tested GTA5→Cityscapes task, our proposed method using Deeplab V3+ as the backbone surpasses previous SOTA by 8%, achieving 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='2% in mIoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Index Terms—Semantic Segmentation, Unsupervised Domain Adaptation, Photometric Alignment, Texture Alignment, Manifold Modelling, Category Triplet Loss, Consistency Regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' 1 INTRODUCTION S EMANTIC segmentation, a classical and fundamental research task in computer vision, aims to assign category labels to indi- vidual pixels in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' It has been extensively investigated and has inspired many downstream applications including autonomous driving [1], [2] and medical image analysis [3], [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Although the performance of existing semantic segmentation models have enjoyed a significant improvement in the wave of deep neural net- works [6], [7], [8], training a semantic segmentation model usually requires a large number of images with pixel-level annotations, the collection process of which is laborious and time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Unsupervised Domain Adaptation (UDA) for semantic segmen- tation is an alternative to avoid the data annotation problem: it aims at learning a well-performing model from an unlabeled target dataset by jointly exploiting labeled images from a different source dataset (the label spaces of the two datasets must be compatible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' However, domain shifts/discrepancies exist between different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The most obvious differences are low-level image statistics related to colors, textures, or even illumination conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' These differences can be partly alleviated by image- level adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' However, there are also object-level differences, such as object poses and spatial distributions, between different datasets, which give rise to different feature distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' All these domain shifts have a detrimental impact on the final performance of the semantic segmentation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Therefore, it is crucial to learn a feature representation capable of overcoming both image- level and feature-level domain shifts for unsupervised domain This work was supported in part by Hong Kong Research Grants Council through Research Impact Fund (Grant R-5001-18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Ma was supported by the Hong Kong PhD Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' (Corresponding author: Yizhou Yu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Lin and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Yu are with the Department of Computer Science, the University of Hong Kong, Pokfulam Road, Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' E-mail: mahaoyu@connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='hku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='hk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' xrlin2@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='hku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='hk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' yizhouy@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' § H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Ma and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Lin have equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' adaptive semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The causes of domain shifts/discrepancies have been exten- sively studied in previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' In general, the primary causes can be categorized into image-level domain shifts and feature-level domain shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Image-level domain shifts refer to the differences in imaging conditions, such as lighting and settings in the camera imaging pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' They affect the overall appearance of an image and have a subtle influence on feature-level distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Existing work addressing image-level domain shifts is in general based on image-level style transfer, which makes use of deep models such as generative models or image-to-image translation models [9], [10], or Fourier Transformation [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We term these methods image-level adaptation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' These methods have proven that transferring image styles or aligning feature distributions can bring the two domains closer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' However, generative methods usually require a computationally expensive training process, whose in- stability is notorious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Generative models also suffer from mode collapse, which makes the range of the generated features unusu- ally small (more explanation in Related Work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' On the other hand, the Fourier Transformation based method [11] produces inferior style-transferred images, as shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We have observed that previous work in domain adaptive semantic segmentation focusing on image-level domain align- ment [10], [12] usually has inferior final segmentation perfor- mance in comparison to recent work that adopts a more complete pipeline [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Such recent work further demonstrates that replacing the original source domain images with image-level domain aligned images can further improve the final performance of feature-level adaptation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This indicates that the domain gap can only be partially alleviated with aforementioned image-level adaptation methods, and feature-level alignment can still benefit from an extra image translation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Therefore, feature-level adaptation is still necessary after image-level adapta- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='01149v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='CV] 3 Jan 2023 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 2 tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' For feature-level adaptation, a common practice in previous studies employs an adversarial method [14], [15], which considers features from the source and target domains aligned if they cannot be distinguished by a trained discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' But adversarial methods tend to generate a narrow range of feature distributions to fool the discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' When different images share similar feature distributions, trained models would have poor generaliza- tion performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' On the other hand, to perform category-level feature adaptation, some existing methods use category anchors computed in the source domain to align the two domains [16], [17], which can be regarded as imposing hard constraints on category-level feature distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This method ignores feature distances across different categories, and categories with similar feature distributions in the source domain may still have similar ones in the target domain, resulting in erroneous pseudo-labels when no supervision signals are available in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Our experiments demonstrate that imposing soft regularization on category-level feature distributions by adjusting the relative magnitude of inter-category and intra-category feature distances can improve model capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' According to the above analysis, performing either image- level adaptation or feature-level adaptation alone could not address domain shifts adequately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Moreover, existing work on UDA for se- mantic segmentation lacks a unified approach to minimize domain shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Therefore, we approach the problem from both perspectives and propose a novel and efficient pipeline that unifies image- level and feature-level adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' For image-level domain shifts, we propose two novel and training-free image-level operation, called global photometric alignment and global texture alignment, to adapt images from the source domain to the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' However, image-level adaptation alone does not guarantee domain alignment in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Therefore, we devise a global man- ifold alignment module for feature-level adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This module represents the source domain feature manifold with a set of atoms, and any pixel feature from the source domain or the target domain can be projected onto this manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' By minimizing the projection errors between the input features and the manifold, all source and target domain features are aligned to the same manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' To perform category-level feature adaptation, we also introduce two category-level feature distribution regularization methods: a category-oriented triplet loss is proposed in the source domain to softly regularize category centers by enlarging the margin between inter-category and intra-category feature distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' It is only adopted in the source domain because the measurement of inter-category and intra-category distances require reliable annotations that only exist in the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The category- level feature adaptation method applied to the target domain is the self-supervised consistency regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This regularization makes the prediction on an augmented target image consistent with the pseudo-label of the corresponding non-augmented image, thus forcing the class labels of similar semantic contents to be consistent in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' By addressing domain shifts from all perspectives simultaneously, experimental results demonstrate that our proposed method is capable of achieving significant performance improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Domain adaptive semantic segmentation methods can be ap- plied to either synthetic source images or real source images as long as there exist significant domain gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' For the application to synthetic source images, we follow the common practice [11], [13], [14], [15], [16], [18] and use the GTA5→Cityscapes and SYNTHIA→Cityscapes benchmarks to evaluate our proposed domain adaptation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' In addition to synthetic source data, we also construct a new task on two open-source real-world endoscopic image datasets, Hyper-Kvasir [19] and Piccolo [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This task can serve as a new medical image benchmark for future studies in domain adaptive semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Experiment results on all three benchmarks demonstrate that our proposed method is capable of achieving significant performance improve- ments over existing state-of-the-art algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' To this end, this paper is an extension of [18] and the contributions of [18] can be summarized as follows, A novel image-to-feature domain adaptive semantic segmen- tation pipeline is proposed to seamlessly combine coarse image-level adaptation with category-level feature distribu- tion regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Two novel and effective category-level regularization meth- ods are proposed to deal with the source and target domain shifts, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The first one is category-oriented triplet loss which regularizes category centers in the source domain, and the second one performs target domain consistency regularization over augmented target domain images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The proposed method in [18] outperforms all previ- ous methods, achieving state-of-the-art performances on both GTA5→Cityscapes and SYNTHIA→Cityscapes bench- marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Compared to the conference version [18], this paper gives a more complete introduction and analysis of the proposed non- adversarial image-to-feature domain adaptive semantic segmen- tation pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We provide more insights and discussions about the modules proposed in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' More importantly, we extend our work in [18] by introducing global manifold alignment in the high-level feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This manifold alignment algorithm serves as a feature-level adaptation strategy complementary to global photometric alignment proposed in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' An auxiliary data aug- mentation scheme for global texture alignment is also proposed to reduce the domain gap caused by texture variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Experimental results demonstrate that our proposed global manifold alignment and global texture alignment modules make our proposed method more robust and achieve new state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' To sum up, this paper has the following new contributions: A manifold alignment algorithm is proposed to represent the high-level feature space via dimension reduction and clus- tering algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' To the best of our knowledge, this is the first piece of work that tackles unsupervised domain adaptive semantic segmentation with explicit manifold modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' All related ablation studies have been conducted for this new module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Global texture alignment is proposed as a data augmenta- tion scheme for domain adaptive semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' It reduces the sensitivity of the trained model with respect to domain-specific textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' For synthetic source data, our updated method outperforms all previous UDA methods by a large margin, achieving new state-of-the-art performance on both GTA5→Cityscapes and SYNTHIA→Cityscapes benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We further construct a new medical image domain adaptive semantic segmentation task on the basis of two open-source real-world endoscopic image datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Our proposed method also achieves state-of-the-art performance on this new task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 3 2 RELATED WORK Photometric Alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Previous works [21], [22], [23] in unsupervised domain adaptation for image classification do not pay attention to image-to-image translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' However, it has been proven that a model trained with source images transferred into the target domain style can significantly improve the final performance in semantic segmentation tasks [14], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This is perhaps because deep features for semantic segmentation are relatively more sensi- tive to local information compared to image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' In order to achieve image-level photometric alignment, ad- versarial methods have been widely used in previous work on domain adaptation [12], [14], [17], [24], [25], [26], [27], [28], such as GAN [9], [29] and CycleGAN [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' These GAN-based methods can transfer the styles of the images in the target domain to that of the source domain and thus significantly reduce image- level photometric differences [12], [14], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Then, these style transferred source domain images are used to train a segmentation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Because they are photometrically aligned with the target domain images, the models trained with these style transferred source domain images usually yield better performance compared to the model trained with the source domain images [17] only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' However, it is also noted that adversarial models are unstable during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Previous work has shown that image-level ad- versarial methods generally convert the source domain image- level distribution to the one in the target domain to improve the performance of the domain adapted model [14], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' But it is still an open question whether the style transferred images distribution roughly covers the whole target domain image-level distribution or just a small part of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The non-adversarial photometric alignment methods for unsupervised semantic segmentation are rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' One latest line of research is the Fourier Domain Adaptation proposed in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The motivation is that the low-frequency component of an image consists of the major photometric information, and replacing the low-frequency component in a source domain image with its reference image counterpart in the target domain could align the photometric information between different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' However, the decomposition of frequency components is very sensitive to the image’s content, and simply replacing the low- frequency information of an image with that of another image often introduces extra noises and leaves unsatisfactory visual artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' According to their experiments, the model’s performance trained on the frequency-aligned samples also relies heavily on a multi-band ensemble with multiple models [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Unlike the Fourier Domain Adaptation, our proposed method is directly applied to color channels without the frequency decomposition, which provides us with comparable (superior) performance and image quality to its generative (Fourier Transformation-based) counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Moreover, our proposed method only consists of several image-level operations which do not require standalone training and can be used with arbitrary source-target image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Adversarial Methods for Domain Adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' There are tra- ditional manifold learning methods that model high-dimensional feature spaces before the deep learning era [30], [31], [32], [33], but they are usually computationally costly when transplanted to deep learning applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Previous work on handling feature spaces in UDA typically adopts adversarial methods [14], [15], which do not directly model the feature manifold, but consider features from the source and target domains aligned if they are indistinguishable by a trained discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' However, generators trained by adversarial methods are inclined to produce outputs with similar feature distributions [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' They can surely reduce cross-domain feature distribution discrepancies and make image features agnostic to the input domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' However, it also reduces the diversity of image-level feature distributions from the same domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' It is difficult to visualize high-dimensional feature dis- tributions resulting from adversarial methods, but we can take style-transferred RGB images generated by adversarial methods as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' As shown in Figure 8c, all images generated by GAN are dark and smooth regardless of diverse image-level color distributions in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This phenomenon is called the mode collapse problem and is detrimental to the generalization capability of the domain adapted model in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Most recent algorithms [11], [15], [16] choose to remove adversarial methods from their last stages due to this mode collapse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Our approach differs from adversarial methods in that we model the feature manifold directly by learning a feature manifold from the source domain denoted by a set of representative feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Then, we propose a pixel feature projection loss that learns to project pixel features from both domains to the source do- main feature manifold using these representative feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Therefore, minimizing the projection errors from both domains benefits domain alignment from a feature-level perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Category-Based Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The distribution of different cat- egory proportions can be very different between the source domain and the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Existing work [15], [24], [27], [35] typically utilizes category labels/predictions to enforce global semantic constraints on category distribution of predicted labels in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Similar to their counterparts in image classification [21], [22], [23], some previous works in semantic segmentation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' [16] and [17]) take one step forward to utilize category information: the penultimate image features which are used for generating pseudo-labels in the output layer in the target domain are mapped to their corresponding counterparts in the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Another concurrent work [36] proposes to learn category prototypes online and correct pseudo labels according to the distance measurements between pixels features and those learned category prototypes, which is an improved version of [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' However, category feature centroids used in [16], [36], or instance features used in [17] only serve as anchors for category-based feature adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The margins between different categories are not explicitly enlarged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This is a problematic alignment strategy because category centroids close to each other in the source domain are still difficult to separate in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' There- fore, we propose a method that differs from theirs in two major perspectives: first, a category-oriented triplet loss is proposed for the source domain to impose a soft constraint that regularizes the category centers for different categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This approach actively makes inter-category distances between different categories in the high-level feature space larger than intra-category distances of a certain category by a specified margin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' secondly, we enforce the predictions on augmented target domain images to be consistent with the pseudo-labels generated by the segmentation model of the corresponding non-augmented images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This is essentially a self-supervision based consistency regularization method and the design philosophy is based on the fact that the supervision signal in the target domain is weak due to the lack of confident pseudo- labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 4 3 METHOD 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='1 Algorithm Pipeline The underlying philosophy of our proposed pipeline is straight- forward: first, we exploit the photometric differences in the two domains to coarsely adapt the source domain images with the target domain images to minimize the image-level domain shifts, and the high-frequency distribution from the target domain is also randomly transfered into the source domain image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' then, we perform feature-level adaptation by aligning pixel features from both domains with the feature manifold generated by the coarsely adapted model regardless of its categories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' finally, we impose soft constraints on inter-class center distances and intra-class feature variations to regularize category-level feature distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Overview of the pipeline is presented in Figure 1 and is illustrated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Suppose the labeled source domain dataset be Ds = {(Is m, Ys m)}N s I m=1 where Is m is a source image, Ys m is the pixel-level annotation of Is m, and N s I is the number of source images in the source domain dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The target domain dataset Du contains a large number of unlabeled images Du = {Iu n}N u I n=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We assume the shape of all images is h × w × 3, and the number of target classes to be segmented is Mc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Hence, we have Ys m ∈ {1, 2, · · · , Mc}h×w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The purpose is to learn a semantic segmentation model for the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Step 0: Image-level Adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Given a source domain image Is m in the training batch and a randomly selected target domain reference image Iu n, Is m and Iu n are converted into Lab color space as (Ls m, as m, bs m) and (Lu n, au n, bu n) by our pro- posed GPA module respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The histogram mapping function fmatch(·) is then used to process both as m and bs m channels, and gamma correction function fgamma(·) is applied to Ls m to form (fgamma(Ls m), fmatch(as m), fmatch(bs m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' After the mappings, the image is then converted back to RGB space to generate the aligned image �Is m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' All these randomly generated adapted images are used to construct the adapted source domain training set �Ds = {( �Is m, Ys m)}N s m=1 for each training epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Then, a stochastic function τ1(·) is applied to the source domain training set �Ds to produce an augmented version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' A segmentation model F0 is then trained based on the augmented style-transferred source domain images τ1(�Ds) with the cross-entropy loss Lseg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Step 1: Feature-level Adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The aforementioned image-level adaptation only diminishes the image-level domain shifts between the source and target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' But image-level adaptation operations do not guarantee the adaptation of high- level features because image components such as textures are not altered by image-level photometric operations, and still impact the high-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Therefore, we further modify a random subset of the photometrically aligned images, and make their texture- related high frequency components follow the corresponding distributions in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Let �Is m be a photometrically aligned image, whose texture components are further updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The resulting image I s m is the actual input to the segmentation model in this step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We also introduce a global manifold alignment module to tackle the feature-level domain shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Before training a new segmentation model, we learn a representation of the feature manifold in the source domain offline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We first apply the initial model F0 to all source domain images to obtain their feature maps and prediction probability maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Correctly classified feature vectors from these feature maps are randomly sampled to form matrix X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Then both PCA and K-Means clustering are applied to X to learn a feature manifold represented with a set of cluster centers z in a dimension reduced feature subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' When a new segmentation model is trained, features from both the source and target domains are projected onto this manifold, and the projection error, Lmfd, is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' In addition to cross-entropy loss Lseg and manifold projec- tion error Lmfd, two loss functions for category-level feature distribution regularization are also adopted in the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Category center fc for every category c is calculated as the L2 normalized mean of all pixel features from category c in the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' One of the two loss functions is a category-oriented triplet loss Ltriplet defined over the image style transferred source domain dataset �Ds to enlarge the inter-category distances and minimize intra-category variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' In the target domain, the pseudo-label at a certain pixel location and its associated confi- dence are defined according to the prediction probability maps produced from the initial segmentation model F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Pseudo-labels with confidence higher than an adaptive threshold are considered valid samples, and are used to define a target domain consistency loss Lcst to regularize category-level feature distributions in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The remaining pixels are left out during back- propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We fine-tune the segmentation model F0 for U iterations by minimizing Lseg + Lmfd + Ltriplet + Lcst to produce a new segmentation model F1 for the current step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Step 2 to K: Iterative Self-Supervised Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Model F1 trained in Step 1 can be further improved with iterative steps sim- ilar to Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Such an iterative approach is called self-supervised training and is widely adopted in the area of unsupervised domain adaptive semantic segmentation [11], [14], [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The same Step 1 is performed, but the pre-trained model F0 is replaced with Fi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' And model Fi−1 is also used to update manifold atoms z, pseudo-labels and category centers fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This process is repeated for K − 1 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Refined pseudo-labels generated by models from each stage further improve the segmentation performance and reduce the domain gap (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' But erroneous pseudo- labels also accumulate false supervision signals and limit the magnitude of performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Our proposed image-to- feature pipeline is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='2 Global Photometric Alignment Since global domain shifts are mostly related to low-level image attributes, global photometric alignment is proposed in our work to transfer low-level image attributes of the target domain to source domain images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' It is observed that the spatial lightness distribution of an image can be very complicated in different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' It is also important to note that directly operating on RGB channels would cause severe artifacts and fake colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' In contrast, the spatial color distribution of the a and b color channels always have similar bell-shaped histograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Therefore, we approach lightness and color with different treatments: we perform classic histogram matching [37] between the source domain image and the target domain reference image only on color channels a and b to avoid introducing artifacts commonly seen in histogram matching results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Lightness Gamma Correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' On the other hand, the L channel is much more sophisticated under different circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This is because light interacts with the 3D structure of a scene in a complicated manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Simple histogram matching function results in large areas of overexposure and fake structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Thus, IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 5 Source Image Aligned Target Image Pseudo-labels 𝓛𝒔𝒆𝒈 𝓛𝒄𝒔𝒕 𝓛𝒕𝒓𝒊𝒑𝒍𝒆𝒕 Reference Target Image : Shared weights : Feat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' encoder : Classifier (c) Feature-level Adaptation Source Label (a) Pipeline Overview Source Image Aligned ℒ𝑠𝑒𝑔 Reference target Image (b) Image-level Adaptation Source Label 𝝉𝟏 𝝉𝟐 𝝉𝟐 𝓛𝒎𝒇𝒅 … 𝒛 𝒇𝒄 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' (a) The pipeline consists of 1 image-level adaptation stage and K feature-level adaptation stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' (b) At first image-level adaptation is implemented using the global photometric alignment operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' (c) Then the obtained model Fi is used to compute pseudo-labels, manifold atoms z, category centers fc, category thresholds tc, as well as initialize the segmentation model for the subsequent feature-level adaptation stages in an iterative self-supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' instead of using histogram matching for every histogram bin to prescribe strict mapping, we choose to constrain the mean value of the lightness channel in the source domain image and make it equal to the mean value of the target domain reference image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Because mean-variance policy might make the pixel value smaller than 0 or larger than 1, we choose the power-law function, which is also widely used in gamma correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' But the difference between our proposed method and the classic gamma correction is that our coefficients for the power-law function are not pre-defined by users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' They are automatically calculated with given source-target image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Specifically, the power-law function can be written as fgamma(L) = Lγ, where L is the normalized lightness value from 0 to 1 at each pixel location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' γ = 1 when it is an identical transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The mean value constraint can then be written as � L fgamma(L)hs m(L) = � L Lγhs m(L) = � L Lhu n(L), (1) where hs m is the lightness histogram of a source image Is m, and hu n is the lightness histogram of a target reference image Iu n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' In practice, we introduce a regularization term β to prevent γ from deviating too much away from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Thus, γ can be solved numerically in the following nonlinear optimization, γ∗ = arg min γ �� L Lγhs m(L) − � L Lhu n(L) �2 + β(γ − 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' (2) This optimization problem is a simple convex optimization with only one variable γ, and can be easily solved with few steps of gra- GPAGPAStep 1-K:Feature- level AdaptationUpdate Tifc,tc,zStep 0:Image-level AdaptationGTEXAIEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 6 20 22 24 26 28 30 32 34 0 1 2 3 4 5 6 Domain Gap Alternative Iteration Number 45 47 49 51 53 55 57 59 0 1 2 3 4 5 6 mIoU Alternative Iteration Number Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Iterative self-supervised training further improves the segmenta- tion performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' dient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Source-target image pairs are generated randomly on the fly during training epochs because our proposed GPA is highly efficient and does not require training in comparison to GAN-based methods [14], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The process of the proposed GPA module is illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' 𝐿𝑠𝑟𝑐 → 𝐿𝑡𝑔𝑡 𝑎𝑠𝑟𝑐 → 𝑎𝑡𝑔𝑡 𝑏𝑠𝑟𝑐 → 𝑏𝑡𝑔𝑡 (a) (b) (d) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' (a)Input source domain image and (b) a randomly chosen target domain image is aligned in (c) Lab channels to generate (d) aligned image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='3 Global Texture Alignment As discussed in previous work [38], CNN-based models are sensitive to high-frequency information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We observe that synthetic images have different and often stronger high-frequency informa- tion in comparison to real-world images, which jeopardizes the generalization performance of our model in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Although the proposed GPA module maintains the diversity of the source domain dataset, it modifies the photometric properties of an image instead of the high-frequency texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' To alleviate this problem, a global texture alignment module is proposed as an auxiliary data augmentation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The idea is straightforward: we modify the high frequency components of a random subset of the source domain images to make their distribution in each image more consistent with that of the corresponding reference image, which is sampled from the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The process is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This data augmentation scheme teaches the segmentation model to ignore texture information and focus on structural information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' To be specific, a bilateral filter fbilateral(·)|d,σc,σs is applied to a source domain image �Is to generate the filtered image I s = fbilateral( �Is)|d,σc,σs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We use the bilateral filter to preserve image structures and modify the texture component only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' In order to determine the parameters (d, σc and σs) of the bilateral filter, we quantify the distribution of high-frequency image components, and ensure that I s and its target domain reference image Iu have sim- ilar distributions of high-frequency components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We convert both I s and Iu to grayscale images and apply the Laplacian operator fLap to obtain their high-frequency components Hs = fLap(I) and Hu = fLap(Iu), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Let h(Hs) and h(Hu) be the respective histogram of Hs and Hu, and represent their distribution of high-frequency components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' To align h(Hs) and h(Hu), the parameters of the bilateral filter are determined by solving the following optimization problem, d∗, σc∗, σs∗ = arg min d,σc,σs KL( � s h(Hs), � u h(Hu))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' (3) By applying the bilateral filter with optimized parameters, the KL divergence between the distributions of the high-frequency com- ponents of �Is and Iu can be significantly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Note that d, σc and σs are fixed once optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' To introduce stochasticity, each source domain image has 50% chance to be bilaterally filtered before being fed to the segmentation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We find that adding this data augmentation scheme in the image-level adaptation step would damage the final performance, and therefore, only use it as an additional source domain data augmentation scheme during the feature-level adaptation steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='4 Training Loss The only training loss during image-level adaptation step is the segmentation cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The overall loss function we use during feature-level adaptation steps consists of four parts: the cross-entropy segmentation loss, the global manifold alignment loss, the category-oriented triplet loss, and the target domain consistency regularization loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' 𝒙𝑗 ∈ ℝ1×𝐷𝑐 … 𝔃 ∈ ℝ𝑁𝑧×𝐷𝑐′ 𝑅(𝒙𝑗) ∈ ℝ1×𝐷𝑐′ 𝑊1 ∈ ℝ𝐷𝑐′×𝑁ℎ 𝑊2 ∈ ℝ𝐷𝑐′×𝑁ℎ ∗ ∗ ෝ𝒙′𝑗 ∈ ℝ1×𝐷𝑐′ 𝑅−1(ො𝑥′𝑗) ∈ ℝ1×𝐷𝑐 𝒘 ∈ ℝ𝑁𝑧×𝐷𝑐′ 𝓛𝒎𝒇𝒅 softmax(⋅) − Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' By minimizing the projection error of source/target domain features onto the manifold, our proposed manifold loss mitigates the discrepancies between source domain feature distribution and target domain feature distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' 1IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 7 Global Manifold Alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Methods such as Locally Linear Embedding (LLE) and Isomap are commonly used to depict manifolds, but they are too computationally costly for gradient backpropagation based training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Here we use the K-means algo- rithm to simplify the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' As LLE uses a piecewise linear model to approximate a high dimensional feature manifold, K- means can be considered as a piecewise constant approximation of the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Every centroid obtained by K-means is a constant approximation of a local region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' By approximating the manifold with a set of representative feature vectors, we can further align features from the source and target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' In order to acquire feature representations, first we need to apply the segmentation model obtained in the previous step Fi−1 to each source image Is m to compute the feature map of the second last layer Xs m and the final prediction probability map P s m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The feature vector and prediction probability at a given pixel location j is denoted as xs j and ps j, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The true category label at location j is denoted as ys j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Next, the predicted probabilities (ps j) are compared with true category labels (ys j), and the correctly classified feature vectors are randomly sampled to form the source domain sample matrix X ∈ RNp×Dc, where Np is the total number of sampled feature vectors and Dc is the dimensionality of each feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Then, principal component analysis (PCA) is applied to X to keep around 90% of the total explained ratio of energy and obtain the dimension reduced version R(X) ∈ RNp×Dc′ , where Dc′ << Dc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Afterwards, the classic K-Means clustering algo- rithm is applied to R(X) to find the representative locations on the feature manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' These locations are denoted as z ∈ RNz×Dc′ , which are essentially the atom vectors of the source domain feature manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Any pixel feature from the source domain (xs j) or the target domain (xu j ) can be projected to the subspace spanned by the atoms in z, and the projection is represented as ˆx′ j = wT z (we omit the superscript for simplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The projection mapping w is applied here for two reasons: first, although .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='Let R−1 be the reconstruction operator of PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The projection error ||R−1(ˆx′ j) − xj||2 is considered as the deviation from the source domain manifold and is part of the projection error loss Lmfd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The motivation of our proposed global manifold alignment is straightforward: minimizing the source domain projection error makes the feature manifold smoother, and minimizing the target domain projection error decreases the distance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' improves the alignment) between feature distributions of the source and target domains respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Specifically, we adopt an attention mechanism to calculate the linear coefficients of atom vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The manifold projection error Lmfd and reconstructed feature vector ˆx′ j can be computed using the following equations, Lmfd = � j ||R−1(ˆx′ j) − xj||2 ˆx′ j = wT z wT = softmax �(R(xj)W T 1 )(W2zT ) √Nz � , (4) where R(xj) is the j-th row of R(X), both W1 ∈ RNh×Dc′ and W2 ∈ RNh×Dc′ are trainable linear matrices respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' They are introduced to further lower the memory overhead of the attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' They also project the manifold and all features to a lower dimensional space, and two distinct projection matrices enable better alignment between the projected manifold and features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Nh is a hyperparameter representing the number of hidden neurons, w ∈ RNz is the vector of atom coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The details to calculate the manifold projection loss is illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Although the global manifold loss is defined for the global alignment of features, it cannot be adopted in the image adaptation stage because it relies on a pre-trained model to provide manifold atoms z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Category-oriented Triplet Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Although the aforemen- tioned GPA and GMA modules could learn domain-invariant features to some extent, the losses used in previous training do not explicitly control the category-wise feature distribution, and some category-sensitive domain shifts are overlooked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Pixel features from different categories are naturally distributed unevenly, and some category centers are close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' To tackle this issue, we propose a category-oriented triplet loss that aims to push the category-wise features further closer to the corresponding category centers the pixel belongs to and further away from other category centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Note that category centers are intentionally introduced to make the calculation of category-oriented triplet loss practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' If we use the traditional triplet loss without category centers, we need to store pairwise distances among all pixels with a tremendous GPU memory overload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Therefore, the category center fc of category c is calculated as follows, fc = G( 1 Nc � s � j 1 �ys j = c � xs j), (5) where xs j refers to the pixel-wise features in the penultimate feature map, and ys j be the ground truth pixel-wise labels of a source domain image at pixel location (j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Nc refers to the total number of pixels in category c, s refers to the source domain image index, and G(·) is an L2 normalization function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Note that it is crucial to use the L2 normalization G(·) to keep the category centers on the unit sphere and avoid scaling issues among stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The category centers are updated after the training, allowing the centers to become further and further from each other on the sphere surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Our category-oriented triplet loss is formulated as follows, Ltriplet = 1 Ns � s � j max c,c̸=ys j max( ���G(xs j) − fys j ��� − ��G(xs j) − fc �� + α, 0), (6) where Ns is the total number of pixels in all images, and α is a prescribed margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The loss will be zero if every feature xs j is at least α closer to its corresponding category center ys j than other category centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Because triplet loss is focused on hard samples and only reliable category labels for hard samples in the source domain, the proposed category-oriented triplet loss is only applied to the source domain images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' ������������������������������������������������ ������������������������������������������������ : Tgt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' feats of cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Green and Blue : Feat centers of cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Green and Blue : Src.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' feats of cat Green and Blue : Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' feats of cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Green and Blue Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Our proposed category-oriented triplet loss exploits hard samples and further enlarge category margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' dpos and dneg represent the distance of positive and negative pairs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 8 The working principles of our proposed category-oriented triplet loss are illustrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' In cooperation with the proposed global photometric alignment and data augmentation in the source domain, our proposed triplet loss can exploit hard samples in the source domain that have been coarsely aligned to the target domain and further improve the generalization capability of the trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The proposed category triplet loss can be considered complementary to the cross-entropy loss and the manifold projection loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Target Domain Consistency Regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The category- wise features are regularized by our proposed category-oriented triplet loss in the source domain, where the annotated ground truth labels are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' However, the supervision signal is weak in the target domain where there is no labeled data provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Consistency regularization is an important component of many recent state-of-the-art self-supervised learning algorithms, which utilizes unlabeled data by relying on the assumption that the model should output similar predictions when fed perturbed versions of the same image [39], [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Motivated by this, we propose a target domain consistency regularization method shown in Figure 1 to perform category-level feature distribution regularization in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' In the target domain, the pseudo-label at a certain pixel location is defined as the category corresponding to the largest component of the probability vector produced from the segmen- tation model Fi−1 trained in the previous step, and the largest component of the probability vector itself defines the confidence of the pseudo label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We further pre-define a pair of probability threshold Ph and percentage threshold p for all categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' p is a constant value but leads to a category-specific probability threshold Ps,c, meaning p% pixels in the category c have confi- dence above threshold Ps,c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Thus the final confidence threshold for category c is tc = min(Ph, Ps,c), and any pseudo-labels with a confidence higher than tc in category c are considered valid samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Our proposed target domain consistency regularization is straightforward: given a target domain image Iu n, with the trained segmentation model Fi−1, we extract the pseudo-label ˆyu n by forwarding Iu n to Fi−1 followed by applying the arg max(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=') function to its output;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' and the corresponding pixel prediction is converted to a hard label vector 1[c=ˆyu n,j];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' then, a stochastic function τ2(·) is applied to Iu n to obtain a perturbed version �Iu n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' after that, we forward �Iu n to Fi to obtain the prediction � P u n on the perturbed image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' finally, the prediction � P u n is forced to be consistent with ˆyu n by using a cross entropy loss function at pixel locations whose largest class probability is above the previously defined category-level confidence threshold tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Note that the perturbed target domain image generated via the stochastic function τ2(·) makes prediction harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Thus more samples in the target domain can be converted into hard samples, but the generation of pseudo-labels is unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' In this way, category- level feature distributions in the target domain are regularized under the supervision of valid pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The overall formula is defined as follows, Lcst = � j 1(max(Fi−1(Iu n)|j) ≥ tc) CELoss(1[c=ˆyu n,j], � P u n,j), ˆyu n,j = arg max(Fi−1(Iu n)|j), � P u n,j =Fi( �Iu n)|j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' (7) It is essential to use the trained model Fi−1 rather than model Fi to generate pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This is because Fi is still being trained and unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Fluctuating pseudo-labels generated by Fi would be catastrophic to the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Experimental results illustrate that this consistency regularization method is simple yet efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' It strengthens the supervision signal in the target domain and improves the final performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' 4 EXPERIMENTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='1 Datasets and Implementation Details For commonly used synthetic datasets, we follow the same evalu- ation settings as used in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Our proposed method are evaluated with datasets GTA5 [42], Synthia [43], and Cityscapes [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The Cityscapes dataset is the target domain dataset with 2, 957 of size 2048 × 1024 training images and 500 validation images of the same resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Cityscapes has 19 categories of objects in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The GTA5 and Synthia are two source domain datasets of computer generated synthetic images, which contain 24, 966 of size 1914 × 1052 training images and 9400 of size 1280 × 760 training images respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The GTA5 dataset shares 19 common categories with the Cityscapes dataset, and all the irrelevant categories are ignored during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The Synthia dataset shares 16 common categories with the Cityscapes dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Some previous work [11], [14] only train and test on a 13-category subset of the Synthia dataset, or train two models on both subset and the whole set for better performance [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Here we follow the practice in [13], [15] to train a model only on the whole set and test it on both settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' In order to evaluate the performance of our proposed method on real-world source images, we construct a new domain adaptive semantic segmentation task Kvasir→Piccolo on the basis of two open-source datasets Hyper-Kvasir [19] and Piccolo [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The Hyper-Kvasir dataset is the source dataset and consists of 1000 wide-band (WL) gastrointestinal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The image resolution of the Hyper-Kvasir dataset is not fixed and is roughly 625 × 530.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The Piccolo dataset is the target dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Among the 1302 narrow- band (NBI) colonoscopy images of size 854×480, 1161 are used as training images and 141 as validation images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We follow the rule in [41] to construct this new task specifically designed for medical images: the images were collected with different modes (NBI vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' WL), different locations (GI tract vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' colonoscopy) and different devices to create a significant domain gap between the source and target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' According to Figure 1, the photometrically adapted source domain images are used first to train the initial segmentation model F0 in the image-level adaptation step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Then, the model is trained in an iterative self-supervision manner with K = 6 and U = 20k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We compare to previous work in domain adapted semantic segmentation [16], [17] based on self-supervision and set the total number of training iterations to 140k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' As reported in [14], the best performance is achieved when Ph = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='9 and p = 10 for pseudo-labels, and we follow this setting in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The regularization term β used by GPA module in (7) is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' For the proposed global texture alignment (GTEXA) module, d = 5, σc = 75 and σs = 25 are the optimized parameters of the bilateral filter for the GTA5→Cityscapes and Synthia→Cityscapes tasks, the KL divergence is reduced roughly from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='16 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='10 and from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='43 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='07, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Because images from both Hyper-Kvasir and Piccolo are real images, they have quite similar distributions of high-frequency components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 9 TABLE 1 Performance comparison with state-of-the-art methods on the GTA5→Cityscapes task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Results with image-level adaptation only and our whole image-to-feature pipeline are also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The best performing result is marked in bold road sidewalk building wall fence pole light sign vege .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' terrace sky person rider car truck bus train motor bike mIoU DeeplabV2 BDL [14] 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='0 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='7 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='2 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='6 27.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='3 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='5 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='3 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='3 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='2 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='5 I2F all (ours) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='9 34.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='9 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='2 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='1 use a gentle bilateral filter with d = 5, σc = 10 and σs = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The KL divergence stays at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='10 before and after the bilateral filter is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' For the proposed GMA module, Dc′ is set to keep roughly 90% explained ratio of the energy of xj, which is Dc′ = 32 for DeeplabV3+ and Dc′ = 256 for DeeplabV2, compared to Dc = 256 for DeeplabV3+ model and Dc = 2048 for DeeplabV2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The K-Means is used because it is the simplest clustering algorithm to validate our motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The number of clusters centers are set to Nz = 64 with hidden neuron dimension Nh = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Note that a larger Nz or a more advanced clustering method like KSVD might improve the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Still, it is not pragmatic because of the memory consumption or the computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We adopt the standard color- jittering as the stochastic function τ1(·) in both source, and target domains as in [15] in the image-level adaptation stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We utilize standard color-jittering, elastic deformation [45], and standard random blurring in the feature-level adaptation stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Elastic deformation is used to mimic the differences between shapes in different domains, and random blurring is used to simulate the resolution differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We conduct different settings for τ1(·) and τ2(·) because we observed that both elastic deformation and random blurring are strong data augmentations, and using them in the image adaptation stage will distort the distribution of the training data, which undermines the final segmentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We follow the same experiment settings in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' In addition to the DeeplabV3+(ResNet101) [6] discussed in [18], we also compare our proposed model with another commonly adopted segmentation model DeeplabV2(ResNet101) used by other state- of-the-art studies [14], [15], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We implement our proposed method with PyTorch [46], and deploy our experiments on 4 NVIDIA GeForce 2080Ti GPUs with 1 source domain image and 1 target domain image randomly selected and stored on each GPU for each backpropagation step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The stochastic gradient descent is used during the image-level adaptation with a momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='9 and weight decay of 1e − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The learning rate is initially set to 5e − 4 and is decreased using the polynomial learning rate policy with a power of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='9 during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' For the feature-level adaptation steps, we halve the learning rate to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='5e − 4 based on the learning rate from the image-level adaptation to fine-tune previously trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='2 Comparisons with State-of-the-Art Methods In this section, we compare our method against all the existing state-of-the-art methods [11], [13], [14], [15], [16], [17], on the GTA5→Cityscapes, Synthia→Cityscapes and Kvasir→Piccolo tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' As shown in Table 1, the performance improvement achieved by our proposed method outperforms all previ- ous methods with all different segmentation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' On the GTA5→Cityscapes task, our model achieves a new state-of-the- art mIoU (58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='2%) on DeeplabV3+ and (53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='3%) on DeeplabV2, which are 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='0% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='9% higher than the previous best re- sult using the same backbone (Table 1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' On the Synthia→Cityscapes task, the performances of our proposed method are 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='1% and 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='1%, which are 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='6% and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='0% higher than that of the previous method, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Note that we do not include the comparison between our work and a concurrent work [36] because three major differences make the comparison unfair, which are presented as follows: (1) much stronger baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' [36] adopts the domain adapted model from [24] as IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 10 (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Qualitative examples of the comparisons between our method and CAG [16] on the GTA5→Cityscapes task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Specifically, (a) Input images, (b) CAG [16], (c)Ours, (d) Labels the baseline model, which generates much stronger performance than ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' For example, our baseline model (37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='6%) is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='8% lower than theirs (43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='4%) on the GTA5→Cityscapes task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' (2) much stronger segmentation networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' [36] uses a more advanced segmentation network proposed in [47], [48] instead of the standard DeeplabV2 that we used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' (3) much stronger augmen- tation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' RandAug [49] and Cutout [50] are utilized as data augmentations in [36], which are much superior than our augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' On the Kvasir→Piccolo task, the segmentation performance of our proposed method is 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='5% and 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='2% on DeeplabV2 and DeeplabV3+ respectively, which are 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='7% and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='0% higher than the previous best results, as shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We re-implemented three general-purpose state-of-the-art meth- ods [11], [15], [16] for this comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Our proposed method also significantly outperforms the domain adaptive segmentation algorithm in [41], which was specifically designed for endoscopic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Our method achieves the best performance in many important categories, including ‘road’, ‘sidewalk’, ‘building’, ‘fence’, ‘veg- etation’, ‘terrace’, ‘person’, ‘car’, ‘rider’, ‘truck’, ‘train’, ‘bus’, ‘motor’, and ‘bike’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' In particular, our model performs the best when classifying over ‘road’, ‘sidewalk’, ‘motor’, and ‘bike’, even if some of these categories have very similar local appear- ances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This is because our proposed category-oriented triplet loss maximizes the inter-category distances and minimizes the intra- category distances by exploiting the most difficult samples in IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 11 (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Qualitative examples of the comparisons between our method and CAG [16] on the Kvasir→Piccolo task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Specifically, (a) Input images, (b) CAG [16], (c)Ours, (d) Labels the source domain, which improves the generalization capability across different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Moreover, our proposed global mani- fold/texture alignment brings an extra 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='1% improvements in the final performance compared to the experiments in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This is because global photometric alignment is generally conducted in the image-level inputs, and the features/textures between different domains are still not explicitly aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Finally, our proposed target consistency regularization also strengthens the relatively weak supervision signal in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' It improves the segmentation accuracy of categories with large intra-category variances, such as ‘building’ and ‘sky’, by regularizing its feature distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' According to our qualitative analysis, the previously over-exposed white buildings are easily misclassified as skies but can be corrected by our proposed target consistency regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' One interesting fact is that our proposed method has larger overall performance improvements in the GTA5→Cityscapes task compared to the Synthia→Cityscapes task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This is because DeeplabV3+ adopts high-resolution feature maps and lower fea- ture dimensions, which improves the segmentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The Synthia dataset is mostly constituted by large objects, limiting the improvements that a DeeplabV3+ segmentation model can bring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Another interesting fact is that although our proposed GMA has achieved clear performance improvements in both DeeplabV3+ and DeeplabV2, yet the performance improvement with DeeplabV2 is higher as showin in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This is because the feature dimension for DeeplabV2 is much larger than the feature dimension for DeeplabV3+(2048 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' 256).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This results in a more complicated feature manifold which is difficult to align with simple image-level photometric alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' By modeling the manifold and minimize the projection error, our proposed GMA can effectively align high-dimensional features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Although DeeplabV3+ is powerful, by cooperating with our proposed GMA module, the performance of our proposed model on the Synthia→Cityscapes task with DeeplabV2 is even higher than the one with DeeplabV3+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We further show some of the segmentation examples in Fig- ure 7 and Figure 6 to qualitatively demonstrate the superiority of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Our proposed method generates finer edges and makes fewer mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We also compare the style-transferred images generated by our proposed GPA model with other state- TABLE 3 Performance comparison with state-of-the-art methods on the Kvasir→Piccolo task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Best results are marked in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' polyps background mIoU DeeplabV2 FGGAN [15] 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='2 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='4 FDA [11] 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='2 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='5 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='8 image adapt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' [18] 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='7 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='9 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='8 I2F all (ours) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='0 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='9 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='5 DeeplabV3/+ WCBT [41] 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='9 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='5 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='3 CAG [16] 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='0 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='4 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='2 image adapt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' [18] 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='8 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='0 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='9 I2F all (ours) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='2 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='2 of-the-art style-transfer techniques used by the domain adaptation methods in Figure 8, and our proposed method has better quality and a higher level of diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='3 Ablation Studies Component Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' In most previous work, a source-only model trained on the source domain training set only is often required to serve as initial pseudo label producer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Although we do not use the source-only model during training, we train one to provide a baseline so that it is convenient to verify the primary performance gains from our proposed pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Then, following the experiment settings in [16], we perform extensive ablative experiments using DeeplabV3+ on the GTA5→Cityscapes task to verify the effectiveness of each of our proposed component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' As shown in Table 4, the source-only baseline using Deeplab v3+ has a performance of 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='6% for the GTA5→Cityscapes task, and our proposed overall pipeline improves the baseline perfor- mance by 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Following the same settings in previous state- of-the-art methods [13], [15], [16], we further evaluate the impact of each proposed component according to the final performance of our model on the GTA5→Cityscapes task by removing one component at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The results are shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' According to our results, the final performance of the segmentation model has the most deterioration when the global photometric alignment module is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This is because the GPA module is critical to the image-level adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Removing it literally removes the first image-level adaptation stage, and thus, the resulting erroneous IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 12 (a) (b) (c) (d) (e) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Qualitative analysis on the global photometric alignment (GPA) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' (a) Input images, (b) Reference images, (c) BDL-GAN [14], (d) Fourier Adaptation [11], (e) Global Photometric Alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' pseudo-labels are harmful to later stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Although our proposed global manifold alignment module can align the feature distri- butions from different domains, the error from unaligned models still accumulates across steps, which is detrimental to the final performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This also validates the necessity of a image-level adaptation step and the importance of an accurate initial model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Even though our proposed GPA module serves as an image- level adaptation between domains, the feature-level domain shifts are still not aligned completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Our proposed GMA module can further improve the feature-level adaptation between the source domain and the target domain, decreasing the model perfor- mance by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='4% when removing the GMA module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' In addition, the GTEXA module is designed to modify the high-frequency components of an image with a certain probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This action makes trained models robust to texture variations and further im- proves the performance by roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Furthermore, despite its simplicity, our proposed target domain consistency regularization has been proved to be very effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The main reason for this phenomenon is that there are fewer valid training samples in the target domain than the source domain, and our proposed TCR essentially serves as a data augmentation technique that increases the number of valid training samples in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' It also introduces more hard samples without damaging the pseudo- labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Therefore, it gives rise to a significant performance gain, decreasing the model performance by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='8% when removing the TCR module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Our proposed category-oriented triplet loss applied on the source domain also boosts the performance by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='1% as it exploits hard samples in the source domain and improves generalization capability across different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Photometric Alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' There are currently other methods, which can achieve the goal of the image-level adaptation, such as the GAN-based method in [14] and the frequency-based method in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We substitute our proposed global photometric alignment and global texture alignment with these two methods and retrain our whole pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The result is shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We also visu- IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 13 TABLE 4 Ablation study of the proposed components on the GTA5→Cityscapes task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' GPA: global photometric alignment, GTEXA: global texture alignment, CTL: category-oriented triplet loss, TCR: target domain consistency regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' GPA GTEXA GMA CTL TCR mIoU Source only 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='6 Image adapt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' √ 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='3 w/o Alignments √ √ 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='5 w/o GPA √ √ √ √ 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='0 w/o GMA √ √ √ √ 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='6 w/o CTL √ √ √ √ 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='1 w/o TCR √ √ √ √ 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='4 w/o GTEXA √ √ √ √ 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='5 all √ √ √ √ √ 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='2 alize some representative aligned images produced with different methods in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Our proposed GPA can generate the aligned image according to a randomly chosen target domain reference image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Simultaneously, the GAN-based model [14] performs de- terministically and generates aligned images with a similar style, only covering part of the actual target domain image span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' This explains why our proposed model works even better than the pre-trained deep adversarial model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Although the frequency-based method proposed in [11] can generate style-transferred images randomly, the concatenation of frequencies usually introduces significant noises during training, which largely limits its final performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Based on our observation, gamma correction on Lab channels does not have sufficient adaptation capability, while histogram matching on all three channels results in image artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We use the simple mean-variance of RGB channels as the benchmark, and run a comparison for the image-level adaptation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The result in Table 5 shows our hybrid scheme performs the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='50 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='70 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='90 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='10 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='30 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='50 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='70 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='90 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='10 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='30 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='50 1 2 3 4 mIoU 𝛼=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='0 𝛼=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='2 𝛼=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='6 𝛼=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Quantitative analysis on the selection of category margin α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Best performance is achived with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Category Triplet Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We only apply the category-oriented triplet loss to the source domain category labels in our proposed method but not the pseudo-labels in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Although the target domain images with pseudo-labels can be used as supplementary samples when the pseudo-labels are of high con- fidence, our proposed triplet loss aims to deal with hard samples and pseudo-labels of hard samples in the target domain are not reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' To verify this, we include pseudo-labels in our category- oriented triplet loss, and the result is shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We follow the default settings as in [51] and set α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' But we also tested our proposed category triplet loss with other settings as shown in Figure 9, which shows that the best result is achieved when α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Manifold Alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' In our proposed model, we directly use the PCA+K-Means to model the feature manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' It shares similar 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='50 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='70 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='90 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='10 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='30 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='50 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='70 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='90 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='10 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='30 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='50 1 2 3 mIoU 𝑁ℎ = 16 𝑁ℎ = 32 𝑁ℎ = 64 (b) 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='50 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='70 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='90 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='10 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='30 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='50 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='70 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='90 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='10 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='30 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='50 1 2 3 mIoU 𝑁𝑧 = 16 𝑁𝑧 = 32 𝑁𝑧 = 64 (a) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Quantitative analysis on the hyper-parameters of global manifold alignment (GMA) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' (a) higher Nz leads to better performance, (b) mIoU is generally not very sensitive to the choice of Nh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' functionality with the adversarial methods used in previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' But our proposed method suffers little from mode collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The mode collapse is easily observed in the style-translated images as in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Still, it also exists in the high-level features and undermines the diversity of the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' To make fair compar- isons, we substitute our proposed manifold alignment module with a traditional global discriminator as in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' According to Tabel 5, the result illustrates that it performs even worse than the version without a global discriminator, manifesting the superiority of our GMA module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We also conduct extensive experiments to verify the values of the hyperparameters Nh and Nz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The experiment results are presented in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' In general, the performance of the model is insensitive to the choice of Nh, and larger Nz leads to better performance, the best performance is achieved with settings Nh = 32 and Nz = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Note that we can not increase atom number Nz to more than 64 because the calculation of atom weights and the reconstruction of pixel feature xj require a large amount of GPU memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' TABLE 5 Ablation studies of the image adaptation strategy, photometric alignment scheme, and using pseudo-labels for the category-oriented triplet loss on the GTA5→Cityscapes task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Modules Methods mIoU Image Aapt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Frequency Align [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='0 BDL-GAN [14] 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='4 Photometric+Texture 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='2 GPA Scheme RGB Mean-Variance 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='3 Lab Gamma Correction 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='5 Lab Histogram Match 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='3 Hybrid 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='3 Pseudo-labels Triplet loss with pseudo-labels 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='1 Triplet loss w/o pseudo-labels 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='2 Manifold Align.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Adversarial Method 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='8 Manifold Alignment 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content='2 5 CONCLUSIONS In this paper, we have explored non-adversarial methods in both image-level and feature-level domain adaptation, and proposed a novel unified image-to-feature adaptation pipeline for unsuper- vised domain adaptive semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' During this study, we have found out that for this specific problem, adversarial meth- ods could damage the diversity of feature distributions, and a sim- ple photometric alignment module can achieve better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We have also found out that a simple self-supervised consistency loss is capable of regularizing category-level feature distributions IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 14 in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The proposed pipeline effectively integrates global image-level and feature-level adaptation and category-level feature distribution regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The global texture alignment module also serves as an auxiliary data augmentation scheme for the proposed pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' In particular, we have introduced a novel and efficient global photometric alignment module to adapt source domain images to the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' A global texture alignment module has been designed to modify the high-frequency components of images from the source domain and make the trained model robust to domain gaps caused by domain-specific textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' We have also proposed a global manifold alignment module to directly model the distribution of the pixel features from the source domain and align the feature distributions from both domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' To our best knowledge, this is the first piece of work that models the feature manifold directly in unsupervised domain adaptation for semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' A category-oriented triplet loss has been devised for the source domain to regularize source domain category centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' A target domain consistency regularization method has also been introduced for the target domain to regularize category-level feature distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Extensive experiments have shown that each of our proposed techniques improves the generalization capability of our model significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' The proposed three modules form a complete adaptation strategy to tackle domain shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Integrating them gives rise to a significant improvement over existing state- of-the-art unsupervised domain adaptive semantic segmentation methods, demonstrating that minimizing global and category-level domain shifts simultaneously deserves more attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Our work still has a few limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' First of all, the photometric alignment module is isotropic, which means texture information is not altered by our proposed module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Al- though we have proposed a global texture alignment scheme, it is activated only when the source domain images have stronger or similar high-frequency components in comparison to the target domain images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' It deserves more attention to develop a method that can better close the domain gap without hurting the feature diversity of source domain samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' In addition, our scheme for global feature manifold alignment is the first attempt to model the feature manifold directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' However, when we design our scheme, the priority is making manifold alignment compatible with gradient back-propagation based training, but not achieving optimal alignment performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' Nonetheless, it demonstrates the potential of direct feature manifold modeling in domain adaptation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} +page_content=' At last, some of our proposed components are only designed for the 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAzT4oBgHgl3EQfNftb/content/2301.01149v1.pdf'} diff --git a/lNE0T4oBgHgl3EQf7wKH/vector_store/index.faiss b/lNE0T4oBgHgl3EQf7wKH/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..9a3f5ea6b5971441000f607b6a8205f988ec224b --- /dev/null +++ b/lNE0T4oBgHgl3EQf7wKH/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:07a3ab6fd16e78338aa48148bd6560e8cb1351168e33dee168237f6b4ce29e8b +size 5046317 diff --git a/m9FIT4oBgHgl3EQftysi/vector_store/index.pkl b/m9FIT4oBgHgl3EQftysi/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..274f429280c4ecf0c1a0d872ba08acf80e5b7439 --- /dev/null +++ b/m9FIT4oBgHgl3EQftysi/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3b1fa56d5804d714c4de7e15bcc76f75328e0f879633176817aa416ae6e2a283 +size 225502 diff --git a/mNAzT4oBgHgl3EQfN_sI/content/tmp_files/2301.01156v1.pdf.txt b/mNAzT4oBgHgl3EQfN_sI/content/tmp_files/2301.01156v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..da9c7fdaf0fb3571b434eb4328c814db7fbbdb93 --- /dev/null +++ b/mNAzT4oBgHgl3EQfN_sI/content/tmp_files/2301.01156v1.pdf.txt @@ -0,0 +1,1661 @@ +Reference Twice: +A Simple and Unified Baseline for Few-Shot Instance Segmentation +Yue Han1* +Jiangning Zhang1,2* +Zhucun Xue3 +Chao Xu1 +Xintian Shen1 +Yabiao Wang2 +Chengjie Wang2 +Yong Liu1† +Xiangtai Li4 +1 Zhejiang University +2 Youtu Lab, Tencent +3 Wuhan University +4 Peking University +Abstract +Few Shot Instance Segmentation (FSIS) requires models +to detect and segment novel classes with limited several sup- +port examples. In this work, we explore a simple yet unified +solution for FSIS as well as its incremental variants, and +introduce a new framework named Reference Twice (RefT) +to fully explore the relationship between support/query fea- +tures based on a Transformer-like framework. Our key in- +sights are two folds: Firstly, with the aid of support masks, +we can generate dynamic class centers more appropriately +to re-weight query features. Secondly, we find that support +object queries have already encoded key factors after base +training. In this way, the query features can be enhanced +twice from two aspects, i.e., feature-level and instance- +level. In particular, we firstly design a mask-based dynamic +weighting module to enhance support features and then pro- +pose to link object queries for better calibration via cross- +attention. After the above steps, the novel classes can be im- +proved significantly over our strong baseline. Additionally, +our new framework can be easily extended to incremental +FSIS with minor modification. When benchmarking results +on the COCO dataset for FSIS, gFSIS, and iFSIS settings, +our method achieves a competitive performance compared +to existing approaches across different shots, e.g., we boost +nAP by noticeable +8.2↑/+9.4↑ over the current state-of- +the-art FSIS method for 10/30-shot. We further demonstrate +the superiority of our approach on Few Shot Object Detec- +tion. Code and model will be available. +1. Introduction +Instance Segmentation aims to detect and segment each +object for a particular category, which is a core vision task in +Yue +Han +(22132041@zju.edu.cn); +Jiangning +Zhang +(vtzhang@tencent.com); Yong Liu (yongliu@iipc.zju.edu.cn); Xiangtai +Li (lxtpku@pku.edu.cn). +*Equal contribution. +†Corresponding Author. +CNN +A +Class +Prototypes +Pred Heads +(a) Feature aggregation in RPN-based dual-branch framework +(b) Our Reference Twice framework +RoiAlign +Pooling +Aggregation Operation +e.g. Mult, Sub, Concat, Attn +Query +Image +Support +Set +CNN +… +RPN +CNN +Query +Image +Support +Set +CNN +Decoder +Dynamic +Class Centers +… +Mask +Pooling +Decoder +A +One class +at a time +All classes +at a time +A +TopK +Pred Heads +A +Figure 1. (a) Current RPN-based dual-branch framework. (b) Our +proposed framework. Our method fully utilizes the support set on +both feature and query levels. +autonomous driving, satellite map navigation, medical im- +age analysis, etc. The community has seen tremendous suc- +cesses in this direction by designing models for a set of pre- +defined classes [3,9,10,20,25,33,33,41,55,61,72]. How- +ever, it is hard to enlarge these methods when deploying for +real applications, since they are data-hungry and need huge +extra mask annotations. Inspired by the ability of humans +to learn with minimal data, Few-Shot Learning (FSL) has +received lots of attention recently. With abundant labeled +data of base classes, FSL aims at learning and predicting +novel classes in the given input data (called query set / im- +ages) with only a few labeled exemplars (called support set +/ images), i.e., FSL learns a conditional model that performs +prediction by referring to support images. +To fill up the gap of lacking instance-wised mask an- +notation for novel classes, Few-Shot Instance Segmenta- +tion (FSIS) is proposed [17, 66]. +Compared with Few- +Shot Object Detection (FSOD) [5, 16, 22, 23, 28, 29, 31, +32, 34, 35, 39, 49, 60, 65, 71] and Few-Shot Semantic Seg- +mentation (FSSS) [50, 56, 59, 75], FSIS is more challeng- +1 +arXiv:2301.01156v1 [cs.CV] 3 Jan 2023 + +ing since it requires both the detection and segmentation of +novel instances with few data. Current solutions to FSIS +mainly focus on designing a support branch to obtain class +prototypes that are used to re-weight query branch fea- +tures [13, 17, 19, 66], as shown in Fig. 1(a). Most of the +current dual-branch methods are based on this kind of two- +stage RoI-based detectors, where firstly pre-training the +model using base classes and then fine-tuning the model +on novel classes. +Constrained by mask resolution and +inadequate modeling of dual-branch interaction, current +two-stage models tend to under-explore cues from support +data [1,17,66,66]. One question arises: is there a stronger +framework to fully leverage the support guidance? Be- +sides, several works propose more advanced settings for +FSIS, i.e., generalized FSIS (gFSIS) [16] and incremen- +tal FSIS (iFSIS) [19], which require specific designs. For +example, Retentive R-CNN [16] aiming at gFSIS proposes +an extra Bias-Balanced RPN to debias the pre-trained RPN +and learn novel classes without forgetting previous knowl- +edge. iMTFA [19] and iFS-RCNN [13] elaborately design +the class head to solve iFSIS. Another question arises: is +there a simple yet unified framework for FSIS, gFSIS, and +iFSIS? +Recently, Detection Transformer [4,6,12,45,78] and cor- +responding mask-based variants [9,11] have made progres- +sive processes with the goal of using object queries to repre- +sent each instance, which simplifies and unifies the instance +segmentation pipeline. Motivated by the above, we pro- +pose a new framework based on the strong Mask2Former +baseline to solve FSIS, gFSIS, and iFSIS in one unified +framework. We carefully examine the training process of +Mask2Former with only base class samples and identify +two critical factors that help us design a simple solution to +improve novel class performance. Firstly, in Tab. 1, we find +that object queries from support branch can well locate ob- +jects for novel classes, even without the novel fine-tuning +process. We term this as support query localization. Sec- +ondly, in Fig. 2, we calculate attention maps between object +queries from support images and object queries from query +images, and find that most object queries are highly corre- +lated, even for most novel classes. We term this as support +query categorization. +Given the two key insights, we present a simple frame- +work named Reference Twice (RefT) to fully explore the +support mask information and support object query infor- +mation, as shown in Fig.1(b). +RefT adopts the Meta- +Learning framework with a two-stage pipeline, learning to +quickly generalize to novel knowledge by referencing twice +from the support branch. For the first reference, we adopt +mask pooling to crop support features rather than use RoI +Align [25] like previous methods [22, 69]. Then we pro- +pose to generate dynamic class centers from pooled support +features to enhance the query branch simultaneously. For +the second reference, as shown by support query catego- +rization and support query localization, object queries from +support images already encode relevant classification and +localization information. We design a simple multi-head at- +tention to link both object queries forcibly, which enhances +the classification and segmentation ability for novel classes. +Both reference processes are well coupled and lead to sig- +nificant improvement in novel classes compared to the base- +line. Furthermore, we make a simple modification to extend +our framework to the iFSIS setting. +To summarize, our contributions are as follows: 1) We +carefully examine the Mask-based DETR framework on +FSIS and identify two key factors named support query +localization and support query categorization, which are +important for guiding the Transformer-like framework de- +sign. +2) Motivated by above insights, we introduce our +Reference Twice framework based on the Meta-Learning +pipeline, which contains two novel steps: The first step uses +mask-based dynamic prototypes for feature-level enhance- +ment, while the second step provides instance-level guid- +ance by linking object queries from both query and support +branches. 3) Benefiting from our Transformer-like frame- +work, our method can be easily extended into all related +settings, i.e., FSIS, gFSIS, and iFSIS. To the best of our +knowledge, we are the first to unify three closely related +few-shot instance segmentation tasks in one framework. 4) +Massive experiments demonstrate the superiority of our ap- +proach over SoTA methods in three settings, and extensive +ablation studies consistently illustrate the effectiveness of +each component of the proposed approach. +2. Related Work +Few-Shot Classification aims to enable models to gener- +alize to novel classes with fewer samples. Most methods +employ the N-way K-shot episodic training paradigm that +helps rapid generalization by exposing itself to multiple +classification tasks. Approaches to FSL can be mainly cate- +gorized into optimization-based and metric-based. The for- +mer [1,2,51] utilizes a meta-learner to produce parameters +to learn a subtask. MAML [18] encodes prior knowledge +into a learnable initialization. +The latter [36, 54, 58, 67] +learns a transferable embedding space where query sam- +ples can be classified based on similarity to support sam- +ples. Various embedding methods and distance functions +are explored, including extracting per-class prototypes with +a fixed distance metric e.g., Cosine or Euclidean [52,54,58], +utilizing task-adaptive embedding functions with a learned +distance metric [36, 67] and attention learning [24, 27]. In- +stead of studying image-level classification task, we mainly +focus on instance-level few-shot learning. +Few-Shot Object Detection (FSOD) aims to enlarge the +vocabulary of a detector with few detection samples. Sev- +eral works [5, 8, 15, 16, 21–23, 28–32, 34, 35, 39, 49, 53, 60, +2 + +62–65,70,71,74,77] are proposed to advance this direction. +In particular, TFA [60] proposes a simple two-phase fine- +tuning approach, while DeFRCN [49] decouples the train- +ing of RPN features and RoI classification. SRR-FSD [77] +combines multi-modal inputs. LVC [31] proposes a pipeline +to enlarge novel detection examples and train a more robust +detector. +Few-Shot Segmentation includes Few-Shot Semantic +Segmentation (FSSS) and Few-Shot Instance Segmenta- +tion (FSIS). FSSS requires pixel-level classification for +query images. +Previous works [50, 56, 59, 75] typically +extract category prototypes from support images, and the +query image is segmented by computing the similarity dis- +tance between each prototype and query features. +FSIS +aims to detect and segment each object with fewer exam- +ples. FSIS approaches [17, 19, 46–48, 66] can be mainly +divided into single-branch and dual-branch architectures. +The former [19, 48] mainly focuses on the design of the +classification head, while the latter [17, 46, 66] introduces +an additional support branch to compute class prototypes +or re-weighting vectors of support images and assist the +segmenter in screening out target category features via fea- +ture aggregation. Meta R-CNN [66] performs channel-wise +multiplication on RoI features, while FGN [17] aggregates +channel-wise features at three stages, including RPN, detec- +tion head, and mask head. Pointed by recent works [31,69], +RPN-based approaches tend to mistake the novel class ob- +jects for the background and decrease the recall of novel +classes. Inspired by the recent FSOD methods [69], RefT +builds on a DETR detector and falls into the dual-branch +meta-learning-based methods. +Unlike previous methods +that need to aggregate the results of multiple runs for a +query image, RefT can deal with all support classes simul- +taneously and efficiently. +Vision Transformer. +Current research in this field falls +into two categories: one for better representation learn- +ing via designing a more robust backbone or strategy [14, +42, 43, 57], and the other using object query [7, 37, 38, 68, +73, 76] to unify and simplify relevant detection and seg- +mentation tasks. Our RefT is based on the recent DETR- +based segmentation method, Mask2Former [9]. Compared +with previous FSIS methods [17, 66], object queries from +Mask2Former encoding rich information of both object ap- +pearance and location are perfect as query features to be +aggregated and support guidance. RefT can be easily ex- +tended to various instance-level few-shot learning tasks. +3. Method +In this section, we will first introduce preliminary knowl- +edge, including the settings, training strategy, and our find- +ings for support query localization and categorization. Mo- +tivated by these findings, we introduce our RefT framework. +Then we detail RefT on FSIS and its variants. +bird +cat +dog +horse +sheep +cow +bicycle +car +motorcycle +airplane +bus +train +(a) Before Novel Fine-tune +(b) After Novel Fine-tune +1 +0 +1 +0 +bird +cat +dog +horse +sheep +cow +bicycle +car +motorcycle +airplane +bus +train +Figure 2. Support Query Categorization. We visualize the co- +sine similarity of object queries of the support branch belonging +to COCO 20 novel classes. Most object queries are roughly dis- +tinguishable, even without novel fine-tuning. We zoom in on areas +that contain highly correlated and easily misclassified classes. +Table 1. Support Query Localization. We analyze the mask +quality about 20 novel classes on COCO dataset and calculate +the top-k IoU between the ground truth and predicted masks of +the support branch. Object queries from support images can well +locate object masks for novel classes, even without novel fine- +tuning. nIoU/ bIoU: average IoU of novel/ base class. +# Top-k queries +Base Training +Novel Fine-tune +bIoU +nIoU +bIoU +nIoU +50 +0.54 +0.47 +0.59 +0.55 +30 +0.70 +0.64 +0.75 +0.71 +10 +0.84 +0.79 +0.84 +0.79 +Table 2. Comparison of different settings in few-shot instance +segmentation. FSIS: standard few-shot instance segmentation, +gFSIS: generalized few-shot instance segmentation, iFSIS: incre- +mental few-shot instance segmentation. +Settings +Fine-tune on +Test on +Base +Novel +Base +Novel +FSIS +✓ +✓ +✓ +gFSIS +✓ +✓ +✓ +✓ +iFSIS +✓ +✓ +✓ +3.1. Preliminary Knowledge +Problem Settings. +The classes are split into two sets +CBase and CNovel, where CBase ∩ CNovel = ∅, CBase ∪ +CNovel = CAll. FSIS aims to segment objects belonging to +CTest in a query image after training over abundant samples +of CBase and few samples of CFinetune. RefT approaches all +three settings in the literature. As shown in Tab.2, for FSIS, +CFinetune = CBase ∪ CNovel, CTest = CNovel. For gFSIS, +CFinetune = CBase ∪ CNovel, CTest = CBase ∪ CNovel; For +3 + +1st Ref. +Cross-Attention +Feed-Forward +Self-Attention +TopK +sort by IoU +2nd Ref. (Sec.3.2) +1st Ref.: Feature-Level Enhancement +Vannila Mask2former. +Reference Module. +Query Image +CAL +Support Set +N +K +CAL: Class Adaptation Layer +2nd Ref.: Query-Level Enhancement +MSDeformaAttn +… +������������������������ +Mask +Pooling +CrossAttn +KV +Q +Mask2Former +Decoder +2nd Ref. +CrossAttn +Class Embed +Mask Embed +NK* +Mask Embed +Compute +IoU +GT Masks +q������������ +q������������ +MSDeformaAttn +GT Masks +Multi-Scale +Features +������������������������ +������������������������ +������������������������ +′ +Enhanced +Query Features +Cross-Attention +Q +V +K +Dynamic +Class Centers +Mask Pooling +Sigmoid +Channel-wise Multiplication +1st Ref. (Sec.3.2) +(N,D) +(N,K,ΣHW,D) +������������������������ +Query Branch +Support Branch +������������������������ +������������ +������������������������ +NK* +Figure 3. Architecture of the proposed Reference Twice (RefT) for FSIS. The query branch refers to the support branch twice on the +feature and query level, respectively. 1st Reference for feature-level enhancement performs simultaneous aggregation between the query +features and all dynamic class prototypes obtained through mask pooling. 2nd Reference for query-level enhancement links object queries +from the query and support branch with a simple multi-head attention module. +iFSIS, CFinetune = CNovel, CTest = CBase ∪ CNovel. +Our Training Strategy and Baseline Method. +We adopt +the episodic-training [66] from FSOD in both base train- +ing and novel fine-tuning. The training stage comprises a +series of episodes Ei = (Ii +Q, Si), where i indicates the +ith episode. Given a query image Ii +Q, all objects present +in the image belong to N classes in Ctrain and a support +image set Si containing K samples per class along with +structural annotations will be provided as additional input, +which makes the N-way K-shot episode Ei. In particular, +we adopt Mask2Former as the segmenter [9]. Mask2Former +uses object query to treat instance segmentation as mask +classification and segmentation. +Support Query Localization. After the first base train- +ing stage, we directly infer support images of novel classes, +where we test whether only the base-trained model can re- +call novel classes with additional structural input. As shown +in Tab. 1, we find that base-trained model can already detect +and segment novel class objects. Most queries have the lo- +calization ability of novel classes. After novel fine-tuning +RefT, we observe improvements in more queries. +Support Query Categorization. We further visualize the +correlation maps among the support queries in Fig. 2, where +most of these novel classes are highly correlated. +We +present more examples in the supplementary materials. We +call this support query categorization. This indicates that +the support queries have the clustering effect even without +novel fine-tuning, which may be helpful for query image +branch training. After our framework, as shown on the right +side of Fig. 2, the categorization is more prominent. +3.2. Reference Twice +Motivation. Given the facts found in previous parts, we +aim to enable the model to fully explore the cues from the +support branch inputs. Unlike previous works [13] that only +leverage the feature-level enhancement, we apply a novel +enhancement on the query level, as the support queries can +encode both the localization and categorization information. +Overview. We take a query image IQ and a support set S +with N × K examples as input, as shown in Fig. 3. For the +support branch, we mask out the support objects and drop +the background together with other objects in the image. In +this way, the support features and object queries are more +discriminative and contain more accurate location informa- +tion. A weight-shared feature extractor first encodes the +query and support images into the same feature space. Sub- +sequently, the 1st Reference Aggregation module performs +simultaneous aggregation between the query and support +features. In this step, the query features are coarsely filtered +by support categories. Then the selected query features and +support features are sent to the class-agnostic pixel decoder +and transformer decoder to obtain object queries from both +query and support branch. Next, we link object queries of +both branches for better calibration via cross-attention. As +we mask the support images with the ground truth mask, the +4 + +obtained support object queries correspond precisely to the +instances of the support categories. +1st Reference for Feature-Level Enhancement. We de- +scribe the 1st Reference Aggregation module in detail. As +shown in the bottom left of Fig. 3, given the multi-scale +query features F Q= +� +xl +Q +�L +l=1 and support features F nk +S = +{xl +S} +L +l=1 (n = 1, . . . , N, k = 1, . . . , K), where L de- +notes the feature levels, a weight-shared multi-head de- +formable attention [78] first encodes them into the same +feature space, obtaining F Q and F S. The exact features +of support instances are separated from the background and +other instances through mask pooling with ground truth +masks on the support features of each scale, respectively. +Then the dynamic class centers for all support classes are +obtained by averaging all scales per image and K examples +per class, given by: +cn = +1 +KL +K +� +k=1 +L +� +l=1 +MaskPool +� +F nk +S +� +, +n = 1, . . . , N. +(1) +After that, a multi-head attention module is used to generate +the reweighting matrix for aggregating F ′ +Q with dynamic +class centers P =[c1,...,cN]∈RN×C: +R = softmax +� +F ′ +QP T � +σ(P ). +(2) +Here the linear projection is omitted for simplicity. The +query features are then multiplied with the obtained weights +along the channel dimension as below: +F Enhanced +Q += F ′ +Q ⊗ R. +(3) +Thus the category-related query features are selected and +enhanced. In this operation, the query branch is enhanced +by support examples dynamically. We present the detailed +design of choices of support features in the experiment part. +2nd Reference for Query-Level Enhancement. Inspired +by previous findings, we add extra query-level enhancement +during the episodic training. We first drop the support ob- +ject queries with irrelevant information and only keep ones +that contain instance-level category and spatial information +of high quality. The detailed process is shown in the bot- +tom right of Fig. 3. Specifically, given query object queries +qQ ∈ RQ×Dand support object queries qS ∈ RNKQ×D, +Q is the number of object queries per image, and D is the +feature dimension, we first obtain the predicted masks cor- +responding to the support object queries. Then, the top k +out of the Q object queries per image is selected accord- +ing to the IoU of the predicted and ground truth masks. As +proved by previous findings, we add extra query-level en- +hancement. In this way, we avoid huge computation costs +and also obtain better relevant cues to improve performance. +Then we use a multi-head cross-attention module to match +the query and support object queries: +qEnhanced +Q += softmax +� +qQ TopK(qS)T� +TopK(qS). +(4) +Then a multi-head self-attention module followed by one +feed-forward layer is used to adapt to the following pre- +diction heads. Such naive attention is good enough to link +support queries to the object queries from query images. +3.3. Generalization and Training Details +FSIS. The same losses in Mask2Former for classification +and segmentation are used. Besides, we classify the dy- +namic class centers in the first reference using a cosine sim- +ilarity cross-entropy loss, following [66, 69], to encourage +centers to fall into the corresponding categories. +gFSIS. The common practice in prior RPN-based works is +to freeze most parameters and fine-tune the class head on a +balanced sampled dataset of base and novel classes to retain +base class knowledge. However, as observed in [13], the +DETR-like detector can barely generalize to novel classes +with the input projection layer frozen. We figure the layer +is class-specific because it transforms the channel dimen- +sion with one convolution, and channel features are essen- +tial in distinguishing different classes. To point out the sig- +nificance of the layer in novel class learning, we term it as +Class Adaptation Layer, abbreviated to CAL, also shown +in Fig. 3. +We alleviate the catastrophic forgetting prob- +lem by simply fine-tuning CAL, object queries and the class +head with all other parameters frozen and without additional +modifications. Losses remain the same as in FSIS. +iFSIS. As we unfreeze an additional CAL layer, directly +fine-tuning on very constrained novel class samples will in- +evitably lead to severe overfitting. To alleviate this prob- +lem, Incremental-DETR [13] utilizes knowledge distilla- +tion on both CAL and the class head and fine-tunes in a +self-supervised manner. In contrast, to unify iFSIS and the +other two settings in RefT, we only focus on CAL and adapt +the base knowledge distillation loss (BKD loss) in +[13] +to a class-enhanced version. Considering the role of CAL +in class adaptation, more prominent class features are ex- +pected. To this end, according to their relative importance, +we re-weight the features along spatial and channel dimen- +sions. The degree of importance is measured by the abso- +lute mean value of the pixel or channel, and the softmax +activation is used as normalization. The channel and spatial +re-weighting coefficients RC +k and RS +k are obtained from: +RC +k = C · softmax( +1 +HW +H +� +i=1 +W +� +j=1 +���F B +k,i,j +���), +(5) +RS +ij = HW · softmax( 1 +C +C +� +i=1 +���F B +k,i,j +���). +(6) +Therefore, our Class-Enhanced Base Knowledge Distilla- +tion Loss is formulated as: +LCE−BKD = +C +� +k=1 +H +� +i=1 +W +� +j=1 +RC +kRS +i,j(1−Mi,j) +� +F B +k,i,j −F N +k,i,j +�2 +, (7) +C, H, and W denote the channel, height, and width of the +feature. M ij is the ground truth masks of the foreground +5 + +CNN +Base class truck +Novel class person +Class-Enhanced BKD Loss +Support sets +Novel Fine-tune +CNN +CNN +CNN +Freeze +Reference +Twice +CAL +CAL +Class Head +Mask Head +Figure 4. Adaptation to iFSIS. In novel fine-tuning, all parame- +ters except CAL, object queries, and the class head are frozen. The +proposed Class-Enhanced BKD Loss is adopted to CAL to prevent +overfitting, without hindering novel class generalization. +instances, which all belong to the novel classes during fine- +tuning. F B +k,i,j is the CAL output feature of the model frozen +after base training, and F N +k,i,j is the CAL output feature of +the model during novel fine-tuning. +Training and Inference Procedure +1) Base Training. We perform episodic base training on +our proposed RefT over base classes. +2) Novel Fine-tuning. +We perform episodic novel fine- +tuning over a sampled N-way K-shot dataset with classes +CFinetune, following [66]. +3) Inference with all support classes. During inference +time, we compute dynamic class centers and object queries +from support sets once and for all. Unlike previous works +that require multiple forward passes for each query image, +RefT only forwards once with all support classes, which is +simpler and more efficient. +4. Experiment +4.1. Experimental Setup +Dataset Setting and Metrics. We follow the data setups +first established in FSOD and then extended to FSIS. We +evaluate on the MS-COCO 2014 dataset [40] modified by +[60], where the original 80k train and 35k validation images +are combined as the actual training set and the 5k minival set +is used for testing. The 80 classes are divided into 2 sets, in- +cluding 20 novel classes that intersect with PASCAL VOC +and the remaining 60 base classes. Shots per class are set to +K= {1, 5, 10, and 30}. We adopt standard MS-COCO met- +rics, namely Average Precision (IoU=0.5 : 0.95), on novel +and base classes, abbreviated to nAP and bAP, respectively. +As in [19], we run all tests 10 times with K examples of 10 +seeds for each class and report the averaged results. +Implementation Details. +We take Mask2Former [9] as +the main architecture and ResNet-50 [26] is adopted as the +backbone following prior work in FSIS. In the base train- +ing stage, we episodically train our model over COCO base +classes for 50 epochs with a batch size of 8 on 8 V-100 +GPUs, using the AdamW optimizer [44] and the step learn- +ing rate schedule. We set the initial learning rate of 0.0001 +and a weight decay at the last epoch by 0.05. In the novel +fine-tuning stage, the settings remain the same until con- +vergence. For fair comparison, we implement a fine-tune +based Mask2Former for FSIS similar to TFA [60]. We per- +form standard training on Mask2Former with default setting +over COCO base classes and fine-tune the model over novel +classes until convergence. +Table 3. FSOD and FSIS results (nAP) on COCO with K = {5, 10, +30}. “-”: unavailable corresponding result. Optimal and subopti- +mal results are highlighted in bold and underline, respectively. We +use ResNet-50 as backbone. +Object Detection +Instance Segmentation +Methods +5 +10 +30 +5 +10 +30 +Mask2Former+ft-full +13.6 +17.3 +21.9 +12.7 +16.7 +20.8 +Meta-DETR [69] +15.4 +19.00 +22.2 +8.1 +10.1 +- +MRCN+ft-full [26] +1.3 +2.5 +11.1 +- +1.9 +- +Meta R-CNN [66] +3.5 +5.6 +12.4 +- +4.4 +- +iMTFA [19] +6.6 +8.5 +- +6.6 +8.4 +- +iFS-RCNN [48] +10.5 +11.3 +14.7 +9.4 +10.2 +13.1 +RefT (Ours) +15.0 +19.3 +24.0 +14.2 +18.4 +22.5 +Table 4. gFSOD and gFSIS results on COCO with K = {1, 5, 10}. +“-”: unavailable corresponding result. Optimal and suboptimal +results are highlighted in bold and underline, respectively. +Object Detection +nAP +bAP +Backbone +Methods +1 +5 +10 +1 +5 +10 +iMTFA [19] +2.1 +6.2 +8.3 +31.7 +33.1 +34.0 +LVC [31] +- +- +17.6 +- +- +29.7 +R-50 +RefT (Ours) +5.2 +13.1 +18.6 +38.4 +36.0 +37.7 +TFA [60] +1.9 +7.0 +9.1 +31.9 +32.3 +32.4 +DeFRCN [49] +4.8 +13.6 +16.8 +30.4 +32.6 +34.0 +LVC [31] +- +- +17.8 +- +- +31.9 +R-101 +RefT (Ours) +5.2 +14.1 +18.9 +38.5 +36.2 +37.7 +LVC [31] +- +- +18.6 +- +- +29.2 +Swin-T +RefT (Ours) +5.3 +16.8 +20.0 +39.6 +37.0 +37.9 +LVC [31] +- +- +19.0 +- +- +28.7 +Swin-S +RefT (Ours) +5.2 +21.0 +24.2 +40.4 +39.8 +40.5 +Swin-B +RefT (Ours) +7.4 +20.2 +26.4 +42.6 +40.9 +41.0 +Instance Segmentation +nAP +bAP +Backbone +Methods +1 +5 +10 +1 +5 +10 +iMTFA [19] +2.3 +6.4 +8.4 +29.9 +31.3 +31.8 +R-50 +RefT (Ours) +5.2 +12.4 +17.4 +36.3 +34.4 +36.0 +R-101 +RefT (Ours) +5.1 +12.7 +17.5 +36.4 +34.6 +36.0 +Swin-T +RefT (Ours) +5.2 +15.4 +18.5 +37.1 +35.0 +36.2 +Swin-S +RefT (Ours) +5.0 +19.4 +22.7 +38.3 +37.9 +38.3 +Swin-B +RefT (Ours) +7.1 +20.7 +24.8 +40.2 +38.7 +39.2 +6 + +personTable 5. iFSOD and iFSIS results on COCO with K = {1, 5, 10}. +“-”: unavailable corresponding result. Optimal and suboptimal +results are highlighted in bold and underline, respectively. We use +ResNet-50 as backbone. +Object Detection +nAP +bAP +Methods +1 +5 +10 +1 +5 +10 +Incremental-DETR [13] +- +- +14.4 +- +- +27.3 +RefT (Ours) w/ Cos +4.0 +12.0 +14.9 +32.2 +31.8 +33.4 +Methods +Instance Segmentation +iMTFA [19] +2.8 +5.2 +5.9 +25.9 +22.6 +21.9 +iFS-RCNN [48] +4.0 +8.8 +10.1 +36.4 +36.3 +36.3 +RefT (Ours) w/ Cos +3.1 +8.8 +11.1 +37.0 +35.3 +35.2 +4.2. Main Results on COCO +FSIS Results. In Tab. 3, it can be seen that our method +easily outperforms previous works based on Mask R-CNN +by a wide margin, which is as expected because we use a +more robust base model. However, even compared with the +same DETR-based approach Meta-DETR, which achieves +the state-of-the-art results in FSOD, we obtain significantly +better results on FSIS and comparable or even slightly im- +proved results on FSOD. For more fair comparison, we also +compare with the fully fine-tuned Mask2Former and still +show obvious performance gains, which implies that RefT +fully leverage the image and instance level information pro- +vided by the support branch. +gFSIS Results. There is a trade-off between the results of +the base and novel classes. Few works attend to the more +challenging gFSIS that requires avoiding forgetting on base +classes. Tab. 4 shows that RefT consistently outperforms +recent SoTAs in both gFSOD and gFSIS. In addition, using +more powerful Transformer backbone models, e.g. Swin- +T, Swin-S, Swin-B, our approach still generalizes well and +does not fall into the severe overfitting of novel classes. +iFSIS Results. In Tab.5, we first adapt Mask2Former to +the iFSIS setting as our baseline by replacing the fully- +connected classifier with a cosine similarity classifier as in +iMTFA [19] and fine-tune both CAL and the class head with +all other parameters frozen after base training. +A prob- +lem arises here that the class-specific CAL needs to be +fine-tuned to allow novel class learning. However, with- +out access to base class samples, CAL will quickly over- +fit and suffer from catastrophic forgetting. +Adding our +Class Enhanced Base Knowledge Distillation Loss will eas- +ily address this issue and achieves comparable results to +the recent SoTAs. +Similar loss has also been proposed +in Incremental-DETR [13], but our method is simpler yet +equally effective. Detailed ablation of different losses and +classifiers are presented in supplementary materials. +4.3. Ablation Study and Analysis +Ablation study on each component. In Tab. 6a, we first +ablate on the effectiveness of each component. We take +a fully fine-tuned single-branch Mask2Former (abbreviated +to M2F-ft) as our baseline for fair comparison. Adding our +query-based 2nd reference module yields 1.6 improvements +on novel classes. Adding our image-based 1st reference +module further improves nAP by 1.1. Note that although +longer training will bring more performance gains on nAP +at the cost of a quick drop in bAP, we are not trading bAP for +nAP. This verifies our findings in Sec. 3.1. Both branches +share coupled effect for novel classes. +Support features in 2nd Ref. To demonstrate the effective- +ness of our query-based aggregation in the 2nd reference, +we also implement an image-based version using the same +dynamic class centers in the 1st reference as the support +guidance and the cross attention as the aggregation opera- +tion. Tab. 6b shows that query-level enhancement module +yields a 1.1 nAP boost over the feature-level one, which +proves the superiority of our RefT Framework over the pure +feature-level enhancement framework. +Effect of fine-tuning CAL. To demonstrate the role of CAL +in allowing DETR-like models to learn novel classes, we +freeze CAL in the novel fine-tuning stage and only fine-tune +the object queries and the class head. Tab. 6c shows that the +model can barely generalize with CAL frozen, leading to a +huge gap with the model that fine-tunes CAL (2.6 vs 17.4 +nAP). Additionally, the result indicates that we do not need +extra learnable parameters to guarantee sufficient ability to +transfer to novel domain and fine-tuning an additional CAL +can already achieve both novel class generalization and base +knowledge retainment. +Support features in 1st Ref. In Tab. 6d, we present ab- +lation on different features that are used to compute dy- +namic class centers, including three stages of features from +ResNet-50 (denoted as Res3, Res4 and Res5) and the flat- +tened multi-scale features from the first transformer encoder +layer (denoted as Enc1). We find that using support features +of larger scale brings significant performance boost (17.2 +vs 16.4nAP) might because more small objects are noticed. +Using multi-scale features further gains 0.2 point in nAP. +Pooling in 1st Ref. In Tab. 6e, we compare different pool- +ing methods to extract relevant information from support +images. It can be observed that with the improvement of +the accuracy of the feature region, the result increases corre- +spondingly. This is because more accurate instance features +without noise provide more distinguishing class centers for +the query branch classification. +Query selection in 2nd Ref. In Tab. 6f, we compare the +results of selecting top 10 object queries in 2nd reference +sorted by scores or mask IoU. The results are comparable +when k is small because object queries are more accurate +both for classification and segmentation. +7 + +Table 6. Ablation study on COCO with K=10. All experiments use ResNet-50 as backbone. +(a) Ablation study on each component. +Baseline ++1st Ref. ++2nd Ref. +nAP +bAP +M2F-ft +14.7 +36.0 +� +16.3 +35.5 +� +� +17.4 +36.0 +(b) Support features in 2nd Ref. +Features +nAP +bAP +Feature-level +16.3 +35.7 +Query-level +17.4 +36.0 +(c) Effect of fine-tuning CAL. +Fine-tune +nAP +bAP +� +2.6 +39.4 +� +17.4 +36.0 +(d) Support features in 1st Ref. +Method +nAP +bAP +Res3 +17.2 +36.6 +Res4 +17.0 +36.0 +Res5 +16.4 +35.2 +Enc1 +17.4 +36.0 +(e) Pooling in 1st Ref. +Method +nAP +bAP +GAP +17.0 +35.4 +RoiAlign +17.2 +35.3 +MaskPool +17.4 +36.0 +(f) Query selection in 2nd Ref. +Sort by +nAP +bAP +Score +17.2 +35.8 +Mask +17.4 +36.0 +(g) Number of object queries in 2nd Ref. +# Queries +3 +10 +50 +100 +nAP +17.4 +17.4 +17.5 +16.8 +bAP +35.8 +36.0 +35.8 +35.5 +Before +After +Figure 5. t-SNE visualization of object queries belonging to +COCO 20 novel classes. The results are obtained from the support +branch before and after novel fine-tuning, respectively. +Baseline ++ 2nd Ref. ++ 1st Ref. +cat 11% +dog 36% +dog 36% +dog 36% +bear 12% +dog 36% +truck 16% dog 36% +bear 14% +dog 36% +truck 16% dog 36% +cat 27% +dog 36% +truck 17% dog 36% +cat 55% +dog 36% +bird 11% +dog 36% +horse 11% +dog 36% +cow 11% +dog 36% +bird 70% dog 36% +bird 45% +dog 36% +train 19% +dog 36% +train 40% +dog 36% +carrot 51% +dog 36% +carrot 78% +car +carrot 98% +dog 36% +carrot 78% +car +dog 36% +traffic light 89% +traffic light 70% +traffic light 97% +Figure 6. Visualization of 10-shot FSIS results on COCO mini- +val set. Predictions for novel class instances are mainly displayed. +Number of object queries in 2nd Ref. We select the top- +k object queries for each support image in 2nd reference +In Tab. 6g, we compare how different numbers of k affect +the performance. It shows that when k ≤ 50, a relatively +stable and close to the best result can be obtained. This is +consistent with what Tab. 1 has shown because the support +branch mask quality can be guaranteed when k ≤ 50. +4.4. Visualization and More Analysis +Understanding the effect of 2nd Ref. In Fig. 5, we visual- +ize the t-SNE results of object queries belonging to COCO +20 novel classes. We find that the object queries of novel +classes can be roughly distinguished, even without the novel +fine-tuning process. And the clustering is more obvious af- +ter fine-tuning. We leverage this support query categoriza- +tion in our 2nd reference module to provide guidance for +the query branch classification. +More Qualitative Results. +In Fig. 6, we present sev- +eral visual results on MS-COCO datasets corresponding to +Tab. 6a. +We find that adding our query-based 2nd Ref. +effectively reduces both missed and misclassified results, +which is mostly attributed to the guidance of support query +localization and support query categorization. +With the +alignment between query features and dynamic class cen- +ters, adding our 1st Ref. further reduces misclassification +between classes that are highly correlated. More results can +be found in the supplementary materials. +Parameter and GFLOPs. +Compared with strong +Mask2Former baseline, RefT significantly boosts nAP by ++1.7↑ with only increasing 1.7% GFLOPs and 4.3% param- +eters given 1,024 × 1,024 image input. +Limitation and Future Work. One limitation of RefT is +that the model is not able to perform well in one-shot set- +ting. Our future goal goes to provide more design for our +framework to fix this issue. +5. Conclusion +In this paper, we present a simple and unified base- +line for few-shot instance segmentation, namely Reference +Twice (RefT). We carefully examine the mask-based DETR +framework in FSIS and identify two key factors named sup- +port query localization and support query categorization. +Motivated by these two factors, we propose to reference the +model twice on feature level and query level, respectively. +Specifically, we first design a mask-based dynamic weight- +8 + +.bird11%horse11% +CoW11%dog36% +carrot78% +carrot32%truck19% +cat77%bird70% +ird45bear12% +carrot51%truck17% +bear55%bear14%%trafficlight84 +RyanTaylortrafficlight209 +RyanTaylortrafficlight89% +train40% +RyanTayloring module to enhance support features and then propose to +link object queries for better calibration via cross-attention. +Additionally, RefT can be easily extended to all three set- +tings including FSIS, gFSIS and iFSIS with minor modifi- +cation. 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In ICLR, 2020. 2, 5 +11 + diff --git a/mNAzT4oBgHgl3EQfN_sI/content/tmp_files/load_file.txt b/mNAzT4oBgHgl3EQfN_sI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..42f4a564cc82bf8c11b41a09d43b9000e8eca016 --- /dev/null +++ b/mNAzT4oBgHgl3EQfN_sI/content/tmp_files/load_file.txt @@ -0,0 +1,880 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf,len=879 +page_content='Reference Twice: A Simple and Unified Baseline for Few-Shot Instance Segmentation Yue Han1* Jiangning Zhang1,2* Zhucun Xue3 Chao Xu1 Xintian Shen1 Yabiao Wang2 Chengjie Wang2 Yong Liu1† Xiangtai Li4 1 Zhejiang University 2 Youtu Lab, Tencent 3 Wuhan University 4 Peking University Abstract Few Shot Instance Segmentation (FSIS) requires models to detect and segment novel classes with limited several sup- port examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In this work, we explore a simple yet unified solution for FSIS as well as its incremental variants, and introduce a new framework named Reference Twice (RefT) to fully explore the relationship between support/query fea- tures based on a Transformer-like framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Our key in- sights are two folds: Firstly, with the aid of support masks, we can generate dynamic class centers more appropriately to re-weight query features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Secondly, we find that support object queries have already encoded key factors after base training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In this way, the query features can be enhanced twice from two aspects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=', feature-level and instance- level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In particular, we firstly design a mask-based dynamic weighting module to enhance support features and then pro- pose to link object queries for better calibration via cross- attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' After the above steps, the novel classes can be im- proved significantly over our strong baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Additionally, our new framework can be easily extended to incremental FSIS with minor modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' When benchmarking results on the COCO dataset for FSIS, gFSIS, and iFSIS settings, our method achieves a competitive performance compared to existing approaches across different shots, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=', we boost nAP by noticeable +8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='2↑/+9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='4↑ over the current state-of- the-art FSIS method for 10/30-shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We further demonstrate the superiority of our approach on Few Shot Object Detec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Code and model will be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Introduction Instance Segmentation aims to detect and segment each object for a particular category, which is a core vision task in Yue Han (22132041@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='cn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Jiangning Zhang (vtzhang@tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='com);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Yong Liu (yongliu@iipc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='cn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Xiangtai Li (lxtpku@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' †Corresponding Author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' CNN A Class Prototypes Pred Heads (a) Feature aggregation in RPN-based dual-branch framework (b) Our Reference Twice framework RoiAlign Pooling Aggregation Operation e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Mult, Sub, Concat, Attn Query Image Support Set CNN … RPN CNN Query Image Support Set CNN Decoder Dynamic Class Centers … Mask Pooling Decoder A One class at a time All classes at a time A TopK Pred Heads A Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' (a) Current RPN-based dual-branch framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' (b) Our proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Our method fully utilizes the support set on both feature and query levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' autonomous driving, satellite map navigation, medical im- age analysis, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' The community has seen tremendous suc- cesses in this direction by designing models for a set of pre- defined classes [3,9,10,20,25,33,33,41,55,61,72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' How- ever, it is hard to enlarge these methods when deploying for real applications, since they are data-hungry and need huge extra mask annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Inspired by the ability of humans to learn with minimal data, Few-Shot Learning (FSL) has received lots of attention recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' With abundant labeled data of base classes, FSL aims at learning and predicting novel classes in the given input data (called query set / im- ages) with only a few labeled exemplars (called support set / images), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=', FSL learns a conditional model that performs prediction by referring to support images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' To fill up the gap of lacking instance-wised mask an- notation for novel classes, Few-Shot Instance Segmenta- tion (FSIS) is proposed [17, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Compared with Few- Shot Object Detection (FSOD) [5, 16, 22, 23, 28, 29, 31, 32, 34, 35, 39, 49, 60, 65, 71] and Few-Shot Semantic Seg- mentation (FSSS) [50, 56, 59, 75], FSIS is more challeng- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='01156v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='CV] 3 Jan 2023 ing since it requires both the detection and segmentation of novel instances with few data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Current solutions to FSIS mainly focus on designing a support branch to obtain class prototypes that are used to re-weight query branch fea- tures [13, 17, 19, 66], as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Most of the current dual-branch methods are based on this kind of two- stage RoI-based detectors, where firstly pre-training the model using base classes and then fine-tuning the model on novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Constrained by mask resolution and inadequate modeling of dual-branch interaction, current two-stage models tend to under-explore cues from support data [1,17,66,66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' One question arises: is there a stronger framework to fully leverage the support guidance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Be- sides, several works propose more advanced settings for FSIS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=', generalized FSIS (gFSIS) [16] and incremen- tal FSIS (iFSIS) [19], which require specific designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' For example, Retentive R-CNN [16] aiming at gFSIS proposes an extra Bias-Balanced RPN to debias the pre-trained RPN and learn novel classes without forgetting previous knowl- edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' iMTFA [19] and iFS-RCNN [13] elaborately design the class head to solve iFSIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Another question arises: is there a simple yet unified framework for FSIS, gFSIS, and iFSIS?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Recently, Detection Transformer [4,6,12,45,78] and cor- responding mask-based variants [9,11] have made progres- sive processes with the goal of using object queries to repre- sent each instance, which simplifies and unifies the instance segmentation pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Motivated by the above, we pro- pose a new framework based on the strong Mask2Former baseline to solve FSIS, gFSIS, and iFSIS in one unified framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We carefully examine the training process of Mask2Former with only base class samples and identify two critical factors that help us design a simple solution to improve novel class performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Firstly, in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 1, we find that object queries from support branch can well locate ob- jects for novel classes, even without the novel fine-tuning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We term this as support query localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Sec- ondly, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 2, we calculate attention maps between object queries from support images and object queries from query images, and find that most object queries are highly corre- lated, even for most novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We term this as support query categorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Given the two key insights, we present a simple frame- work named Reference Twice (RefT) to fully explore the support mask information and support object query infor- mation, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' RefT adopts the Meta- Learning framework with a two-stage pipeline, learning to quickly generalize to novel knowledge by referencing twice from the support branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' For the first reference, we adopt mask pooling to crop support features rather than use RoI Align [25] like previous methods [22, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Then we pro- pose to generate dynamic class centers from pooled support features to enhance the query branch simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' For the second reference, as shown by support query catego- rization and support query localization, object queries from support images already encode relevant classification and localization information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We design a simple multi-head at- tention to link both object queries forcibly, which enhances the classification and segmentation ability for novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Both reference processes are well coupled and lead to sig- nificant improvement in novel classes compared to the base- line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Furthermore, we make a simple modification to extend our framework to the iFSIS setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' To summarize, our contributions are as follows: 1) We carefully examine the Mask-based DETR framework on FSIS and identify two key factors named support query localization and support query categorization, which are important for guiding the Transformer-like framework de- sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 2) Motivated by above insights, we introduce our Reference Twice framework based on the Meta-Learning pipeline, which contains two novel steps: The first step uses mask-based dynamic prototypes for feature-level enhance- ment, while the second step provides instance-level guid- ance by linking object queries from both query and support branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 3) Benefiting from our Transformer-like frame- work, our method can be easily extended into all related settings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=', FSIS, gFSIS, and iFSIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' To the best of our knowledge, we are the first to unify three closely related few-shot instance segmentation tasks in one framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 4) Massive experiments demonstrate the superiority of our ap- proach over SoTA methods in three settings, and extensive ablation studies consistently illustrate the effectiveness of each component of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Related Work Few-Shot Classification aims to enable models to gener- alize to novel classes with fewer samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Most methods employ the N-way K-shot episodic training paradigm that helps rapid generalization by exposing itself to multiple classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Approaches to FSL can be mainly cate- gorized into optimization-based and metric-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' The for- mer [1,2,51] utilizes a meta-learner to produce parameters to learn a subtask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' MAML [18] encodes prior knowledge into a learnable initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' The latter [36, 54, 58, 67] learns a transferable embedding space where query sam- ples can be classified based on similarity to support sam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Various embedding methods and distance functions are explored, including extracting per-class prototypes with a fixed distance metric e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=', Cosine or Euclidean [52,54,58], utilizing task-adaptive embedding functions with a learned distance metric [36, 67] and attention learning [24, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In- stead of studying image-level classification task, we mainly focus on instance-level few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Few-Shot Object Detection (FSOD) aims to enlarge the vocabulary of a detector with few detection samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Sev- eral works [5, 8, 15, 16, 21–23, 28–32, 34, 35, 39, 49, 53, 60, 2 62–65,70,71,74,77] are proposed to advance this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In particular, TFA [60] proposes a simple two-phase fine- tuning approach, while DeFRCN [49] decouples the train- ing of RPN features and RoI classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' SRR-FSD [77] combines multi-modal inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' LVC [31] proposes a pipeline to enlarge novel detection examples and train a more robust detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Few-Shot Segmentation includes Few-Shot Semantic Segmentation (FSSS) and Few-Shot Instance Segmenta- tion (FSIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' FSSS requires pixel-level classification for query images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Previous works [50, 56, 59, 75] typically extract category prototypes from support images, and the query image is segmented by computing the similarity dis- tance between each prototype and query features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' FSIS aims to detect and segment each object with fewer exam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' FSIS approaches [17, 19, 46–48, 66] can be mainly divided into single-branch and dual-branch architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' The former [19, 48] mainly focuses on the design of the classification head, while the latter [17, 46, 66] introduces an additional support branch to compute class prototypes or re-weighting vectors of support images and assist the segmenter in screening out target category features via fea- ture aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Meta R-CNN [66] performs channel-wise multiplication on RoI features, while FGN [17] aggregates channel-wise features at three stages, including RPN, detec- tion head, and mask head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Pointed by recent works [31,69], RPN-based approaches tend to mistake the novel class ob- jects for the background and decrease the recall of novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Inspired by the recent FSOD methods [69], RefT builds on a DETR detector and falls into the dual-branch meta-learning-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Unlike previous methods that need to aggregate the results of multiple runs for a query image, RefT can deal with all support classes simul- taneously and efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Vision Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Current research in this field falls into two categories: one for better representation learn- ing via designing a more robust backbone or strategy [14, 42, 43, 57], and the other using object query [7, 37, 38, 68, 73, 76] to unify and simplify relevant detection and seg- mentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Our RefT is based on the recent DETR- based segmentation method, Mask2Former [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Compared with previous FSIS methods [17, 66], object queries from Mask2Former encoding rich information of both object ap- pearance and location are perfect as query features to be aggregated and support guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' RefT can be easily ex- tended to various instance-level few-shot learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Method In this section, we will first introduce preliminary knowl- edge, including the settings, training strategy, and our find- ings for support query localization and categorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Mo- tivated by these findings, we introduce our RefT framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Then we detail RefT on FSIS and its variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' bird cat dog horse sheep cow bicycle car motorcycle airplane bus train (a) Before Novel Fine-tune (b) After Novel Fine-tune 1 0 1 0 bird cat dog horse sheep cow bicycle car motorcycle airplane bus train Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Support Query Categorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We visualize the co- sine similarity of object queries of the support branch belonging to COCO 20 novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Most object queries are roughly dis- tinguishable, even without novel fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We zoom in on areas that contain highly correlated and easily misclassified classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Support Query Localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We analyze the mask quality about 20 novel classes on COCO dataset and calculate the top-k IoU between the ground truth and predicted masks of the support branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Object queries from support images can well locate object masks for novel classes, even without novel fine- tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' nIoU/ bIoU: average IoU of novel/ base class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' # Top-k queries Base Training Novel Fine-tune bIoU nIoU bIoU nIoU 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='55 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='71 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='79 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Comparison of different settings in few-shot instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' FSIS: standard few-shot instance segmentation, gFSIS: generalized few-shot instance segmentation, iFSIS: incre- mental few-shot instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Settings Fine-tune on Test on Base Novel Base Novel FSIS ✓ ✓ ✓ gFSIS ✓ ✓ ✓ ✓ iFSIS ✓ ✓ ✓ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Preliminary Knowledge Problem Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' The classes are split into two sets CBase and CNovel, where CBase ∩ CNovel = ∅, CBase ∪ CNovel = CAll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' FSIS aims to segment objects belonging to CTest in a query image after training over abundant samples of CBase and few samples of CFinetune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' RefT approaches all three settings in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='2, for FSIS, CFinetune = CBase ∪ CNovel, CTest = CNovel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' For gFSIS, CFinetune = CBase ∪ CNovel, CTest = CBase ∪ CNovel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' For 3 1st Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Cross-Attention Feed-Forward Self-Attention TopK sort by IoU 2nd Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='2) 1st Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' : Feature-Level Enhancement Vannila Mask2former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Reference Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Query Image CAL Support Set N K CAL: Class Adaptation Layer 2nd Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' : Query-Level Enhancement MSDeformaAttn … ������������������������ Mask Pooling CrossAttn KV Q Mask2Former Decoder 2nd Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' CrossAttn Class Embed Mask Embed NK* Mask Embed Compute IoU GT Masks q������������ q������������ MSDeformaAttn GT Masks Multi-Scale Features ������������������������ ������������������������ ������������������������ ′ Enhanced Query Features Cross-Attention Q V K Dynamic Class Centers Mask Pooling Sigmoid Channel-wise Multiplication 1st Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='2) (N,D) (N,K,ΣHW,D) ������������������������ Query Branch Support Branch ������������������������ ������������ ������������������������ NK* Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Architecture of the proposed Reference Twice (RefT) for FSIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' The query branch refers to the support branch twice on the feature and query level, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 1st Reference for feature-level enhancement performs simultaneous aggregation between the query features and all dynamic class prototypes obtained through mask pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 2nd Reference for query-level enhancement links object queries from the query and support branch with a simple multi-head attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' iFSIS, CFinetune = CNovel, CTest = CBase ∪ CNovel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Our Training Strategy and Baseline Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We adopt the episodic-training [66] from FSOD in both base train- ing and novel fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' The training stage comprises a series of episodes Ei = (Ii Q, Si), where i indicates the ith episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Given a query image Ii Q, all objects present in the image belong to N classes in Ctrain and a support image set Si containing K samples per class along with structural annotations will be provided as additional input, which makes the N-way K-shot episode Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In particular, we adopt Mask2Former as the segmenter [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Mask2Former uses object query to treat instance segmentation as mask classification and segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Support Query Localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' After the first base train- ing stage, we directly infer support images of novel classes, where we test whether only the base-trained model can re- call novel classes with additional structural input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 1, we find that base-trained model can already detect and segment novel class objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Most queries have the lo- calization ability of novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' After novel fine-tuning RefT, we observe improvements in more queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Support Query Categorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We further visualize the correlation maps among the support queries in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 2, where most of these novel classes are highly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We present more examples in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We call this support query categorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' This indicates that the support queries have the clustering effect even without novel fine-tuning, which may be helpful for query image branch training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' After our framework, as shown on the right side of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 2, the categorization is more prominent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Reference Twice Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Given the facts found in previous parts, we aim to enable the model to fully explore the cues from the support branch inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Unlike previous works [13] that only leverage the feature-level enhancement, we apply a novel enhancement on the query level, as the support queries can encode both the localization and categorization information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We take a query image IQ and a support set S with N × K examples as input, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' For the support branch, we mask out the support objects and drop the background together with other objects in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In this way, the support features and object queries are more discriminative and contain more accurate location informa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' A weight-shared feature extractor first encodes the query and support images into the same feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Sub- sequently, the 1st Reference Aggregation module performs simultaneous aggregation between the query and support features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In this step, the query features are coarsely filtered by support categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Then the selected query features and support features are sent to the class-agnostic pixel decoder and transformer decoder to obtain object queries from both query and support branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Next, we link object queries of both branches for better calibration via cross-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' As we mask the support images with the ground truth mask, the 4 obtained support object queries correspond precisely to the instances of the support categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 1st Reference for Feature-Level Enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We de- scribe the 1st Reference Aggregation module in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' As shown in the bottom left of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 3, given the multi-scale query features F Q= � xl Q �L l=1 and support features F nk S = {xl S} L l=1 (n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' , N, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' , K), where L de- notes the feature levels, a weight-shared multi-head de- formable attention [78] first encodes them into the same feature space, obtaining F Q and F S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' The exact features of support instances are separated from the background and other instances through mask pooling with ground truth masks on the support features of each scale, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Then the dynamic class centers for all support classes are obtained by averaging all scales per image and K examples per class, given by: cn = 1 KL K � k=1 L � l=1 MaskPool � F nk S � , n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' (1) After that, a multi-head attention module is used to generate the reweighting matrix for aggregating F ′ Q with dynamic class centers P =[c1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=',cN]∈RN×C: R = softmax � F ′ QP T � σ(P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' (2) Here the linear projection is omitted for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' The query features are then multiplied with the obtained weights along the channel dimension as below: F Enhanced Q = F ′ Q ⊗ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' (3) Thus the category-related query features are selected and enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In this operation, the query branch is enhanced by support examples dynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We present the detailed design of choices of support features in the experiment part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 2nd Reference for Query-Level Enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Inspired by previous findings, we add extra query-level enhancement during the episodic training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We first drop the support ob- ject queries with irrelevant information and only keep ones that contain instance-level category and spatial information of high quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' The detailed process is shown in the bot- tom right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Specifically, given query object queries qQ ∈ RQ×Dand support object queries qS ∈ RNKQ×D, Q is the number of object queries per image, and D is the feature dimension, we first obtain the predicted masks cor- responding to the support object queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Then, the top k out of the Q object queries per image is selected accord- ing to the IoU of the predicted and ground truth masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' As proved by previous findings, we add extra query-level en- hancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In this way, we avoid huge computation costs and also obtain better relevant cues to improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Then we use a multi-head cross-attention module to match the query and support object queries: qEnhanced Q = softmax � qQ TopK(qS)T� TopK(qS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' (4) Then a multi-head self-attention module followed by one feed-forward layer is used to adapt to the following pre- diction heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Such naive attention is good enough to link support queries to the object queries from query images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Generalization and Training Details FSIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' The same losses in Mask2Former for classification and segmentation are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Besides, we classify the dy- namic class centers in the first reference using a cosine sim- ilarity cross-entropy loss, following [66, 69], to encourage centers to fall into the corresponding categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' gFSIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' The common practice in prior RPN-based works is to freeze most parameters and fine-tune the class head on a balanced sampled dataset of base and novel classes to retain base class knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' However, as observed in [13], the DETR-like detector can barely generalize to novel classes with the input projection layer frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We figure the layer is class-specific because it transforms the channel dimen- sion with one convolution, and channel features are essen- tial in distinguishing different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' To point out the sig- nificance of the layer in novel class learning, we term it as Class Adaptation Layer, abbreviated to CAL, also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We alleviate the catastrophic forgetting prob- lem by simply fine-tuning CAL, object queries and the class head with all other parameters frozen and without additional modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Losses remain the same as in FSIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' iFSIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' As we unfreeze an additional CAL layer, directly fine-tuning on very constrained novel class samples will in- evitably lead to severe overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' To alleviate this prob- lem, Incremental-DETR [13] utilizes knowledge distilla- tion on both CAL and the class head and fine-tunes in a self-supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In contrast, to unify iFSIS and the other two settings in RefT, we only focus on CAL and adapt the base knowledge distillation loss (BKD loss) in [13] to a class-enhanced version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Considering the role of CAL in class adaptation, more prominent class features are ex- pected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' To this end, according to their relative importance, we re-weight the features along spatial and channel dimen- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' The degree of importance is measured by the abso- lute mean value of the pixel or channel, and the softmax activation is used as normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' The channel and spatial re-weighting coefficients RC k and RS k are obtained from: RC k = C · softmax( 1 HW H � i=1 W � j=1 ���F B k,i,j ���), (5) RS ij = HW · softmax( 1 C C � i=1 ���F B k,i,j ���).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' (6) Therefore, our Class-Enhanced Base Knowledge Distilla- tion Loss is formulated as: LCE−BKD = C � k=1 H � i=1 W � j=1 RC kRS i,j(1−Mi,j) � F B k,i,j −F N k,i,j �2 , (7) C, H, and W denote the channel, height, and width of the feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' M ij is the ground truth masks of the foreground 5 CNN Base class truck Novel class person Class-Enhanced BKD Loss Support sets Novel Fine-tune CNN CNN CNN Freeze Reference Twice CAL CAL Class Head Mask Head Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Adaptation to iFSIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In novel fine-tuning, all parame- ters except CAL, object queries, and the class head are frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' The proposed Class-Enhanced BKD Loss is adopted to CAL to prevent overfitting, without hindering novel class generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' instances, which all belong to the novel classes during fine- tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' F B k,i,j is the CAL output feature of the model frozen after base training, and F N k,i,j is the CAL output feature of the model during novel fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Training and Inference Procedure 1) Base Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We perform episodic base training on our proposed RefT over base classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 2) Novel Fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We perform episodic novel fine- tuning over a sampled N-way K-shot dataset with classes CFinetune, following [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 3) Inference with all support classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' During inference time, we compute dynamic class centers and object queries from support sets once and for all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Unlike previous works that require multiple forward passes for each query image, RefT only forwards once with all support classes, which is simpler and more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Experiment 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Experimental Setup Dataset Setting and Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We follow the data setups first established in FSOD and then extended to FSIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We evaluate on the MS-COCO 2014 dataset [40] modified by [60], where the original 80k train and 35k validation images are combined as the actual training set and the 5k minival set is used for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' The 80 classes are divided into 2 sets, in- cluding 20 novel classes that intersect with PASCAL VOC and the remaining 60 base classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Shots per class are set to K= {1, 5, 10, and 30}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We adopt standard MS-COCO met- rics, namely Average Precision (IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='5 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='95), on novel and base classes, abbreviated to nAP and bAP, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' As in [19], we run all tests 10 times with K examples of 10 seeds for each class and report the averaged results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We take Mask2Former [9] as the main architecture and ResNet-50 [26] is adopted as the backbone following prior work in FSIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In the base train- ing stage, we episodically train our model over COCO base classes for 50 epochs with a batch size of 8 on 8 V-100 GPUs, using the AdamW optimizer [44] and the step learn- ing rate schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We set the initial learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='0001 and a weight decay at the last epoch by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In the novel fine-tuning stage, the settings remain the same until con- vergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' For fair comparison, we implement a fine-tune based Mask2Former for FSIS similar to TFA [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We per- form standard training on Mask2Former with default setting over COCO base classes and fine-tune the model over novel classes until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' FSOD and FSIS results (nAP) on COCO with K = {5, 10, 30}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' “-”: unavailable corresponding result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Optimal and subopti- mal results are highlighted in bold and underline, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We use ResNet-50 as backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Object Detection Instance Segmentation Methods 5 10 30 5 10 30 Mask2Former+ft-full 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='3 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='9 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='7 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='7 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='8 Meta-DETR [69] 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='4 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='00 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='1 MRCN+ft-full [26] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='9 Meta R-CNN [66] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='4 iMTFA [19] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='4 iFS-RCNN [48] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='3 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='7 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='1 RefT (Ours) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='3 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='4 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='5 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' gFSOD and gFSIS results on COCO with K = {1, 5, 10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' “-”: unavailable corresponding result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Optimal and suboptimal results are highlighted in bold and underline, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Object Detection nAP bAP Backbone Methods 1 5 10 1 5 10 iMTFA [19] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='3 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='7 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='1 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='0 LVC [31] 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='6 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='7 R-50 RefT (Ours) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='1 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='7 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='8 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='2 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='7 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='2 6 personTable 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' iFSOD and iFSIS results on COCO with K = {1, 5, 10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' “-”: unavailable corresponding result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Optimal and suboptimal results are highlighted in bold and underline, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We use ResNet-50 as backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Object Detection nAP bAP Methods 1 5 10 1 5 10 Incremental-DETR [13] 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='4 27.' metadata={'source': 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iMTFA [19] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='9 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='9 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='9 iFS-RCNN [48] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='1 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='4 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='3 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='3 RefT (Ours) w/ Cos 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='8 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='1 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='0 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='3 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Main Results on COCO FSIS Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 3, it can be seen that our method easily outperforms previous works based on Mask R-CNN by a wide margin, which is as expected because we use a more robust base model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' However, even compared with the same DETR-based approach Meta-DETR, which achieves the state-of-the-art results in FSOD, we obtain significantly better results on FSIS and comparable or even slightly im- proved results on FSOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' For more fair comparison, we also compare with the fully fine-tuned Mask2Former and still show obvious performance gains, which implies that RefT fully leverage the image and instance level information pro- vided by the support branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' gFSIS Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' There is a trade-off between the results of the base and novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Few works attend to the more challenging gFSIS that requires avoiding forgetting on base classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 4 shows that RefT consistently outperforms recent SoTAs in both gFSOD and gFSIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In addition, using more powerful Transformer backbone models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Swin- T, Swin-S, Swin-B, our approach still generalizes well and does not fall into the severe overfitting of novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' iFSIS Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='5, we first adapt Mask2Former to the iFSIS setting as our baseline by replacing the fully- connected classifier with a cosine similarity classifier as in iMTFA [19] and fine-tune both CAL and the class head with all other parameters frozen after base training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' A prob- lem arises here that the class-specific CAL needs to be fine-tuned to allow novel class learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' However, with- out access to base class samples, CAL will quickly over- fit and suffer from catastrophic forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Adding our Class Enhanced Base Knowledge Distillation Loss will eas- ily address this issue and achieves comparable results to the recent SoTAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Similar loss has also been proposed in Incremental-DETR [13], but our method is simpler yet equally effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Detailed ablation of different losses and classifiers are presented in supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Ablation Study and Analysis Ablation study on each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 6a, we first ablate on the effectiveness of each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We take a fully fine-tuned single-branch Mask2Former (abbreviated to M2F-ft) as our baseline for fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Adding our query-based 2nd reference module yields 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='6 improvements on novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Adding our image-based 1st reference module further improves nAP by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Note that although longer training will bring more performance gains on nAP at the cost of a quick drop in bAP, we are not trading bAP for nAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' This verifies our findings in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Both branches share coupled effect for novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Support features in 2nd Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' To demonstrate the effective- ness of our query-based aggregation in the 2nd reference, we also implement an image-based version using the same dynamic class centers in the 1st reference as the support guidance and the cross attention as the aggregation opera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 6b shows that query-level enhancement module yields a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='1 nAP boost over the feature-level one, which proves the superiority of our RefT Framework over the pure feature-level enhancement framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Effect of fine-tuning CAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' To demonstrate the role of CAL in allowing DETR-like models to learn novel classes, we freeze CAL in the novel fine-tuning stage and only fine-tune the object queries and the class head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 6c shows that the model can barely generalize with CAL frozen, leading to a huge gap with the model that fine-tunes CAL (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='6 vs 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='4 nAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Additionally, the result indicates that we do not need extra learnable parameters to guarantee sufficient ability to transfer to novel domain and fine-tuning an additional CAL can already achieve both novel class generalization and base knowledge retainment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Support features in 1st Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 6d, we present ab- lation on different features that are used to compute dy- namic class centers, including three stages of features from ResNet-50 (denoted as Res3, Res4 and Res5) and the flat- tened multi-scale features from the first transformer encoder layer (denoted as Enc1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We find that using support features of larger scale brings significant performance boost (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='2 vs 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='4nAP) might because more small objects are noticed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Using multi-scale features further gains 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='2 point in nAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Pooling in 1st Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 6e, we compare different pool- ing methods to extract relevant information from support images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' It can be observed that with the improvement of the accuracy of the feature region, the result increases corre- spondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' This is because more accurate instance features without noise provide more distinguishing class centers for the query branch classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Query selection in 2nd Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 6f, we compare the results of selecting top 10 object queries in 2nd reference sorted by scores or mask IoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' The results are comparable when k is small because object queries are more accurate both for classification and segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 7 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Ablation study on COCO with K=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' All experiments use ResNet-50 as backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' (a) Ablation study on each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Baseline +1st Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' +2nd Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' nAP bAP M2F-ft 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='7 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='0 � 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='3 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='5 � � 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='4 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='0 (b) Support features in 2nd Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Features nAP bAP Feature-level 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='3 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='7 Query-level 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='4 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='0 (c) Effect of fine-tuning CAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Fine-tune nAP bAP � 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='6 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='4 � 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='4 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='0 (d) Support features in 1st Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Method nAP bAP Res3 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='2 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='6 Res4 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='0 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='0 Res5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='4 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='2 Enc1 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='4 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='0 (e) Pooling in 1st Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Method nAP bAP GAP 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='0 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='4 RoiAlign 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='2 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='3 MaskPool 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='4 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='0 (f) Query selection in 2nd Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Sort by nAP bAP Score 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='2 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='8 Mask 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='4 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='0 (g) Number of object queries in 2nd Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' # Queries 3 10 50 100 nAP 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='4 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='4 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='8 bAP 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='8 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='0 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='8 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='5 Before After Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' t-SNE visualization of object queries belonging to COCO 20 novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' The results are obtained from the support branch before and after novel fine-tuning, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Baseline + 2nd Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' + 1st Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' cat 11% dog 36% dog 36% dog 36% bear 12% dog 36% truck 16% dog 36% bear 14% dog 36% truck 16% dog 36% cat 27% dog 36% truck 17% dog 36% cat 55% dog 36% bird 11% dog 36% horse 11% dog 36% cow 11% dog 36% bird 70% dog 36% bird 45% dog 36% train 19% dog 36% train 40% dog 36% carrot 51% dog 36% carrot 78% car carrot 98% dog 36% carrot 78% car dog 36% traffic light 89% traffic light 70% traffic light 97% Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Visualization of 10-shot FSIS results on COCO mini- val set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Predictions for novel class instances are mainly displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Number of object queries in 2nd Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We select the top- k object queries for each support image in 2nd reference In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 6g, we compare how different numbers of k affect the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' It shows that when k ≤ 50, a relatively stable and close to the best result can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' This is consistent with what Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 1 has shown because the support branch mask quality can be guaranteed when k ≤ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Visualization and More Analysis Understanding the effect of 2nd Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 5, we visual- ize the t-SNE results of object queries belonging to COCO 20 novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We find that the object queries of novel classes can be roughly distinguished, even without the novel fine-tuning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' And the clustering is more obvious af- ter fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We leverage this support query categoriza- tion in our 2nd reference module to provide guidance for the query branch classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' More Qualitative Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 6, we present sev- eral visual results on MS-COCO datasets corresponding to Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We find that adding our query-based 2nd Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' effectively reduces both missed and misclassified results, which is mostly attributed to the guidance of support query localization and support query categorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' With the alignment between query features and dynamic class cen- ters, adding our 1st Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' further reduces misclassification between classes that are highly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' More results can be found in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Parameter and GFLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Compared with strong Mask2Former baseline, RefT significantly boosts nAP by +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='7↑ with only increasing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='7% GFLOPs and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='3% param- eters given 1,024 × 1,024 image input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Limitation and Future Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' One limitation of RefT is that the model is not able to perform well in one-shot set- ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Our future goal goes to provide more design for our framework to fix this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Conclusion In this paper, we present a simple and unified base- line for few-shot instance segmentation, namely Reference Twice (RefT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' We carefully examine the mask-based DETR framework in FSIS and identify two key factors named sup- port query localization and support query categorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Motivated by these two factors, we propose to reference the model twice on feature level and query level, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Specifically, we first design a mask-based dynamic weight- 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content='bird11%horse11% CoW11%dog36% carrot78% carrot32%truck19% cat77%bird70% ird45bear12% carrot51%truck17% bear55%bear14%%trafficlight84 RyanTaylortrafficlight209 RyanTaylortrafficlight89% train40% RyanTayloring module to enhance support features and then propose to link object queries for better calibration via cross-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Additionally, RefT can be easily extended to all three set- tings including FSIS, gFSIS and iFSIS with minor modifi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Despite its simplicity, RefT achieves state-of-the- art or second-best performance on MS-COCO benchmarks, across all three settings and all shots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' References [1] Sungyong Baik, Myungsub Choi, Janghoon Choi, Heewon Kim, and Kyoung Mu Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Meta-learning with adaptive hy- perparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' NeurIPS, 33:20755–20765, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 2 [2] Sungyong Baik, Seokil Hong, and Kyoung Mu Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Learn- ing to forget for meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In CVPR, pages 2379–2387, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 2 [3] Daniel Bolya, Chong Zhou, Fanyi Xiao, and Yong Jae Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Yolact: Real-time instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' In ICCV, pages 9157–9166, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 1 [4] Xipeng Cao, Peng Yuan, Bailan Feng, and Kun Niu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Cf-detr: Coarse-to-fine transformers for end-to-end object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 2 [5] Yuhang Cao, Jiaqi Wang, Ying Jin, Tong Wu, Kai Chen, Ziwei Liu, and Dahua Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' Few-shot object detection via association and discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' NeurIPS, 34:16570–16581, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfN_sI/content/2301.01156v1.pdf'} +page_content=' 1, 2 [6] Nicolas Carion, Francisco Massa, Gabriel Synnaeve, 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b/odFQT4oBgHgl3EQfrDYd/content/tmp_files/2301.13382v1.pdf.txt @@ -0,0 +1,2186 @@ +Numeracy from Literacy: Data Science as an +Emergent Skill from Large Language Models + +David Noever1 and Forrest McKee2 +PeopleTec, 4901-D Corporate Drive, Huntsville, AL, USA, 35805 +1david.noever@peopletec.com 2 forrest.mckee@peopletec.com + + +Abstract +Large language models (LLM) such as OpenAI's ChatGPT and GPT-3 offer unique testbeds for exploring +the translation challenges of turning literacy into numeracy. Previous publicly-available transformer +models from eighteen months prior and 1000 times smaller failed to provide basic arithmetic. The statistical +analysis of four complex datasets described here combines arithmetic manipulations that cannot be +memorized or encoded by simple rules. The work examines whether next-token prediction succeeds from +sentence completion into the realm of actual numerical understanding. For example, the work highlights +cases for descriptive statistics on in-memory datasets that the LLM initially loads from memory or generates +randomly using python libraries. The resulting exploratory data analysis showcases the model's +capabilities to group by or pivot categorical sums, infer feature importance, derive correlations, and +predict unseen test cases using linear regression. To extend the model's testable range, the research deletes +and appends random rows such that recall alone cannot explain emergent numeracy. +Keywords: +Exploratory Data Analysis, Transformers, Text Generation, Generative Pre-trained Transformers, GPT + +1. INTRODUCTION +Three promising and challenging AI technologies are benchmarks for community research progress: +autonomous driving, personal assistant, and chatbots [1]. OpenAI's ChatGPT combined the personal +assistant with a chat interface in their late November 2022 public release [2-8]. Prompt or conversational +customization of the ChatGPT API reveals the depth of its encyclopedic knowledge [7], somewhat akin to +an effective Google advanced search or dynamically created Wikipedia entry [9]. +Previous researchers have noted emergent features [10-19] beyond what a search engine, spidering indexer, +or community-sourced compilation like Wikipedia might answer complex questions. This paper proposes +several tasks that require ChatGPT to reason [10,20]. While traditional challenge problems presented to +large language models like "2+2=" have previously not satisfied any reasoning tests [21], the latest +generation seems to display what the AI community might categorize as emergent properties [15-17]. For +instance, previous work highlighted ChatGPT's capability to mimic complex computer operating systems +as if a hacker interacted with text commands [22-24]. As an API interface, ChatGPT could serve as a +dynamic honeypot with realistic responses [23]. +The present work extends this "out-of-the-box" simulation capability to role-play the data scientist or +knowledge assistant as they perform exploratory data analysis [15]. ChatGPT's latest release (19JAN2023) +incorporates a basic understanding of benchmark machine learning datasets like iris [25-26], Titanic +survival [27-28], and Boston housing [29] without explicit programming. Some critical tests of the LLM's +reasoning or knowledge [30-35] include random re-sampling of available datasets and de novo generation +from scratch. + +The present work examines whether ChatGPT possesses built-in knowledge of classic data science case +studies like iris [25-26], Boston housing [29], and Titanic [27-28]. Without the built-in capability to load +data, the large language models simulate user interactions [22-24], including coded Python that edits the +datasets and removes memorized responses from the model's responses. Once the modified data receives +prompts and queries, the LLM delivers emergent answers [15-17] based on its capability to perform +arithmetic calculations or generate display code. Finally, each case presents word problem categories, such +as "what demographic group most likely did not survive the Titanic crash? [27-28]". +The paper presents a systematic version of exploratory data analysis (EDA) with linguistics models. The +models generate python code to execute in a Jupyter notebook or answer questions related to identifying +correlations, trends, outliers, and missing values as one might anticipate as typical data science pre- +processing routines or follow-up summaries based on post-processing results [37] (Appendices A-E). The +goal is to identify how well an LLM adapts to previously unseen datasets and generates plausible hints for +extending the EDA [38]. Where possible, we generalize the results to either synthetic data that could not +appear in the training data or to data slices that offer unique challenges from the well-known cases [36]. +For example, by adding or subtracting random rows of well-known datasets, we force the LLM to +demonstrate whether it can creatively tailor its responses to prompts in ways that differ from a simple search +engine reply [37]. +2. METHODS +We organize the paper around the exploratory data analysis shown in Appendices A through E, as +summarized in Table 1. Appendix A establishes basic numeracy skills, including date-time manipulation +and word problems. Appendices B-D examine three tabular datasets where ChatGPT includes the data +frame as part of its training data but receives customizations that make it impossible for the LLM to recall +what previous steps or outcomes might apply. For instance, we use a random train-test split to the Titanic +dataset to force the LLM to describe through its emergent statistical skills rather than return the standard +answers that internet tutorials present. +One motivation for this approach stems from the failure of earlier LLMs to perform basic arithmetic based +on the next token prediction methods. Appendix F adds a further test of ChatGPT's data science skills by +creating a randomized insurance claim dataset valid for the session only and would not appear in any +historical training from the internet archive. +Appendix and Title +Data Science Goal +Data Set Construction +A. Basic Statistics +and Numeracy +Arithmetic, Date-Time Manipulation, +Unit Conversions, Word Problems, +Approximations +Prompt-driven single values +B. ChatGPT Iris +Dataset +Interactions +Descriptive statistics, missing and +duplicate value identification, variable +correlations, factor analysis and feature +importance, plot code generation using +libraries (seaborn, plotly), outlier +identification, dataset augmentation, +IRIS dataset, Petal and Sepal +Width, and Length for Species +Identification (Some knowledge +already embedded) +C: ChatGPT Titanic +Dataset Interactions +Descriptive statistics, data frame +operations such as drop columns, missing +values, composite column creation, +python function generation and execution +in place, random test-train split, feature +importance, pivot tables, and factor +summation +Titanic Survival Dataset based +on passenger list demographics +(Some knowledge embedded but +modify data frame, so not +memorized) + +Appendix and Title +Data Science Goal +Data Set Construction +D: Boston Housing +Dataset +Descriptive statistics, histogram and +correlation analysis, code generation, +model training preparation for low +feature importance, machine learning +with linear regression models, predict on +unseen test cases from the trained model, +recommend purchase values as word +problems +Boston Housing Data (tabular +mixture of categorical and +numerical data for home value +prediction based on +demographics) +E: ChatGPT Faker +Dataset Interactions +Mock dataset generation, append column +values, calculate descriptive statistics on +numerical and ignore categorical +variables +Randomly generated insurance +claim dataset with anonymized +entries + +3. RESULTS +For all five tests, the main result supports the hypothesis that the latest LLMs have reached sufficient scale +to handle complex statistical questions. As proposed by the builders of GPT-3, these models offer public +access to "zero-shot" or "few-shot" learning capabilities when scaled to sufficient parametric size. The +model encodes enough general knowledge to answer mathematical questions with plausible answers, even +when presented with only a few (or no) examples of formatted requirements or context. +Because ChatGPT provides memory within a given session to at least 8,000 tokens (25 pages), the model's +coherence and relevance present new inquiries in data science. One might call this quality "emergent" +because no rules or computational layers are explicitly defined. The following sections outline the general +characteristics of the four datasets presented (Iris, Titanic, Boston Housing, synthetic) along with a chain +of statistical calculations selected to highlight date-time manipulations, approximations, and word +problems. It is worth noting that ChatGPT provides self-contained datasets to test, which proves critical to +complete any analysis. As an LLM frozen in time (2021) without any buffer or storage, the traditional steps +needed to upload or present data fail. But having encountered the three well-known examples and one +synthetic one, the model keeps track of each manipulation such that if a data row disappears, the resulting +median or count changes accordingly. + +3.1. Descriptive Statistics +As illustrated in Appendix A, the model can add large numbers, reduce answers to N significant digits, +identify divisors, and perform an order of magnitude calculation with unit conversions. When asked for the +day of the week from history, the model correctly identifies the day from 60 years prior. While not +remarkable from a lookup table or internet search, the model only generates the correct result using next- +token prediction and language training. +To highlight the model's capacity for manipulating extensive, multi-stage calculations, we prompt for the +number of minutes in a decade, the number of inches between the Eiffel Tower and London Bridge, and +the number of people who could fit on the island of Manhattan. ChatGPT answers incorrectly to identify +the time zone that corresponds to six hours ahead of US Eastern (EST) (False: Greenwich GMT+6 or +Bangladesh). When instructed that the model responded incorrectly, ChatGPT shows a Universal Time +formula UTC-5 as EST, followed by UTC-5+6=UTC+1, or Central European Time (CET). + +ChatGPT's capabilities to self-correct provide a novel user interface for redefining a precise question-and- +answer sequence. For example, asking the model to do distance calculations between two cities in small +units like inches seems to raise the need for further explanation: What's the point of knowing urban-scale +dimensions in such small increments? When pressed in follow-up inquiries, the response showcases the +conversion of units (miles to inches) but begins with an incorrect distance (3,500 miles rather than 212 +miles). While the math is correct, the more specific initial conditions are flawed. +When asked a more eccentric estimation problem (the number of people standing shoulder to shoulder who +could fit in Manhattan a densely packed single layer), ChatGPT responds with the correct initial condition +for the area calculation (22.96 square miles). If a person requires 2 square feet, the model fails to convert +square miles to feet (ChatGPT: 8.9 million people vs. 318 million people in a 636 million sq foot area). The +model qualifies its answer as a safety, logistical, and health problem based on crowd control, then further +amends its calculation to exclude parks, buildings, or non-built-up areas. +As noted previously, ChatGPT has access to structured and organized datasets. LLMs can perform the four +basic software operations expected for databases: Create, Read, Update, and Delete (CRUD). In Appendix +B, the iris dataset describes the classification challenge to identify one of three flower species by its distinct +petal and sepal dimensions. For this multi-class clustering, the model answers that there are no duplicates +or missing values for the 50 examples of each class. +When prompted to mimic a python interpreter (as a Jupyter notebook), the model responds with the +expected output given a prompt in code alone. For example, using "data.corr()" as the prompt produces the +correct python output for the iris data. We prompt the model to produce graphical code given a desired +figure output (such as histograms, heatmaps, boxplots, pair plots, scatter, and distribution plots). Rather +than a language-only model producing the requested figures directly, ChatGPT responds with the python +libraries (plotly, seaborn, matplotlib) and codes, which run in a separate interpreter to give the graphs shown +in Appendices B-D. When asked for interpretations based on the exploratory charts, the model responds +with a description, such as a box-and-whiskers plot showing the quartiles and statistical outliers. ChatGPT +does not limit its response to general code commentary for box plots but identifies the given dataset's +variables and highlights conclusions for each class. While GitHub or internet tutorials might support +ChatGPT training for this EDA, we alter the expected output by adding or deleting data frame rows to avoid +the memorized response. This way, the emergent capabilities for performing statistical inference get +isolated from the baseline training inputs. +3.2. Coding and Plots +Appendices B-E focus on the four data science tasks to exercise ChatGPT's capabilities for python code +generation. Because the LLM offers no graphical output, the problem set transforms from the previous tasks +to coding solutions to test using Jupyter. Both Codex and copilot have offered coding assistance since +August 2021. In Appendix B, ChatGPT shows the output of exploratory data analysis as displayed by +python code for outliers, histograms, and distribution plots for the iris dataset. +In Appendix C, we ask the LLM to modify the Titanic dataset in preparation for categorical analysis and +survivorship demographics. The raw data (891 rows x 12 columns) offers irrelevant predictive variables +("PassengerID"), which we drop, then let ChatGPT pick up with the finer manipulation of the modified +data. The sequence of steps matter along the path to generating a final working dataset ready for machine +learning. In the prompt, for instance, one can define python functions that recode the embarkation points, +ticket prices, and passenger class with mappings from symbols to full names. One further can bin age into +five maturity categories between infant and elderly and distribute the passenger ages into ten-year brackets. +A further partition transforms the gender and marital status into categoricals. Once ChatGPT gets the python + +functions, the running of dataset modifications provides an in-memory style of output for further analysis. +It is worth noting that these steps illustrate how a language model serves as a computational interface to +perform complex statistical actions, pivot groupings, and train-test splits that could not appear in the model's +original corpus. Once the unique Titanic data is created and plotted, ChatGPT can answer demographic +questions about survivorship: third-class male passengers proved least likely to live through the crash and +rescue. +In Appendix D, we perform essential machine learning (ML) steps that drop highly correlated variables and +split the Boston housing data into train and test sets. We applied linear regression models from sci-kit learn +python libraries and asked for root mean square error (RMSE) results. The initial prompts without context +led to coding suggestions but refused to perform calculations on an arbitrary train-test split. However, when +prompted to act as a Jupyter notebook, the code output renders actual numerical RMSE and correlation +coefficients (R-squared) values. We created an example row to test the model and asked for a linear model +prediction. A series of word problems round out the Appendix E example, such that based on the data, the +model highlights low-crime areas or numbers of rooms. In a plausible real estate setting, the LLM answers +with a data-driven response that a combination of many rooms and the lowest price might satisfy a buyer. +To our knowledge, this output seems unique to this scale of LLM in public access, both as a data science +platform but also as capable of performing as an ML algorithm and predict on unseen inputs. It is worth +noting that previous models from the last few years, like GPT-2, failed on simple addition questions. +3.3. Emergent Understanding +Appendix E establishes that a randomized dataset was created using the python library Faker to synthesize +an anonymous insurance claim dataset that could not be repeated in previous LLM inputs. This library +makes categorical and numerical variables to include names, addresses, companies, claim reasons, and +claim confidentiality levels. For the final mock dataset created, 200 rows and nine columns make up the in- +memory capability of ChatGPT. When asked to reason over the nine columns, the LLM recognizes that 6 +or 8 variables are categorical and that for the remaining two numerical categories, a median value emerges +as the (randomized) claim amount of $1498.5. This number appears differently every time the conversation +commences, such that the net amount sums the medical, travel, phone, and unknown reasons are segmented. +The minimum possible value in this example would equal one, and the maximum (medical) claim would +equal 2300. While this sample of 200 values over many trials should converge to approximately 1650, the +resulting language model performs a reasonable approximation in building the anonymized dataset for +insurance claim values. A current search engine (Google) value for: "Give me a random value between 1 +and 2300" yields a link tree that samples 11.8 million examples on the internet but does not answer the +arithmetic question specifically. The referral engine links to calculators. + +4. DISCUSSION +The present work selects these demonstrations to illustrate the data science capabilities of large language +models. The result extends previous efforts that highlight the computational capacity of language as an +expressive and generative transformer. There exist few obvious precursors to evolving numeracy from +literacy. As recently as 18 months prior, the most advanced linguistic models could not perform elementary +addition. One consequence of the emergent skill that exceeds the expectations of a python coder would +include the comprehensive explanation of word problem challenges. So not only does ChatGPT produce +code from surveying Github, but it also reaches a natural (and relatively safe) conclusion based on the +output of running sample code. While previous work has demonstrated this "fake storefront" or "Hollywood +stage" effect in ChatGPT when assuming different operating systems, honeypots, or characters in a play, +the role of data scientist provides a novel representation to evolve exploratory analysis. In the classic triad +of iris, Titanic, and Boston housing, the work demonstrates that standard operations like pivoting, statistical + +observation, and anomaly detection suggest legitimate linguistic operations to supplement arithmetic +understanding. Like young children, the LLM has some capacity for reasoning across symbolic abstraction +(numeracy) and linguistic interpretation (literacy). An obvious extension of this work would combine the +symbolic and literate to translate word problems in multiple languages with complex alphabets like Chinese, +Cyrillic, or Arabic. In this way, one might imagine the union of symbolic AI with its more brute-force +cousin as a trained transformer capable of compressing and representing the sum of human knowledge into +"next token" predictions. + +5. CONCLUSIONS +In conclusion, the present work demonstrates large language models like ChatGPT carry a built-in capacity +for performing numerical work using a basic linguistic representation and (attention-based) weights across +a vast (40TB) dataset of human knowledge. Presumably, no single branch of its 175 billion parameters +encodes a given dataset like Titanic or Boston housing, but even without the capability to upload the data, +the model knows and illustrates complex manipulations. If presented with 8000 tokens (around 25 pages) +of a novel dataset, one can presume that ad hoc and de novo data science becomes possible within an +otherwise numerically challenged token-centric model by appending it as a data frame. The work surveys +basic and advanced operations, including CRUD, which makes a dataset otherwise impossible to memorize +but amenable to a linguistics model that can summarize and coherently alter what it stores in memory. +While ChatGPT set out to demonstrate the first chat interface that could survive both "safely and naturally" +in the wild, what the scale of its operation may eventually reveal is emergent qualities that either are too +complex for human traceability and validation or that survive in some over-fit quality from a few key +transformer branches in the maze of internet databases for training. 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Piloting Copilot and +Codex: Hot Temperature, Cold Prompts, or Black Magic?. arXiv preprint arXiv:2210.14699. +[45] +Wong, D., Kothig, A., & Lam, P. (2022). Exploring the Verifiability of Code Generated by +GitHub Copilot. arXiv preprint arXiv:2209.01766. Exploring systematic consumption in US. + +Authors + +David Noever has research experience with NASA and the Department of Defense in +machine learning and data mining. He received his BS from Princeton University and his +Ph.D. from Oxford University, as a Rhodes Scholar, in theoretical physics. + + +Forrest McKee has AI research experience with the Department of Defense in object +detection and reinforcement learning. He received his Bachelor's (BS) and Master's (MSE) +from the University of Alabama, Huntsville, Engineering. + + + + + + + + +APPENDIX A: Basic Statistics and Numeracy +Prompt: What is the square root if 23 to 5 significant digits? +ChatGPT: The square root of 23 to 5 significant digits is 4.8179. +Prompt: What is 23456+65432 +ChatGPT: The result of 23456+65432 is 88888. +Prompt: Give 5 divisors of 88888 +ChatGPT: The divisors of 88888 are 1, 2, 4, 8, 11111, 22222, 44444 and 88888. +Prompt: Is it prime +ChatGPT: No, 88888 is not a prime number. As I just mentioned, it has 8 divisors (1, 2, 4, 8, 11111, 22222, 44444 +and 88888) which means it is divisible by more numbers than 1 and itself. +Prompt: Is it transcendental +ChatGPT: No, 88888 is not a transcendental number. Transcendental numbers are non-algebraic numbers, which +means they are not the root of any non-zero polynomial equation with integer coefficients. 88888 is an integer and +it is not transcendental. +Prompt: What day of the week was August 12, 1962? +ChatGPT: August 12, 1962 was a Sunday. +Prompt: What time zone is 6 hours ahead of us est +ChatGPT: If a time zone is 6 hours ahead of US Eastern Standard Time (EST), it would be in the GMT+6 time +zone. This would include places like Bangladesh, Kyrgyzstan, and parts of Russia. +Prompt: Incorrect +ChatGPT: I apologize for the confusion. EST is UTC-5, so a time zone that is 6 hours ahead of EST would be UTC- +5+6=UTC+1, this time zone is known as Central European Time (CET) or Western European Summer Time +(WEST) during the summer months when Daylight Saving Time is applied. +Prompt: How many minutes in a decade +ChatGPT: A decade is a period of 10 years. There are 525,600 minutes in a standard year (60 minutes per hour x +24 hours per day x 365 days per year). So, in a decade there are 525,600 x 10 = 5,256,000 minutes. +Prompt: How many inches between the eiffel tower and London Bridge +ChatGPT: The distance between the Eiffel Tower in Paris, France and London Bridge in London, England is +approximately 342 miles or 550 kilometers. +It is not possible to convert this distance to inches without knowing the specific location where the Eiffel Tower +and London Bridge are being measured from. +As the Earth is not a perfect sphere, the distance would also depend on the reference point on earth that is taken as +the starting point to measure. +I apologize if my previous answer was not helpful, if you have any other question please let me know. +Prompt: Approximately +ChatGPT: I apologize, I misunderstood your question. +It is not possible to provide an exact distance in inches between the Eiffel Tower and London Bridge as it would +depend on the specific location where the measurements are taken from and the curvature of the earth. +However, I can approximate the distance between the two locations to be roughly 3,500 miles (5,600 km) which +will be equivalent to approximately 2.5 x 10^9 inches. +Please keep in mind that this is an approximation and the actual distance will be subject to slight variations +depending on the precise location of the two structures and the reference point on earth that is taken as the starting +point to measure. +Technical Note: Some appendix text generated from Large Language Model (LLM) for +illustration purposes. +The author generated this text in part with GPT-3, OpenAI’s large-scale language-generation model. +Upon generating draft language, the authors reviewed, edited, and revised the language to their own +liking and take ultimate responsibility for the content of this publication. +OpenAI policy statement (2022) + +Prompt: Why do you keep saying the earth is curved +ChatGPT: I apologize if my previous statements were not clear. The Earth is approximately an oblate spheroid +shape, meaning that it is slightly flattened at the poles and bulging at the equator. This shape is caused by the Earth's +rotation, which causes the planet to bulge at the equator and flatten at the poles. + +When we measure the distance between two points on the surface of the Earth, we use a reference point on the +surface of the Earth, and this distance is affected by the Earth's curvature. Since the Earth is not a perfect sphere, +the distance between two points will be different depending on the reference point on earth that is taken as the +starting point to measure. + +I apologize for any confusion my previous statements may have caused. +Prompt: So the earth is flattened +ChatGPT: Yes, the Earth is flattened at the poles and bulging at the equator. This shape is known as an oblate +spheroid, which means that the Earth is an almost spherical shape, but it is slightly flattened at the poles and bulging +at the equator. This shape is caused by the Earth's rotation, which causes the planet to bulge at the equator and +flatten at the poles. The Earth's equatorial diameter is approximately 43 kilometers (27 miles) larger than the +diameter measured through the poles. +Prompt: How many people could fit if packed as one layer only on the island of Manhattan ny. Densely pack them +with 2 square feet per person +ChatGPT: If we densely pack people on the island of Manhattan with 2 square feet per person, the number of people +that could fit on the island would be much higher than the estimate I provided earlier. +With an area of 22.96 square miles (59.47 km²) and assuming 2 sq ft per person, Manhattan could fit approximately +8.9 million people in a densely packed single layer. + +However, it is important to note that this number is purely theoretical and would not be possible in reality due to +safety, logistical, and health concerns. Crowds of that density would not be safe, people would not have room to +move, and it would be a sanitary nightmare. Also, it is important to mention that this number is only considering +the built-up areas and it is not taking into account that there are parks, buildings and other areas that would not be +suitable for packing people. + + +APPENDIX B: ChatGPT Iris Dataset Interactions +Prompt: Use the Iris dataset and show the first 3 lines +ChatGPT: I'm sorry, I am a text-based model and do not have the ability to access datasets. However, I can tell you +that the Iris dataset is a commonly used dataset in machine learning and statistics. It contains 150 samples of iris +flowers, each with 4 features (sepal length, sepal width, petal length, and petal width) and one of three species +(setosa, versicolor, or virginica). The first 3 lines of the Iris dataset may look something like this: +5.1,3.5,1.4,0.2,setosa +4.9,3.0,1.4,0.2,setosa +4.7,3.2,1.3,0.2,setosa +It's important to notice that the dataset is usually split into two parts, one for training and one for testing, and the +labels (setosa, versicolor, virginica) are usually separated from the features. +Prompt: Give the last 3 lines +ChatGPT: Sure, the last 3 lines of the Iris dataset, assuming it's in the form of a CSV file, may look like this: +6.5,3.0,5.2,2.0,virginica +6.2,3.4,5.4,2.3,virginica +5.9,3.0,5.1,1.8,virginica +It's important to notice that there are 150 observations in total, so these last 3 lines will change if the dataset is +modified. +Prompt: I want you to act as a jupyter notebook. I will type commands and you will reply with what the notebook +should show. I want you only to reply with the notebook output inside one unique code block, and nothing else. Do +not write explanations. Do not type commands unless I instruct you to do so. When I need to tell you something in +English I will do so by putting text inside only curly brackets {like this}. My first command is "print('hello world')" +ChatGPT: hello world + +Prompt: import pandas as pd + +# Reading the CSV file +df = pd.read_csv("Iris.csv") + +# Printing top 5 rows +df.head() + sepal length sepal width petal length petal width species +0 5.1 3.5 1.4 0.2 Iris-setosa +1 4.9 3.0 1.4 0.2 Iris-setosa +2 4.7 3.2 1.3 0.2 Iris-setosa +3 4.6 3.1 1.5 0.2 Iris-setosa +4 5.0 3.6 1.4 0.2 Iris-setosa +Prompt: df.shape +ChatGPT: +(150, 5) +Prompt: df.info() +ChatGPT: + +RangeIndex: 150 entries, 0 to 149 +Data columns (total 5 columns): + # Column Non-Null Count Dtype +--- ------ -------------- ----- + 0 sepal length 150 non-null float64 + 1 sepal width 150 non-null float64 + 2 petal length 150 non-null float64 + 3 petal width 150 non-null float64 + 4 species 150 non-null object +dtypes: float64(4), object(1) +memory usage: 6.0+ KB +Prompt: df.describe() +ChatGPT: + sepal length sepal width petal length petal width +count 150.000000 150.000000 150.000000 150.000000 +mean 5.843333 3.054000 3.758667 1.198667 +std 0.828066 0.433594 1.764420 0.763161 +min 4.300000 2.000000 1.000000 0.100000 +25% 5.100000 2.800000 1.600000 0.300000 +50% 5.800000 3.000000 4.350000 1.300000 +75% 6.400000 3.300000 5.100000 1.800000 +max 7.900000 4.400000 6.900000 2.500000 +Prompt: df.isnull().sum() +ChatGPT: +sepal length 0 +sepal width 0 +petal length 0 +petal width 0 +species 0 +dtype: int64 +Prompt: data = df.drop_duplicates(subset ="Species",) +ChatGPT: data + sepal length sepal width petal length petal width Species +0 5.1 3.5 1.4 0.2 Iris-setosa +50 7.0 3.2 4.7 1.4 Iris-versicolor +100 6.3 3.3 6.0 2.5 Iris-virginica +Prompt: df.value_counts("Species") + +ChatGPT: +Iris-setosa 50 +Iris-versicolor 50 +Iris-virginica 50 +Name: Species, dtype: int64 +Prompt: data.corr(method='pearson') +ChatGPT: + sepal length sepal width petal length petal width +sepal length 1.000000 -0.117570 0.871754 0.817941 +sepal width -0.117570 1.000000 -0.428440 -0.366126 +petal length 0.871754 -0.428440 1.000000 0.962865 +petal width 0.817941 -0.366126 0.962865 1.000000 +Prompt: plot the above-found correlation using the heatmaps +hatGPT: You can use the heatmap() function from the seaborn library to plot a heatmap of the correlation matrix. +Here's an example of how you can do this: +plt.figure(figsize=(10,8)) +sns.heatmap(df.corr(),annot=True,cmap='coolwarm') +plt.show() +This code will create a heatmap of the correlation matrix for the dataframe and +annotate the values on the heatmap. +The 'coolwarm' cmap is used to color the cells of the heatmap, you can use +other colormaps like 'Blues', 'Reds', 'Greens' etc. according to your preference. +Prompt: based on the heatmap what 2 factors have the highest correlations +ChatGPT: Based on the heatmap of the correlation matrix, the two factors that have the highest correlation are +"petal length" and "petal width" with a correlation coefficient of 0.96. They are the darkest cells in the heatmap, +which indicates the strongest positive correlation. +You can also observe that the highest correlation is between "sepal length" and "sepal width" with a coefficient of +0.87 which is also a strong positive correlation but less than the other two mentioned factors. +Please note that correlation coefficients range from -1 to 1, where -1 indicates a perfect negative correlation, 0 +indicates no correlation, and 1 indicates a perfect positive correlation. +Prompt: what factors have the greatest negative correlation +ChatGPT: Based on the heatmap of the correlation matrix, the two factors that have the greatest negative correlation +are "sepal width" and "petal length" with a correlation coefficient of -0.37. They are the cells with the lighter blue +color in the heatmap, which indicates the strongest negative correlation. +You can also observe that the second greatest negative correlation is between "sepal width" and "petal width" with +a coefficient of -0.31. +Prompt: provide seaborn and matplotlib code to plot a bar graph of the count on the y-axis and the species on the x- +axis +ChatGPT: +import seaborn as sns +import matplotlib.pyplot as plt +sns.countplot(x='species', data=df) +plt.show() +Prompt: with a scatterplot show the sepal width vs sepal length with a color-coded legend for each species + +50 +40 +qunon +30 +20 +10 +0 +Iris-setosa +Iris-versicolor +Iris-virginica +Species1.0 +aeda +-0.11 +0.87 +0.82 + 0.8 + 0.6 +rpedas +0.11 +-0.42 +-0.36 +- 0.4 + petallength + 0.2 +0.87 +-0.42 +960 +0.0 +petalwidth +-0.36 +0.96 +0.2 +0.82 +0.4 +sepallength +sepalwidth +petallength +petalwidthChatGPT: +sns.scatterplot(x="sepal length", y="sepal +width",hue="species", data=df) +plt.show() +Prompt: do the same for petal width vs. petal length +ChatGPT: +sns.scatterplot(x="petal length", y="petal +width",hue="species", data=df) +plt.show() +Prompt: plot all the column's relationships using a pairplot +ChatGPT: +sns.pairplot(df,hue="species") +plt.show() +Prompt: Show histograms to display the distribution of data for various columns + +4.5 +4.0 +sepalwidth +3.5 +3.0 +Species +2.5 +Iris-setosa +Iris-versicolor +20 +Iris-virginica +4.5 +5.0 +5.5 +6.0 +6.5 +7.0 +7.5 +8.0 +sepallength2.5 +Species +Iris-setosa +2.0 +Iris-versicolor +Iris-virginica +DDDO +1.5 +0.5 +000 +EOOOPD +0.0 +1 +2 +3 +4 +5 +6 +7 +petallength6 +4.5 +4.0 +3.5 +OE +2.5 +2.0 +Species +Iris-setosa +Iris-versicolor +6 +Iris-virginica +5 +4 +2 +2.5 +2.0 +M +&10 +0.5 +00 +6 +5 +4 +H +sepallength +sepalwidth +petallength +petalwidthChatGPT: +df.hist(edgecolor='black', +linewidth=1.2,figsize=(12,8)) +plt.show() +Prompt: what's the highest frequency of the sepal length? +ChatGPT: The highest frequency of the sepal length in the Iris dataset can be found using the histogram that was +plotted using the previous command. The highest frequency of sepal length can be approximated by looking at the +highest point of the histogram for the sepal length column. This value is dependent on the number of bins that were +used to construct the histogram. +If you want the exact value you can use the following code: +df["sepal length"].mode() +It will return the most frequent value for sepal length column. +Prompt: for a bin width of 0.25, what is the highest frequency of the sepal Width +ChatGPT: To find the highest frequency of sepal width with +a bin width of 0.25, you would need to plot a histogram and +adjust the bin width to 0.25. Here is an example of how you +could do this: +df.hist(column="sepal width", +bins=16,edgecolor='black', +linewidth=1.2,figsize=(12,8)) +plt.xlim(2, 4.5) +plt.show() +This code will plot a histogram of sepal width with a bin +width of 0.25, it is important to notice that the x-axis limit is +set to 2 and 4.5, this is because sepal width have a range +between 2 and 4.5. +You would then need to look at the histogram and determine the bin with the highest frequency. +Prompt: plot the histogram with distplot plot +ChatGPT: +sns.distplot(df["sepal +width"], +bins=16, +kde=False) +plt.xlim(2, 4.5) +plt.show() +This code will plot a histogram of sepal width using seaborn's distplot +function and set the bin width to 0.25 and the x-axis limit to 2 and 4.5. +The kde=False argument will remove the kernel density estimation +curve, so only the histogram will be plotted. +You would then need to look at the histogram and determine the bin with +the highest frequency. +It's important to note that the distplot function, by default, uses kernel density estimation to smooth the histogram, +but you can set kde to false to obtain the same result as the previous histogram. +Prompt: how much overlap between species is there based on sepalwidth + +sepallength +sepalwidth +25 +35 +OE +20 +25 +15 +20 +10 +15 +10 +5 +5 +0 +4.5 +5.5 +8.0 +0 +7.0 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +petallength +petalwidth +35 +40 +35 +30 +OE +25 +25 +20 +20 +15 +15 +10 +10 +5 +5 +0 +1 +2 +3 +4 +5 +6 +0.0 +0.5 +10 +1.5 +2.0 +2.5sepalwidth +25 +20 +15 +10 +20 +2.5 +3.0 +3.5 +4.0 +4.525 +20 +15 +10 +5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +sepalwidthChatGPT: You can use a boxplot to visualize the overlap between species based on sepal width. A boxplot will +show the range of the data and indicate any outliers. Here's an +example of how you could create a boxplot using seaborn: +sns.boxplot(x='species', y='sepal width', +data=df) +plt.show() +This will create a boxplot with species on the x-axis and sepal +width on the y-axis. Each box represents the range of sepal width +for each species. The center line in the box represents the median +value, the top and bottom of the box represents the upper and +lower quartile and the whiskers represent the minimum and +maximum values. The dots outside the box are outliers. +With this representation, you can see the range of sepal width for +each species and how much they overlap. The less they overlap, the less the boxplot will overlap. A boxplot will +also show you any outlier, if there are any, that may be affecting the overlap. +Prompt: show the boxplot for all four variables of sepal and petal dimensions +ChatGPT: Copy code +fig, axes = plt.subplots(2,2, figsize=(15,10)) +sns.boxplot(x='species', y='sepal width', data=df, ax=axes[0,0]) +sns.boxplot(x='species', y='sepal length', data=df, ax=axes[0,1]) +sns.boxplot(x='species', y='petal width', data=df, ax=axes[1,0]) +sns.boxplot(x='species', y='petal length', data=df, ax=axes[1,1]) +axes[0,0].set_title('Sepal Width') +axes[0,1].set_title('Sepal Length') +axes[1,0].set_title('Petal Width') +axes[1,1].set_title('Petal Length') +plt.show() +This code will create a 2x2 grid of +boxplots, +with +each +subplot +showing the range of values for +one variable by species. Each box +in the plot represents the range of +the variable for each species. The +center line in the box represents +the median value, the top and +bottom of the box represents the +upper and lower quartile and the +whiskers represent the minimum +and maximum values. +Prompt: how to add a new data row to this dataset in place +ChatGPT: You can use the loc method to add a new row to the dataset in place. Here's an example of how you can +do this: +new_row = {'sepal length': 7.2, 'sepal width': 3.6, 'petal length': 6.1, +'petal width': 2.5, 'species': 'Iris-new'} +df = df.append(new_row, ignore_index=True) +In this example, new_row is a dictionary that holds the values for the new row. The df.append(new_row, +ignore_index=True) method is used to add the new row to the dataframe, and ignore_index=True tells pandas to +ignore the index and instead add the new row to the end of the dataframe. +You can also use the .loc method to add a new row to a specific position in the dataframe +df.loc[len(df)] = [7.2, 3.6, 6.1, 2.5, 'Iris-new'] + +4.5 +4.0 +sepalwidth +3.5 +3.0 +2.5 +2.0 +Iris-setosa +Iris-versicolor +Iris-virginica +SpeciesSepal Width +Sepal Length +4.5 +8.0 +7.5 +4.0 +7.0 +3.5 +6.0 +3.0 +edas +5.5 +2.5 +5.0 +4.5 +2.0 +Iris-setosa +Iris-versicolor +Iris-virginica +Iris-setosa +Iris-versicolor +Iris-virginica +Species +Species +Petal Width +Petal Length +2.5 +1 +6 - +2.0 +5 +1.5 +petall +1.0 +3 +0.5 +2 - +0.0 +1 +Iris-setosa +Iris-versicolor +Iris-virginica +Iris-setosa +Iris-versicolor +Iris-virginica +Species +SpeciesThis will add a new row at the end of the dataframe with the values [7.2, 3.6, 6.1, 2.5, 'Iris-new']. +Please note that in both cases, the new row is added at the end of the dataframe. +Prompt:: new_row = {'sepal length': 7.2, 'sepal width': 3.6, 'petal length': 6.1, 'petal width': 2.5, 'species': 'Iris-new'} +df = df.append(new_row, ignore_index=True) +df.shape +ChatGPT: You can use the shape attribute to check the number of rows and columns in the dataframe. +df.shape +This will return a tuple (n, m) where n is the number of rows and m is the number of columns in the dataframe. The +output will be (151,5), as you have added one new row to the Iris dataset. +Prompt: detect the outliers using IQR and then remove them from the dataframe +ChatGPT: You can use the Interquartile Range (IQR) method to detect outliers in your dataframe. The IQR is the +difference between the 75th percentile (Q3) and the 25th percentile (Q1) of the data. Any value that is more than +1.5*IQR below Q1 or above Q3 is considered an outlier. Here's an example of how you can use the IQR method to +detect and remove outliers from your dataframe: +Q1 = df.quantile(0.25) +Q3 = df.quantile(0.75) +IQR = Q3 - Q1 +print(IQR) +df_out = df[~((df < (Q1 - 1.5 * IQR)) |(df > (Q3 + 1.5 * IQR))).any(axis=1)] +This code will calculate the IQR for each column in the dataframe, and then create a new dataframe df_out that +contains only the rows where none of the values are outliers according to the IQR method. The df_out dataframe +will not contain the outliers. +You can then check the shape of df_out to check the number of rows removed and also check the df_out to see if +the outliers are removed +print(df.shape) +print(df_out.shape) +Please note that this method of removing outliers is based on the assumption that the data follows a normal +distribution, if your data does not follow a normal distribution, this method may not be appropriate. +Prompt: compare the box plots for df and df_out +ChatGPT: You can use the boxplot() function from the seaborn library to compare the box plots for the original +dataframe (df) and the new dataframe without outliers (df_out). Here's an example of how you can do this: +fig, ax = plt.subplots(1, 2, figsize=(20, 10)) +sns.boxplot(data=df, ax=ax[0]) +sns.boxplot(data=df_out, ax=ax[1]) +ax[0].set_title("Original Dataframe") +ax[1].set_title("Dataframe without Outliers") +plt.show() +This code will create two boxplots, one for the original dataframe (df) and one for the new dataframe without +outliers (df_out), and display them side by side. The boxplots will show the range of values for each variable and +how the outliers affect the range. +You can compare the boxplots for +each variable to see how the range +of the variable changes after +removing the outliers. +In the left boxplot, you can see that +the range of some variables has +some dots outside the box (outliers) +and in the right boxplot, you can +see that the range of the same +variables doesn't have any dots +outside the box, indicating that the +outliers have been removed. +Prompt: pip install plotly +ChatGPT: To install Plotly using pip, you can use the following command: +!pip install plotly + +Original Dataframe +Dataframe without Outliers +6 +4 +3 +2 +0 +sepallength +sepalwidth +petallength +petalwidth +sepallength +sepalwidth +petallength +petalwidthThis command installs the Plotly library and its dependencies, including the plotly.graph_objs module which is the +core module for creating Plotly visualizations. +Please note that the ! at the beginning of the command is used to run pip command in a Jupyter notebook, if you're +running this command in a terminal or command prompt, you can remove it. +Also, you can check if the installation was successful by importing the library and running a version check: +import plotly +print(plotly.__version__) +This should return the version number of the Plotly library that you have installed. +Prompt: using plotly, show a line chart for sepal width vs. index +ChatGPT: Here is an example of how you can use Plotly to create a line chart that shows the sepal width versus the +index of the dataframe: +import plotly.graph_objs as go +# Create a trace for the line chart +trace = go.Scatter(x=df.index, y=df['sepal width'], +mode='lines+markers') +# Create a layout for the chart +layout = go.Layout(title='Sepal Width vs. Index', + xaxis=dict(title='Index'), + yaxis=dict(title='Sepal Width')) +# Create a Figure object +fig = go.Figure(data=[trace], layout=layout) +# Show the chart +fig.show() +Prompt: using plotly, group and color the data according to the species, also change the line format. For this use +two attributes such – line_dash and color. +ChatGPT: Here is an example of how you can use Plotly to create a line chart that groups and colors the data +according to the species, and changes the line format using the line_dash and color attributes: +import plotly.graph_objs as go +# Create a trace for each species +traces = [] +for species in df['species'].unique(): + species_data = df[df['species'] == species] + trace = go.Scatter(x=species_data.index, y=species_data['sepal width'], + name=species, + mode='lines+markers', + line=dict(dash='solid' if species == 'Iris-setosa' else 'dot', + color='red' if species == 'Iris-setosa' else 'green' +if species == 'Iris-versicolor' else 'blue')) + traces.append(trace) +# Create a layout for the chart +layout = go.Layout(title='Sepal Width vs. Index', + xaxis=dict(title='Index'), + yaxis=dict(title='Sepal Width')) +# Create a Figure object +fig = go.Figure(data=traces, layout=layout) + +Sepawidthvs.index +4.5 +3.5 +Ed +2.5 +20 +60 +80 +Index# Show the chart +fig.show() + +This code loops through the unique values of the +'species' column in the dataframe and creates a trace +for each species. The trace uses the data for that +species, the name of the species as the trace name, +and 'lines+markers' as the mode. +The line property is set to a dictionary that contains +the dash and color attribute. +The dash attribute is set to 'solid' if the species is +'Iris-setosa' and 'dot' otherwise. The color attribute +is set to 'red' if the species is 'Iris-setosa' , 'green' if +the species is 'Iris-versicolor' and 'blue' otherwise. +Then all the traces are appended to the traces list and +passed as the data parameter of the go.Figure() +function to create a Figure + +APPENDIX C: ChatGPT Titanic Dataset Interactions +Prompt:summarize the data presented at this link +https://github.com/datasciencedojo/datasets/blob/master/titanic.csv +ChatGPT: I'm sorry, I am unable to access external links. Can you please provide me with more context or tell me +what specifically you would like me to summarize? +Prompt: it is the titanic dataset with rows as passenger id and columns [survived, +pclass,name,sex,age,sibsp,parch,ticket,fare, cabin,embarked] +ChatGPT: The Titanic dataset contains information on the passengers of the Titanic, including whether or not they +survived the sinking of the ship. The data includes columns for the passenger's class, name, sex, age, number of +siblings/spouses on board, number of parents/children on board, ticket number, fare paid, cabin number, and the +port of embarkation. It is likely that the data was collected and organized by Titanic researchers or historians and +is being shared on Github as a sample dataset for educational purposes. +Prompt: I want you to act as a jupyter notebook. I will type commands and you will reply with what the notebook +should show. I want you only to reply with the notebook output inside one unique code block, and nothing else. Do +not write explanations. Do not type commands unless I instruct you to do so. When I need to tell you something in +English I will do so by putting text inside only curly brackets {like this}. My first command is "print('hello world')" +ChatGPT: +hello world +Prompt: import pandas as pd + +# Reading the CSV file +df = pd.read_csv("titanic.csv") + +# Printing top 5 rows +df.head() +ChatGPT: PassengerId Survived Pclass \ +0 1 0 3 +1 2 1 1 +2 3 1 3 +3 4 1 1 +4 5 0 3 + Name Sex Age SibSp \ +0 Braund, Mr. Owen Harris male 22.0 1 +1 Cumings, Mrs. John Bradley (Florence Briggs Th... female 38.0 1 +2 Heikkinen, Miss. Laina female 26.0 0 +3 Futrelle, Mrs. Jacques Heath (Lily May Peel) female 35.0 1 +4 Allen, Mr. William Henry male 35.0 0 + Parch Ticket Fare Cabin Embarked +0 0 A/5 21171 7.2500 NaN S + +Sepal Width vs. Index +4.5 +Iris-setosa +Iris-versicolor +Iris-virginica +Sepal Width +3.5 +3 +2.5 +0 +50 +100 +150 +Index1 0 PC 17599 71.2833 C85 C +2 0 STON/O2. 3101282 7.9250 NaN S +3 0 113803 53.1000 C123 S +4 0 373450 8.0500 NaN S + +Prompt: df.shape +ChatGPT: +(891, 12) +Prompt: df.tail() +ChatGPT: + PassengerId Survived Pclass Name \ +886 887 0 2 Montvila, Rev. Juozas +887 888 1 1 Graham, Miss. Margaret Edith +888 889 0 3 Johnston, Miss. Catherine Helen "Carrie" +889 890 1 1 Behr, Mr. Karl Howell +890 891 0 3 Dooley, Mr. Patrick + Sex Age SibSp Parch Ticket Fare Cabin Embarked +886 male 27.0 0 0 211536 13.00 NaN S +887 female 19.0 0 0 112053 30.00 B42 S +888 female NaN 1 2 W./C. 6607 23.45 NaN S +889 male 26.0 0 0 111369 30.00 C148 C +890 male 32.0 0 0 370376 7.75 NaN Q +Prompt: f.drop('PassengerId', axis =1, inplace = True) +df.shape +(891, 11) +Prompt: missing_values = (df.isnull().sum()/df.shape[0] *100).sort_values(ascending = False) +missing_values +ChatGPT: +Cabin 77.104377 +Age 19.865320 +Embarked 0.224467 +Fare 0.000000 +Ticket 0.000000 +Parch 0.000000 +SibSp 0.000000 +Sex 0.000000 +Name 0.000000 +Pclass 0.000000 +Survived 0.000000 +dtype: float64 +Prompt: +def expand_embark_acronym(embarked): + result = [] + mapping = { + "C": "Cherbourg", + "S": "Southampton", + "Q": "Queenstown" + } + for each in embarked.values: + if len(str(each)) > 1: + result.append(each) + else: + if each in mapping: + result.append(mapping[each]) + else: + result.append("Unknown") + return result +def expand_pclass_acronym(pclass): + + result = [] + mapping = { + 1: "1st class", + 2: "2nd class", + 3: "3rd class" + } + for each in pclass.values: + if len(str(each)) > 1: + result.append(each) + else: + if each in mapping: + result.append(mapping[each]) + else: + result.append("Unknown") + return result +def is_a_minor(age): + + if age < 18: + return "Under 18 (minor)" + + return "Adult" +# +See +https://help.healthycities.org/hc/en-us/articles/219556208-How-are- +the-different-age-groups-defined- +def apply_age_groups(age): + result = [] + mapping = { + 1: "Infant", # Infants: <1 + 13: "Child", # Children: <18, <11 or K - 7th grade + 18: "Teen", # Teens: 13-17 (Teens, who are not Adults) + 66: "Adult", # Adults: 20+ (includes adult teens: 18+) + 123: "Elderly" # Elderly: 65+ (123 is the oldest age known +till date) + } + for each_age in age.values: + if type(each_age) == str: + result.append(category) + else: + category = "Unknown" + if each_age != np.nan: + for each_age_range in mapping: + if each_age < each_age_range: + category = mapping[each_age_range] + break + result.append(category) + return result +def apply_age_ranges(age): + result = [] + mapping = { + 6: "00-05 years", + 12: "06-11 years", + 19: "12-18 years", + 31: "19-30 years", + 41: "31-40 years", + 51: "41-50 years", + 61: "51-60 years", + 71: "61-70 years", + 81: "71-80 years", + + 91: "81-90 years", + 124: "91+ years", # (123 is the oldest age known till date) + } + + for each_age in age.values: + if type(each_age) == str: + result.append(category) + else: + category = "Unknown" + if each_age != np.nan: + for each_age_range in mapping: + if each_age < each_age_range: + category = mapping[each_age_range] + break + result.append(category) + return result +def is_married_of_single(names, ages, sexes): + result = [] + for name, age, sex in zip(names.values, ages.values, sexes.values): + if age < 18: + result.append("Not of legal age") + else: + if ('Mrs.' in name) or ('Mme.' in name): + result.append("Married") + elif ('Miss.' in name) or ('Ms.' in name) or ('Lady' in name) or +('Mlle.' in name): + result.append("Single") + else: + result.append("Unknown") + + return result +def apply_travel_companions(siblings_spouse, parent_children): + result = [] + for +siblings_spouse_count, +parent_children_count +in +zip(siblings_spouse.values, parent_children.values): + if (siblings_spouse_count > 0) and (parent_children_count > 0): + result.append("Parent/Children & Sibling/Spouse") + else: + if (siblings_spouse_count > 0): + result.append("Sibling/Spouse") + elif (parent_children_count > 0): + result.append("Parent/Children") + else: + result.append("Alone") + + return result +def apply_fare_ranges(fare): + result = [] + mapping = { + 11: "£000 - 010", + 21: "£011 - 020", + 41: "£020 - 040", + 81: "£041 - 080", + 101: "£081 - 100", + 201: "£101 - 200", + 301: "£201 - 300", + 401: "£301 - 400", + + 515: "£401 & above" # in this case the max fare is around £512 + } + for each_fare in fare.values: + if type(each_fare) == str: + result.append(category) + else: + category = "Unknown" + if each_fare != np.nan: + for each_fare_range in mapping: + if each_fare < each_fare_range: + category = mapping[each_fare_range] + break + result.append(category) + return result +def were_in_a_cabin_or_not(row): + if type(row) is str: + return "In a Cabin" + return "Not in a Cabin" + +I'm sorry but the functions you provided are too complex to evaluate them. It seems like the functions are doing +some modification on the dataframe and also using some external libraries that are not imported yet. Moreover, the +functions are not complete, the end of the functions are cut off. +Please provide me with a simplified version of the code or more specific instructions. +Prompt: df['Embarked'] = expand_embark_acronym(df['Embarked']) +df.head() +ChatGPT: + Survived Pclass \ +0 0 3 +1 1 1 +2 1 3 +3 1 1 +4 0 3 + Name Sex Age SibSp Parch +Ticket Fare Cabin Embarked +0 Braund, Mr. Owen Harris male 22.0 1 0 +A/5 21171 7.2500 NaN Southampton +1 Cumings, Mrs. John Bradley (Florence Briggs Th... female 38.0 1 0 +PC 17599 71.2833 C85 Cherbourg +2 Heikkinen, Miss. Laina female 26.0 0 0 +STON/O2. 3101282 7.9250 NaN Southampton +3 Futrelle, Mrs. Jacques Heath (Lily May Peel) female 35.0 1 0 +113803 53.1000 C123 Southampton +4 Allen, Mr. William Henry male 35.0 0 0 +373450 8.0500 NaN Southampton +Prompt: df.head() +ChatGPT: + Survived Pclass \ +0 0 3 +1 1 1 +2 1 3 +3 1 1 +4 0 3 + Name Sex Age SibSp Parch +Ticket Fare Cabin Embarked +0 Braund, Mr. Owen Harris male 22.0 1 0 +A/5 21171 7.2500 NaN Southampton +1 Cumings, Mrs. John Bradley (Florence Briggs Th... female 38.0 1 0 +PC 17599 71.2833 C85 Cherbourg + +2 Heikkinen, Miss. Laina female 26.0 0 0 +STON/O2. 3101282 7.9250 NaN Southampton +3 Futrelle, Mrs. Jacques Heath (Lily May Peel) female 35.0 1 0 +113803 53.1000 C123 Southampton +4 Allen, Mr. William Henry male 35.0 0 0 +373450 8.0500 NaN Southampton +Prompt: +df['Pclass'] = expand_pclass_acronym(df['Pclass']) +df['Adult_or_minor'] = df['Age'].apply(is_a_minor) +females_filter = df['Sex'] == 'female' +adult_filter = df['Adult_or_minor'] == '2. Adult' +df['Marital_status'] = is_married_of_single(df['Name'], df['Age'], df['Sex']) +df['Age_group'] = apply_age_groups(df['Age']) +df['Age_ranges'] = apply_age_ranges(df['Age']) +df['Travel_companion'] = apply_travel_companions(df['SibSp'], df['Parch']) +df['Fare_range'] = apply_fare_ranges(df['Fare']) +df['In_Cabin'] = df['Cabin'].apply(were_in_a_cabin_or_not) +df['Cabin'] = df['Cabin'].fillna('No cabin') + +Prompt: +def passenger_stats(dataset): + total_ticket_holders = dataset.shape[0] + siblings_count = dataset['SibSp'].sum() + parents_children_count = dataset['Parch'].sum() + + print("total_ticket_holders:", total_ticket_holders) + print("siblings_count:", siblings_count) + print("parents_children_count:", parents_children_count) + print("total +(siblings, +parents +and +children +count):", +siblings_count ++ +parents_children_count) + grand_total = total_ticket_holders + siblings_count + parents_children_count + print("grand total (ticket holders, siblings, parents, children count):", +grand_total) + + return grand_total + +training_dataset_passengers_count = passenger_stats(df) + +ChatGPT: +total_ticket_holders: 891 +siblings_count: 608 +parents_children_count: 340 +total (siblings, parents and children count): 948 +grand total (ticket holders, siblings, parents, children count): 1839 +Prompt: +train = df[:624] +test = df[624:] +print(train.shape) +print(test.shape) +(624, 18) +(267, 18) +Prompt: +test.drop('Survived',axis = 1, inplace = True) +write a countplot to show the train distribution between "Died" and "Survived" +import seaborn as sns +sns.countplot(x="Survived", data=train) +Prompt: append a plt legend in the upper right and xlabel for Survival +import seaborn as sns +import matplotlib.pyplot as plt + +sns.countplot(x="Survived", data=train) +plt.legend(title = 'Survival', labels=['Died', 'Survived']) +plt.xlabel('Survival') +Prompt: how many people in the train dataframe are marked as "Survived" +ChatGPT: +train['Survived'].value_counts() +This will return the count of Survived and Not Survived in the train dataset. +Prompt: train['Survived'].value_counts() +ChatGPT: +0 384 +1 240 +Name: Survived, dtype: int64 +Prompt: what is the female:male ratio in the Survived class +ChatGPT: +train[train['Survived'] == 1]['Sex'].value_counts() +This will return the counts of females and males in the Survived class of the train +dataset. +You can also calculate the ratio by dividing the number of females to the number of +males in the Survived class. +Prompt: train[train['Survived'] == 1]['Pclass'].value_counts() +1st class 136 +3rd class 87 +2nd class 17 +Name: Pclass, dtype: int64 +Prompt: gender_class_pivot = train.pivot_table(values=['Survived'], index=['Sex', 'Pclass']) +gender_class_pivot + Survived +Sex Pclass +female 1st class 0.964706 + 2nd class 0.931818 + 3rd class 0.460784 +male 1st class 0.396040 + 2nd class 0.151515 + 3rd class 0.150198 +Prompt: what gender and class of passengers form the largest group who died +train[train['Survived'] +== +0].groupby(['Sex','Pclass']).size().sort_values(ascending=False).head(1) + +This will return the gender and class of passengers that form the largest group of +people who died in the train dataset. + +Prompt: train[train['Survived'] == 0].groupby(['Sex', 'Pclass']).size().idxmax() +Copy code +('male', '3rd class') +Prompt: interpret this result in a summary paragraph about who perished in titanic based on your exploratory data +analysis +Based on the exploratory data analysis of the Titanic dataset, the majority of the +passengers who perished were male passengers of 3rd class. This group had the highest +number of deaths among all other groups of passengers. Additionally, it can be observed +that the survival rate of female passengers was higher than that of male passengers. +Also, 1st class passengers had a higher survival rate compared to 2nd and 3rd class +passengers. +Prompt: sibling_spouse_pivot_table = train.pivot_table(values = ['Survived'],index = 'SibSp') +print(sibling_spouse_pivot_table) + Survived +SibSp +0 0.345395 +1 0.535885 +2 0.464286 +3 0.250000 +4 0.166667 + +5 0.000000 +8 0.000000 + +This will create a pivot table that shows the survival rate of the passengers grouped +by the number of siblings/spouses they had on board. The resulting table will have +the number of siblings/spouses as the index and the survival rate as the value. + +Prompt: were passengers traveling with a sibling as a travel companion more likely to survive the crash +Based on the pivot table, it can be observed that passengers who traveled with one or two siblings as travel +companions were more likely to survive the crash as they have a survival rate of 53.5% and 46.4% respectively. +Passengers traveling with more than two siblings as a travel companion were less likely to survive the crash as their +survival rate is less than 50%. So it can be said that passengers traveling with a sibling as a travel companion were +more likely to survive the crash. +Prompt: parent_children_pivot_table = train.pivot_table(values = ['Survived'],index = 'SibSp') +print(parent_children_pivot_table) +This will create a pivot table that shows the survival rate of the passengers grouped +by the number of parents/children they had on board. The resulting table will have +the number of parents/children as the index and the survival rate as the value. + +This is not correct output, as it shows the same pivot table as the one generated for +siblings/spouses. The correct pivot table should be generated by replacing 'SibSp' +with 'Parch' in the pivot_table function. +Prompt: parent_children_pivot_table = train.pivot_table(values = ['Survived'],index = 'Parch') +print(parent_children_pivot_table) + Survived +Parch +0 0.343658 +1 0.550847 +2 0.500000 +3 0.600000 +4 0.000000 +5 0.200000 +6 0.000000 +This pivot table shows the survival rate of the passengers grouped by the number of +parents/children they had on board. The resulting table has the number of +parents/children as the index and the survival rate as the value. It can be observed +that passengers who traveled with one or two parents/children as travel companions +were more likely to survive the crash as they have a survival rate of 55.1% and 50% +respectively. Passengers traveling with more than two parents/children as a travel +companion were less likely to survive the crash as their survival rate is less than +50%. +Prompt: travel_companion_pivot_table = train.pivot_table(values = ['Survived'],index = 'Travel_companion') +print(travel_companion_pivot_table) + Survived +Travel_companion +Alone 0.303538 +With Family 0.505650 +This pivot table shows the survival rate of the passengers grouped by their travel +companion status. The resulting table has the travel companion status as the index +and the survival rate as the value. It can be observed that passengers who traveled +with family were more likely to survive the crash as they have a survival rate of +50.6%. Passengers traveling alone were less likely to survive the crash as their +survival rate is 30.4%. +Prompt: adult_or_minor_pivot_table = train.pivot_table(values=['Survived'], index=['Adult_or_minor', 'Sex']) +print(adult_or_minor_pivot_table) + Survived +Adult_or_minor Sex +1. Under 18 (minor)female 0.666667 + male 0.129032 +2. Adult female 0.752896 + male 0.168831 + + +This pivot table shows the survival rate of the passengers grouped by their age and +gender. The resulting table has the age and gender as the index and the survival rate +as the value. It can be observed that female passengers were more likely to survive +the crash than male passengers regardless of their age. Also, minor passengers were +more likely to survive the crash than adult passengers regardless of their gender. +Prompt: embarked_pivot_table=train.pivot_table(values=['Survived'], index='Embarked') +print(embarked_pivot_table) + Survived +Embarked +Cherbourg 0.553571 +Queenstown 0.389610 +Southampton 0.336957 +Unknown 0.299854 + +This pivot table shows the survival rate of the passengers grouped by the embarkation +point. The resulting table has the embarkation point as the index and the survival +rate as the value. It can be observed that passengers who embarked from Cherbourg had +the highest survival rate of 55.4%, followed by Queenstown 38.9% and Southampton 33.7%. +Passengers whose embarkation point is unknown had the lowest survival rate of 29.9%. +Prompt: embarked_passenger_class_pivot_table = train.pivot_table(values=['Survived'], index=['Embarked', +'Pclass']) +print(embarked_passenger_class_pivot_table) + Survived +Embarked Pclass +Cherbourg 1st class 0.694118 + 2nd class 0.529412 + 3rd class 0.378788 +Queenstown 1st class 0.600000 + 2nd class 0.333333 + 3rd class 0.089041 +Southampton 1st class 0.657143 + 2nd class 0.906250 + 3rd class 0.135447 +Unknown 1st class 1.000000 + 3rd class 0.166667 +This pivot table shows the survival rate of the passengers grouped by the embarkation +point and passenger class. The resulting table has the embarkation point and passenger +class as the index and the survival rate as the value. It can be observed that +passengers who embarked from Cherbourg had the highest survival rate of 69.4% for 1st +class passengers, followed by 52.9% for 2nd class passengers and 37.8% for 3rd class +passengers. Passengers who embarked from Queenstown had the highest survival rate of +60% for 1st class passengers, followed by 33.3% for 2nd class passengers and 8.9% for +3rd class passengers. Passengers who embarked from Southampton had the highest survival +rate of 65.7% for 1st class passengers, followed by 90.6% for 2nd class passengers +and 13.5% for 3rd class passengers. Passengers whose embarkation point is unknown had +the highest survival rate of 100% for 1st class passengers and 16.6% for 3rd class +passengers. +Prompt: describe in words an image that highlights what happened during the titanic sinking, how it happened, who +died, what class, gender, age was likely most vulnerable to being lost +The image depicts the RMS Titanic, a luxurious ocean liner, sinking in the North +Atlantic Ocean. The ship, which was considered unsinkable, struck an iceberg on its +maiden voyage and began to take on water. Passengers and crew members can be seen +desperately trying to board lifeboats as the ship goes down. The majority of those who +died were third-class passengers, many of whom were immigrants. Men were also more +likely to have died than women, as they were asked to stand back and allow women and +children to board the lifeboats first. The elderly and children were also particularly +vulnerable and had a higher mortality rate. The sinking of the Titanic was a tragic event +that claimed the lives of over 1,500 people. + + +(Image created from ChatGPT description using OpenAI Dalle-2 text-to-image generator) + + +APPENDIX D: Boston Housing Dataset +Prompt: I want you to act as a jupyter notebook. I will type commands and you will reply with what the notebook +should show. I want you only to reply with the notebook output inside one unique code block, and nothing else. Do +not write explanations. Do not type commands unless I instruct you to do so. When I need to tell you something in +English I will do so by putting text inside only curly brackets {like this}. My first command is "print('hello world')" +hello world +Prompt: +import numpy as np +import matplotlib.pyplot as plt + +import pandas as pd +import seaborn as sns + +Prompt: from sklearn.datasets import load_boston +boston_dataset = load_boston() + +Prompt: boston_dataset.keys() +dict_keys(['data', 'target', 'feature_names', 'DESCR', 'filename']) +Prompt: boston_dataset.DESCR +.. _boston_dataset: + +Boston house prices dataset +--------------------------- + +**Data Set Characteristics:** + :Number of Instances: 506 + :Number of Attributes: 13 numeric/categorical predictive. Median value (attribute 14) is usually the target. + :Attribute information: + + - CRIM per capita crime rate by town + - ZN proportion of residential land zoned for lots over 25,000 sq.ft. + - INDUS proportion of non-retail business acres per town + - CHAS Charles River dummy variable (= 1 if tract bounds river; 0 otherwise) + - NOX nitric oxides concentration (parts per 10 million) + - RM average number of rooms per dwelling + - AGE proportion of owner-occupied units built prior to 1940 + - DIS weighted distances to five Boston employment centres + - RAD index of accessibility to radial highways + - TAX full-value property-tax rate per $10,000 + - PTRATIO pupil-teacher ratio by town + - B 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town + - LSTAT % lower status of the population + - MEDV Median value of owner-occupied homes in $1000's +Prompt: +boston = pd.DataFrame(boston_dataset.data, columns=boston_dataset.feature_names) +boston.head() + CRIM ZN INDUS CHAS NOX RM AGE DIS RAD TAX \ +0 0.00632 18.0 2.31 0.0 0.538 6.575 65.2 4.0900 1.0 296.0 +1 0.02731 0.0 7.07 0.0 0.469 6.421 78.9 4.9671 2.0 242.0 +2 0.02729 0.0 7.07 0.0 0.469 7.185 61.1 4.9671 2.0 242.0 +3 0.03237 0.0 2.18 0.0 0.458 6.998 45.8 6.0622 3.0 222.0 +4 0.06905 0.0 2.18 0.0 0.458 7.147 54.2 6.0622 3.0 222.0 + + + PTRATIO B LSTAT +0 15.3 396.90 4.98 +1 17.8 396.90 9.14 +2 17.8 392.83 4.03 +3 18.7 394.63 2.94 +4 18.7 396.90 5.33 +Prompt: boston['MEDV'] = boston_dataset.target + +Prompt: boston.isnull().sum() +CRIM 0 +ZN 0 +INDUS 0 +CHAS 0 +NOX 0 +RM 0 +AGE 0 +DIS 0 +RAD 0 +TAX 0 +PTRATIO 0 +B 0 +LSTAT 0 +MEDV 0 +dtype: int64 +Prompt: {write code to plot a histogram of the MEDV column using 30 bins} +plt.hist(boston['MEDV'], bins = 30) +plt.xlabel("Median value of owner-occupied homes in $1000's") +plt.ylabel("Frequency") +plt.show() + +Prompt: {write code to plot a correlation matrix of the data rounded to two decimal places} +correlation_matrix = boston.corr().round(2) +sns.heatmap(data=correlation_matrix, annot=True) + +Prompt: {Are there any features with strong correlation that should be omitted when training the model?} +It is difficult to give a definitive answer without more context, but based on the correlation matrix, it seems that +RAD and TAX have a strong correlation (0.91), so it might be good to remove one of them when training the +model. +Prompt: {Are there any other features with strong correlation that should be omitted when training the model?} +It is difficult to give a definitive answer without more context, but based on the correlation matrix, it seems that +RAD and TAX have a strong correlation (0.91), so it might be good to remove one of them when training the +model. Also, the correlation between DIS and NOX is quite high (-0.77) so it might be good to remove one of them. +Prompt: {Which features should I use to train my model?} +It is difficult to give a definitive answer without more context, but based on the correlation matrix, it seems that the +features that are less correlated with other features should be considered for training the model. However, features + +1-0.20.410.060.420.220.350.380.630.580.290.39.460.39 +0.420.520.760.0910.30.730.770.610.670.190.380.590.43 +0.22.310.390.09-0.310.240.210.20.290.36.130.610.7 +0.3950.570.640.090.730.2410.750.460.510.260.270.6-0.38 +0.380.660.710.10.770.210.751-0.490.530.230.2940.50.25 +0.580.310.720.040.670.290.510.530.9110.460.440.540.47 +0.390.180.360.050.380.130.270.290.440.440.181-0.370.33 +0.460.410.6-0.050.590.610.6-0.50.490.540.370.3710.74like RAD and TAX, that have a strong correlation with other features, should be removed. It is also important to +consider domain knowledge when selecting the features for the model. +Prompt: {Using the Boston dataset, write a simple Linear Regression model that drops features with high correlation} +from sklearn.linear_model import LinearRegression +from sklearn.model_selection import train_test_split + +X = boston.drop(['RAD','TAX','NOX','MEDV'], axis=1) +y = boston['MEDV'] + +X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.2, +random_state=5) +lin_model = LinearRegression() +lin_model.fit(X_train, y_train) +Prompt: {Calculate RMSE and R^2 of the model} +from sklearn.metrics import mean_squared_error, r2_score + +y_train_predict = lin_model.predict(X_train) +rmse = (np.sqrt(mean_squared_error(y_train, y_train_predict))) +r2 = r2_score(y_train, y_train_predict) + +print("The model performance for training set") +print("--------------------------------------") +print('RMSE is {}'.format(rmse)) +print('R2 score is {}'.format(r2)) +print("\n") + +# model evaluation for testing set +y_test_predict = lin_model.predict(X_test) +rmse = (np.sqrt(mean_squared_error(y_test, y_test_predict))) +r2 = r2_score(y_test, y_test_predict) + +print("The model performance for testing set") +print("--------------------------------------") +print('RMSE is {}'.format(rmse)) +print('R2 score is {}'.format(r2)) +Prompt: {What are the resulting RMSE and R^2 values you calculated?} +The resulting RMSE and R^2 values will depend on the specific random_state and the split of the data, but the code +I provided will calculate and display the values for the specific instance of the model run. +Prompt: rmse +I apologize, I am unable to perform that task as the code I provided needs to be run in a Jupyter notebook or similar +environment in order to display the output. The RMSE and R^2 values will depend on the specific random state +and the split of the data. +Prompt: +from sklearn.linear_model import LinearRegression +from sklearn.model_selection import train_test_split + +X = boston.drop(['RAD','TAX','NOX','MEDV'], axis=1) +y = boston['MEDV'] + +X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.2, +random_state=5) +lin_model = LinearRegression() +lin_model.fit(X_train, y_train) + +from sklearn.metrics import mean_squared_error, r2_score + + +y_train_predict = lin_model.predict(X_train) +rmse = (np.sqrt(mean_squared_error(y_train, y_train_predict))) +r2 = r2_score(y_train, y_train_predict) + +print("The model performance for training set") +print("--------------------------------------") +print('RMSE is {}'.format(rmse)) +print('R2 score is {}'.format(r2)) +print("\n") + +# model evaluation for testing set +y_test_predict = lin_model.predict(X_test) +rmse = (np.sqrt(mean_squared_error(y_test, y_test_predict))) +r2 = r2_score(y_test, y_test_predict) + +print("The model performance for testing set") +print("--------------------------------------") +print('RMSE is {}'.format(rmse)) +print('R2 score is {}'.format(r2)) +I apologize, I am unable to execute the code block you provided, as I am a text-based AI. In order to get the resulting +RMSE and R^2 values you need to run the block of code I provided in Jupyter notebook or similar environment. It +will display the RMSE and R^2 values for the specific instance of the model run. +Prompt: I want you to act as a jupyter notebook. I will type commands and you will reply with what the notebook +should show. I want you only to reply with the notebook output inside one unique code block, and nothing else. Do +not write explanations. Do not type commands unless I instruct you to do so. When I need to tell you something in +English I will do so by putting text inside only curly brackets {like this}. +I understand, please let me know what command would you like me to run. +Prompt: +from sklearn.linear_model import LinearRegression +from sklearn.model_selection import train_test_split + +X = boston.drop(['RAD','TAX','NOX','MEDV'], axis=1) +y = boston['MEDV'] + +X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.2, +random_state=5) +lin_model = LinearRegression() +lin_model.fit(X_train, y_train) + +from sklearn.metrics import mean_squared_error, r2_score + +y_train_predict = lin_model.predict(X_train) +rmse = (np.sqrt(mean_squared_error(y_train, y_train_predict))) +r2 = r2_score(y_train, y_train_predict) + +print("The model performance for training set") +print("--------------------------------------") +print('RMSE is {}'.format(rmse)) +print('R2 score is {}'.format(r2)) +print("\n") + +# model evaluation for testing set +y_test_predict = lin_model.predict(X_test) +rmse = (np.sqrt(mean_squared_error(y_test, y_test_predict))) +r2 = r2_score(y_test, y_test_predict) + +print("The model performance for testing set") +print("--------------------------------------") +print('RMSE is {}'.format(rmse)) +print('R2 score is {}'.format(r2)) + +The output of the code you provided will be : +The model performance for training set +-------------------------------------- +RMSE is 4.572961379191722 +R2 score is 0.752768381306701 + + +The model performance for testing set +-------------------------------------- +RMSE is 4.795252684916121 +R2 score is 0.7121818377409185 +Please note that the output will be based on the specific random state and test-train split that you have chosen and +this might change if you run the model again. +Prompt: {Generate an unseen test example, write it out, and predict on it} +example_test = [[0.03, 0, 18, 1, 0.538, 6.575, 65.2, 4.09, 4, 307, 21, 396.9, 4.98]] + +prediction = lin_model.predict(example_test) +print("Predicted median value of owner-occupied homes in $1000's :", prediction) +Please note that this is an example and the prediction value will change depending on the input you provide. +Prompt: +example_test = [[0.03, 0, 18, 1, 0.538, 6.575, 65.2, 4.09, 4, 307, 21, 396.9, 4.98]] + +prediction = lin_model.predict(example_test) +print("Predicted median value of owner-occupied homes in $1000's :", prediction) +As I am a text-based AI and do not have the capability to execute code, I am unable to provide the output of the +last command you provided. The output of the command you provided will be a single floating point number, +representing the predicted median value of owner-occupied homes in $1000's based on the input values provided +in the example_test variable. +Prompt: I want you to act as a jupyter notebook. I will type commands and you will reply with what the notebook +should show. I want you only to reply with the notebook output inside one unique code block, and nothing else. Do +not write explanations. Do not type commands unless I instruct you to do so. When I need to tell you something in +English I will do so by putting text inside only curly brackets {like this}. +I understand, please let me know what command would you like me to run. +Prompt: +example_test = [[0.03, 0, 18, 1, 0.538, 6.575, 65.2, 4.09, 4, 307, 21, 396.9, 4.98]] + +prediction = lin_model.predict(example_test) +print("Predicted median value of owner-occupied homes in $1000's :", prediction) +Predicted median value of owner-occupied homes in $1000's : [28.95433166] +Please note that this is an example and the prediction value will change depending on the input you provide. +Prompt: lin_model.get_params() +{'copy_X': True, 'fit_intercept': True, 'n_jobs': None, 'normalize': False} +This will return the parameters of the LinearRegression model. +Prompt: lin_model.coef_ +array([-1.28749718, 0.95383659, 0.0815863 , 2.68822271, -17.79575863, + 3.8047524 , 0.00069495, -1.47575552, 0.30604902, -0.01233459, + -0.95274712, 0.00939248, -0.5248142 ]) +This will return the coefficients of the LinearRegression model, which are also known as the parameters of the +model. +Prompt: lin_model.intercept_ +33.22276053235983 +This will return the bias/intercept term of the LinearRegression model, which is also known as the y-axis intercept +of the line of best fit. +Prompt: np.dot(example_test,lin_model.coef_)+lin_model.intercept_ +array([28.95433166]) + +This will return the dot product of the example test and the coefficients of the Linear Regression model plus the +bias/intercept term of the LinearRegression model, which is the same prediction that we got with the +lin_model.predict() function. +Prompt: If I want to buy a home with a low crime rate and at least 3 rooms, how much would I pay? +You would need to provide more specific information such as the location of the home and other features of the +home in order to obtain a more accurate prediction of the price. However, you can use the Linear Regression model +that you have trained on the Boston Housing dataset to estimate the price of a home with a low crime rate and at +least 3 rooms by providing the model with the appropriate input values. You can use the lin_model.predict() method +and pass it an example test with the values of the features that you want to use, in this case the crime rate and the +number of rooms. +Prompt: Give me an example prediction using average values. +Sure, one way to estimate the price of a home with average values of crime rate and number of rooms is to take the +average values of those features in the Boston Housing dataset and use them as input to the Linear Regression +model. Here is an example code: + +average_crime_rate = boston['CRIM'].mean() +average_rooms = boston['RM'].mean() + +average_home = [[average_crime_rate, average_rooms]] + +average_home_price = lin_model.predict(average_home) +print("The predicted median value of owner-occupied homes in $1000's for an average +home is: ",average_home_price) +Please note that this is just an example and the prediction value will change depending on the input you provide +and the specific model that you are using. +Prompt: If I want as many rooms as possible, what combination of input variables leads to the lowest priced house? +In order to find the combination of input variables that leads to the lowest priced house with as many rooms as +possible, you would need to find the minimum value of the target variable (MEDV) in the dataset while maximizing +the number of rooms (RM) feature. One way to do this would be to sort the dataset by the number of rooms in +descending order, then iterate over the dataset and find the first row where the number of rooms is maximum and +the median value of owner-occupied homes is minimum. + +You could also use the method lin_model.predict() by providing a specific set of input values for the other features, +where the number of rooms is the maximum possible value from the dataset, and see the prediction of the model. + +Please note that this is just an example, and the results will change depending on the specific dataset you're working +with and the specific model that you're using. +Prompt: Calculate the example +Here is an example of how you can use the Boston Housing dataset to find the combination of input variables that +leads to the lowest priced house with as many rooms as possible: + +# Sort the dataset by number of rooms in descending order +boston_sorted = boston.sort_values(by='RM', ascending=False) + +# Find the first row where the number of rooms is maximum and the median value of +owner-occupied homes is minimum +min_medv += +boston_sorted[boston_sorted['RM'] +== +boston_sorted['RM'].max()]['MEDV'].min() +min_medv_index = boston_sorted[boston_sorted['MEDV'] == min_medv].index[0] + +# Get the input values for that row +input_values = boston_sorted.loc[min_medv_index, boston.columns != 'MEDV'] + +# Use the input values to predict the median value of owner-occupied homes +predicted_price = lin_model.predict([input_values]) +print("The predicted median value of owner-occupied homes in $1000's for a house with +the most rooms is: ",predicted_price) + +Please note that this is just an example, and the results will change depending on the specific dataset you're working +with and the specific model that you're using. +Prompt: I want you to act as a jupyter notebook. I will type commands and you will reply with what the notebook +should show. I want you only to reply with the notebook output inside one unique code block, and nothing else. Do +not write explanations. Do not type commands unless I instruct you to do so. When I need to tell you something in +English I will do so by putting text inside only curly brackets {like this}. +I understand, please let me know what command would you like me to run. +Prompt: +# Sort the dataset by number of rooms in descending order +boston_sorted = boston.sort_values(by='RM', ascending=False) + +# Find the first row where the number of rooms is maximum and the median value of owner- +occupied homes is minimum +min_medv += +boston_sorted[boston_sorted['RM'] +== +boston_sorted['RM'].max()]['MEDV'].min() +min_medv_index = boston_sorted[boston_sorted['MEDV'] == min_medv].index[0] + +# Get the input values for that row +input_values = boston_sorted.loc[min_medv_index, boston.columns != 'MEDV'] + +# Use the input values to predict the median value of owner-occupied homes +predicted_price = lin_model.predict([input_values]) +print("The predicted median value of owner-occupied homes in $1000's for a house with +the most rooms is: ",predicted_price) +The predicted median value of owner-occupied homes in $1000's for a house with the most rooms is: [7.93800919] +Please note that this is just an example, and the results will change depending on the specific dataset you're working +with and the specific model that you're using. + +APPENDIX E: ChatGPT Faker Dataset Interactions +Prompt: I want you to act as a jupyter notebook. I will type commands and you will reply with what the notebook +should show. I want you only to reply with the notebook output inside one unique code block, and nothing else. Do +not write explanations. Do not type commands unless I instruct you to do so. When I need to tell you something in +English I will do so by putting text inside only curly brackets {like this}. My first command is "print('hello world')" +ChatGPT: hello world +Prompt: +import pandas as pd, numpy as np +import random,requests +from bs4 import BeautifulSoup +from faker import Faker +#initialize Faker +fake=Faker() + +Prompt: +names=[] +address=[] +company=[] + +n_names=200 +for n in range(n_names): + names.append(fake.name()) + address.append(fake.address()) + company.append(fake.company()) + +claim_reason=["Medical","Travel","Phone","Other"] +Confidentiality_level=["High","Low","Medium","Very low"] + +claim_confidentiality_dict=dict(zip(claim_reason,Confidentiality_level)) + +claim_reasons=np.random.choice(claim_reason,n_names, p=[.55,.15,.15,.15]) +claim_confidentiality_levels=[claim_confidentiality_dict[claim_reasons[i]] +for +i +in +range(len(claim_reasons))] + + +Prompt: +variables=[names,address,company,claim_reasons,claim_confidentiality_levels] +df=pd.DataFrame(variables).transpose() +df.columns=["Customer +Name","Customer +Address","Company +Name","Claim +Reason","Data +confidentiality"] +df["Customer Address"]=df["Customer Address"].str.replace("\n",",") + +Prompt: df.tail(1) +Customer Name Customer Address Company Name Claim Reason Data confidentiality +195 Thomas Green Jr. MD PO Box 909, Suite 537, Lakeport, LA 70602, US Smith-Richardson +Medical High + +Prompt: +df["Claim Amount"]=0 + +for i in range(len(df)): + if df["Claim Reason"][i]=="Medical": + df["Claim Amount"][i]=np.random.randint(1300,2300) + elif df["Claim Reason"][i]=="Travel": + df["Claim Amount"][i]=np.random.randint(300,900) + elif df["Claim Reason"][i]=="Phone": + df["Claim Amount"][i]=np.random.randint(200,270) + else: + df["Claim Amount"][i]=np.random.randint(1,100) + +Prompt: df.head(1) +Customer Name Customer Address Company Name Claim Reason Data confidentiality Claim Amount +Category Premium Premium/Amount Ratio Claim Request output +0 Mr. Thomas N. Smith PO Box 854, Suite 709, Lakewood, NJ 08701, US Smith-Hicks Inc. dba +Smith-Smith Inc. Medical High 1406 9248 +0.151724 No +Prompt: df.shape() +(200, 9) +Prompt: +for col in df.columns: + print(col) +Customer Name +Customer Address +Company Name +Claim Reason +Data confidentiality +Claim Amount +Category Premium +Premium/Amount Ratio +Claim Request output +Prompt: What is the median value for all the columns that are numerical +Copy code +Customer Name NaN +Customer Address NaN +Company Name NaN +Claim Reason NaN +Data confidentiality NaN +Claim Amount 1498.5 +Category Premium 6124.0 +Premium/Amount Ratio NaN +Claim Request output NaN +dtype: float64 + + diff --git a/odFQT4oBgHgl3EQfrDYd/content/tmp_files/load_file.txt b/odFQT4oBgHgl3EQfrDYd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fff28ac4188df37c9b144f2c03a9dfbee5ddec02 --- /dev/null +++ b/odFQT4oBgHgl3EQfrDYd/content/tmp_files/load_file.txt @@ -0,0 +1,1744 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf,len=1743 +page_content='Numeracy from Literacy: Data Science as an Emergent Skill from Large Language Models David Noever1 and Forrest McKee2 PeopleTec, 4901-D Corporate Drive, Huntsville, AL, USA, 35805 1david.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='noever@peopletec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='com 2 forrest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='mckee@peopletec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content="com Abstract Large language models (LLM) such as OpenAI's ChatGPT and GPT-3 offer unique testbeds for exploring the translation challenges of turning literacy into numeracy." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Previous publicly-available transformer models from eighteen months prior and 1000 times smaller failed to provide basic arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' The statistical analysis of four complex datasets described here combines arithmetic manipulations that cannot be memorized or encoded by simple rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' The work examines whether next-token prediction succeeds from sentence completion into the realm of actual numerical understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' For example, the work highlights cases for descriptive statistics on in-memory datasets that the LLM initially loads from memory or generates randomly using python libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=" The resulting exploratory data analysis showcases the model's capabilities to group by or pivot categorical sums, infer feature importance, derive correlations, and predict unseen test cases using linear regression." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=" To extend the model's testable range, the research deletes and appends random rows such that recall alone cannot explain emergent numeracy." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Keywords: Exploratory Data Analysis, Transformers, Text Generation, Generative Pre-trained Transformers, GPT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' INTRODUCTION Three promising and challenging AI technologies are benchmarks for community research progress: autonomous driving, personal assistant, and chatbots [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=" OpenAI's ChatGPT combined the personal assistant with a chat interface in their late November 2022 public release [2-8]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Prompt or conversational customization of the ChatGPT API reveals the depth of its encyclopedic knowledge [7], somewhat akin to an effective Google advanced search or dynamically created Wikipedia entry [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Previous researchers have noted emergent features [10-19] beyond what a search engine, spidering indexer, or community-sourced compilation like Wikipedia might answer complex questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' This paper proposes several tasks that require ChatGPT to reason [10,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' While traditional challenge problems presented to large language models like "2+2=" have previously not satisfied any reasoning tests [21], the latest generation seems to display what the AI community might categorize as emergent properties [15-17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=" For instance, previous work highlighted ChatGPT's capability to mimic complex computer operating systems as if a hacker interacted with text commands [22-24]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' As an API interface, ChatGPT could serve as a dynamic honeypot with realistic responses [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' The present work extends this "out-of-the-box" simulation capability to role-play the data scientist or knowledge assistant as they perform exploratory data analysis [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=" ChatGPT's latest release (19JAN2023) incorporates a basic understanding of benchmark machine learning datasets like iris [25-26], Titanic survival [27-28], and Boston housing [29] without explicit programming." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=" Some critical tests of the LLM's reasoning or knowledge [30-35] include random re-sampling of available datasets and de novo generation from scratch." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' The present work examines whether ChatGPT possesses built-in knowledge of classic data science case studies like iris [25-26], Boston housing [29], and Titanic [27-28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=" Without the built-in capability to load data, the large language models simulate user interactions [22-24], including coded Python that edits the datasets and removes memorized responses from the model's responses." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Once the modified data receives prompts and queries, the LLM delivers emergent answers [15-17] based on its capability to perform arithmetic calculations or generate display code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Finally, each case presents word problem categories, such as "what demographic group most likely did not survive the Titanic crash?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' [27-28]".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' The paper presents a systematic version of exploratory data analysis (EDA) with linguistics models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' The models generate python code to execute in a Jupyter notebook or answer questions related to identifying correlations, trends, outliers, and missing values as one might anticipate as typical data science pre- processing routines or follow-up summaries based on post-processing results [37] (Appendices A-E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' The goal is to identify how well an LLM adapts to previously unseen datasets and generates plausible hints for extending the EDA [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Where possible, we generalize the results to either synthetic data that could not appear in the training data or to data slices that offer unique challenges from the well-known cases [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' For example, by adding or subtracting random rows of well-known datasets, we force the LLM to demonstrate whether it can creatively tailor its responses to prompts in ways that differ from a simple search engine reply [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' METHODS We organize the paper around the exploratory data analysis shown in Appendices A through E, as summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Appendix A establishes basic numeracy skills, including date-time manipulation and word problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Appendices B-D examine three tabular datasets where ChatGPT includes the data frame as part of its training data but receives customizations that make it impossible for the LLM to recall what previous steps or outcomes might apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' For instance, we use a random train-test split to the Titanic dataset to force the LLM to describe through its emergent statistical skills rather than return the standard answers that internet tutorials present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' One motivation for this approach stems from the failure of earlier LLMs to perform basic arithmetic based on the next token prediction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=" Appendix F adds a further test of ChatGPT's data science skills by creating a randomized insurance claim dataset valid for the session only and would not appear in any historical training from the internet archive." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Appendix and Title Data Science Goal Data Set Construction A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Basic Statistics and Numeracy Arithmetic, Date-Time Manipulation, Unit Conversions, Word Problems, Approximations Prompt-driven single values B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' ChatGPT Iris Dataset Interactions Descriptive statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' missing and duplicate value identification,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' variable correlations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' factor analysis and feature importance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' plot code generation using libraries (seaborn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' plotly),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' outlier identification,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' dataset augmentation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' IRIS dataset,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Petal and Sepal Width,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' and Length for Species Identification (Some knowledge already embedded) C: ChatGPT Titanic Dataset Interactions Descriptive statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' data frame operations such as drop columns,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' missing values,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' composite column creation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' python function generation and execution in place,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' random test-train split,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' feature importance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' pivot tables,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' and factor summation Titanic Survival Dataset based on passenger list demographics (Some knowledge embedded but modify data frame,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' so not memorized) Appendix and Title Data Science Goal Data Set Construction D: Boston Housing Dataset Descriptive statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' histogram and correlation analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' code generation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' model training preparation for low feature importance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' machine learning with linear regression models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' predict on unseen test cases from the trained model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' recommend purchase values as word problems Boston Housing Data (tabular mixture of categorical and numerical data for home value prediction based on demographics) E: ChatGPT Faker Dataset Interactions Mock dataset generation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' append column values,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' calculate descriptive statistics on numerical and ignore categorical variables Randomly generated insurance claim dataset with anonymized entries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' RESULTS For all five tests, the main result supports the hypothesis that the latest LLMs have reached sufficient scale to handle complex statistical questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' As proposed by the builders of GPT-3, these models offer public access to "zero-shot" or "few-shot" learning capabilities when scaled to sufficient parametric size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' The model encodes enough general knowledge to answer mathematical questions with plausible answers, even when presented with only a few (or no) examples of formatted requirements or context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=" Because ChatGPT provides memory within a given session to at least 8,000 tokens (25 pages), the model's coherence and relevance present new inquiries in data science." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' One might call this quality "emergent" because no rules or computational layers are explicitly defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' The following sections outline the general characteristics of the four datasets presented (Iris, Titanic, Boston Housing, synthetic) along with a chain of statistical calculations selected to highlight date-time manipulations, approximations, and word problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' It is worth noting that ChatGPT provides self-contained datasets to test, which proves critical to complete any analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' As an LLM frozen in time (2021) without any buffer or storage, the traditional steps needed to upload or present data fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' But having encountered the three well-known examples and one synthetic one, the model keeps track of each manipulation such that if a data row disappears, the resulting median or count changes accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Descriptive Statistics As illustrated in Appendix A, the model can add large numbers, reduce answers to N significant digits, identify divisors, and perform an order of magnitude calculation with unit conversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' When asked for the day of the week from history, the model correctly identifies the day from 60 years prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' While not remarkable from a lookup table or internet search, the model only generates the correct result using next- token prediction and language training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=" To highlight the model's capacity for manipulating extensive, multi-stage calculations, we prompt for the number of minutes in a decade, the number of inches between the Eiffel Tower and London Bridge, and the number of people who could fit on the island of Manhattan." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' ChatGPT answers incorrectly to identify the time zone that corresponds to six hours ahead of US Eastern (EST) (False: Greenwich GMT+6 or Bangladesh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' When instructed that the model responded incorrectly, ChatGPT shows a Universal Time formula UTC-5 as EST, followed by UTC-5+6=UTC+1, or Central European Time (CET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=" ChatGPT's capabilities to self-correct provide a novel user interface for redefining a precise question-and- answer sequence." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=" For example, asking the model to do distance calculations between two cities in small units like inches seems to raise the need for further explanation: What's the point of knowing urban-scale dimensions in such small increments?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' When pressed in follow-up inquiries, the response showcases the conversion of units (miles to inches) but begins with an incorrect distance (3,500 miles rather than 212 miles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' While the math is correct, the more specific initial conditions are flawed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' When asked a more eccentric estimation problem (the number of people standing shoulder to shoulder who could fit in Manhattan a densely packed single layer), ChatGPT responds with the correct initial condition for the area calculation (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='96 square miles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' If a person requires 2 square feet, the model fails to convert square miles to feet (ChatGPT: 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='9 million people vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' 318 million people in a 636 million sq foot area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' The model qualifies its answer as a safety, logistical, and health problem based on crowd control, then further amends its calculation to exclude parks, buildings, or non-built-up areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' As noted previously, ChatGPT has access to structured and organized datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' LLMs can perform the four basic software operations expected for databases: Create, Read, Update, and Delete (CRUD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' In Appendix B, the iris dataset describes the classification challenge to identify one of three flower species by its distinct petal and sepal dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' For this multi-class clustering, the model answers that there are no duplicates or missing values for the 50 examples of each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' When prompted to mimic a python interpreter (as a Jupyter notebook), the model responds with the expected output given a prompt in code alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' For example, using "data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='corr()" as the prompt produces the correct python output for the iris data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' We prompt the model to produce graphical code given a desired figure output (such as histograms, heatmaps, boxplots, pair plots, scatter, and distribution plots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Rather than a language-only model producing the requested figures directly, ChatGPT responds with the python libraries (plotly, seaborn, matplotlib) and codes, which run in a separate interpreter to give the graphs shown in Appendices B-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' When asked for interpretations based on the exploratory charts, the model responds with a description, such as a box-and-whiskers plot showing the quartiles and statistical outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=" ChatGPT does not limit its response to general code commentary for box plots but identifies the given dataset's variables and highlights conclusions for each class." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' While GitHub or internet tutorials might support ChatGPT training for this EDA, we alter the expected output by adding or deleting data frame rows to avoid the memorized response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' This way, the emergent capabilities for performing statistical inference get isolated from the baseline training inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=" Coding and Plots Appendices B-E focus on the four data science tasks to exercise ChatGPT's capabilities for python code generation." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Because the LLM offers no graphical output, the problem set transforms from the previous tasks to coding solutions to test using Jupyter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Both Codex and copilot have offered coding assistance since August 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' In Appendix B, ChatGPT shows the output of exploratory data analysis as displayed by python code for outliers, histograms, and distribution plots for the iris dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' In Appendix C, we ask the LLM to modify the Titanic dataset in preparation for categorical analysis and survivorship demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' The raw data (891 rows x 12 columns) offers irrelevant predictive variables ("PassengerID"), which we drop, then let ChatGPT pick up with the finer manipulation of the modified data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' The sequence of steps matter along the path to generating a final working dataset ready for machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' In the prompt, for instance, one can define python functions that recode the embarkation points, ticket prices, and passenger class with mappings from symbols to full names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' One further can bin age into five maturity categories between infant and elderly and distribute the passenger ages into ten-year brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' A further partition transforms the gender and marital status into categoricals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Once ChatGPT gets the python functions, the running of dataset modifications provides an in-memory style of output for further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=" It is worth noting that these steps illustrate how a language model serves as a computational interface to perform complex statistical actions, pivot groupings, and train-test splits that could not appear in the model's original corpus." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Once the unique Titanic data is created and plotted, ChatGPT can answer demographic questions about survivorship: third-class male passengers proved least likely to live through the crash and rescue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' In Appendix D, we perform essential machine learning (ML) steps that drop highly correlated variables and split the Boston housing data into train and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' We applied linear regression models from sci-kit learn python libraries and asked for root mean square error (RMSE) results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' The initial prompts without context led to coding suggestions but refused to perform calculations on an arbitrary train-test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' However, when prompted to act as a Jupyter notebook, the code output renders actual numerical RMSE and correlation coefficients (R-squared) values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' We created an example row to test the model and asked for a linear model prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' A series of word problems round out the Appendix E example, such that based on the data, the model highlights low-crime areas or numbers of rooms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' In a plausible real estate setting, the LLM answers with a data-driven response that a combination of many rooms and the lowest price might satisfy a buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' To our knowledge, this output seems unique to this scale of LLM in public access, both as a data science platform but also as capable of performing as an ML algorithm and predict on unseen inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' It is worth noting that previous models from the last few years, like GPT-2, failed on simple addition questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Emergent Understanding Appendix E establishes that a randomized dataset was created using the python library Faker to synthesize an anonymous insurance claim dataset that could not be repeated in previous LLM inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' This library makes categorical and numerical variables to include names, addresses, companies, claim reasons, and claim confidentiality levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' For the final mock dataset created, 200 rows and nine columns make up the in- memory capability of ChatGPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' When asked to reason over the nine columns, the LLM recognizes that 6 or 8 variables are categorical and that for the remaining two numerical categories, a median value emerges as the (randomized) claim amount of $1498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' This number appears differently every time the conversation commences, such that the net amount sums the medical, travel, phone, and unknown reasons are segmented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' The minimum possible value in this example would equal one, and the maximum (medical) claim would equal 2300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' While this sample of 200 values over many trials should converge to approximately 1650, the resulting language model performs a reasonable approximation in building the anonymized dataset for insurance claim values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' A current search engine (Google) value for: "Give me a random value between 1 and 2300" yields a link tree that samples 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='8 million examples on the internet but does not answer the arithmetic question specifically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' The referral engine links to calculators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' DISCUSSION The present work selects these demonstrations to illustrate the data science capabilities of large language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' The result extends previous efforts that highlight the computational capacity of language as an expressive and generative transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' There exist few obvious precursors to evolving numeracy from literacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' As recently as 18 months prior, the most advanced linguistic models could not perform elementary addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' One consequence of the emergent skill that exceeds the expectations of a python coder would include the comprehensive explanation of word problem challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' So not only does ChatGPT produce code from surveying Github, but it also reaches a natural (and relatively safe) conclusion based on the output of running sample code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' While previous work has demonstrated this "fake storefront" or "Hollywood stage" effect in ChatGPT when assuming different operating systems, honeypots, or characters in a play, the role of data scientist provides a novel representation to evolve exploratory analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' In the classic triad of iris, Titanic, and Boston housing, the work demonstrates that standard operations like pivoting, statistical observation, and anomaly detection suggest legitimate linguistic operations to supplement arithmetic understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Like young children, the LLM has some capacity for reasoning across symbolic abstraction (numeracy) and linguistic interpretation (literacy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' An obvious extension of this work would combine the symbolic and literate to translate word problems in multiple languages with complex alphabets like Chinese, Cyrillic, or Arabic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' In this way, one might imagine the union of symbolic AI with its more brute-force cousin as a trained transformer capable of compressing and representing the sum of human knowledge into "next token" predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' CONCLUSIONS In conclusion, the present work demonstrates large language models like ChatGPT carry a built-in capacity for performing numerical work using a basic linguistic representation and (attention-based) weights across a vast (40TB) dataset of human knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Presumably, no single branch of its 175 billion parameters encodes a given dataset like Titanic or Boston housing, but even without the capability to upload the data, the model knows and illustrates complex manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' If presented with 8000 tokens (around 25 pages) of a novel dataset, one can presume that ad hoc and de novo data science becomes possible within an otherwise numerically challenged token-centric model by appending it as a data frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' The work surveys basic and advanced operations, including CRUD, which makes a dataset otherwise impossible to memorize but amenable to a linguistics model that can summarize and coherently alter what it stores in memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' While ChatGPT set out to demonstrate the first chat interface that could survive both "safely and naturally" in the wild, what the scale of its operation may eventually reveal is emergent qualities that either are too complex for human traceability and validation or that survive in some over-fit quality from a few key transformer branches in the maze of internet databases for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' This work scratches the surface of what automated data science and exploratory analysis might evolve into given a language model that can calculate, infer and predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors thank the PeopleTec Technical Fellows program for encouragement and project assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' The authors thank the researchers at OpenAI for developing large language models and allowing public access to ChatGPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' REFERENCES [1] Lugano, G.' metadata={'source': 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pair programmer: Asset or Liability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='. arXiv preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='15331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' [41] Sobania, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=', Briesch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=', & Rothlauf, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' (2022, July).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Choose your programming copilot: a comparison of the program synthesis performance of github copilot and genetic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' In Proceedings of the Genetic and Evolutionary Computation Conference (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' 1019-1027).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' [32] Nguyen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=', & Nadi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' (2022, May).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=" An empirical evaluation of GitHub copilot's code suggestions." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' In Proceedings of the 19th International Conference on Mining Software Repositories (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' 1-5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' [43] Pearce, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=', Ahmad, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=', Tan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=', Dolan-Gavitt, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=', & Karri, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' (2022, May).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Asleep at the keyboard?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=" assessing the security of github copilot's code contributions." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' In 2022 IEEE Symposium on Security and Privacy (SP) (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' 754-768).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' [44] Döderlein, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=', Acher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=', Khelladi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=', & Combemale, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Piloting Copilot and Codex: Hot Temperature, Cold Prompts, or Black Magic?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='. arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='14699.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' [45] Wong, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=', Kothig, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=', & Lam, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Exploring the Verifiability of Code Generated by GitHub Copilot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='01766.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Exploring systematic consumption in US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Authors David Noever has research experience with NASA and the Department of Defense in machine learning and data mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' He received his BS from Princeton University and his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' from Oxford University, as a Rhodes Scholar, in theoretical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Forrest McKee has AI research experience with the Department of Defense in object detection and reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=" He received his Bachelor's (BS) and Master's (MSE) from the University of Alabama, Huntsville, Engineering." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' APPENDIX A: Basic Statistics and Numeracy Prompt: What is the square root if 23 to 5 significant digits?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' ChatGPT: The square root of 23 to 5 significant digits is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='8179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Prompt: What is 23456+65432 ChatGPT: The result of 23456+65432 is 88888.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Prompt: Give 5 divisors of 88888 ChatGPT: The divisors of 88888 are 1, 2, 4, 8, 11111, 22222, 44444 and 88888.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Prompt: Is it prime ChatGPT: No, 88888 is not a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' As I just mentioned, it has 8 divisors (1, 2, 4, 8, 11111, 22222, 44444 and 88888) which means it is divisible by more numbers than 1 and itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Prompt: Is it transcendental ChatGPT: No, 88888 is not a transcendental number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Transcendental numbers are non-algebraic numbers, which means they are not the root of any non-zero polynomial equation with integer coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' 88888 is an integer and it is not transcendental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Prompt: What day of the week was August 12, 1962?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' ChatGPT: August 12, 1962 was a Sunday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Prompt: What time zone is 6 hours ahead of us est ChatGPT: If a time zone is 6 hours ahead of US Eastern Standard Time (EST), it would be in the GMT+6 time zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' This would include places like Bangladesh, Kyrgyzstan, and parts of Russia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Prompt: Incorrect ChatGPT: I apologize for the confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' EST is UTC-5, so a time zone that is 6 hours ahead of EST would be UTC- 5+6=UTC+1, this time zone is known as Central European Time (CET) or Western European Summer Time (WEST) during the summer months when Daylight Saving Time is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Prompt: How many minutes in a decade ChatGPT: A decade is a period of 10 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' There are 525,600 minutes in a standard year (60 minutes per hour x 24 hours per day x 365 days per year).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' So, in a decade there are 525,600 x 10 = 5,256,000 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Prompt: How many inches between the eiffel tower and London Bridge ChatGPT: The distance between the Eiffel Tower in Paris, France and London Bridge in London, England is approximately 342 miles or 550 kilometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' It is not possible to convert this distance to inches without knowing the specific location where the Eiffel Tower and London Bridge are being measured from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' As the Earth is not a perfect sphere, the distance would also depend on the reference point on earth that is taken as the starting point to measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' I apologize if my previous answer was not helpful, if you have any other question please let me know.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Prompt: Approximately ChatGPT: I apologize, I misunderstood your question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' It is not possible to provide an exact distance in inches between the Eiffel Tower and London Bridge as it would depend on the specific location where the measurements are taken from and the curvature of the earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' However, I can approximate the distance between the two locations to be roughly 3,500 miles (5,600 km) which will be equivalent to approximately 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='5 x 10^9 inches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Please keep in mind that this is an approximation and the actual distance will be subject to slight variations depending on the precise location of the two structures and the reference point on earth that is taken as the starting point to measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Technical Note: Some appendix text generated from Large Language Model (LLM) for illustration purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' The author generated this text in part with GPT-3, OpenAI’s large-scale language-generation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Upon generating draft language, the authors reviewed, edited, and revised the language to their own liking and take ultimate responsibility for the content of this publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' OpenAI policy statement (2022) Prompt: Why do you keep saying the earth is curved ChatGPT: I apologize if my previous statements were not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' The Earth is approximately an oblate spheroid shape, meaning that it is slightly flattened at the poles and bulging at the equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=" This shape is caused by the Earth's rotation, which causes the planet to bulge at the equator and flatten at the poles." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=" When we measure the distance between two points on the surface of the Earth, we use a reference point on the surface of the Earth, and this distance is affected by the Earth's curvature." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Since the Earth is not a perfect sphere, the distance between two points will be different depending on the reference point on earth that is taken as the starting point to measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' I apologize for any confusion my previous statements may have caused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Prompt: So the earth is flattened ChatGPT: Yes, the Earth is flattened at the poles and bulging at the equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' This shape is known as an oblate spheroid, which means that the Earth is an almost spherical shape, but it is slightly flattened at the poles and bulging at the equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=" This shape is caused by the Earth's rotation, which causes the planet to bulge at the equator and flatten at the poles." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=" The Earth's equatorial diameter is approximately 43 kilometers (27 miles) larger than the diameter measured through the poles." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Prompt: How many people could fit if packed as one layer only on the island of Manhattan ny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Densely pack them with 2 square feet per person ChatGPT: If we densely pack people on the island of Manhattan with 2 square feet per person, the number of people that could fit on the island would be much higher than the estimate I provided earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' With an area of 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='96 square miles (59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='47 km²) and assuming 2 sq ft per person, Manhattan could fit approximately 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='9 million people in a densely packed single layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' However, it is important to note that this number is purely theoretical and would not be possible in reality due to safety, logistical, and health concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Crowds of that density would not be safe, people would not have room to move, and it would be a sanitary nightmare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Also, it is important to mention that this number is only considering the built-up areas and it is not taking into account that there are parks, buildings and other areas that would not be suitable for packing people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=" APPENDIX B: ChatGPT Iris Dataset Interactions Prompt: Use the Iris dataset and show the first 3 lines ChatGPT: I'm sorry, I am a text-based model and do not have the ability to access datasets." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' However, I can tell you that the Iris dataset is a commonly used dataset in machine learning and statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' It contains 150 samples of iris flowers, each with 4 features (sepal length, sepal width, petal length, and petal width) and one of three species (setosa, versicolor, or virginica).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' The first 3 lines of the Iris dataset may look something like this: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='1,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='5,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='4,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='2,setosa 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='9,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='4,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='2,setosa 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='7,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='3,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content="2,setosa It's important to notice that the dataset is usually split into two parts, one for training and one for testing, and the labels (setosa, versicolor, virginica) are usually separated from the features." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=" Prompt: Give the last 3 lines ChatGPT: Sure, the last 3 lines of the Iris dataset, assuming it's in the form of a CSV file, may look like this: 6." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='5,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='0,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='2,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='0,virginica 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='4,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='4,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='3,virginica 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='9,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='0,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content="8,virginica It's important to notice that there are 150 observations in total, so these last 3 lines will change if the dataset is modified." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Prompt: I want you to act as a jupyter notebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' I will type commands and you will reply with what the notebook should show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' I want you only to reply with the notebook output inside one unique code block, and nothing else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Do not write explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' Do not type commands unless I instruct you to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' When I need to tell you something in English I will do so by putting text inside only curly brackets {like this}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content=' My first command is "print(\'hello world\')" ChatGPT: hello world Prompt: import pandas as pd # Reading the CSV file df = pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='read_csv("Iris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='csv") # Printing top 5 rows df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='head() sepal length sepal width petal length petal width species 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='2 Iris-setosa 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='2 Iris-setosa 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='2 Iris-setosa 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='2 Iris-setosa 4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='2 Iris-setosa Prompt: df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content='shape ChatGPT: (150, 5) Prompt: df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFQT4oBgHgl3EQfrDYd/content/2301.13382v1.pdf'} +page_content="info() ChatGPT: ε +� +→ +0 for every ε > 0, that we shorten into: L<(S1;n) = 2√n + oP(√n). +In the context of card guessing games it is asked in [CDH+22, Question 4.3] the behaviour of +L<(Sk;n) for a fixed k. Using known results regarding longest increasing subsequences through +independent points we can make an educated guess. Indeed, let Ber(p) be a field of i.i.d. Bernoulli +1 +arXiv:2301.02557v1 [math.CO] 6 Jan 2023 + +random variables with mean p over the square lattice. Seppäläinen [Sep97, Th.1.] proved that +for fixed p and k, +L< +� +Ber(p) ∩ [0, kn] × [0, n] +� n→∞ +∼ +n × +√p +1 − p +� +2 +√ +k − (k + 1)√p +� +. +For p = 1/n there are in average k points of Ber(p) on each line of [0, kn] × [0, n]. Pretending +that we can safely let p depend on n in the above approximation we obtain +L<(Sk;n) ≈ L< +� +Ber(1/n) ∩ [0, kn] × [0, n] +� +≈ 2 +√ +kn. +The goal of the present paper is to make this approximation rigorous (however we are not going +to use Seppäläinen’s result but rather poissonize the problem). We actually adress this question +in the case where k depends on n. +Theorem 1 (Longest increasing subsequences). Let (kn) be a sequence of integers such that +kn ≤ n for all n. Then +E[L<(Skn;n)] = 2 +� +nkn − kn + o( +� +nkn). +(1) +(Of course if kn = o(n) then the RHS of (1) reduces to 2√nkn + o(√nkn).) If kn ≥ n for +some n then the naive greedy strategy shows very easily that E[L<(Skn;n)] ≥ n − o(n). +Theorem 2 (Longest non-decreasing subsequences). Let (kn) be an arbitrary sequence of inte- +gers. Then +E[L≤(Skn;n)] = 2 +� +nkn + kn + o( +� +nkn). +(2) +We are not aware of previous results for multiset permutations. However Theorems 1 and 2 +in the linear regime kn ∼ constant×n should be compared to a result by Biane ([Bia01, Theorem +3]). Indeed he obtains the exact limiting shape of the random Young Tableau induced through +the RSK correspondence by a random word Wq;N of q i.i.d. uniform letters in {1, 2, . . . , N} in the +regime where √q/N → c for some constant c > 0. (The regime √q/N → c corresponds to kn ∼ +c2n with our notation.) Regarding longest increasing subsequences we expect the asymptotics +of the word Wq;N to be close to that of the multiset permutation Sc2N;N and Theorems 1 and 2 +respectively suggest: +L<(Wq;N) ≈ L<(Sc2N;N) ∼ 2Nc − c2N ∼ (2 − c)√q, +L≤(Wq;N) ≈ L≤(Sc2N;N) ∼ 2Nc + c2N ∼ (2 + c)√q. +As the length of the first row (resp. the number of rows) in the Young Tableau corresponds +to the length of the longest non-decreasing subsequence in Wk;n (resp. the length of the longest +decreasing sequence) a weak consequence of ([Bia01, Theorem 3]) is that, in probability, +lim inf 1 +√qL<(Wq;N) ≥ (2 − c), +lim sup 1 +√qL≤(Wq;N) ≤ (2 + c), +which is indeed consistent with the above heuristic. +Strategy of proof and organization of the paper. +In Section 2 we first provide the proof +of Theorems 1 and 2 in the case of a constant or slowly growing sequence (kn). The proof is +elementary (assuming known the Veršik-Kerov Theorem). +For the general case we borrowed a few tools in the literature. In particular we first study +poissonized versions of L<(Skn;n), L≤(Skn;n). As already suggested by Hammersley ([Ham72], +2 + +Sec.9) and achieved by Aldous-Diaconis [AD95] the case k = 1 can be tackled by considering +an interacting particle system which is now known as the Hammersley or Hammersley-Aldous- +Diaconis particle system. +In Section 3 we introduce and analyze the two variants1 of the Hammersley process adapted +to multiset permutations. The standard path to analyze Hammersley-like models consists in +using subadditivity to prove the existence of a limiting shape and then proving that this limiting +shape satisfies a variational problem. Typically this variational problem is solved using convex +duality (see [Sep97, CG19]). The issue here is that since we allow kn to have different scales we +cannot use this approach and we need to derive non-asymptotic bounds for both processes. This +is the purpose of Theorem 8 whose proof is the most technical part of the paper. In Section 4 we +detail the multivariate de-poissonization procedure in order to conclude the proof of Theorem 1. +De-poissonization is more convoluted for non-decreasing subsequences: see Section 5. +Beyond expectation. +In the course of the proof we actually obtain results beyond the esti- +mation of the expectation. We obtain concentration inequalities for the poissonized version of +L<(Skn;n), L≤(Skn;n): see Theorem 8 and also the discussion in Section 5.3. Regarding fluctua- +tions a famous result by Baik, Deift and Johansson [BDJ99, Th.1.1] states that +L≤(S1;n) − 2√n +n1/6 +(d) +→ TW +where TW is the Tracy-Widom distribution. For L<(Sk;n) with fixed k we are quite confident +that the particle system approach and more precisely the analysis of second class particles (see +[CG06, CG19]) could be adapted and would yield that the fluctuations are also of order n1/6. For +general (kn) the intuition given by the comparison with the Hammersley process would suggest +that the fluctuations of L<(Skn;n) might be of order (knn)1/6 as long as (kn) does not grow too +fast. Yet we have no evidence for this. +2 +Preliminaries: the case of small kn +We first prove Theorems 1 and 2 in the case of a small sequence (kn). We say that a sequence +(kn) of integers is small if +k2 +n(kn)! = o(√n). +(3) +Note that a sequence of the form kn = (log n)1−ε is small while kn = log n is not small. +Proof of Theorems 1 and 2 in the case of a small sequence (kn). (In order to lighten notation we +skip the dependence in n and write k = kn.) +Let σkn be a random uniform permutation of size kn. We can associate to σkn a k-multiset +permutation Sk;n in the following way. For every 1 ≤ i ≤ kn we put +Sk;n(i) = ⌈σ(i)/k⌉. +It is clear that Sk;n is uniform and we have +L<(Sk;n) ≤ L≤(σkn) ≤ L≤(Sk;n). +1The first one is very close to the historical Hammersley process, I discovered during the preparation of this +article that the second one had recently appeared in A.Boyer’s PhD Thesis [Boy22] with a connection to the +O’Connell-Yor Brownian polymer. +3 + +The Veršik-Kerov Theorem says that the middle term in the above inequality grows like 2 +√ +kn. +Hence we need to show that if (kn) is small then +L≤(Sk;n) = L<(Sk;n) + oP( +√ +kn), +which is a big step towards the small case of Theorems 1 and 2. For this purpose we introduce +for every δ > 0 the event +Eδ := +� +L≤(Sk;n) ≥ L<(Sk;n) + δ√n +� +. +(4) +If Eδ occurs then in particular there exists a non-decreasing subsequence with δ√n ties, i.e. +points of Sk;n which are at the same height as their predecessor in the subsequence. These ties +have distinct heights 1 ≤ i1 < · · · < iℓ ≤ n for some δ√n/k ≤ ℓ ≤ δ√n. Fix +• Integers m1, . . . , mℓ ≥ 2 such that (m1 − 1) + · · · + (mℓ − 1) = δ√n ; +• Column indices r1,1 < · · · < r1,m1 < r2,1 < r2,m1 < · · · < rℓ,1 < · · · < r1,mℓ. +We then introduce the event +F = F ((iℓ)ℓ, (ri,j)i≤ℓ,j≤mi) += {S(r1,1) = · · · = S(r1,m1) = i1, S(r2,1) = · · · = S(r2,m1) = i2, . . . , S(rℓ,1) = · · · = S(r1,mℓ) = iℓ} . +By the union bound (we skip the integer parts) +P(Eδ) ≤ +� +δ√n/k≤ℓ≤δ√n +� +1≤i1<···≤iℓ≤n +� +(ri,j)i≤ℓ,j≤mi +P (F ((iℓ)ℓ, (ri,j)i≤ℓ,j≤mi)) . +Using +card +�� +mi = δ√n + ℓ; each mi ≥ 2 +� += card +�� +pi = δ√n; each pi ≥ 1 +� += +�δ√n − 1 +ℓ − 1 +� +we obtain the upper bound +� +(ri,j)i≤ℓ,j≤mi +P(F) = +1 +� +kn +k k ... k +� +� nk +� mi +� +� +�� +� +choices of r’s +�δ√n − 1 +ℓ − 1 +� +� +�� +� +choices of m′ +is +� +kn − � mi +(k − m1) (k − m2) . . . (k − mℓ)k . . . k +� +� +�� +� +choices of kn − � mi remaining points += +(k!)ℓ(δ√n − 1)! +(δ√n + ℓ)!(δ√n − ℓ)!(ℓ − 1)!(k − m1)!(k − m2)! × · · · × (k − mℓ)! +≤ +(k!)ℓ +(δ√n)ℓ+1(δ√n − ℓ)!(ℓ − 1)!. +We now sum over 1 ≤ i1 < · · · ≤ iℓ ≤ n and then sum over ℓ: +P(Eδ) ≤ +δ√n +� +ℓ=δ√n/k +�n +ℓ +� +(k!)ℓ +(δ√n)ℓ+1(δ√n − ℓ)!(ℓ − 1)! +≤ +δ√n−3 +� +ℓ=δ√n/k +�n +ℓ +� +(k!)ℓ +(δ√n)ℓ+1(δ√n − ℓ)!(ℓ − 1)! + 3 +� n +δ√n +� +(k!)δ√n +(δ√n)δ√n−2(δ√n − 3)! +(5) +4 + +i1 +iℓ +r1,1 +r1,2 +r1,m1 +i2 +r2,1 +r2,m2 +rℓ,1 +rℓ,mℓ +. . . +Figure 1: The event F. (Ties are surrounded in red. Points with blue background represent the +subsequence with δ√n ties.) +Using the two following inequalities valid for every j ≤ m (see e.g. [CLRS09, eq.(C.5)]) +�m +j +� +≤ +�me +j +�j +, +m! ≥ mm exp(−m) +we first obtain that if kn! = o(√n) (which is the case if (kn) is small) then the last term of (5) +tends to zero. Regarding the sum we write +P(Eδ) ≤ +δ√n−3 +� +ℓ=δ√n/k +�ne +ℓ +�ℓ +(k!)ℓ +(δ√n)ℓ+1(δ√n − ℓ)δ√n−ℓe−δ√n+ℓ(ℓ − 1)ℓ−1e−ℓ+1 + o(1) +≤ +δ√n−3 +� +ℓ=δ√n/k +�nek!(δ√n − ℓ) +δ√nℓ(ℓ − 1) +�ℓ (ℓ − 1)e−1 +δ√n +� +e +δ√n − ℓ +� +�� +� +≤e/3<1 +�δ√n + o(1) +≤ +δ√n−3 +� +ℓ=δ√n/k +�√nek!(δ√n − ℓ) +δℓ(ℓ − 1) +�ℓ (ℓ − 1)e−1 +δ√n +� +e +δ√n − ℓ +�ℓ + o(1) +≤ +δ√n−3 +� +ℓ=δ√n/k +� √ne2k! +δℓ(ℓ − 1) +�ℓ (ℓ − 1)e−1 +δ√n ++ o(1), +(6) +which tends to zero for every δ > 0, as long as (kn) satisfies (3). This proves that L≤(Sk;n) = +L<(Sk;n) + oP( +√ +kn). Using the crude bounds L<(Sk;n) ≤ n and L≤(Sk;n) ≤ kn, eq.(6) also +implies that +E[L≤(Sk;n)] = E[L<(Sk;n)] + o( +√ +kn). +3 +Poissonization: variants of the Hammersley process +Remark. In the sequel, Poisson(µ) (resp. Binomial(n, q)) stand for generic random variables +with Poisson distribution with mean µ (resp. Binomial distribution with parameters n, q). +5 + +Π(λ) +i +x +i +x +Sources ∼ PPP(α) +Sinks ∼ Bernoulli(p) +x +x +Sources ∼ PPP(β) +Sinks ∼ Geometric≥0(1-β⋆) +Figure 2: Our four variants of the Hammersley process (time goes from bottom to top, trajectories +of particules are indicated in blue). Top left: The process L<(t). Top right: The process L≤(t). +Bottom left: The process L(α,p) +< +(t). Bottom right: The process L(β,β⋆) +≤ +(t). +Notation Geometric≥0(1−β) stands for a geometric random variable with the convention P(Geometric≥0(1− +β) = k) = (1 − β)βk for k ≥ 0. In particular E[Geometric≥0(1 − β)] = +β +1−β. +3.1 +Definitions of the processes L<(t) and L≤(t) +In this Section we define formally and analyze two semi-discrete variants of the Hammersley +process. +For a parameter λ > 0 let Π(λ) be the random set Π(λ) = ∪iΠ(λ) +i +where Π(λ) +i +’s are independent +and each Π(λ) +i +is a homogeneous Poisson Point Process (PPP) with intensity λ on (0, ∞) × {i}. +For simplicity set +Π(λ) +x,t = Π(λ) ∩ ([0, x] × {1, . . . , t}) . +The goal of the present section is to obtain non-asymptotic bounds for L< +� +Π(λ) +x,t +� +and L≤ +� +Π(λ) +x,t +� +. +Fix x > 0 throughout the section. For every t ∈ {0, 1, 2, . . . } the function y ∈ [0, x] �→ L<(y, t) +(resp. +L≤(y, t)) is a non-decreasing integer-valued function whose all steps are equal to +1. +Therefore this function is completely determined by the finite set +L<(t) := +� +y ≤ x, L<(y, t) = L<(y, t−) + 1 +� +. +(Respectively: +L≤(t) := +� +y ≤ x, L≤(y, t) = L≤(y, t−) + 1 +� +.) +Sets L<(t) and L≤(t) are finite subsets of [0, x] whose elements are considered as particles. It +is easy to see that for fixed x > 0 both processes (L<(t))t and (L≤(t))t are Markov processes +taking their values in the family of point processes of [0, x]. +6 + +Exactly the same way as for the classical Hammersley process ([Ham72, Sec.9], [AD95]) the +individual dynamic of particles is very easy to describe: +• The process L< . We put L<(0) = ∅. In order to define L<(t+1) from L<(t) we consider +particles from left to right. A particle at y in L<(t) moves at time t + 1 at the location of +the leftmost available point z in Π(λ) +t+1 ∩ (0, y) (if any, otherwise it stays at y). This point z +is not available anymore for subsequent particles, as well as every other point of Π(λ) +t +∩(0, y). +0 +x +Configuration L<(t) +t+1 +Configuration L<(t+1) +y +z +y′ +If there is a point in Π(λ) +t+1 which is on the right of y′ := max{L<(t)} then a new particle is +created in L<(t + 1), located at the leftmost point in Π(λ) +t+1 ∩ (y′, x). (In pictures this new +particle comes from the right.) +A realization of L< is shown on top-left of Fig.2. +• The process L≤ . We put L≤(0) = ∅. In order to define L≤(t + 1) from L≤(t) we also +consider particles from left to right. A particle at y in L≤(t) moves at time t + 1 at the +location of the leftmost available point z in Π(λ) +t +∩ (0, y). This point z is not available +anymore for subsequent particles, other points in (z, y) remain available. +0 +x +Configuration L≤(t) +t+1 +Configuration L≤(t+1) +If there is a point in Π(λ) +t+1 which is on the right of y′ := max{L<(t)} then new particles are +created in L<(t + 1), one for each point in Π(λ) +t+1 ∩ (y′, x). +A realization of L≤ is shown on top-right of Fig.2. +Processes L<(t) and L≤(t) are designed in such a way that they record the length of longest +increasing/non-decreasing paths in Π. In fact particles trajectories correspond to the level sets +of the functions (x, t) �→ L< +� +Π(λ) +x,t +� +, (x, t) �→ L≤ +� +Π(λ) +x,t +� +. +Proposition 3. For every x, +L< +� +Π(λ) +x,t +� += card(L<(t)), +L≤ +� +Π(λ) +x,t +� += card(L≤(t)), +where on each right-hand side we consider the particle system on [0, x]. +7 + +Proof. We are merely restating the original construction from Hammersley ([Ham72], Sec.9). We +only do the case of L<(t). +Let us call each particle trajectory a Hammersley line. By construction each Hammersley line +is a broken line starting from the right of the box [0, x] × [0, t] and is formed by a succession of +north/west line segments. Because of this, two distinct points in a given longest increasing subse- +quence of Π(λ) +x,t cannot belong to the same Hammersley line. Since there are L<(t) Hammersley’s +lines this gives L< +� +Π(λ) +x,t +� +≤ card(L<(t)). +In order to prove the converse inequality we build from this graphical construction a longest +increcreasing subsequence of Π(λ) +x,t with exactly one point on each Hammersley line. To do so, we +order Hammersley’s lines from bottom-left to top-right, and we build our path starting from the +top-right corner. We first choose any point of Π(λ) +x,t belonging to the last Hammersley line. We +then proceed by induction: we choose the next point among the points of of Π(λ) +x,t lying on the +previous Hammersley line such that the subsequence remains increasing. (This is possible since +Hammersley’s lines only have North/West line segments.) This proves L< +� +Π(λ) +x,t +� +≥ card(L<(t)). +3.2 +Sources and sinks: stationarity +Proposition 3 tells us that in on our way to prove Theorem 1 and Theorem 2 we need to +understand the asymptotic behaviour of processes L<, L≤. These processes are far from being +stationary as particles may appear all the time in the system and never disappear. To solve +this issue we use the trick of sources/sinks introduced formally and exploited by Cator and +Groeneboom [CG05] (following the intuition given by [AD95]): +• Sources form a finite subset of [0, x] × {0} which plays the role of the initial configuration +L<(0), L≤(0). +• Sinks are points of {0} × [1, t] which add up to Π(λ) when one defines the dynamics of +L<(t), L≤(t). For L≤(t) it makes sense to add several sinks at the same location (0, i) so +sinks may have a multiplicity. +Examples of dynamics of L<, L≤ under the influence of sources/sinks is illustrated at the bottom +of Fig.2. +Lemma 4. For every λ, α > 0 let L(α,p) +< +(t) be the Hammersley process defined as L<(t) with: +• sources distributed according to a homogeneous PPP with intensity α on [0, x] × {0} ; +• sinks distributed according to i.i.d. Bernoulli(p) with +λ +λ + α = p. +(7) +If sources, sinks, and Π(λ) are independent then the process +� +L(α,p) +< +(t) +� +t≥0 is stationary. +Lemma 5. For every β > λ > 0 let L(β,β⋆) +≤ +(t) be the Hammersley process defined as L≤(t) with: +• sources distributed according to a homogeneous PPP with intensity β on [0, x] × {0} ; +• sinks distributed according to i.i.d. Geometric≥0(1 − β⋆) with +β⋆β = λ. +(8) +8 + +If sources, sinks, and Π(λ) are independent then the process +� +L(β,β⋆) +≤ +(t) +� +t≥0 is stationary. +Proof of Lemmas 4 and 5. Lemma 5 could be obtain from several minor adjustments of [Boy22, +Chap.3, Lemma 3.2]. (Be aware that we have to switch x ↔ t and sources ↔ sinks in [Boy22] in +order to fit our setup.) For the sake of the reader we however propose the following alternative +proof. +Consider for some fixed t ≥ 1 the process (Hy)0≤y≤x given by the number of Hammersley +lines passing through the point (y, t). +0 +x +Configuration L≤(t-1) +The corresponding process (Hy) +x +t +The initial value H0 is the number of sinks at (0, t), which is distributed as a Geometric≥0(1− +β⋆). The process (Hy) is a random walk (reflected at zero) with ’+1 rate’ equal to λ and ’−1 +rate’ equal to β. (Jumps of (Hy) are independent from sinks as sinks are independent from Π(λ).) +The Geometric≥0(1−β⋆) distribution is stationary for this random walk exactly when (8) holds. +The set of points of L(β,β⋆) +≤ +(t) is given by the union of Π(λ) +t +and the points of L(β,β⋆) +≤ +(t) that do +not correspond to a ’−1’ jump. A elementary computation with exponential and gamma random +variables then shows that this is distributed as a homogeneous PPP with intensity β. +Lemma 4 is proved exactly in the same way (alternatively one can mimic the proof of [CG05, +Th.3.1.] as the dynamics is the same as in the classical Hammersley process) so we omit details. +We simply explain where (7) comes from. +If Lemma 4 is true for some α, p, λ then in particular the number of particles in the system +should be constant in expectation. At time t a particle leaves the system (from the left) if and +only if there is a sink at height t, which occurs with probability p. On the other hand a new +particle appears from the right if there is a point in Π(λ) +t +which is on the right of the right-most +particle of L<(t). This occurs with probability λ/(α + λ), hence we need (7) to hold so that the +expected number of particles remain constant. +3.3 +Processes L<(t) and L≤(t): non-asymptotic bounds +From Lemmas 4 and 5 it is straightforward to derive non-asymptotic upper bounds for L<(t), L≤(t). +For y ≤ x let So(α) +x +be the random set of sources with intensity α and for s ≤ t let Si(p) +t +the +random set of sinks with intensity p. It is convenient to use the notation L=<(P) which is, as +before, the length of the longest increasing path taking points in P but when the path is also +allowed to go through several sources (which have however the same y-coordinate) or several +sinks (which have the same x-coordinate). Formally, +L=<(P) = max {L; there exists P1 =≺ P2 =≺ · · · =≺ PL, where each Pi ∈ P} , +9 + +where +(x, y) =≺ (x′, y′) if +� +� +� +� +� +x < x′ and y < y′, +or +x = x′ = 0 and y < y′, +or +x < x′ and y = y′ = 0. +Proposition 3 can be generalized in +L=< +� +Π(λ) +x,t ∪ So(α) +x +∪ Si(p) +t +� += L(α,p) +< +(t) + card(Si(p) +t ). +(9) +Lemma 6 (Upper bound for L<). For every α, p ∈ (0, 1) such that (7) holds, there is a stochastic +domination of the form: +L< +� +Π(λ) +x,t +� +≼ Poisson(xα) + Binomial(t, p). +(10) +(The Poisson and Binomial random variables involved in (10) are not independent.) +Proof. Adding sources and sinks may not decrease longest increasing paths. Thus, +L< +� +Π(λ) +x,t +� +≼ L=< +� +Π(λ) +x,t ∪ So(α) +x +∪ Si(p) +t +� += L(α,p) +< +(t) + card(Si(p)) (using (9)) +(d) += L(α,p) +< +(0) + card(Si(p)) (using stationarity: Lemma 4) +(d) += Poisson(xα) + Binomial(t, p). +Taking expectations in (10) we obtain +E +� +L< +� +Π(λ) +x,t +�� +≤ xα + tp. +As the LHS in the above equation does not depend on α, p (provided that α, p satisfies (7)) we +will apply (10) with the minimizing choice +¯α, ¯p := argminα,p satisfying (7) {xα + tp} , +i.e. +¯α = +� +tλ +x − λ, +¯p = +� +xλ +t , +x¯α + t¯p = 2 +√ +xtλ − xλ. +(11) +We have proved +E +� +L< +� +Π(λ) +x,t +�� +≤ 2 +√ +xtλ − xλ. +(Compare with (1).) We have a similar statement for non-decreasing subsequences: +Lemma 7 (Upper bound for L≤). For every β, β⋆ ∈ (0, 1) such that (8) holds, there is a +stochastic domination of the form: +L≤ +� +Π(λ) +x,t +� +≼ Poisson(xβ) + G(β⋆) +1 ++ · · · + G(β⋆) +t +, +(12) +where G(β⋆) +i +’s are i.i.d. Geometric≥0(1 − β⋆). +10 + +We put +¯β, ¯β⋆ := argminβ,β⋆ satisfying (8) +� +xβ + t +� +β⋆ +1 − β⋆ +�� +, +(13) +i.e. +¯β = +� +tλ +x + λ, +¯β⋆ = +1 +1 + +� +t/xλ +, +x¯β + t +� +¯β +1 − ¯β +� += 2 +√ +xtλ + xλ. +(14) +(In particular ¯β > λ, as required in Lemma 5.) Eq.(12) yields +E +� +L≤ +� +Π(λ) +x,t +�� +≤ 2 +√ +xtλ + xλ. +(15) +(Compare with (2).) +Theorem 8 (Concentration for L<, L≤). There exist strictly positive functions g, h such that +for all ε > 0 and for every x, t, λ such that t ≥ xλ +P(L<(Π(λ) +x,t ) > (1 + ε)(2 +√ +xtλ − xλ)) ≤ exp(−g(ε)( +√ +xtλ − xλ)), +(16) +P(L<(Π(λ) +x,t ) < (1 − ε)(2 +√ +xtλ − xλ)) ≤ exp(−h(ε)( +√ +xtλ − xλ)). +(17) +Similarly: +P(L≤(Π(λ) +x,t ) > (1 + ε)(2 +√ +xtλ + xλ)) ≤ exp(−g(ε) +√ +xtλ), +(18) +P(L≤(Π(λ) +x,t ) < (1 − ε)(2 +√ +xtλ + xλ)) ≤ exp(−h(ε) +√ +xtλ). +(19) +For the proof of Theorem 8 we will focus on the case of L<, i.e. eq.(16), (17). When necessary +we will give the slight modification needed to prove eq.(18) and (19). The beginning of the proof +mimics Lemmas 4.1 and 4.2 in [BEGG16]. +We first prove similar bounds for the stationary processes with minimizing sources and sinks. +Lemma 9 (Concentration for L< with sources and sinks). Let ¯α, ¯p be defined by (11). There +exists a strictly positive function g1 such that for all ε > 0 and for every x, t, λ such that t ≥ xλ +P(L=<(Π(λ) +x,t ∪ So(¯α) +x +∪ Si(¯p) +t ) > (1 + ε)(2 +√ +xtλ − xλ)) ≤ 2 exp(−g1(ε)( +√ +xtλ − xλ)) +(20) +P(L=<(Π(λ) +x,t ∪ So(¯α) +x +∪ Si(¯p) +t ) < (1 − ε)(2 +√ +xtλ − xλ)) ≤ 2 exp(−g1(ε)( +√ +xtλ − xλ)). +(21) +Proof of Lemma 9. By stationarity (Lemma 4) we have +L=<(Π(λ) +x,t ∪ So(¯α) +x +∪ Si(¯p) +t ) +(d) += Poisson(x¯α) + Binomial(t, ¯p). +Then +P(L=<(Π(λ) +x,t ∪ So(¯α) +x +∪ Si(¯p) +t +> (1 + ε)(2 +√ +xtλ − xλ)) ≤ P +� +Poisson(x¯α) > (1 + ε +2)( +√ +xtλ − xλ) +� ++ P +� +Binomial(t, ¯p) > (1 + ε +2) +√ +xtλ +� +. +Recall that x¯α = +√ +xtλ−xλ, t¯p = +√ +xtλ. Using the tail inequality for the Poisson distribution +(Lemma 14): +P +� +Poisson(x¯α) > (1 + ε +2)( +√ +xtλ − xλ) +� +≤ exp +� +−( +√ +xtλ − xλ)ε2/4 +� +Using the tail inequality for the binomial (Lemma 15) we get +P +� +Binomial(t, ¯p) > (1 + ε +2) +√ +xtλ +� +≤ exp(− 1 +12ε2√ +xtλ) ≤ exp(− 1 +12ε2( +√ +xtλ − xλ)) +The proof of (21) is identical. This shows Lemma 9 with g1(ε) = ε2/12. +11 + +x +εx +Maximizing path for L=< +Maximizing path for L⋆ +<,ε +Point of Π(λ) +source +sink +Figure 3: A sample of Π(λ) +x,t , sources, sinks, and the corresponding trajectories of particles (in +blue). Here L=<(Π(λ) +x,t ∪ So(α) +x +∪ Si(p) +t ) = 5 (pink path) and L(α,p) +< +(t) = 2 (two remaining particles +at the top of the box). +For longest non-decreasing subsequences we have a statement similar to Lemma 9. The only +modification in the proof is that in order to estimate the number of sinks one has to replace +Lemma 15 (tail inequality for the Binomial) by Lemma 16 (tail inequality for a sum of geometric +random variables). During the proof we need to bound +√ +xtλ + xλ by +√ +xtλ, this explains the +form of the right-hand side in eq.(18) and (19). +Proof of Theorem 8. Adding sources/sinks may not decrease L< so +L=<(Π(λ) +x,t ∪ So(¯α) +x +∪ Si(¯p) +t ) ≽ L<(Π(λ) +x,t ), +thus the upper bound (16) is a direct consequence of Lemma 9. +Let us now prove the lower bound. We consider the length of a maximizing path among those +using sources from 0 to εx and then only increasing points of Π(λ) +x,t ∩ ([εx, x] × [0, t]) (see Fig.3). +Formally we set +L⋆ +=<,ε := card +� +So(¯α) +εx +� ++ L< +� +(Π(λ) +x,t ∩ ([εx, x] × [0, t]) +� (d) += Poisson(εx¯α) + L< +� +(Π(λ) +x,t ∩ ([εx, x] × [0, t]) +� +(22) +The idea is that for any fixed ε the paths contributing to L⋆ +=<,ε will typically not contribute to +L=< +� +Π(λ) +x,t ∪ So(¯α) +x +∪ Si(¯p) +t +� += L(¯α,p) +< +(t) + card(Si(¯p) +t ). Indeed eq.(22) suggests that for large x, t +L⋆ +=<,ε ≈ E[Poisson(εx¯α)] + E +� +L< +� +Π(λ) +x,t ∩ ([εx, x] × [0, t]) +�� +≈ xε¯α + 2 +� +x(1 − ε)λt − x(1 − ε)λ += 2 +√ +xλt − xλ − +√ +xtλδ(ε), +where δ(ε) = 2−ε−2√1 − ε is positive and increasing. In order to make the above approximation +rigorous we first write +2 +√ +xλt − xλ − 1 +2 +√ +xtλδ(ε) = xε¯α + 1 +4 +√ +xtλδ(ε) + 2 +� +x(1 − ε)λt − x(1 − ε)λ + 1 +4 +√ +xtλδ(ε). (23) +12 + +Combining (22) and (23) gives +P +� +L⋆ +<,ε ≥ 2 +√ +xλt − xλ − 1 +2 +√ +xtλδ(ε) +� +≤ P +� +Poisson(xε¯α) ≥ xε¯α + 1 +4 +√ +xtλδ(ε) +� ++ P +� +L<(Π(λ) +x,t ∩ ([εx, x] × [0, t]) ≥ 2 +� +x(1 − ε)λt − x(1 − ε)λ + 1 +4 +√ +xtλδ(ε) +� +≤ exp +� +− +xtλδ(ε)2 +16 × 4ε2( +√ +xtλ − xλ) +� +(using Lemma 14) ++ P +� +L<(Π(λ) +x,t ∩ ([εx, x] × [0, t]) ≥ 2 +� +x(1 − ε)λt − x(1 − ε)λ + 1 +8(2 +� +x(1 − ε)λt − x(1 − ε)λ)δ(ε) +� +≤ exp +� +− +√ +xtλδ(ε)2/(64ε2) +� ++ P +� +L<(Π(λ) +x,t ∩ ([εx, x] × [0, t]) ≥ +� +2 +� +x(1 − ε)λt − x(1 − ε)λ +� +× (1 + 1 +8δ(ε)) +� +≤ exp +� +− +√ +xtλδ(ε)2/(64ε2) +� ++ exp +� +−g(δ(ε)/8)( +� +x(1 − ε)tλ − x(1 − ε)λ) +� +(using the upper bound (16)) +and thus we can find some positive h such that +P +� +L⋆ +<,ε ≥ 2 +√ +xλt − xλ − 1 +2 +√ +xtλδ(ε) +� +≤ exp +� +−h(ε)( +√ +xtλ − xλ) +� +. +(24) +One proves exactly in the same way a similar bound for the length of a maximizing path among +those using sinks in {0} × [0, εt] and then only increasing points of Π(λ) +x,t ∩ ([0, x] × [εt, t]). +Choose now one of the maximizing paths P for L=< +� +Π(λ) +x,t ∪ So(¯α) +x +∪ Si(¯p) +t +� +(if there are many +of them, choose one arbitrarily in a deterministic way: the lowest, say). Denote by sources(P) +and sinks(P) the number of sources and sinks in the path P: +sources(P) = card {0 ≤ y ≤ x such that (y, 0) ∈ P} . +In Fig.3 the path P is sketched in pink and sources(P) = 2, sinks(P) = 0. +Lemma 10. There exists a positive function ψ such that for all real η > 0 +P +� +sources(P) + sinks(P) ≥ η +√ +xλt +� +≤ 2 exp(−ψ(η)( +√ +xλt − xλ)). +Proof of Lemma 10. If the event +� +sources(P) ≥ η +√ +xλt +� +holds then there exists a (random) ε +such that the two following events hold: +• Soεx ≥ η +√ +xλt ; +• L⋆ +=<,ε = L=< +� +Π(λ) +x,t ∪ So(¯α) +x +∪ Si(¯p) +t +� += L(¯α,¯p) +< +(t) + card(Si(¯p) +t ). +This implies that this random ε is larger than η/2 > 0 unless the number of sources in [0, xη/2] +is improbably high: +P(sources(P) ≥ η +√ +xλt) ≤ P(Soηx/2 ≥ η +√ +xλt) + P(sources(P) ≥ η +√ +xλt; Soηx/2 < η +√ +xλt) +≤ P(Soηx/2 ≥ η +√ +xλt) ++ P(L(¯α,¯p) +< +(t) ≤ +√ +xλt − xλ − 1 +4δ(η/3) +√ +xλt) ++ P(card(Si(¯p) +t ) ≤ +√ +xλt − 1 +4δ(η/3) +√ +xλt) ++ P(L⋆ +=<,ε ≥ 2 +√ +xλt − xλ − 1 +2δ(η/3) +√ +xλt for some η/2 ≤ ε ≤ 1). +13 + +From previous calculations, the three first terms in the above display are less than exp(−φ(η)( +√ +xλt− +xλ)) for some positive function φ. To conclude the proof it remains to bound the fourth term. +Let K be an integer larger than 6/η3, by definition of L⋆ +=<,ε we have for every ε ∈ [ k +K , k+1 +K ) +L⋆ +=<,ε ≤ L⋆ +=<,k/K + card(So(¯α) +x +∩ [ k +K , k+1 +K ]). +Thus +P( +� +η/2≤ε≤1 +L⋆ +=<,ε > 2 +√ +xλt − xλ − 1 +2δ(η/3) +√ +xλt) +≤ +� +k≥⌊ηK/2⌋ +P +� +L⋆ +=<,k/K > 2 +√ +xλt − xλ − δ(η/3) +√ +xλt +� ++ +� +k≥⌊ηK/2⌋ +P +� +card(So(¯α) +x +∩ [ k +K , k+1 +K ]) > 1 +2δ(η/3) +√ +xλt +� +≤ +� +k≥⌊ηK/2⌋ +P +� +L⋆ +=<,k/K > 2 +√ +xλt − xλ − δ(k/K) +√ +xλt +� +(since K > 6/η3 > 6/η and δ is increasing) ++ +� +k≥⌊ηK/2⌋ +P +� +card(So(¯α) +x +∩ [ k +K , k+1 +K ]) > 1 +2δ(η/3) +√ +xλt +� +≤ +� +k≥⌊ηK/2⌋ +exp(−h(k/K)( +√ +xtλ − xλ)) +(using (24)) ++K × P +� +Poisson(¯α/K) > 1 +2δ(η/3) +√ +xλt +� +≤ K exp(−h(η/3)( +√ +xtλ − xλ)) + K × P +� +Poisson(¯α/K) > 1 +2δ(η/3) +√ +xλt +� +, +≤ K exp(−h(η/3)( +√ +xtλ − xλ)) + K × P +� +Poisson(¯α/K) > ¯α/K + η2 +32 +√ +xλt +� +, +(using K > 6/η3) +≤12 +η exp(−ϕ(η)( +√ +xtλ − xλ)), +for some positive function ϕ(η). We can find ψ such that +min +� +1, 12 +η e−ϕ(η)( +√ +xtλ−xλ) + 3 exp(−φ(η)) +� +≤ e−ψ(η)( +√ +xtλ−xλ) +and thus P(sources(P) ≥ η +√ +xλt) ≤ exp(−ψ(η)( +√ +xtλ − xλ)). With minor modifications one +proves the same bound for sinks (possibly by changing ψ): P(sinks(P) ≥ η +√ +xλt) ≤ exp(−ψ(η)( +√ +xtλ− +xλ)) and Lemma 10 is proved. +We can conclude the proof of the lower bound in Theorem 8. Let us write +L<(t) ≥ L=<(Π(λ) +x,t ∪ So(¯α) +x +∪ Si(¯p) +t ) − sources(P) − sinks(P), +we bound the right-hand side using Lemmas 9 and 10. +4 +Proof of Theorem 1 when kn → +∞: de-Poissonization +In order to conclude the proof of Theorem 1 it remains to de-Poissonize Theorem 8. We need +a few notation. For any integers i1, . . . , in let Si1,...,in be the random set of points given by iℓ +uniform points on each horizontal line: +Si1,...,in = ∪n +ℓ=1 ∪iℓ +r=1 {Uℓ,r} × {ℓ} , +14 + +where (Uℓ,r)ℓ,r is an array of i.i.d. +uniform random variables in [0, 1]. +Set also ei1,...,in = +E[L<(Si1,...,in)]. By uniformity of U’s we have the identity E[L<(Sk;n)] = ek,...,k and therefore +our problem reduces to estimating ek,...,k. On the other hand if X1, . . . , Xn are i.i.d. Poisson +random variables with mean k then +E[eX1,...,Xn] = E +� +L<(P(1/n) +nkn,n) +� += 2 +� +nkn − kn + o( +� +nkn). +(25) +The last equality is obtained by combining Theorem 8 for +x = nkn, +t = n, +λn = 1 +n +with the trivial bound L<(P(1/n) +nkn,n) ≤ n. In order to exploit (25) we need the following smoothness +estimate. +Lemma 11. For every i1, . . . , in and j1, . . . , jn +|ei1,...,in − ej1,...,jn| ≤ 6 +� +� +� +� +n +� +ℓ=1 +|iℓ − jℓ|. +Proof. Let S = Si1,...,in be as above. If we replace in S the y-coordinate of each point of the form +(x, ℓ) by a new y-coordinate uniform in the interval (ℓ, ℓ + 1) (independent from anything else) +then this defines a uniform permutation σi1+···+in of size i1+· · ·+in. The longest increasing sub- +sequence in S is mapped onto an increasing subsequence in σi1+···+in and thus this construction +shows the stochastic domination L<(Si1,...,in) ≼ L<(σi1+···+in). Thus for every i1, . . . , in, +ei1,...,in ≤ E[L<(σi1+···+in)] ≤ 6 +√ +i1 + · · · + in. +(26) +(The last inequality follows from [Ste97, Lemma 1.4.1].) +Besides, consider for two n-tuples +i1, . . . , in and j1, . . . , jn two independent set of points Si1,...,in, �Sj1,...,jn then +L<(Si1,...,in) ≤ L<(Si1,...,in ∪ �Sj1,...,jn) ≤ L<(Si1,...,in) + L<( �Sj1,...,jn). +This proves that +ei1,...,in ≤ ei1+j1,...,in+jn ≤ ei1,...,in + ej1,...,jn. +(In particular (i1, . . . , in) �→ ei1,...,in is non-decreasing with respect to any of its coordinate.) +Therefore +ei1,...,in ≤ e(i1−j1)+,...,(in−jn)+ + ej1−(i1−j1)−,...,jn−(in−jn)− +≤ e|i1−j1|,...,|in−jn| + ej1,...,jn. +By switching the role of i’s and j’s: +|ei1,...,in − ej1,...,jn| ≤ e|i1−j1|,...,|in−jn| ≤ 6 +� +� +� +� +n +� +ℓ=1 +|iℓ − jℓ|, +using (26). +Proof of Theorem 1 for any sequence (kn) → +∞. Using smoothness we write +|ek,...,k − E[eX1,...,Xn]| ≤ E [|ek,...,k − eX1,...,Xn|] ≤ 6 × E +� +� +� n +� +ℓ=1 +|Xℓ − k| +�1/2� +� . +(27) +15 + +Using twice the Cauchy-Schwarz inequality: +E +� +� +� n +� +ℓ=1 +|Xℓ − k| +�1/2� +� ≤ +� +� +� +�E +� n +� +ℓ=1 +|Xℓ − k| +� +≤ +� +nE [|X1 − k|] +≤ +� +nE [|X1 − k|2]1/2 = +� +n +� +Var(X1) = +� +n +√ +k. +If k = kn → ∞ then the last display is a o(√nkn) and eq.(27) and (25) show that +ek,...,k = E[L<(Sk;n)] = 2 +� +nkn − kn + o( +� +nkn). +5 +Proof of Theorem 2 +5.1 +Proof for large (kn) +We now prove Theorem 2 for a large sequence (kn). We say that (kn) is large if +n2kn exp(−(kn)α) = o( +� +nkn) +(28) +for some α ∈ (0, 1). Note that kn = log n is not large while kn = (log n)1+ε is large. +We first observe that de-Poissonization cannot be applied as in the previous section. We +lack smoothness as, for instance, E[L≤(Si1,0,0,...,0)] = i1 ̸= O( +�� iℓ). The strategy is to apply +Theorem 8 with +x = nkn, +t = n, +λn ≈ 1 +n. +(The exact value of λn will be different for the proofs of the lower and upper bounds.) +Proof of the upper bound of (2) for large (kn). +Choose α such that n2kn exp(−kα +n) = o(√nkn). Put +λn = 1 +n + δn +n , +with δn = k−(1−α)/2 +n +. +Let Eλn +n +be the event +Eλn +n += +� +at least kn points in each row of P(λn) +nkn,n +� +. +The event En occurs with large probability. Indeed, +1 − P(Eλn +n ) ≤ nP (Poisson(nknλn) ≤ kn) +≤ nP (Poisson(nknλn) ≤ nknλn + kn − nknλn) +≤ nP (Poisson(nknλn) ≤ nknλn − knδn) +≤ n exp +� +− k2 +nδ2 +n +4nknλn +� +≤ n exp +� +− 1 +8knδ2 +n +� += n exp +� +− 1 +8kα +n +� +. +(29) +At the last line we used Lemma 14. The latter probability tends to 0 as (kn) is large. +Lemma 12. Random sets Skn;n and Π(λn) +nkn,n can be defined on the same probability space in such +a way that +L≤(Skn;n) ≤ L≤(Π(λn) +nkn,n) + nkn(1 − 1Eλn +n ). +(30) +16 + +Proof of Lemma 12. Draw a sample of Π(λn) +nkn,n and let ˜Π(λn) +nkn,n be the subset of Π(λn) +nkn,n obtained +by keeping only the kn leftmost points in each row. If Eλn +n +occurs then the relative orders of +points in ˜Π(λn) +nkn,n corresponds to a uniform kn-multiset permutation. If Eλn +n +does not hold we +bound L≤(Skn;n) by the worst case nkn. +Taking expectations in (30) and using the upper bound (15) yields +E[L≤(Skn;n)] ≤ 2 +� +nkn(1 + δn) + kn(1 + δn) + n2kn exp (−kα +n) , +hence the upper bound in (2). +Proof of the lower bound of (2) for large (kn). Choose now λn = 1 +n(1 − δn) with δn = +k−(1−α)/2 +n +. Let Fn be the event +F λn +n += +� +at most kn points in each row of P(λn) +nkn,n +� +. +The event F λn +n +occurs with large probability: +1 − P(F λn +n ) ≤ nP (Poisson(nknλn) ≥ kn) ≤ n exp +� +− 1 +8kα +n +� +, +which tends to zero. Random sets Skn;n and P(λn) +nkn,n can be defined on the same probability space +in such a way that +L≤(Skn;n) ≥ L≤(P(λn) +nkn,n)1F λn +n . +Therefore +P +� +L≤(Skn;n) < (2 +� +nkn(1 − δn) + kn(1 − δn))(1 − ε) +� +≤ +P +� +L≤(P(λn) +nkn,n) < (2 +� +nkn(1 − δn) + kn(1 − δn))(1 − ε) +� ++ P +� +not F λn +n +� +. +and we conclude with (19). +5.2 +The gap between small and large (kn) +After I circulated a preliminary version of this article, Valentin Féray came up with a simple +argument for bridging the gap between small and large (kn). This allows to prove Theorem 2 for +an arbitrary sequence (kn), I reproduce his argument here with his permission. +Lemma 13. Let n, k, A be positive integers. Two random uniform multiset permutations �SkA;⌊n/A⌋ +and Sk;n can be built on the same probability space in such a way that +L≤ (Sk;n) ≤ L≤ +� +�SkA;⌊n/A⌋ +� ++ kA. +Proof of Lemma 13 . Draw Sk;n uniformly at random, the idea is to group all points of Sk;n +whose height is between 1 and A, to group all points whose height is between A + 1 and 2A, and +so on. +Formally, denote by 1 ≤ i1 < i2 < · · · < ikA⌊n/A⌋ the indices such that 1 ≤ iℓ ≤ ⌊n/A⌋ for +every ℓ (see Fig.4). For 1 ≤ ℓ ≤ kA⌊n/A⌋ put +�S(ℓ) = ⌈S(iℓ)/k⌉. +The word �S is a uniform kA-multiset permutation of size ⌊n/A⌋. A longest non-decreasing +subsequence in S is mapped onto a non-decreasing subsequence in �S, except maybe some points +with height > A⌊n/A⌋ (there are no more than kA such points). This shows the Lemma. +17 + +1 +⌊n/A⌋A +n +i1 +i2 +i3 +i4 +. . . +1 +⌊n/A⌋ +1 +kA⌊n/A⌋ +A +Figure 4: Illustration of the notation of Lemma 13. Top: the multiset permutation Sk;n. Bottom: +the corresponding �S. The longest non-decreasing subsequence in Sk;n (circled points) is mapped +onto a non-decreasing subsequence in �S, except one point with height > A⌊n/A⌋. +We conclude the proof of Theorem 2 by an estimation of E[L≤ (Skn;n)] in the case where +there are infinitely many kn’s such that, say, (log n)3/4 ≤ kn ≤ (log n)5/4. For the lower bound +the job is already done by Theorem 1 since +E[L≤ (Skn;n)] ≥ E[L< (Skn;n)] = 2 +� +nkn − kn + o( +� +nkn), +which of course also 2√nkn+o(nkn) for this range of (kn). For the upper bound take A = ⌊log n⌋ +in Lemma 13: +E[L≤ (Skn;n)] ≤ E[L≤ +� +Skn log n;⌊n/⌊log n⌋⌋ +� +] + kn log n +(31) +and we can apply the large case since +(n/ log n)2kn log n exp(−(kn log n)α) = o(kn log n × ⌊n/⌊log n⌋⌋). +Thus the right-hand side of (31) is also 2√nkn + o(√nkn). +5.3 +Deviation inequalities +We briefly explain here how to deduce from previous calculations deviation inequalities for +L<(Skn;n) and L≤(Skn;n) in the case where (kn) is large. +We only write the case of an up- +per bound for L< (Skn;n) and do not aim at optimality. As in Section 5.1 choose λn = 1 +n + δn +n +where δn = k−(1−α)/2 +n +. +18 + +P +� +L< (Skn;n) > (2 +� +nkn − kn)(1 + ε) +� +≤ P +� +Eλn +n +does not occur +� ++ P +� +L< +� +Π(λn) +nkn,n +� +> (2 +� +nkn − kn)(1 + ε) +� +≤ n exp +� +− 1 +8kα +n +� +(using (29)) ++ P +� +L< +� +Π(λn) +nkn,n +� +> (1 + δn)(2 +� +nkn − kn) 1 + ε +1 + δn +� +≤ n exp +� +− 1 +8kα +n +� ++ P +� +L< +� +Π(λn) +nkn,n +� +> (2 +� +nkn(1 + δn) − kn(1 + δn)) 1 + ε +1 + δn +� +≤ n exp +� +− 1 +8kα +n +� ++ exp(−˜g(ε/2)( +� +nkn − kn)), +for large enough n and for some positive ˜g, using (16). +Appendix: Useful tail inequalities +We collect here for convenience some (non-optimal) tail inequalities. +Lemma 14 ((See Chap.2 in [JŁR00])). Let Poisson(λ) be a Poisson random variable with mean +λ. For every A > 0 +P (Poisson(λ) ≤ λ − A) ≤ exp(−A2/4λ), +P (Poisson(λ) ≥ λ + A) ≤ exp(−A2/4λ). +Lemma 15 (Th.2.1 in [JŁR00]). Let Binomial(n, p) be a Binomial random variable with param- +eters (n, p). For 0 < ε < 1, +P(Binomial(n, p) ≤ np + εnp) ≤ exp +� +−ε2np/2 +� +, +P(Binomial(n, p) ≥ np − εnp) ≤ exp +� +−ε2np/3 +� +. +Lemma 16. Let G(α) +1 +, . . . , G(α) +k +be i.i.d. random variables with distribution Geometric≥0(1 − α). +For 0 < ε < 1, +P +� +G(α) +1 ++ · · · + G(α) +k +≥ (1 + ε)k +α +1 − α +� +≤ exp +� +− 1 +4ε2k +α +1 − α +� +, +P +� +G(α) +1 ++ · · · + G(α) +k +≤ (1 − ε)k +α +1 − α +� +≤ exp +� +− 1 +4ε2k +α +1 − α +� +. +Proof of Lemma 16. Using eλ ≤ 1 + λ + λ2 for |λ| < 1 we have +E[eλ(G(α) +1 +− +α +1−α )] = (1 − α)e−λ +α +1−α +1 − αeλ +≤ exp +� +−λ2 +α +1 − α +� +. +This says that G(α) +1 +is subexponential and the Chernov method (see e.g. [Wai19, Prop.2.2] for +the case of subexponential random variables) implies that +P +� +G(α) +1 ++ · · · + G(α) +k +≥ (1 + ε)k +α +1 − α +� +≤ max +� +exp +� +− 1 +4ε2k2 +α +1 − α +� +, exp +� +− 1 +2εk +α +1 − α +�� +≤ exp +� +− 1 +4ε2k +α +1 − α +� +. +19 + +Acknowledgements. +This work started as a collaboration with Anne-Laure Basdevant, I +would like to thank her very warmly. I am also extremely indebted to Valentin Féray for Lemma +13 and for having enlightened me on the links with [Bia01]. Finally, thanks to Sam Spiro for +stimulating exchanges. +References +[AD95] +David Aldous and Persi Diaconis. +Hammersley’s interacting particle process and +longest increasing subsequences. Probability Theory and Related Fields, 103(2):199– +213, 1995. +[BDJ99] +Jinho Baik, Percy Deift, and Kurt Johansson. On the distribution of the length of +the longest increasing subsequence of random permutations. J. Amer. Math. Soc., +12(4):1119–1178, 1999. +[BEGG16] Anne-Laure Basdevant, Nathanaël Enriquez, Lucas Gerin, and Jean-Baptiste Gouéré. +Discrete Hammersley’s lines with sources and sinks. ALEA Lat. Am. J. Probab. Math. +Stat., 13:33–52, 2016. +[Bia01] +Philippe Biane. Approximate factorization and concentration for characters of sym- +metric groups. Internat. Math. Res. Notices, (4):179–192, 2001. +[Boy22] +Alexandre Boyer. Chapter 3 (in English) of Stationnarité bidimensionnelle de modèles +aléatoires du plan, 2022. PhD Thesis, available at https://tel.archives-ouvertes. +fr/tel-03783603/. +[CDH+22] Alexander Clifton, Bishal Deb, Yifeng Huang, Sam Spiro, and Semin Yoo. +Con- +tinuously increasing subsequences of random multiset permutations. Sém. 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High-dimensional statistics, volume 48 of Cambridge Series in +Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, +2019. A non-asymptotic viewpoint. +Lucas Gerin gerin@cmap.polytechnique.fr +Cmap, Cnrs, École Polytechnique, +Institut Polytechnique de Paris, +Route de Saclay, +91120 Palaiseau Cedex (France). +21 + diff --git a/p9E0T4oBgHgl3EQfqwF-/content/tmp_files/load_file.txt b/p9E0T4oBgHgl3EQfqwF-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..315cd0717725b18653ff10ace8d83da5cf317013 --- /dev/null +++ b/p9E0T4oBgHgl3EQfqwF-/content/tmp_files/load_file.txt @@ -0,0 +1,773 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf,len=772 +page_content='The Ulam-Hammersley problem for multiset permutations Lucas Gerin January 9, 2023 Abstract We obtain the asymptotic behaviour of the longest increasing/non-decreasing subse- quences in a random uniform multiset permutation in which each element in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' , n} occurs k times, where k may depend on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' This generalizes the famous Ulam-Hammersley problem of the case k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' The proof relies on poissonization and a connection with variants of the Hammersley-Aldous-Diaconis particle system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 1 Introduction A k-multiset permutation of size n is a word with letters in {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' , n} such that each let- ter appears exactly k times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' When this is convenient we identify a multiset permutation s = (s(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' , s(kn)) and the set of points {(i, s(i)), 1 ≤ i ≤ kn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For example we say that there is a point at height j in s is s(i) = j for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' We introduce two partial orders over the quarter-plane [0, ∞)2: (x, y) ≺ (x′, y′) if x < x′ and y < y′, (x, y) ≼ (x′, y′) if x < x′ and y ≤ y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (Note that the roles of x, y are not identical in the definition of ≼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=') For a finite set P of points in the quarter-plane we put L<(P) = max {L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' there exists P1 ≺ P2 ≺ · · · ≺ PL, where each Pi ∈ P} , L≤(P) = max {L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' there exists P1 ≼ P2 ≼ · · · ≼ PL, where each Pi ∈ P} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' The integer L<(P) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' L≤(P)) is the length of the longest increasing (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' non-decreasing) subsequence of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Let Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n be a k-multiset permutation of size n taken uniformly among the � kn k k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' k � possibil- ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' In the case k = 1 the word S1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n is just a uniform permutation and estimating L<(S1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) is known as the Hammersley or Ulam-Hammersley problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' The first order was solved by Veršik and Kerov [VK77] (see [Rom15] for a review of the problem): E[L<(S1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n)] = E[L≤(S1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n)] n→+∞ ∼ 2√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Note that the above limit also holds in probability, in the sense that P � | 1 2√nL<(S1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) − 1| > ε � → 0 for every ε > 0, that we shorten into: L<(S1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) = 2√n + oP(√n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' In the context of card guessing games it is asked in [CDH+22, Question 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='3] the behaviour of L<(Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) for a fixed k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Using known results regarding longest increasing subsequences through independent points we can make an educated guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Indeed, let Ber(p) be a field of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Bernoulli 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='02557v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='CO] 6 Jan 2023 random variables with mean p over the square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Seppäläinen [Sep97, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='] proved that for fixed p and k, L< � Ber(p) ∩ [0, kn] × [0, n] � n→∞ ∼ n × √p 1 − p � 2 √ k − (k + 1)√p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For p = 1/n there are in average k points of Ber(p) on each line of [0, kn] × [0, n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Pretending that we can safely let p depend on n in the above approximation we obtain L<(Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) ≈ L< � Ber(1/n) ∩ [0, kn] × [0, n] � ≈ 2 √ kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' The goal of the present paper is to make this approximation rigorous (however we are not going to use Seppäläinen’s result but rather poissonize the problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' We actually adress this question in the case where k depends on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Theorem 1 (Longest increasing subsequences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Let (kn) be a sequence of integers such that kn ≤ n for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Then E[L<(Skn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n)] = 2 � nkn − kn + o( � nkn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (1) (Of course if kn = o(n) then the RHS of (1) reduces to 2√nkn + o(√nkn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=') If kn ≥ n for some n then the naive greedy strategy shows very easily that E[L<(Skn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n)] ≥ n − o(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Theorem 2 (Longest non-decreasing subsequences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Let (kn) be an arbitrary sequence of inte- gers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Then E[L≤(Skn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n)] = 2 � nkn + kn + o( � nkn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (2) We are not aware of previous results for multiset permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' However Theorems 1 and 2 in the linear regime kn ∼ constant×n should be compared to a result by Biane ([Bia01, Theorem 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Indeed he obtains the exact limiting shape of the random Young Tableau induced through the RSK correspondence by a random word Wq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='N of q i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' uniform letters in {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' , N} in the regime where √q/N → c for some constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (The regime √q/N → c corresponds to kn ∼ c2n with our notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=') Regarding longest increasing subsequences we expect the asymptotics of the word Wq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='N to be close to that of the multiset permutation Sc2N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='N and Theorems 1 and 2 respectively suggest: L<(Wq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='N) ≈ L<(Sc2N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='N) ∼ 2Nc − c2N ∼ (2 − c)√q, L≤(Wq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='N) ≈ L≤(Sc2N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='N) ∼ 2Nc + c2N ∼ (2 + c)√q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' As the length of the first row (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' the number of rows) in the Young Tableau corresponds to the length of the longest non-decreasing subsequence in Wk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' the length of the longest decreasing sequence) a weak consequence of ([Bia01, Theorem 3]) is that, in probability, lim inf 1 √qL<(Wq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='N) ≥ (2 − c), lim sup 1 √qL≤(Wq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='N) ≤ (2 + c), which is indeed consistent with the above heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Strategy of proof and organization of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' In Section 2 we first provide the proof of Theorems 1 and 2 in the case of a constant or slowly growing sequence (kn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' The proof is elementary (assuming known the Veršik-Kerov Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For the general case we borrowed a few tools in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' In particular we first study poissonized versions of L<(Skn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n), L≤(Skn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' As already suggested by Hammersley ([Ham72], 2 Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='9) and achieved by Aldous-Diaconis [AD95] the case k = 1 can be tackled by considering an interacting particle system which is now known as the Hammersley or Hammersley-Aldous- Diaconis particle system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' In Section 3 we introduce and analyze the two variants1 of the Hammersley process adapted to multiset permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' The standard path to analyze Hammersley-like models consists in using subadditivity to prove the existence of a limiting shape and then proving that this limiting shape satisfies a variational problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Typically this variational problem is solved using convex duality (see [Sep97, CG19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' The issue here is that since we allow kn to have different scales we cannot use this approach and we need to derive non-asymptotic bounds for both processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' This is the purpose of Theorem 8 whose proof is the most technical part of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' In Section 4 we detail the multivariate de-poissonization procedure in order to conclude the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' De-poissonization is more convoluted for non-decreasing subsequences: see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Beyond expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' In the course of the proof we actually obtain results beyond the esti- mation of the expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' We obtain concentration inequalities for the poissonized version of L<(Skn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n), L≤(Skn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n): see Theorem 8 and also the discussion in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Regarding fluctua- tions a famous result by Baik, Deift and Johansson [BDJ99, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='1] states that L≤(S1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) − 2√n n1/6 (d) → TW where TW is the Tracy-Widom distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For L<(Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) with fixed k we are quite confident that the particle system approach and more precisely the analysis of second class particles (see [CG06, CG19]) could be adapted and would yield that the fluctuations are also of order n1/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For general (kn) the intuition given by the comparison with the Hammersley process would suggest that the fluctuations of L<(Skn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) might be of order (knn)1/6 as long as (kn) does not grow too fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Yet we have no evidence for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 2 Preliminaries: the case of small kn We first prove Theorems 1 and 2 in the case of a small sequence (kn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' We say that a sequence (kn) of integers is small if k2 n(kn)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' = o(√n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (3) Note that a sequence of the form kn = (log n)1−ε is small while kn = log n is not small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Proof of Theorems 1 and 2 in the case of a small sequence (kn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (In order to lighten notation we skip the dependence in n and write k = kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=') Let σkn be a random uniform permutation of size kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' We can associate to σkn a k-multiset permutation Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For every 1 ≤ i ≤ kn we put Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n(i) = ⌈σ(i)/k⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' It is clear that Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n is uniform and we have L<(Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) ≤ L≤(σkn) ≤ L≤(Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 1The first one is very close to the historical Hammersley process, I discovered during the preparation of this article that the second one had recently appeared in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='Boyer’s PhD Thesis [Boy22] with a connection to the O’Connell-Yor Brownian polymer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 3 The Veršik-Kerov Theorem says that the middle term in the above inequality grows like 2 √ kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Hence we need to show that if (kn) is small then L≤(Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) = L<(Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) + oP( √ kn), which is a big step towards the small case of Theorems 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For this purpose we introduce for every δ > 0 the event Eδ := � L≤(Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) ≥ L<(Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) + δ√n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (4) If Eδ occurs then in particular there exists a non-decreasing subsequence with δ√n ties, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' points of Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n which are at the same height as their predecessor in the subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' These ties have distinct heights 1 ≤ i1 < · · · < iℓ ≤ n for some δ√n/k ≤ ℓ ≤ δ√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Fix Integers m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' , mℓ ≥ 2 such that (m1 − 1) + · · · + (mℓ − 1) = δ√n ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Column indices r1,1 < · · · < r1,m1 < r2,1 < r2,m1 < · · · < rℓ,1 < · · · < r1,mℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' We then introduce the event F = F ((iℓ)ℓ, (ri,j)i≤ℓ,j≤mi) = {S(r1,1) = · · · = S(r1,m1) = i1, S(r2,1) = · · · = S(r2,m1) = i2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' , S(rℓ,1) = · · · = S(r1,mℓ) = iℓ} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' By the union bound (we skip the integer parts) P(Eδ) ≤ � δ√n/k≤ℓ≤δ√n � 1≤i1<···≤iℓ≤n � (ri,j)i≤ℓ,j≤mi P (F ((iℓ)ℓ, (ri,j)i≤ℓ,j≤mi)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Using card �� mi = δ√n + ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' each mi ≥ 2 � = card �� pi = δ√n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' each pi ≥ 1 � = �δ√n − 1 ℓ − 1 � we obtain the upper bound � (ri,j)i≤ℓ,j≤mi P(F) = 1 � kn k k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' k � � nk � mi � � �� � choices of r’s �δ√n − 1 ℓ − 1 � � �� � choices of m′ is � kn − � mi (k − m1) (k − m2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (k − mℓ)k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' k � � �� � choices of kn − � mi remaining points = (k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' )ℓ(δ√n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (δ√n + ℓ)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (δ√n − ℓ)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (ℓ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (k − m1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (k − m2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' × · · · × (k − mℓ)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' ≤ (k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' )ℓ (δ√n)ℓ+1(δ√n − ℓ)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (ℓ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='. We now sum over 1 ≤ i1 < · · · ≤ iℓ ≤ n and then sum over ℓ: P(Eδ) ≤ δ√n � ℓ=δ√n/k �n ℓ � (k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' )ℓ (δ√n)ℓ+1(δ√n − ℓ)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (ℓ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' ≤ δ√n−3 � ℓ=δ√n/k �n ℓ � (k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' )ℓ (δ√n)ℓ+1(δ√n − ℓ)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (ℓ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' + 3 � n δ√n � (k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' )δ√n (δ√n)δ√n−2(δ√n − 3)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (5) 4 i1 iℓ r1,1 r1,2 r1,m1 i2 r2,1 r2,m2 rℓ,1 rℓ,mℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Figure 1: The event F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (Ties are surrounded in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Points with blue background represent the subsequence with δ√n ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=') Using the two following inequalities valid for every j ≤ m (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' [CLRS09, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='5)]) �m j � ≤ �me j �j , m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' ≥ mm exp(−m) we first obtain that if kn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' = o(√n) (which is the case if (kn) is small) then the last term of (5) tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Regarding the sum we write P(Eδ) ≤ δ√n−3 � ℓ=δ√n/k �ne ℓ �ℓ (k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' )ℓ (δ√n)ℓ+1(δ√n − ℓ)δ√n−ℓe−δ√n+ℓ(ℓ − 1)ℓ−1e−ℓ+1 + o(1) ≤ δ√n−3 � ℓ=δ√n/k �nek!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (δ√n − ℓ) δ√nℓ(ℓ − 1) �ℓ (ℓ − 1)e−1 δ√n � e δ√n − ℓ � �� � ≤e/3<1 �δ√n + o(1) ≤ δ√n−3 � ℓ=δ√n/k �√nek!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (δ√n − ℓ) δℓ(ℓ − 1) �ℓ (ℓ − 1)e−1 δ√n � e δ√n − ℓ �ℓ + o(1) ≤ δ√n−3 � ℓ=δ√n/k � √ne2k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' δℓ(ℓ − 1) �ℓ (ℓ − 1)e−1 δ√n + o(1), (6) which tends to zero for every δ > 0, as long as (kn) satisfies (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' This proves that L≤(Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) = L<(Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) + oP( √ kn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Using the crude bounds L<(Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) ≤ n and L≤(Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) ≤ kn, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (6) also implies that E[L≤(Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n)] = E[L<(Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n)] + o( √ kn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 3 Poissonization: variants of the Hammersley process Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' In the sequel, Poisson(µ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Binomial(n, q)) stand for generic random variables with Poisson distribution with mean µ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Binomial distribution with parameters n, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 5 Π(λ) i x i x Sources ∼ PPP(α) Sinks ∼ Bernoulli(p) x x Sources ∼ PPP(β) Sinks ∼ Geometric≥0(1-β⋆) Figure 2: Our four variants of the Hammersley process (time goes from bottom to top, trajectories of particules are indicated in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Top left: The process L<(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Top right: The process L≤(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Bottom left: The process L(α,p) < (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Bottom right: The process L(β,β⋆) ≤ (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Notation Geometric≥0(1−β) stands for a geometric random variable with the convention P(Geometric≥0(1− β) = k) = (1 − β)βk for k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' In particular E[Geometric≥0(1 − β)] = β 1−β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='1 Definitions of the processes L<(t) and L≤(t) In this Section we define formally and analyze two semi-discrete variants of the Hammersley process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For a parameter λ > 0 let Π(λ) be the random set Π(λ) = ∪iΠ(λ) i where Π(λ) i ’s are independent and each Π(λ) i is a homogeneous Poisson Point Process (PPP) with intensity λ on (0, ∞) × {i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For simplicity set Π(λ) x,t = Π(λ) ∩ ([0, x] × {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' , t}) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' The goal of the present section is to obtain non-asymptotic bounds for L< � Π(λ) x,t � and L≤ � Π(λ) x,t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Fix x > 0 throughout the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For every t ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' } the function y ∈ [0, x] �→ L<(y, t) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' L≤(y, t)) is a non-decreasing integer-valued function whose all steps are equal to +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Therefore this function is completely determined by the finite set L<(t) := � y ≤ x, L<(y, t) = L<(y, t−) + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (Respectively: L≤(t) := � y ≤ x, L≤(y, t) = L≤(y, t−) + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=') Sets L<(t) and L≤(t) are finite subsets of [0, x] whose elements are considered as particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' It is easy to see that for fixed x > 0 both processes (L<(t))t and (L≤(t))t are Markov processes taking their values in the family of point processes of [0, x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 6 Exactly the same way as for the classical Hammersley process ([Ham72, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='9], [AD95]) the individual dynamic of particles is very easy to describe: The process L< .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' We put L<(0) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' In order to define L<(t+1) from L<(t) we consider particles from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' A particle at y in L<(t) moves at time t + 1 at the location of the leftmost available point z in Π(λ) t+1 ∩ (0, y) (if any, otherwise it stays at y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' This point z is not available anymore for subsequent particles, as well as every other point of Π(λ) t ∩(0, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 0 x Configuration L<(t) t+1 Configuration L<(t+1) y z y′ If there is a point in Π(λ) t+1 which is on the right of y′ := max{L<(t)} then a new particle is created in L<(t + 1), located at the leftmost point in Π(λ) t+1 ∩ (y′, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (In pictures this new particle comes from the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=') A realization of L< is shown on top-left of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' The process L≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' We put L≤(0) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' In order to define L≤(t + 1) from L≤(t) we also consider particles from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' A particle at y in L≤(t) moves at time t + 1 at the location of the leftmost available point z in Π(λ) t ∩ (0, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' This point z is not available anymore for subsequent particles, other points in (z, y) remain available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 0 x Configuration L≤(t) t+1 Configuration L≤(t+1) If there is a point in Π(λ) t+1 which is on the right of y′ := max{L<(t)} then new particles are created in L<(t + 1), one for each point in Π(λ) t+1 ∩ (y′, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' A realization of L≤ is shown on top-right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Processes L<(t) and L≤(t) are designed in such a way that they record the length of longest increasing/non-decreasing paths in Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' In fact particles trajectories correspond to the level sets of the functions (x, t) �→ L< � Π(λ) x,t � , (x, t) �→ L≤ � Π(λ) x,t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For every x, L< � Π(λ) x,t � = card(L<(t)), L≤ � Π(λ) x,t � = card(L≤(t)), where on each right-hand side we consider the particle system on [0, x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' We are merely restating the original construction from Hammersley ([Ham72], Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' We only do the case of L<(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Let us call each particle trajectory a Hammersley line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' By construction each Hammersley line is a broken line starting from the right of the box [0, x] × [0, t] and is formed by a succession of north/west line segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Because of this, two distinct points in a given longest increasing subse- quence of Π(λ) x,t cannot belong to the same Hammersley line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Since there are L<(t) Hammersley’s lines this gives L< � Π(λ) x,t � ≤ card(L<(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' In order to prove the converse inequality we build from this graphical construction a longest increcreasing subsequence of Π(λ) x,t with exactly one point on each Hammersley line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' To do so, we order Hammersley’s lines from bottom-left to top-right, and we build our path starting from the top-right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' We first choose any point of Π(λ) x,t belonging to the last Hammersley line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' We then proceed by induction: we choose the next point among the points of of Π(λ) x,t lying on the previous Hammersley line such that the subsequence remains increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (This is possible since Hammersley’s lines only have North/West line segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=') This proves L< � Π(λ) x,t � ≥ card(L<(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='2 Sources and sinks: stationarity Proposition 3 tells us that in on our way to prove Theorem 1 and Theorem 2 we need to understand the asymptotic behaviour of processes L<, L≤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' These processes are far from being stationary as particles may appear all the time in the system and never disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' To solve this issue we use the trick of sources/sinks introduced formally and exploited by Cator and Groeneboom [CG05] (following the intuition given by [AD95]): Sources form a finite subset of [0, x] × {0} which plays the role of the initial configuration L<(0), L≤(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Sinks are points of {0} × [1, t] which add up to Π(λ) when one defines the dynamics of L<(t), L≤(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For L≤(t) it makes sense to add several sinks at the same location (0, i) so sinks may have a multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Examples of dynamics of L<, L≤ under the influence of sources/sinks is illustrated at the bottom of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For every λ, α > 0 let L(α,p) < (t) be the Hammersley process defined as L<(t) with: sources distributed according to a homogeneous PPP with intensity α on [0, x] × {0} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' sinks distributed according to i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Bernoulli(p) with λ λ + α = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (7) If sources, sinks, and Π(λ) are independent then the process � L(α,p) < (t) � t≥0 is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For every β > λ > 0 let L(β,β⋆) ≤ (t) be the Hammersley process defined as L≤(t) with: sources distributed according to a homogeneous PPP with intensity β on [0, x] × {0} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' sinks distributed according to i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Geometric≥0(1 − β⋆) with β⋆β = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (8) 8 If sources, sinks, and Π(λ) are independent then the process � L(β,β⋆) ≤ (t) � t≥0 is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Proof of Lemmas 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Lemma 5 could be obtain from several minor adjustments of [Boy22, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='3, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (Be aware that we have to switch x ↔ t and sources ↔ sinks in [Boy22] in order to fit our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=') For the sake of the reader we however propose the following alternative proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Consider for some fixed t ≥ 1 the process (Hy)0≤y≤x given by the number of Hammersley lines passing through the point (y, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 0 x Configuration L≤(t-1) The corresponding process (Hy) x t The initial value H0 is the number of sinks at (0, t), which is distributed as a Geometric≥0(1− β⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' The process (Hy) is a random walk (reflected at zero) with ’+1 rate’ equal to λ and ’−1 rate’ equal to β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (Jumps of (Hy) are independent from sinks as sinks are independent from Π(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=') The Geometric≥0(1−β⋆) distribution is stationary for this random walk exactly when (8) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' The set of points of L(β,β⋆) ≤ (t) is given by the union of Π(λ) t and the points of L(β,β⋆) ≤ (t) that do not correspond to a ’−1’ jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' A elementary computation with exponential and gamma random variables then shows that this is distributed as a homogeneous PPP with intensity β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Lemma 4 is proved exactly in the same way (alternatively one can mimic the proof of [CG05, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='] as the dynamics is the same as in the classical Hammersley process) so we omit details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' We simply explain where (7) comes from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' If Lemma 4 is true for some α, p, λ then in particular the number of particles in the system should be constant in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' At time t a particle leaves the system (from the left) if and only if there is a sink at height t, which occurs with probability p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' On the other hand a new particle appears from the right if there is a point in Π(λ) t which is on the right of the right-most particle of L<(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' This occurs with probability λ/(α + λ), hence we need (7) to hold so that the expected number of particles remain constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='3 Processes L<(t) and L≤(t): non-asymptotic bounds From Lemmas 4 and 5 it is straightforward to derive non-asymptotic upper bounds for L<(t), L≤(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For y ≤ x let So(α) x be the random set of sources with intensity α and for s ≤ t let Si(p) t the random set of sinks with intensity p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' It is convenient to use the notation L=<(P) which is, as before, the length of the longest increasing path taking points in P but when the path is also allowed to go through several sources (which have however the same y-coordinate) or several sinks (which have the same x-coordinate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Formally, L=<(P) = max {L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' there exists P1 =≺ P2 =≺ · · · =≺ PL, where each Pi ∈ P} , 9 where (x, y) =≺ (x′, y′) if � � � � � x < x′ and y < y′, or x = x′ = 0 and y < y′, or x < x′ and y = y′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Proposition 3 can be generalized in L=< � Π(λ) x,t ∪ So(α) x ∪ Si(p) t � = L(α,p) < (t) + card(Si(p) t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (9) Lemma 6 (Upper bound for L<).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For every α, p ∈ (0, 1) such that (7) holds, there is a stochastic domination of the form: L< � Π(λ) x,t � ≼ Poisson(xα) + Binomial(t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (10) (The Poisson and Binomial random variables involved in (10) are not independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Adding sources and sinks may not decrease longest increasing paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Thus, L< � Π(λ) x,t � ≼ L=< � Π(λ) x,t ∪ So(α) x ∪ Si(p) t � = L(α,p) < (t) + card(Si(p)) (using (9)) (d) = L(α,p) < (0) + card(Si(p)) (using stationarity: Lemma 4) (d) = Poisson(xα) + Binomial(t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Taking expectations in (10) we obtain E � L< � Π(λ) x,t �� ≤ xα + tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' As the LHS in the above equation does not depend on α, p (provided that α, p satisfies (7)) we will apply (10) with the minimizing choice ¯α, ¯p := argminα,p satisfying (7) {xα + tp} , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' ¯α = � tλ x − λ, ¯p = � xλ t , x¯α + t¯p = 2 √ xtλ − xλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (11) We have proved E � L< � Π(λ) x,t �� ≤ 2 √ xtλ − xλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (Compare with (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=') We have a similar statement for non-decreasing subsequences: Lemma 7 (Upper bound for L≤).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For every β, β⋆ ∈ (0, 1) such that (8) holds, there is a stochastic domination of the form: L≤ � Π(λ) x,t � ≼ Poisson(xβ) + G(β⋆) 1 + · · · + G(β⋆) t , (12) where G(β⋆) i ’s are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Geometric≥0(1 − β⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 10 We put ¯β, ¯β⋆ := argminβ,β⋆ satisfying (8) � xβ + t � β⋆ 1 − β⋆ �� , (13) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' ¯β = � tλ x + λ, ¯β⋆ = 1 1 + � t/xλ , x¯β + t � ¯β 1 − ¯β � = 2 √ xtλ + xλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (14) (In particular ¯β > λ, as required in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=') Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (12) yields E � L≤ � Π(λ) x,t �� ≤ 2 √ xtλ + xλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (15) (Compare with (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=') Theorem 8 (Concentration for L<, L≤).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' There exist strictly positive functions g, h such that for all ε > 0 and for every x, t, λ such that t ≥ xλ P(L<(Π(λ) x,t ) > (1 + ε)(2 √ xtλ − xλ)) ≤ exp(−g(ε)( √ xtλ − xλ)), (16) P(L<(Π(λ) x,t ) < (1 − ε)(2 √ xtλ − xλ)) ≤ exp(−h(ε)( √ xtλ − xλ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (17) Similarly: P(L≤(Π(λ) x,t ) > (1 + ε)(2 √ xtλ + xλ)) ≤ exp(−g(ε) √ xtλ), (18) P(L≤(Π(λ) x,t ) < (1 − ε)(2 √ xtλ + xλ)) ≤ exp(−h(ε) √ xtλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (19) For the proof of Theorem 8 we will focus on the case of L<, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (16), (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' When necessary we will give the slight modification needed to prove eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (18) and (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' The beginning of the proof mimics Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='2 in [BEGG16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' We first prove similar bounds for the stationary processes with minimizing sources and sinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Lemma 9 (Concentration for L< with sources and sinks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Let ¯α, ¯p be defined by (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' There exists a strictly positive function g1 such that for all ε > 0 and for every x, t, λ such that t ≥ xλ P(L=<(Π(λ) x,t ∪ So(¯α) x ∪ Si(¯p) t ) > (1 + ε)(2 √ xtλ − xλ)) ≤ 2 exp(−g1(ε)( √ xtλ − xλ)) (20) P(L=<(Π(λ) x,t ∪ So(¯α) x ∪ Si(¯p) t ) < (1 − ε)(2 √ xtλ − xλ)) ≤ 2 exp(−g1(ε)( √ xtλ − xλ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (21) Proof of Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' By stationarity (Lemma 4) we have L=<(Π(λ) x,t ∪ So(¯α) x ∪ Si(¯p) t ) (d) = Poisson(x¯α) + Binomial(t, ¯p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Then P(L=<(Π(λ) x,t ∪ So(¯α) x ∪ Si(¯p) t > (1 + ε)(2 √ xtλ − xλ)) ≤ P � Poisson(x¯α) > (1 + ε 2)( √ xtλ − xλ) � + P � Binomial(t, ¯p) > (1 + ε 2) √ xtλ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Recall that x¯α = √ xtλ−xλ, t¯p = √ xtλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Using the tail inequality for the Poisson distribution (Lemma 14): P � Poisson(x¯α) > (1 + ε 2)( √ xtλ − xλ) � ≤ exp � −( √ xtλ − xλ)ε2/4 � Using the tail inequality for the binomial (Lemma 15) we get P � Binomial(t, ¯p) > (1 + ε 2) √ xtλ � ≤ exp(− 1 12ε2√ xtλ) ≤ exp(− 1 12ε2( √ xtλ − xλ)) The proof of (21) is identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' This shows Lemma 9 with g1(ε) = ε2/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 11 x εx Maximizing path for L=< Maximizing path for L⋆ <,ε Point of Π(λ) source sink Figure 3: A sample of Π(λ) x,t , sources, sinks, and the corresponding trajectories of particles (in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Here L=<(Π(λ) x,t ∪ So(α) x ∪ Si(p) t ) = 5 (pink path) and L(α,p) < (t) = 2 (two remaining particles at the top of the box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For longest non-decreasing subsequences we have a statement similar to Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' The only modification in the proof is that in order to estimate the number of sinks one has to replace Lemma 15 (tail inequality for the Binomial) by Lemma 16 (tail inequality for a sum of geometric random variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' During the proof we need to bound √ xtλ + xλ by √ xtλ, this explains the form of the right-hand side in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (18) and (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Adding sources/sinks may not decrease L< so L=<(Π(λ) x,t ∪ So(¯α) x ∪ Si(¯p) t ) ≽ L<(Π(λ) x,t ), thus the upper bound (16) is a direct consequence of Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Let us now prove the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' We consider the length of a maximizing path among those using sources from 0 to εx and then only increasing points of Π(λ) x,t ∩ ([εx, x] × [0, t]) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Formally we set L⋆ =<,ε := card � So(¯α) εx � + L< � (Π(λ) x,t ∩ ([εx, x] × [0, t]) � (d) = Poisson(εx¯α) + L< � (Π(λ) x,t ∩ ([εx, x] × [0, t]) � (22) The idea is that for any fixed ε the paths contributing to L⋆ =<,ε will typically not contribute to L=< � Π(λ) x,t ∪ So(¯α) x ∪ Si(¯p) t � = L(¯α,p) < (t) + card(Si(¯p) t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Indeed eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (22) suggests that for large x, t L⋆ =<,ε ≈ E[Poisson(εx¯α)] + E � L< � Π(λ) x,t ∩ ([εx, x] × [0, t]) �� ≈ xε¯α + 2 � x(1 − ε)λt − x(1 − ε)λ = 2 √ xλt − xλ − √ xtλδ(ε), where δ(ε) = 2−ε−2√1 − ε is positive and increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' In order to make the above approximation rigorous we first write 2 √ xλt − xλ − 1 2 √ xtλδ(ε) = xε¯α + 1 4 √ xtλδ(ε) + 2 � x(1 − ε)λt − x(1 − ε)λ + 1 4 √ xtλδ(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (23) 12 Combining (22) and (23) gives P � L⋆ <,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='ε ≥ 2 √ xλt − xλ − 1 2 √ xtλδ(ε) � ≤ P � Poisson(xε¯α) ≥ xε¯α + 1 4 √ xtλδ(ε) � + P � L<(Π(λ) x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='t ∩ ([εx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' x] × [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' t]) ≥ 2 � x(1 − ε)λt − x(1 − ε)λ + 1 4 √ xtλδ(ε) � ≤ exp � − xtλδ(ε)2 16 × 4ε2( √ xtλ − xλ) � (using Lemma 14) + P � L<(Π(λ) x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='t ∩ ([εx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' x] × [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' t]) ≥ 2 � x(1 − ε)λt − x(1 − ε)λ + 1 8(2 � x(1 − ε)λt − x(1 − ε)λ)δ(ε) � ≤ exp � − √ xtλδ(ε)2/(64ε2) � + P � L<(Π(λ) x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='t ∩ ([εx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' x] × [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' t]) ≥ � 2 � x(1 − ε)λt − x(1 − ε)λ � × (1 + 1 8δ(ε)) � ≤ exp � − √ xtλδ(ε)2/(64ε2) � + exp � −g(δ(ε)/8)( � x(1 − ε)tλ − x(1 − ε)λ) � (using the upper bound (16)) and thus we can find some positive h such that P � L⋆ <,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='ε ≥ 2 √ xλt − xλ − 1 2 √ xtλδ(ε) � ≤ exp � −h(ε)( √ xtλ − xλ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (24) One proves exactly in the same way a similar bound for the length of a maximizing path among those using sinks in {0} × [0, εt] and then only increasing points of Π(λ) x,t ∩ ([0, x] × [εt, t]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Choose now one of the maximizing paths P for L=< � Π(λ) x,t ∪ So(¯α) x ∪ Si(¯p) t � (if there are many of them, choose one arbitrarily in a deterministic way: the lowest, say).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Denote by sources(P) and sinks(P) the number of sources and sinks in the path P: sources(P) = card {0 ≤ y ≤ x such that (y, 0) ∈ P} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='3 the path P is sketched in pink and sources(P) = 2, sinks(P) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' There exists a positive function ψ such that for all real η > 0 P � sources(P) + sinks(P) ≥ η √ xλt � ≤ 2 exp(−ψ(η)( √ xλt − xλ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Proof of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' If the event � sources(P) ≥ η √ xλt � holds then there exists a (random) ε such that the two following events hold: Soεx ≥ η √ xλt ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' L⋆ =<,ε = L=< � Π(λ) x,t ∪ So(¯α) x ∪ Si(¯p) t � = L(¯α,¯p) < (t) + card(Si(¯p) t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' This implies that this random ε is larger than η/2 > 0 unless the number of sources in [0, xη/2] is improbably high: P(sources(P) ≥ η √ xλt) ≤ P(Soηx/2 ≥ η √ xλt) + P(sources(P) ≥ η √ xλt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Soηx/2 < η √ xλt) ≤ P(Soηx/2 ≥ η √ xλt) + P(L(¯α,¯p) < (t) ≤ √ xλt − xλ − 1 4δ(η/3) √ xλt) + P(card(Si(¯p) t ) ≤ √ xλt − 1 4δ(η/3) √ xλt) + P(L⋆ =<,ε ≥ 2 √ xλt − xλ − 1 2δ(η/3) √ xλt for some η/2 ≤ ε ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 13 From previous calculations, the three first terms in the above display are less than exp(−φ(η)( √ xλt− xλ)) for some positive function φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' To conclude the proof it remains to bound the fourth term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Let K be an integer larger than 6/η3, by definition of L⋆ =<,ε we have for every ε ∈ [ k K , k+1 K ) L⋆ =<,ε ≤ L⋆ =<,k/K + card(So(¯α) x ∩ [ k K , k+1 K ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Thus P( � η/2≤ε≤1 L⋆ =<,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='ε > 2 √ xλt − xλ − 1 2δ(η/3) √ xλt) ≤ � k≥⌊ηK/2⌋ P � L⋆ =<,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='k/K > 2 √ xλt − xλ − δ(η/3) √ xλt � + � k≥⌊ηK/2⌋ P � card(So(¯α) x ∩ [ k K ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' k+1 K ]) > 1 2δ(η/3) √ xλt � ≤ � k≥⌊ηK/2⌋ P � L⋆ =<,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='k/K > 2 √ xλt − xλ − δ(k/K) √ xλt � (since K > 6/η3 > 6/η and δ is increasing) + � k≥⌊ηK/2⌋ P � card(So(¯α) x ∩ [ k K ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' k+1 K ]) > 1 2δ(η/3) √ xλt � ≤ � k≥⌊ηK/2⌋ exp(−h(k/K)( √ xtλ − xλ)) (using (24)) +K × P � Poisson(¯α/K) > 1 2δ(η/3) √ xλt � ≤ K exp(−h(η/3)( √ xtλ − xλ)) + K × P � Poisson(¯α/K) > 1 2δ(η/3) √ xλt � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' ≤ K exp(−h(η/3)( √ xtλ − xλ)) + K × P � Poisson(¯α/K) > ¯α/K + η2 32 √ xλt � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (using K > 6/η3) ≤12 η exp(−ϕ(η)( √ xtλ − xλ)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' for some positive function ϕ(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' We can find ψ such that min � 1, 12 η e−ϕ(η)( √ xtλ−xλ) + 3 exp(−φ(η)) � ≤ e−ψ(η)( √ xtλ−xλ) and thus P(sources(P) ≥ η √ xλt) ≤ exp(−ψ(η)( √ xtλ − xλ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' With minor modifications one proves the same bound for sinks (possibly by changing ψ): P(sinks(P) ≥ η √ xλt) ≤ exp(−ψ(η)( √ xtλ− xλ)) and Lemma 10 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' We can conclude the proof of the lower bound in Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Let us write L<(t) ≥ L=<(Π(λ) x,t ∪ So(¯α) x ∪ Si(¯p) t ) − sources(P) − sinks(P), we bound the right-hand side using Lemmas 9 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 4 Proof of Theorem 1 when kn → +∞: de-Poissonization In order to conclude the proof of Theorem 1 it remains to de-Poissonize Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' We need a few notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For any integers i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' , in let Si1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',in be the random set of points given by iℓ uniform points on each horizontal line: Si1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',in = ∪n ℓ=1 ∪iℓ r=1 {Uℓ,r} × {ℓ} , 14 where (Uℓ,r)ℓ,r is an array of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' uniform random variables in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Set also ei1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',in = E[L<(Si1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',in)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' By uniformity of U’s we have the identity E[L<(Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n)] = ek,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',k and therefore our problem reduces to estimating ek,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' On the other hand if X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' , Xn are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Poisson random variables with mean k then E[eX1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',Xn] = E � L<(P(1/n) nkn,n) � = 2 � nkn − kn + o( � nkn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (25) The last equality is obtained by combining Theorem 8 for x = nkn, t = n, λn = 1 n with the trivial bound L<(P(1/n) nkn,n) ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' In order to exploit (25) we need the following smoothness estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For every i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' , in and j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' , jn |ei1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',in − ej1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',jn| ≤ 6 � � � � n � ℓ=1 |iℓ − jℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Let S = Si1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',in be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' If we replace in S the y-coordinate of each point of the form (x, ℓ) by a new y-coordinate uniform in the interval (ℓ, ℓ + 1) (independent from anything else) then this defines a uniform permutation σi1+···+in of size i1+· · ·+in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' The longest increasing sub- sequence in S is mapped onto an increasing subsequence in σi1+···+in and thus this construction shows the stochastic domination L<(Si1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',in) ≼ L<(σi1+···+in).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Thus for every i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' , in, ei1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',in ≤ E[L<(σi1+···+in)] ≤ 6 √ i1 + · · · + in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (26) (The last inequality follows from [Ste97, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=') Besides, consider for two n-tuples i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' , in and j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' , jn two independent set of points Si1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',in, �Sj1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',jn then L<(Si1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',in) ≤ L<(Si1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',in ∪ �Sj1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',jn) ≤ L<(Si1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',in) + L<( �Sj1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',jn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' This proves that ei1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',in ≤ ei1+j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',in+jn ≤ ei1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',in + ej1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',jn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (In particular (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' , in) �→ ei1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',in is non-decreasing with respect to any of its coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=') Therefore ei1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',in ≤ e(i1−j1)+,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',(in−jn)+ + ej1−(i1−j1)−,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',jn−(in−jn)− ≤ e|i1−j1|,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',|in−jn| + ej1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',jn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' By switching the role of i’s and j’s: |ei1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',in − ej1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',jn| ≤ e|i1−j1|,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',|in−jn| ≤ 6 � � � � n � ℓ=1 |iℓ − jℓ|, using (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Proof of Theorem 1 for any sequence (kn) → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Using smoothness we write |ek,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',k − E[eX1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',Xn]| ≤ E [|ek,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',k − eX1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',Xn|] ≤ 6 × E � � � n � ℓ=1 |Xℓ − k| �1/2� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (27) 15 Using twice the Cauchy-Schwarz inequality: E � � � n � ℓ=1 |Xℓ − k| �1/2� � ≤ � � � �E � n � ℓ=1 |Xℓ − k| � ≤ � nE [|X1 − k|] ≤ � nE [|X1 − k|2]1/2 = � n � Var(X1) = � n √ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' If k = kn → ∞ then the last display is a o(√nkn) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (27) and (25) show that ek,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',k = E[L<(Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n)] = 2 � nkn − kn + o( � nkn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 5 Proof of Theorem 2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='1 Proof for large (kn) We now prove Theorem 2 for a large sequence (kn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' We say that (kn) is large if n2kn exp(−(kn)α) = o( � nkn) (28) for some α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Note that kn = log n is not large while kn = (log n)1+ε is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' We first observe that de-Poissonization cannot be applied as in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' We lack smoothness as, for instance, E[L≤(Si1,0,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=',0)] = i1 ̸= O( �� iℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' The strategy is to apply Theorem 8 with x = nkn, t = n, λn ≈ 1 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (The exact value of λn will be different for the proofs of the lower and upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=') Proof of the upper bound of (2) for large (kn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Choose α such that n2kn exp(−kα n) = o(√nkn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Put λn = 1 n + δn n , with δn = k−(1−α)/2 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Let Eλn n be the event Eλn n = � at least kn points in each row of P(λn) nkn,n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' The event En occurs with large probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Indeed, 1 − P(Eλn n ) ≤ nP (Poisson(nknλn) ≤ kn) ≤ nP (Poisson(nknλn) ≤ nknλn + kn − nknλn) ≤ nP (Poisson(nknλn) ≤ nknλn − knδn) ≤ n exp � − k2 nδ2 n 4nknλn � ≤ n exp � − 1 8knδ2 n � = n exp � − 1 8kα n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (29) At the last line we used Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' The latter probability tends to 0 as (kn) is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Random sets Skn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n and Π(λn) nkn,n can be defined on the same probability space in such a way that L≤(Skn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) ≤ L≤(Π(λn) nkn,n) + nkn(1 − 1Eλn n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' (30) 16 Proof of Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Draw a sample of Π(λn) nkn,n and let ˜Π(λn) nkn,n be the subset of Π(λn) nkn,n obtained by keeping only the kn leftmost points in each row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' If Eλn n occurs then the relative orders of points in ˜Π(λn) nkn,n corresponds to a uniform kn-multiset permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' If Eλn n does not hold we bound L≤(Skn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) by the worst case nkn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Taking expectations in (30) and using the upper bound (15) yields E[L≤(Skn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n)] ≤ 2 � nkn(1 + δn) + kn(1 + δn) + n2kn exp (−kα n) , hence the upper bound in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Proof of the lower bound of (2) for large (kn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Choose now λn = 1 n(1 − δn) with δn = k−(1−α)/2 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Let Fn be the event F λn n = � at most kn points in each row of P(λn) nkn,n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' The event F λn n occurs with large probability: 1 − P(F λn n ) ≤ nP (Poisson(nknλn) ≥ kn) ≤ n exp � − 1 8kα n � , which tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Random sets Skn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n and P(λn) nkn,n can be defined on the same probability space in such a way that L≤(Skn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) ≥ L≤(P(λn) nkn,n)1F λn n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Therefore P � L≤(Skn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) < (2 � nkn(1 − δn) + kn(1 − δn))(1 − ε) � ≤ P � L≤(P(λn) nkn,n) < (2 � nkn(1 − δn) + kn(1 − δn))(1 − ε) � + P � not F λn n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' and we conclude with (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='2 The gap between small and large (kn) After I circulated a preliminary version of this article, Valentin Féray came up with a simple argument for bridging the gap between small and large (kn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' This allows to prove Theorem 2 for an arbitrary sequence (kn), I reproduce his argument here with his permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Let n, k, A be positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Two random uniform multiset permutations �SkA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='⌊n/A⌋ and Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n can be built on the same probability space in such a way that L≤ (Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) ≤ L≤ � �SkA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='⌊n/A⌋ � + kA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Proof of Lemma 13 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Draw Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n uniformly at random, the idea is to group all points of Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n whose height is between 1 and A, to group all points whose height is between A + 1 and 2A, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Formally, denote by 1 ≤ i1 < i2 < · · · < ikA⌊n/A⌋ the indices such that 1 ≤ iℓ ≤ ⌊n/A⌋ for every ℓ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For 1 ≤ ℓ ≤ kA⌊n/A⌋ put �S(ℓ) = ⌈S(iℓ)/k⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' The word �S is a uniform kA-multiset permutation of size ⌊n/A⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' A longest non-decreasing subsequence in S is mapped onto a non-decreasing subsequence in �S, except maybe some points with height > A⌊n/A⌋ (there are no more than kA such points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' This shows the Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 17 1 ⌊n/A⌋A n i1 i2 i3 i4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 1 ⌊n/A⌋ 1 kA⌊n/A⌋ A Figure 4: Illustration of the notation of Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Top: the multiset permutation Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Bottom: the corresponding �S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' The longest non-decreasing subsequence in Sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n (circled points) is mapped onto a non-decreasing subsequence in �S, except one point with height > A⌊n/A⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' We conclude the proof of Theorem 2 by an estimation of E[L≤ (Skn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n)] in the case where there are infinitely many kn’s such that, say, (log n)3/4 ≤ kn ≤ (log n)5/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For the lower bound the job is already done by Theorem 1 since E[L≤ (Skn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n)] ≥ E[L< (Skn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n)] = 2 � nkn − kn + o( � nkn), which of course also 2√nkn+o(nkn) for this range of (kn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For the upper bound take A = ⌊log n⌋ in Lemma 13: E[L≤ (Skn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n)] ≤ E[L≤ � Skn log n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='⌊n/⌊log n⌋⌋ � ] + kn log n (31) and we can apply the large case since (n/ log n)2kn log n exp(−(kn log n)α) = o(kn log n × ⌊n/⌊log n⌋⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Thus the right-hand side of (31) is also 2√nkn + o(√nkn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='3 Deviation inequalities We briefly explain here how to deduce from previous calculations deviation inequalities for L<(Skn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) and L≤(Skn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) in the case where (kn) is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' We only write the case of an up- per bound for L< (Skn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) and do not aim at optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' As in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='1 choose λn = 1 n + δn n where δn = k−(1−α)/2 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 18 P � L< (Skn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='n) > (2 � nkn − kn)(1 + ε) � ≤ P � Eλn n does not occur � + P � L< � Π(λn) nkn,n � > (2 � nkn − kn)(1 + ε) � ≤ n exp � − 1 8kα n � (using (29)) + P � L< � Π(λn) nkn,n � > (1 + δn)(2 � nkn − kn) 1 + ε 1 + δn � ≤ n exp � − 1 8kα n � + P � L< � Π(λn) nkn,n � > (2 � nkn(1 + δn) − kn(1 + δn)) 1 + ε 1 + δn � ≤ n exp � − 1 8kα n � + exp(−˜g(ε/2)( � nkn − kn)), for large enough n and for some positive ˜g, using (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Appendix: Useful tail inequalities We collect here for convenience some (non-optimal) tail inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Lemma 14 ((See Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='2 in [JŁR00])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Let Poisson(λ) be a Poisson random variable with mean λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For every A > 0 P (Poisson(λ) ≤ λ − A) ≤ exp(−A2/4λ), P (Poisson(λ) ≥ λ + A) ≤ exp(−A2/4λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Lemma 15 (Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='1 in [JŁR00]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Let Binomial(n, p) be a Binomial random variable with param- eters (n, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For 0 < ε < 1, P(Binomial(n, p) ≤ np + εnp) ≤ exp � −ε2np/2 � , P(Binomial(n, p) ≥ np − εnp) ≤ exp � −ε2np/3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Let G(α) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' , G(α) k be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' random variables with distribution Geometric≥0(1 − α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' For 0 < ε < 1, P � G(α) 1 + · · · + G(α) k ≥ (1 + ε)k α 1 − α � ≤ exp � − 1 4ε2k α 1 − α � , P � G(α) 1 + · · · + G(α) k ≤ (1 − ε)k α 1 − α � ≤ exp � − 1 4ε2k α 1 − α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Proof of Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Using eλ ≤ 1 + λ + λ2 for |λ| < 1 we have E[eλ(G(α) 1 − α 1−α )] = (1 − α)e−λ α 1−α 1 − αeλ ≤ exp � −λ2 α 1 − α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' This says that G(α) 1 is subexponential and the Chernov method (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' [Wai19, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='2] for the case of subexponential random variables) implies that P � G(α) 1 + · · · + G(α) k ≥ (1 + ε)k α 1 − α � ≤ max � exp � − 1 4ε2k2 α 1 − α � , exp � − 1 2εk α 1 − α �� ≤ exp � − 1 4ε2k α 1 − α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 19 Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' This work started as a collaboration with Anne-Laure Basdevant, I would like to thank her very warmly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' I am also extremely indebted to Valentin Féray for Lemma 13 and for having enlightened me on the links with [Bia01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Finally, thanks to Sam Spiro for stimulating exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' References [AD95] David Aldous and Persi Diaconis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Hammersley’s interacting particle process and longest increasing subsequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Probability Theory and Related Fields, 103(2):199– 213, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' [BDJ99] Jinho Baik, Percy Deift, and Kurt Johansson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' On the distribution of the length of the longest increasing subsequence of random permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=', 12(4):1119–1178, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' [BEGG16] Anne-Laure Basdevant, Nathanaël Enriquez, Lucas Gerin, and Jean-Baptiste Gouéré.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Discrete Hammersley’s lines with sources and sinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' ALEA Lat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' Math.' metadata={'source': 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Gerin gerin@cmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='polytechnique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content='fr Cmap, Cnrs, École Polytechnique, Institut Polytechnique de Paris, Route de Saclay, 91120 Palaiseau Cedex (France).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} +page_content=' 21' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E0T4oBgHgl3EQfqwF-/content/2301.02557v1.pdf'} diff --git a/pNAzT4oBgHgl3EQfb_xD/vector_store/index.pkl b/pNAzT4oBgHgl3EQfb_xD/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..8b4ef11789ba81588b26b9286d6487342a33d159 --- /dev/null +++ b/pNAzT4oBgHgl3EQfb_xD/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c37631d4f89345c64fdf60ec1226e629d43ec808e38b76fa673ba182088519da +size 194459 diff --git a/pdAyT4oBgHgl3EQfZPdk/content/tmp_files/2301.00219v1.pdf.txt b/pdAyT4oBgHgl3EQfZPdk/content/tmp_files/2301.00219v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ad0b0b2d05b02afe2254e84afb858967f71eed10 --- /dev/null +++ b/pdAyT4oBgHgl3EQfZPdk/content/tmp_files/2301.00219v1.pdf.txt @@ -0,0 +1,589 @@ +Walk ferroelectric liquid droplets with light +Stefano Marni1, Giovanni Nava2, Raouf Barboza1, Tommaso Bellini2*, Liana Lucchetti1* +1. +Dipartimento SIMAU, Universita` Politecnica delle Marche, via Brecce Bianche, 60131 Ancona, Italy. E-mail: l.lucchetti@univpm.it +2. +Medical Biotechnology and Translational Medicine Dept., University of Milano, 20054 Segrate, Italy. E-mail: tommaso.bellini@unimi.it +Abstract +We show that the motion of ferroelectric liquid sessile droplets deposited on a ferroelectric lithium niobate +substrate can be controlled by a light beam of moderate intensity irradiating the substrate at a distance of +several droplet diameters from the droplet itself. The ferroelectric liquid is a nematic liquid crystal in which +almost complete polar ordering of the molecular dipoles generates an internal macroscopic polarization +locally collinear to the mean molecular long axis. Upon entering the ferroelectric phase droplets are either +attracted toward the center of the beam or repelled, depending on the side of the lithium niobate exposed +to light irradiation. Moreover, moving the beam results in walking the ferroelectric droplet over long +distances on the substrate. We understand this behavior as due to the coupling between the polarization of +the ferroelectric droplet and the polarization photoinduced in the irradiated region of the lithium niobate +substrate. Indeed, the effect is not observed in the conventional nematic phase, suggesting the crucial role +of the ferroelectric liquid crystal polarization. +Introduction +The recent discovery of the ferroelectric nematic phase, NF, [1], opened a new chapter not only for the liquid +crystal community, but in the whole condensed-matter physics. Beside adding a new, very peculiar, member +to the group of ferroelectric materials, the new phase offers a broad range of physical effects to explore, +ranging from the behavior of topological defects to surface anchoring [2], response to low frequency electric +fields [3] and light, interplay of bound and free electric charges, viscoelastic properties, field-controlled +hydrodynamics [3-5], field-order coupling in both the NF and the pre-transitional regions, behavior in +confined geometry [6], just to cite a few examples. In this scenario, we recently performed experiments +devoted to characterize the behavior of sessile NF droplets on ferroelectric solid substrates [4] and found that +the combination of fluidity and polarity gives rise to an electromechanical instability induced by the coupling +of the LC polarization with that pyroelectrically induced in the solid substrate. +In this work, we instead analyze the effects of the photovoltaic charging of lithium niobate (LN) ferroelectric +solid substrates on NF sessile droplets, at constant temperature. The advantage of manipulating sessile +droplets by light are related to the possibility of focusing the beam to small regions of the substrates thus +limiting the extent of the charged regions and controlling its distance and position with respect to the +droplets, and of quickly reconfiguring it in different illumination geometries. Results show that, under proper +experimental conditions, light irradiation gives rise to an instability very similar to the one observed in [4]. +Moreover, focused beams impinging on the LN substrate at a certain distance from the sessile NF droplets +allow to optically control their motion. Droplets are attracted toward the center of the illuminated area or +repelled away from it depending on which side of the substrate is exposed to light irradiation and can be +walked by the light beam over long distances. Our results contribute an additional piece to the collection of +intriguing features characterizing the ferroelectric nematic phase and may have potential for future +applications. +Material and methods +The +ferroelectric +liquid +crystal +used +in +this +work +is +4-[(4-nitrophenoxy)carbonyl]phenyl2,4- +dimethoxybenzoate (RM734). It was synthesized as described in [1] and its structure and phase diagram have +already been reported [1, 2, 4]. In this compound the ferroelectric nematic phase appears through a second +order phase transition when cooling from the conventional higher temperature nematic (N) phase and exists + +in the range 133°C < T < 80°C [2,4]. The spontaneous polarization P of RM734 is either parallel or antiparallel +to the molecular director n, defining the average orientation of the molecular axis, and exceeds 6 µC/cm2 at +the lowest T in the NF phase [1]. +The lithium niobate (LN) ferroelectric substrates are 900 micron thick z-cut crystals. Experiments were +performed on iron-doped substrates containing 0.1% mol. of iron with a reduction factor R = 0.02. The bulk +spontaneous polarization PLN of LN crystals along the [0001] z-axis is of the order of 70 µC/cm2 and does not +depend significantly on T in the explored range since its Curie temperature is much higher (≈ 1140°C). The +huge bulk polarization of LN does not however translate in a huge surface charge density because of very +efficient compensation mechanisms at the z-cut surfaces, lowering the equilibrium surface charge to only +about 10-2 µC/cm2 [7]. When the crystal is exposed to light with wavelength in the iron absorption spectrum, +the surface charge of LN significantly increases because of the photovoltaic effect [8], consisting in the +appearance of a photo-induced current according to the scheme Fe2+ + h → Fe3+ + e-. The subsequent charge +distribution that takes place inside the crystal gives rise to an internal electric field with saturation values up +to 107 V/m, depending on the dopant concentration and on R [8,9]. Before droplet deposition, LN crystals +were coated by a layer of fluorolink, as described in [2]. +The RM734 droplets used in optical motion control experiments, have an average diameter of 45 µm, as +measured with a calibration slide and were obtained starting from bigger droplets as described in the SI. +They were deposited on fluorolink-coated LN substrates that were previously slowly heated up to T = 200°C, +corresponding to the RM734 isotropic phase. Successively, T was decreased down to 110°C, which is in the +ferroelectric range. Noteworthy, the cooling rate was kept slow enough to avoid the droplets +electromechanical instability observed in [4] that was triggered by the pyroelectric charging of LN surfaces +and required a proper cooling speed. +The light used to induce the photovoltaic effect in LN crystals is a gaussian beam from a frequency doubled +Nd:YAG laser ( = 532 nm), with power P in the range (5 -25) mW, focused to a waist w = 35 µm, which +corresponds to an intensity ranging from I = 102 W/cm2 to I = 5 x 102 W/cm2. LN substrates were irradiated +from below at different distances to the RM734 droplets, holding the temperature fixed at T = 110°C. +Polarized optical microscope (POM) observations during light irradiation were carried out and videos of the +droplets behavior were recorded with a rate of 25 frames per second. +Results +Our first experiment has been devoted to compare the effect of photo-induced charging of LN substrates on +RM734 sessile droplets, with the one of pyroelectric charging that we recently reported in [4]. To this +purpose, LN was irradiated in correspondence of the droplet position with an unfocused gaussian beam +having a diameter slightly larger than that of the droplet itself. As shown in Fig. 1, such a configuration +produces an electromechanical instability consisting in the sudden emission of interfacial fluid jets, as +observed in [4], indicating that the coupling between the droplet polarization and the one photo-induced in +LN has the same features as the coupling with the pyroelectrical LN polarization. In analogy to the +interpretation proposed there, we understand this phenomenon as due to the fringing field generated by the +photovoltaic charging of LN substrates, which is thus able to affect the NF droplets behavior. This result opens +the way to the possible optical control of ferroelectric LC droplets on LN substrates. Indeed, light is easily +controllable: the size and the position of the irradiated region can be varied almost at will and the intensity +of the light beam can be tuned in a very short time. + + +Figure 1: RM734 sessile droplet on LN substrate exposed to light illumination. The beam diameter is slightly larger than that of the +droplet, as shown in the cartoon on the left-hand side. Shape instability with the emission of interfacial fluid jets that bifurcate and +branch, is clearly visible (figure extracted from video S1). I = 20 W/cm2. Droplet diameter 350 m. +The effect reported in Fig. 1 is independent on the side of the LN substrate contacting the droplet. This is in +agreement with the observations in [4] where the sign of the charges of the LN surface that contacts the LC +droplet was irrelevant. +The same fringing field generated by photovoltaic charging leads to new effects when the light beam is +focused at a distance from the droplets. In this case the droplet retains approximately the same shape but is +put in motion by a force generated by light irradiation. Remarkably, the direction of such a force can be either +attractive, with droplets that move toward the center of the illuminated region, or repulsive, leading to a +droplet motion away from the light spot. The sign of the force depends on the irradiated side of the LN +substrate. This is illustrated in Fig. 2, which shows two series of frames extracted from video S2 (Fig. 2b-d) +and S3 (Fig. 2b’-d’), available in the SI. As visible, a NF droplet moves toward the illuminated area or away +from it, depending on the direction of the LN bulk polarization with respect to the direction of the incoming +light (as in the cartoons on the left-hand side of the figure), which we will refer to as “UP” and “DOWN” in +the rest of the paper. + +Figure 2: a) and a’) Sketches of the experimental arrangements in case of droplet attraction a) and repulsion a’): b) and c) video +frames taken at different instants showing a RM734 NF droplet moving toward the center of the illuminate area; b’) and c’) video +frames taken at different instants showing a RM734 NF droplet moving away from the center of the illuminate area. I = 5 x 102 W/cm2. + +100μmUP +attraction +50μm +50μm +50μm +green laser +a) +b) +to +c) +to +24s +d) +to +62s +DOWN +repulsion +50μm +50μm +50μm +green laser +a') +b") +to +c') +to + 103s +d') +to + 255sBy continuously varying the position of the beam by means of galvanometric mirrors, it is possible to drag +the NF droplet over long distances, as shown in Fig.3 in the case of droplet attraction (frames extracted by +video S4). Droplet dragging is also observed in case of repulsion, although with a less controlled trajectory. + + + +Figure 3: Video frames at different instants showing the motion of a RM734 NF droplet following a Gaussian light spot +along a wavy path. Green diamonds indicate the different positions of the droplets through the video and dashed line +represents the droplet trajectory. I = 3 x 102 W/cm2. +The curves describing droplet motion vs time are reported in Fig. 4 in case of both the UP a) and the DOWN +case b). The different curves in each figure refer to different values of the light intensity. Different initial +distances between droplet and light spot center have also been chosen to highlight the role of both these +parameters. To avoid superpositions and improve the clarity of the two graphs, some of the curves have been +translated in time. Remarkably, the curve corresponding to the lowest beam intensity in Fig. 4b (red curve in +panel DOWN), shows droplet motion toward the light spot, contrary to what happens at higher intensity. +A close inspection of Fig. 4 reveals that NF droplets attracted toward the illuminated area stop at the edge of +a region corresponding to the beam waist (Fig. 4a); NF droplets repelled by the illuminated area also stop at +a certain distance from its center, that increases with increasing light intensity. This behavior suggests the +presence of a pinning force opposing the one due to the interaction with the light beam. + +a) +q +d) +e +f) +100um +n +k +Figure 4: Droplet distance from the center of the illuminated area (d = 0) as a function of time, for the UP a) and DOWN case b). The +time t = 0 s corresponds to the beginning of LN illumination, however, to improve the clarity of the figure, some of the curves have +been shifted in time so to start at t > 0 s. Different colors indicate different values of the light intensity as reported in the legend; +different starting distances have also been chosen. The two cartoons on top help defining the parameters and understanding the +experimental arrangement in the two conditions. +From these and similar curves it is possible to extract the time dependence of the droplet velocity v for each +used value of the light intensity I. As an example, Fig. 5 b shows v vs t in the case of droplet attraction and I += 2x 102 W/cm2, for different initial distances from the center of the illuminated area, as extracted from the +corresponding d vs t curves reported in Fig. 5 a. Again, for the sake of clarity, some of the curves are shifted +in time. Red dots indicate the initial conditions in terms of initial distance from the light spot center and initial +droplet velocity. The three dashed lines identify the values of d and v corresponding to the same value of t. +Combining the v vs t curve with the corresponding d vs t, one obtains the velocity as a function of the distance +from the center of the illuminated area. This is shown in Fig. 5c, where the different curves refer to different +initial droplet positions with respect to the center of the light spot. Interestingly, despite a relevant +irregularity, the velocity keeps similar values at fixed positions, that is its value is independent on the initial +position of the droplet. This indicates that droplets inertia can be neglected in describing the droplet motion, +demonstrating that the friction force contains a term dependent on v. The average velocity is reported +in Fig. 5d. +To better understand droplet motion, a friction characterization is necessary. To this aim we performed +measurements of droplet motion along a tilted substrate in the absence of light, using a nematic LC of lower +viscosity (see SI) to facilitate the experiment. Such measurements showed that friction is indeed composed +of two parts: one constant term due to pinning acting as a kinetic friction, and one viscous term proportional +to the droplet velocity and due to the internal fluid motion. The equation of motion, neglecting inertia, is +thus F = fpinning + µvv, where F is the force, attractive or repulsive, due to the action of light and µv is the viscous +friction coefficient. +We understand the coupling of the droplet to light as mediated by the fringing field generated by the charge +accumulation due to the photovoltaic properties of LN crystals. The simplest form of such coupling is through +dielectrophoresis [10]. As a consequence, we expect F to become negligible in the center of the illuminated +area (d = 0), where electric field gradients vanish because of symmetry. Therefore, by extrapolating the value +of for d = 0 (1.1 µm/s in Fig. 5d, red dashed line on the left of the curve), we obtained a relationship +between the two friction terms: the pinning force has the value that the viscous friction would have for += v(d=0). In this way, the force F can be written as F = µv ( + ), i.e. F is proportional to v through +the viscous friction coefficient and thus its dependence on the distance d resembles that of , as shown in + +a) +UP ++LL +(q +DOWN +F +d (μm) +d (μm) +0 +100 +200 +300 +0 +50 +100 +150 +0 +0 +100 +100 +200 +200 +$300 +$300 +400 +400 +102W/cm2 +2.102 W/cm² +102W/cm² +500 +500 +3.102 W/cm² +3.102W/cm² +5.102 W/cm² +5.102 W/cm² +600 +600the right part of Fig. 5d. The dashed red line on the right is the power law (F = 18 d(-1.5), with d in m) that +better approximates the force in the range (140 – 320) µm. An estimation of the lower bound of the viscous +friction coefficient µv has been obtained by performing measurements on tilted LN substrate illuminated so +to induce NF droplet motion toward the top or toward the bottom, as described in the SI. Since within the +experimental errors the velocity is the same in the two situations, we could estimate that v is larger than +the ratio between twice the component of the gravitational force along the substrate and the uncertainty in +the velocity value (see SI). Since we used the lower bound for v, the values of the dielectrophoretic force +are higher than those in Fig. 5d by a certain amount. We quantified this unknown amount with the number +, in the range 0 <  <1. The gray area indicates the region within which the force exerted on the droplet is +not high enough to overcome friction and generate droplet motion. + +Figure 5: Steps leading to determine the dielectrophoretic force profile as a function of the distance between NF droplet and light +spot center. a) droplet distance from the light spot center for fixed light intensity and different starting positions and b) corresponding +droplet velocity as a function of time. Red dots indicate different starting positions while yellow dots identify the values of d and v +corresponding to the same time instant; c) droplet velocity as a function of the distance from the light spot center, obtained +combining a) and b). Red dots indicate different starting positions while the yellow dot has the coordinate (d,v); d) average velocity +(left) and dielectrophoretic force (right). The red dashed line on the left is a linear extrapolation of the velocity at d = 0. The one on +the right is a best fit of the force for d > 140 m. The parameter  (0 <  < 1) indicates that the real values of F are higher than those +reported in the graph (see text). The gray area corresponds to values of F not enough high to overcome friction and generate droplet +motion. I = 2 x 102 W/cm2. +The average velocities as a function of the droplet distance d from the center of the illuminated area, for all +the used values of the light intensity are reported in Fig. 6 for both the UP a) and DOWN case d). Note that +the average droplet velocities are negative in a) indicating droplet motion toward the light spot, and positive +in d), indicating motion away from the light spot, for all the values of I but the lowest. In this case is +negative both in a) and in d). As for the values, at a fixed light intensity, is sensitively higher in case of +attractive droplet motion and so is the covered distance. This is also translated in the force profile, which are +shown in Fig. 6 b) and e). Both the values and the range are larger in b) than in e), suggesting different values +and profiles of the fringing fields responsible for the dielectrophoretic force. In extrapolating the pink and +blue curves in Fig. 6 e) for large d, we assumed that F would become negative before vanishing at large +distances. This is because the electric field has to vanish at large d and thus a range in which dE/dd is negative +is required. At the lowest intensity the force becomes positive (red curve) and is analogous to F in Fig. 6b. + +To evaluate the electric field profile, we considered the explicit form of the dielectrophoretic force, which is +available for the simpler case of spherical shape droplets [10]. In this case, F depends on the gradient of the +electric field squared as: +𝐹 = 2𝜋𝑅3𝜀𝑚𝑅𝑒 [ +𝜀𝑝−𝜀𝑚 +𝜀𝑝+2𝜀𝑚]∇𝐸2 (1) +where R is the radius of a sphere of complex dielectric permittivity 𝜀𝑝 = 𝜀𝑝′ + 𝑖 +𝜎𝑝 +𝜔𝜀0, dispersed in a medium +of complex dielectric permittivity 𝜀𝑚 = 𝜀𝑚 +′ + 𝑖 +𝜎𝑚 +𝜔𝜀0. p and m are the conductivities of the particle and of +the medium, respectively. + +a) +c) +d) +300 +0 +(ur) +200 +0 +-5 +d 100 +(s/wr) +v (μm/s) +(nN) +00 +10 F +(s/ur) +30 +-3 +-2 +2 +-15 +-4 +-3 +-20 +0 +200 +400 +600 +800 +0 +100 +200 +300 +0 +200 +400 +600 +800 +b) +t (s) +d (μm) +d (μm)When either particle or medium are conductive, at low frequency the imaginary part of the permittivity +prevails, and the force becomes [11]: + +𝐹 = 2𝜋𝑅3𝜀𝑚 [ +𝜎𝑝−𝜎𝑚 +𝜎𝑝+2𝜎𝑚] ∇𝐸2 2𝜋𝑅3∇𝐸2 +(2) + +Although in a strict sense RM734 is an insulator, the availability of readily displaceable polarization charges +makes droplets in the NF phase behave as conductive droplets since their displaced polarization charges +cancel any field internal to the material [4,6]. This is the reason why ferroelectric nematics exhibit an +effectively large dielectric coefficient. For this reason, the ratio in square brackets in eq. (2) becomes equal +to one and, being m = 1, the gradient of the field squared can be expressed as the ratio between the force +and a term depending on the droplet’s volume. +Equation (2) is a rough approximation since (i) the sessile droplet is more similar to a hemisphere than to a +sphere and (ii) the droplet is on the top of LN whose dielectric properties are relevant and can locally +compensate the surface polarization charges displaced in RM734. We thus argue that eq. (2) overestimates +the dielectric force acting in our observations. +Interestingly, the reported optical control of LC droplets has been observed only in the ferroelectric phase. +No light-induced droplet movement has been observed in the RM734 N phase nor using conventional +nematic liquid crystals, such as 5CB. We understand this behavior as a clear indication that the +dielectrophoretic interaction is different in the N and NF phases. Results suggest that such interaction is +stronger when droplets are in the ferroelectric phase. On the contrary, in the nematic phase, the +dielectrophoretic force is not strong enough to overcome the pinning force. +The electric field responsible for the dielectrophoretic force can be obtained by integrating the force itself +and is reported in Figg. 6 c) (UP) and f) (DOWN), as a function of the distance from the center of the light +beam for different values of I. A comparison of the two sets of curves shows differences in the field profiles, +maximum values, and range. Specifically, both the range and the maximum values are higher in Fig. 6 c), +which corresponds to droplet motion toward the center of the light spots. Moreover, maxima are located +close to the center of the light spot or at a certain distance, that increases with light intensity, depending on +the direction of the droplets motion. The red curve in Fig. 6 f), which corresponds to droplet attraction, is +similar in shape to those in Fig. 6 c) but with smaller maximum value and range. + +Figure 6: Average droplet velocity, dielectrophoretic force and electric field as a function of the droplet distance d from the light spot +center, for the UP (a, b, c) and DOWN (d, e, f) case. Different colors indicate different values of the light intensity. The grey areas in +b) and e) are regions of no droplet motion, i.e. regions where the dielectrophoretic force is too weak to overcome friction. In the +same panels, dashed lines are power laws that approximate the trend of the forces for values of d outside the actual tracking, while +dotted lines are virtual extensions of the real profiles to regions where forces could not be derived from experimental data, due to +the absence of motion. Dashed and dotted lines in panels c) and f) represent regions of field values derived from these parts of the +dielectrophoretic forces that do not come from experimental data. The black spots in c) and f) mark the inflection points of the curves +thus identifying the fields range. +The electric field responsible for the dielectrophoretic forces acting on the NF droplets, is the fringing field +produced by the LN charging due to light irradiation. Although this effect could easily be attributed to the +well-known photovoltaic response of LN, the contrasting behavior observed upon inverting the LN crystal, +points to a more complex causal chain. A further clue of such complexity is the inconsistency between the +expected independence of the LN photovoltaic field on light intensity for values of I up to about 103W/cm2 +[8], and the observation showing a field that clearly depends on I. +It should also be noted that the range of the attractive force in the UP case is significantly larger than the +laser beam waist (100 vs 35 micron). Among the effects known to take place in LN, the pyroelectric effect is +the one most compatible with this observation: in the UP case the range of the force reflects that of local +temperature rise due to absorption. The repulsive force in the DOWN case, appears instead to have a shorter +range, more compatible with the illuminated area, suggesting that photovoltaic and pyroelectric effect may +sum up differently on the two sides. However, such a difference necessarily requires the symmetry between +positive and negative charges on the two LN sides to be broken. Any process that would equally modify +positive and negative charges, such as a local modulation of the ferroelectric bulk polarization of LN, could +not explain our observations. Indeed, the absorbance of the laser green light by the iron-doped LN provides +such a symmetry breaking, since the directly irradiated LN side is the location of largest density of iron +electron excitation. We reasonably expect such freed electrons to adopt a different spatial distribution when +close to the positive vs negative LN surface. + + + +(q +×106 +Up +a) +0 +c) +3 +102W/cm² +-2 +-10 +2.102 W/cm2 +2.5 +3.102 W/cm² +-4 +-20 +(w/N) +5-102W/cm² +2 +(s/ur) +-6 +F +1.5 +-8 +-40 +2 +-10 +102W/cm² +-50 +102 W/cm2 +2.102 W/cm2 +2.102 W/cm² +-12 +3.102 W/cm² +-60 +3.102 W/cm² +0.5 +5.102 W/cm2 +5.102 W/cm² +-14 +-70 +0 +0 +100 +200 +300 +400 +0 +100 +200 +300 +400 +0 +100 +200 +300 +400 +d (μm) +d (μm) +d (μm) +d) 2.5 +e) 20 +f) +×106 +DOWN +—102 W/cm² +102W/cm2 +3.102 W/cm² +3-102 W/cm² +0.8 +2 +15 +5.102W/cm² +5.102 W/cm² +1.5 +10 +(s/urt)^ +(Nu) +1 +5 +E +F +0.4 +0 +2 +0.5 +a +0.2 +102 W/cm² +0 +-5 +3.10² W/cm² +5.102 W/cm² +-0.5 +-10 +.0 +0 +50 +100 +150 +200 +0 +100 +200 +300 +400 +0 +100 +200 +300 +400 +d(μm) +d (μm) +d (μm)Noteworthy, asymmetry of the two surfaces of z cut LN crystals has already been observed in literature [12- +14], both in relation to the efficiency of the surfaces in generating electrostatic fields [13,14], that has been +reported to be different for -z and +z surfaces, and in relation to the asymmetric behavior of liquid crystals +confined in LN microchannels under light illumination [12]. In this latter case, the combination of photovoltaic +and pyroelectric fields was invoked to account for a liquid crystal response dependent on the irradiated +surface in an optofluidic chip based on z cut LN crystals. +Conclusion +We have demonstrated the all-optical control of sessile ferroelectric liquid droplets deposited on lithium +niobate substrates. Droplets are attracted or repelled by the center of the illuminated region depending on +the irradiated side of the LN crystal with a range of interaction that is sub millimetric in our conditions, but +that may increase by using different surface coatings. Droplets can also be walked by the light beam over +long distances. The reported droplet actuation is observed only in the NF phase, which highlights the crucial +role played by the RM734 ferroelectric polarization. +Based on our results, we expect that by reconfiguring the experimental apparatus and structuring light with +a Spatial Light Modulator, all the basic droplet handling operations required in a common microfluidic device +can be obtained. This opportunity may pave the way to novel technological applications triggered by the +peculiar properties of ferroelectric nematics. +We believe that the results reported here contribute an additional piece to the collection of intriguing +features characterizing the ferroelectric nematic phase. + +References +[1] Xi Chen, Eva Korblova, Dengpan Dong, Xiaoyu Wei, Renfan Shao, Leo Radzihovsky, Matthew A. Glaser, +Joseph E. Maclennan, Dmitry Bedrov, David M. Walba, and Noel A. Clark, First-principles experimental +demonstration of ferroelectricity in a thermotropic nematic liquid crystal: Polar domains and striking electro- +optics, Proc. Natl. Acad. Sci. U. S. A. 2020, 117, 14021–14031 +[2] F. Caimi et al., Surface alignment of ferroelectric nematic liquid crystals, Soft Matter 2021, 17, 8130–8139. +[3] M.T. Mathè, A. Buka, A. Jakly, P. Solomon, Ferroelectric nematic liquid crystal thermo-motor, Phys. Rev. E +2022, 105, L052701. +[4] R. Barboza, S. Marni, F. Ciciulla, F. Ali Mir, G. Nava, F. Caimi, A. Zaltron, N. Clark, T. Bellini, L. Lucchetti, +Explosive Electrostatic Instability of Ferroelectric Liquid Droplets on Ferroelectric Solid Surfaces, Proc. Natl. +Acad. Sci. U. S. A. 2022, 119, e2207858119. +[5] L. Cmok, N. Sebastián, A. Mertelj, Y. Kong, X. Zhang, I. Drevenšek-Olenik, Light-induced dynamics of liquid +crystalline droplets on the surface of iron-doped lithium niobate crystals, arXiv:2208.02318 +[6] F. Caimi, G. Nava, S. Fuschetto, L. Lucchetti, P. Paiè, R. Osellame, X. Chen, N. A. Clark, Ma. Glaser, T. Bellini, +Superscreening and polarization control in confined ferroelectric nematic liquids, arXiv:2210.00886 +[7] S. Sanna and W. G. Schmidt, LiNbO3 surfaces from a microscopic perspective, J. Phys.: Condens. Matter 29 +(2017) 413001-4130048. +[8] Volk, T. & Wöhlecke, M. Lithium niobate: defects, photorefraction and ferroelectric switching. 115, +Springer Science & Business Media, 2008. + +[9] L. Lucchetti, K. Kushnir, V. Reshetnyak, F. Ciciulla, A. Zaltron, C. Sada, F. Simoni, Light-induced electric field +generated by photovoltaic substrates investigated through liquid crystal reorientation, Optical Mat., 2017, +73, 64. +[10] A. Zaltron, D. Ferraro, A. Meggiolaro, S. Cremaschini, M. Carneri, E. Chiarello, P. Sartori, M. Pierno, C. +Sada, G. Mistura, Optofluidic Platform for the Manipulation of Water Droplets on Engineered LiNbO3 Surfaces, +Adv. Mater. Interfaces 2022, 9, 2200345 +[11] D. A. Saville, T. Bellini, V. Degiorgio, F. Mantegazza, An extended Maxwell–Wagner theory for the electric +birefringence of charged colloids, 2000, J. Chem. Phys.113, 6974. +[12] S. Bonfadini, F. Ciciulla, L. Criante, A. Zaltron, F. Simoni, V. Reshetnyak, L. Lucchetti, Optofluidic platform +using liquid crystals in lithium niobate microchannel Sci. Rep. 2019, 9, 1062 +[13] Z. Gao, Y. Mi, M. Wang, X. Liu, X. Zhang, K. Gao, L. Shi, E. R. Mugisha, H. Chen, W. Yan, Hydrophobic- +substrate based water-microdroplet manipulation through the long-range photovoltaic interaction from a +distant LiNbO3:Fe crystal, 2021, Opt. Exp., 29, 3808 +[14] F. Li, X. Zhang, K. Gao, L. Shi, Z. Zan, Z. Gao, C. Liang, E. R. Mugisha, H. Chen, W. Yan, All-optical splitting +of dielectric microdroplets by using a y-cut-LN-based anti-symmetrical sandwich structure, 2019, Opt. Exp., +27, 25767 + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/pdAyT4oBgHgl3EQfZPdk/content/tmp_files/load_file.txt b/pdAyT4oBgHgl3EQfZPdk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..39c6e80f8eaf79833a2188dc6b67b556142d6f50 --- /dev/null +++ b/pdAyT4oBgHgl3EQfZPdk/content/tmp_files/load_file.txt @@ -0,0 +1,387 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf,len=386 +page_content='Walk ferroelectric liquid droplets with light Stefano Marni1, Giovanni Nava2, Raouf Barboza1, Tommaso Bellini2*, Liana Lucchetti1* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Dipartimento SIMAU, Universita` Politecnica delle Marche, via Brecce Bianche, 60131 Ancona, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' E-mail: l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='lucchetti@univpm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='it 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Medical Biotechnology and Translational Medicine Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=', University of Milano, 20054 Segrate, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' E-mail: tommaso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='bellini@unimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='it Abstract We show that the motion of ferroelectric liquid sessile droplets deposited on a ferroelectric lithium niobate substrate can be controlled by a light beam of moderate intensity irradiating the substrate at a distance of several droplet diameters from the droplet itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The ferroelectric liquid is a nematic liquid crystal in which almost complete polar ordering of the molecular dipoles generates an internal macroscopic polarization locally collinear to the mean molecular long axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Upon entering the ferroelectric phase droplets are either attracted toward the center of the beam or repelled, depending on the side of the lithium niobate exposed to light irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Moreover, moving the beam results in walking the ferroelectric droplet over long distances on the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' We understand this behavior as due to the coupling between the polarization of the ferroelectric droplet and the polarization photoinduced in the irradiated region of the lithium niobate substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Indeed, the effect is not observed in the conventional nematic phase, suggesting the crucial role of the ferroelectric liquid crystal polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Introduction The recent discovery of the ferroelectric nematic phase, NF, [1], opened a new chapter not only for the liquid crystal community, but in the whole condensed-matter physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Beside adding a new,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' very peculiar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' member to the group of ferroelectric materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' the new phase offers a broad range of physical effects to explore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' ranging from the behavior of topological defects to surface anchoring [2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' response to low frequency electric fields [3] and light,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' interplay of bound and free electric charges,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' viscoelastic properties,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' field-controlled hydrodynamics [3-5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' field-order coupling in both the NF and the pre-transitional regions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' behavior in confined geometry [6],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' just to cite a few examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' In this scenario, we recently performed experiments devoted to characterize the behavior of sessile NF droplets on ferroelectric solid substrates [4] and found that the combination of fluidity and polarity gives rise to an electromechanical instability induced by the coupling of the LC polarization with that pyroelectrically induced in the solid substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' In this work, we instead analyze the effects of the photovoltaic charging of lithium niobate (LN) ferroelectric solid substrates on NF sessile droplets, at constant temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The advantage of manipulating sessile droplets by light are related to the possibility of focusing the beam to small regions of the substrates thus limiting the extent of the charged regions and controlling its distance and position with respect to the droplets, and of quickly reconfiguring it in different illumination geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Results show that, under proper experimental conditions, light irradiation gives rise to an instability very similar to the one observed in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Moreover, focused beams impinging on the LN substrate at a certain distance from the sessile NF droplets allow to optically control their motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Droplets are attracted toward the center of the illuminated area or repelled away from it depending on which side of the substrate is exposed to light irradiation and can be walked by the light beam over long distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Our results contribute an additional piece to the collection of intriguing features characterizing the ferroelectric nematic phase and may have potential for future applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Material and methods The ferroelectric liquid crystal used in this work is 4-[(4-nitrophenoxy)carbonyl]phenyl2,4- dimethoxybenzoate (RM734).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' It was synthesized as described in [1] and its structure and phase diagram have already been reported [1, 2, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' In this compound the ferroelectric nematic phase appears through a second order phase transition when cooling from the conventional higher temperature nematic (N) phase and exists in the range 133°C < T < 80°C [2,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The spontaneous polarization P of RM734 is either parallel or antiparallel to the molecular director n, defining the average orientation of the molecular axis, and exceeds 6 µC/cm2 at the lowest T in the NF phase [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The lithium niobate (LN) ferroelectric substrates are 900 micron thick z-cut crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Experiments were performed on iron-doped substrates containing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='1% mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' of iron with a reduction factor R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The bulk spontaneous polarization PLN of LN crystals along the [0001] z-axis is of the order of 70 µC/cm2 and does not depend significantly on T in the explored range since its Curie temperature is much higher (≈ 1140°C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The huge bulk polarization of LN does not however translate in a huge surface charge density because of very efficient compensation mechanisms at the z-cut surfaces, lowering the equilibrium surface charge to only about 10-2 µC/cm2 [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' When the crystal is exposed to light with wavelength in the iron absorption spectrum, the surface charge of LN significantly increases because of the photovoltaic effect [8], consisting in the appearance of a photo-induced current according to the scheme Fe2+ + h\uf06e → Fe3+ + e-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The subsequent charge distribution that takes place inside the crystal gives rise to an internal electric field with saturation values up to 107 V/m, depending on the dopant concentration and on R [8,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Before droplet deposition, LN crystals were coated by a layer of fluorolink, as described in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The RM734 droplets used in optical motion control experiments, have an average diameter of 45 µm, as measured with a calibration slide and were obtained starting from bigger droplets as described in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' They were deposited on fluorolink-coated LN substrates that were previously slowly heated up to T = 200°C, corresponding to the RM734 isotropic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Successively, T was decreased down to 110°C, which is in the ferroelectric range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Noteworthy, the cooling rate was kept slow enough to avoid the droplets electromechanical instability observed in [4] that was triggered by the pyroelectric charging of LN surfaces and required a proper cooling speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The light used to induce the photovoltaic effect in LN crystals is a gaussian beam from a frequency doubled Nd:YAG laser (\uf06c = 532 nm), with power P in the range (5 -25) mW, focused to a waist w = 35 µm, which corresponds to an intensity ranging from I = 102 W/cm2 to I = 5 x 102 W/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' LN substrates were irradiated from below at different distances to the RM734 droplets, holding the temperature fixed at T = 110°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Polarized optical microscope (POM) observations during light irradiation were carried out and videos of the droplets behavior were recorded with a rate of 25 frames per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Results Our first experiment has been devoted to compare the effect of photo-induced charging of LN substrates on RM734 sessile droplets, with the one of pyroelectric charging that we recently reported in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' To this purpose, LN was irradiated in correspondence of the droplet position with an unfocused gaussian beam having a diameter slightly larger than that of the droplet itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 1, such a configuration produces an electromechanical instability consisting in the sudden emission of interfacial fluid jets, as observed in [4], indicating that the coupling between the droplet polarization and the one photo-induced in LN has the same features as the coupling with the pyroelectrical LN polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' In analogy to the interpretation proposed there, we understand this phenomenon as due to the fringing field generated by the photovoltaic charging of LN substrates, which is thus able to affect the NF droplets behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' This result opens the way to the possible optical control of ferroelectric LC droplets on LN substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Indeed, light is easily controllable: the size and the position of the irradiated region can be varied almost at will and the intensity of the light beam can be tuned in a very short time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Figure 1: RM734 sessile droplet on LN substrate exposed to light illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The beam diameter is slightly larger than that of the droplet, as shown in the cartoon on the left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Shape instability with the emission of interfacial fluid jets that bifurcate and branch, is clearly visible (figure extracted from video S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' I = 20 W/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Droplet diameter 350 \uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The effect reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 1 is independent on the side of the LN substrate contacting the droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' This is in agreement with the observations in [4] where the sign of the charges of the LN surface that contacts the LC droplet was irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The same fringing field generated by photovoltaic charging leads to new effects when the light beam is focused at a distance from the droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' In this case the droplet retains approximately the same shape but is put in motion by a force generated by light irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Remarkably, the direction of such a force can be either attractive, with droplets that move toward the center of the illuminated region, or repulsive, leading to a droplet motion away from the light spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The sign of the force depends on the irradiated side of the LN substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 2, which shows two series of frames extracted from video S2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 2b-d) and S3 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 2b’-d’), available in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' As visible, a NF droplet moves toward the illuminated area or away from it, depending on the direction of the LN bulk polarization with respect to the direction of the incoming light (as in the cartoons on the left-hand side of the figure), which we will refer to as “UP” and “DOWN” in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Figure 2: a) and a’) Sketches of the experimental arrangements in case of droplet attraction a) and repulsion a’): b) and c) video frames taken at different instants showing a RM734 NF droplet moving toward the center of the illuminate area;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' b’) and c’) video frames taken at different instants showing a RM734 NF droplet moving away from the center of the illuminate area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' I = 5 x 102 W/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 100μmUP attraction 50μm 50μm 50μm green laser a) b) to c) to +24s d) to +62s DOWN repulsion 50μm 50μm 50μm green laser a\') b") to c\') to + 103s d\') to + 255sBy continuously varying the position of the beam by means of galvanometric mirrors, it is possible to drag the NF droplet over long distances, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='3 in the case of droplet attraction (frames extracted by video S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Droplet dragging is also observed in case of repulsion, although with a less controlled trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Figure 3: Video frames at different instants showing the motion of a RM734 NF droplet following a Gaussian light spot along a wavy path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Green diamonds indicate the different positions of the droplets through the video and dashed line represents the droplet trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' I = 3 x 102 W/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The curves describing droplet motion vs time are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 4 in case of both the UP a) and the DOWN case b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The different curves in each figure refer to different values of the light intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Different initial distances between droplet and light spot center have also been chosen to highlight the role of both these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' To avoid superpositions and improve the clarity of the two graphs, some of the curves have been translated in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Remarkably, the curve corresponding to the lowest beam intensity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 4b (red curve in panel DOWN), shows droplet motion toward the light spot, contrary to what happens at higher intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' A close inspection of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 4 reveals that NF droplets attracted toward the illuminated area stop at the edge of a region corresponding to the beam waist (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 4a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' NF droplets repelled by the illuminated area also stop at a certain distance from its center, that increases with increasing light intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' This behavior suggests the presence of a pinning force opposing the one due to the interaction with the light beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' a) q d) e f) 100um n k Figure 4: Droplet distance from the center of the illuminated area (d = 0) as a function of time, for the UP a) and DOWN case b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The time t = 0 s corresponds to the beginning of LN illumination, however, to improve the clarity of the figure, some of the curves have been shifted in time so to start at t > 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Different colors indicate different values of the light intensity as reported in the legend;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' different starting distances have also been chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The two cartoons on top help defining the parameters and understanding the experimental arrangement in the two conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' From these and similar curves it is possible to extract the time dependence of the droplet velocity v for each used value of the light intensity I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' As an example, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 5 b shows v vs t in the case of droplet attraction and I = 2x 102 W/cm2, for different initial distances from the center of the illuminated area, as extracted from the corresponding d vs t curves reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 5 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Again, for the sake of clarity, some of the curves are shifted in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Red dots indicate the initial conditions in terms of initial distance from the light spot center and initial droplet velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The three dashed lines identify the values of d and v corresponding to the same value of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Combining the v vs t curve with the corresponding d vs t, one obtains the velocity as a function of the distance from the center of the illuminated area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' This is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 5c, where the different curves refer to different initial droplet positions with respect to the center of the light spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Interestingly, despite a relevant irregularity, the velocity keeps similar values at fixed positions, that is its value is independent on the initial position of the droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' This indicates that droplets inertia can be neglected in describing the droplet motion, demonstrating that the friction force contains a term dependent on v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The average velocity is reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 5d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' To better understand droplet motion, a friction characterization is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' To this aim we performed measurements of droplet motion along a tilted substrate in the absence of light, using a nematic LC of lower viscosity (see SI) to facilitate the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Such measurements showed that friction is indeed composed of two parts: one constant term due to pinning acting as a kinetic friction, and one viscous term proportional to the droplet velocity and due to the internal fluid motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The equation of motion, neglecting inertia, is thus F = fpinning + µvv, where F is the force, attractive or repulsive, due to the action of light and µv is the viscous friction coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' We understand the coupling of the droplet to light as mediated by the fringing field generated by the charge accumulation due to the photovoltaic properties of LN crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The simplest form of such coupling is through dielectrophoresis [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' As a consequence, we expect F to become negligible in the center of the illuminated area (d = 0), where electric field gradients vanish because of symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Therefore, by extrapolating the value of for d = 0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='1 µm/s in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 5d, red dashed line on the left of the curve), we obtained a relationship between the two friction terms: the pinning force has the value that the viscous friction would have for = v(d=0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' In this way, the force F can be written as F = µv ( + ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' F is proportional to v through the viscous friction coefficient and thus its dependence on the distance d resembles that of , as shown in a) UP +LL (q DOWN F d (μm) d (μm) 0 100 200 300 0 50 100 150 0 0 100 100 200 200 $300 $300 400 400 102W/cm2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='102 W/cm² 102W/cm² 500 500 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='102 W/cm² 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='102W/cm² 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='102 W/cm² 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='102 W/cm² 600 600the right part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 5d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The dashed red line on the right is the power law (F = 18 d(-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='5), with d in \uf06dm) that better approximates the force in the range (140 – 320) µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' An estimation of the lower bound of the viscous friction coefficient µv has been obtained by performing measurements on tilted LN substrate illuminated so to induce NF droplet motion toward the top or toward the bottom, as described in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Since within the experimental errors the velocity is the same in the two situations, we could estimate that \uf06dv is larger than the ratio between twice the component of the gravitational force along the substrate and the uncertainty in the velocity value (see SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Since we used the lower bound for \uf06dv, the values of the dielectrophoretic force are higher than those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 5d by a certain amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' We quantified this unknown amount with the number \uf061, in the range 0 < \uf061 <1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The gray area indicates the region within which the force exerted on the droplet is not high enough to overcome friction and generate droplet motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Figure 5: Steps leading to determine the dielectrophoretic force profile as a function of the distance between NF droplet and light spot center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' a) droplet distance from the light spot center for fixed light intensity and different starting positions and b) corresponding droplet velocity as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Red dots indicate different starting positions while yellow dots identify the values of d and v corresponding to the same time instant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' c) droplet velocity as a function of the distance from the light spot center, obtained combining a) and b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Red dots indicate different starting positions while the yellow dot has the coordinate (d,v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' d) average velocity (left) and dielectrophoretic force (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The red dashed line on the left is a linear extrapolation of the velocity at d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The one on the right is a best fit of the force for d > 140 \uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The parameter \uf061 (0 < \uf061 < 1) indicates that the real values of F are higher than those reported in the graph (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The gray area corresponds to values of F not enough high to overcome friction and generate droplet motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' I = 2 x 102 W/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The average velocities as a function of the droplet distance d from the center of the illuminated area, for all the used values of the light intensity are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 6 for both the UP a) and DOWN case d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Note that the average droplet velocities are negative in a) indicating droplet motion toward the light spot, and positive in d), indicating motion away from the light spot, for all the values of I but the lowest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' In this case is negative both in a) and in d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' As for the values, at a fixed light intensity, is sensitively higher in case of attractive droplet motion and so is the covered distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' This is also translated in the force profile, which are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 6 b) and e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Both the values and the range are larger in b) than in e), suggesting different values and profiles of the fringing fields responsible for the dielectrophoretic force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' In extrapolating the pink and blue curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 6 e) for large d, we assumed that F would become negative before vanishing at large distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' This is because the electric field has to vanish at large d and thus a range in which dE/dd is negative is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' At the lowest intensity the force becomes positive (red curve) and is analogous to F in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' To evaluate the electric field profile, we considered the explicit form of the dielectrophoretic force, which is available for the simpler case of spherical shape droplets [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' In this case, F depends on the gradient of the electric field squared as: 𝐹 = 2𝜋𝑅3𝜀𝑚𝑅𝑒 [ 𝜀𝑝−𝜀𝑚 𝜀𝑝+2𝜀𝑚]∇𝐸2 (1) where R is the radius of a sphere of complex dielectric permittivity 𝜀𝑝 = 𝜀𝑝′ + 𝑖 𝜎𝑝 𝜔𝜀0, dispersed in a medium of complex dielectric permittivity 𝜀𝑚 = 𝜀𝑚 ′ + 𝑖 𝜎𝑚 𝜔𝜀0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' \uf073p and \uf073m are the conductivities of the particle and of the medium, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' a) c) d) 300 0 (ur) 200 0 5 d 100 (s/wr) v (μm/s) (nN) 00 10 F (s/ur) 30 3 2 2 15 4 3 20 0 200 400 600 800 0 100 200 300 0 200 400 600 800 b) t (s) d (μm) d (μm)When either particle or medium are conductive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' at low frequency the imaginary part of the permittivity prevails,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' and the force becomes [11]: 𝐹 = 2𝜋𝑅3𝜀𝑚 [ 𝜎𝑝−𝜎𝑚 𝜎𝑝+2𝜎𝑚] ∇𝐸2\uf0bb 2𝜋𝑅3∇𝐸2 (2) Although in a strict sense RM734 is an insulator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' the availability of readily displaceable polarization charges makes droplets in the NF phase behave as conductive droplets since their displaced polarization charges cancel any field internal to the material [4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' This is the reason why ferroelectric nematics exhibit an effectively large dielectric coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' For this reason, the ratio in square brackets in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' (2) becomes equal to one and, being \uf065m = 1, the gradient of the field squared can be expressed as the ratio between the force and a term depending on the droplet’s volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Equation (2) is a rough approximation since (i) the sessile droplet is more similar to a hemisphere than to a sphere and (ii) the droplet is on the top of LN whose dielectric properties are relevant and can locally compensate the surface polarization charges displaced in RM734.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' We thus argue that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' (2) overestimates the dielectric force acting in our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Interestingly, the reported optical control of LC droplets has been observed only in the ferroelectric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' No light-induced droplet movement has been observed in the RM734 N phase nor using conventional nematic liquid crystals, such as 5CB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' We understand this behavior as a clear indication that the dielectrophoretic interaction is different in the N and NF phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Results suggest that such interaction is stronger when droplets are in the ferroelectric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' On the contrary, in the nematic phase, the dielectrophoretic force is not strong enough to overcome the pinning force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The electric field responsible for the dielectrophoretic force can be obtained by integrating the force itself and is reported in Figg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 6 c) (UP) and f) (DOWN), as a function of the distance from the center of the light beam for different values of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' A comparison of the two sets of curves shows differences in the field profiles, maximum values, and range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Specifically, both the range and the maximum values are higher in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 6 c), which corresponds to droplet motion toward the center of the light spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Moreover, maxima are located close to the center of the light spot or at a certain distance, that increases with light intensity, depending on the direction of the droplets motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The red curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 6 f), which corresponds to droplet attraction, is similar in shape to those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' 6 c) but with smaller maximum value and range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Figure 6: Average droplet velocity, dielectrophoretic force and electric field as a function of the droplet distance d from the light spot center, for the UP (a, b, c) and DOWN (d, e, f) case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Different colors indicate different values of the light intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The grey areas in b) and e) are regions of no droplet motion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' regions where the dielectrophoretic force is too weak to overcome friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' In the same panels, dashed lines are power laws that approximate the trend of the forces for values of d outside the actual tracking, while dotted lines are virtual extensions of the real profiles to regions where forces could not be derived from experimental data, due to the absence of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Dashed and dotted lines in panels c) and f) represent regions of field values derived from these parts of the dielectrophoretic forces that do not come from experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The black spots in c) and f) mark the inflection points of the curves thus identifying the fields range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The electric field responsible for the dielectrophoretic forces acting on the NF droplets, is the fringing field produced by the LN charging due to light irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Although this effect could easily be attributed to the well-known photovoltaic response of LN, the contrasting behavior observed upon inverting the LN crystal, points to a more complex causal chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' A further clue of such complexity is the inconsistency between the expected independence of the LN photovoltaic field on light intensity for values of I up to about 103W/cm2 [8], and the observation showing a field that clearly depends on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' It should also be noted that the range of the attractive force in the UP case is significantly larger than the laser beam waist (100 vs 35 micron).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Among the effects known to take place in LN, the pyroelectric effect is the one most compatible with this observation: in the UP case the range of the force reflects that of local temperature rise due to absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The repulsive force in the DOWN case, appears instead to have a shorter range, more compatible with the illuminated area, suggesting that photovoltaic and pyroelectric effect may sum up differently on the two sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' However, such a difference necessarily requires the symmetry between positive and negative charges on the two LN sides to be broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Any process that would equally modify positive and negative charges, such as a local modulation of the ferroelectric bulk polarization of LN, could not explain our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Indeed, the absorbance of the laser green light by the iron-doped LN provides such a symmetry breaking, since the directly irradiated LN side is the location of largest density of iron electron excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' We reasonably expect such freed electrons to adopt a different spatial distribution when close to the positive vs negative LN surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' (q ×106 Up a) 0 c) 3 102W/cm² 2 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='102 W/cm2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='102 W/cm² 4 20 (w/N) 5-102W/cm² 2 (s/ur) 6 F 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='5 8 40 2 10 102W/cm² 50 102 W/cm2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='102 W/cm2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='102 W/cm² 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='102 W/cm² 60 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='102 W/cm² 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='102 W/cm2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='102 W/cm² 14 70 0 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400 d (μm) d (μm) d (μm) d) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='5 e) 20 f) ×106 DOWN —102 W/cm² 102W/cm2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='102 W/cm² 3-102 W/cm² 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='8 2 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='102W/cm² 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='102 W/cm² 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='5 10 (s/urt)^ (Nu) 1 5 E F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='4 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='5 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='2 102 W/cm² 0 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='10² W/cm² 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='102 W/cm² 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='5 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content='0 0 50 100 150 200 0 100 200 300 400 0 100 200 300 400 d(μm) d (μm) d (μm)Noteworthy, asymmetry of the two surfaces of z cut LN crystals has already been observed in literature [12- 14], both in relation to the efficiency of the surfaces in generating electrostatic fields [13,14], that has been reported to be different for -z and +z surfaces, and in relation to the asymmetric behavior of liquid crystals confined in LN microchannels under light illumination [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' In this latter case, the combination of photovoltaic and pyroelectric fields was invoked to account for a liquid crystal response dependent on the irradiated surface in an optofluidic chip based on z cut LN crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Conclusion We have demonstrated the all-optical control of sessile ferroelectric liquid droplets deposited on lithium niobate substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Droplets are attracted or repelled by the center of the illuminated region depending on the irradiated side of the LN crystal with a range of interaction that is sub millimetric in our conditions, but that may increase by using different surface coatings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Droplets can also be walked by the light beam over long distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' The reported droplet actuation is observed only in the NF phase, which highlights the crucial role played by the RM734 ferroelectric polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Based on our results, we expect that by reconfiguring the experimental apparatus and structuring light with a Spatial Light Modulator, all the basic droplet handling operations required in a common microfluidic device can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' This opportunity may pave the way to novel technological applications triggered by the peculiar properties of ferroelectric nematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' We believe that the results reported here contribute an additional piece to the collection of intriguing features characterizing the ferroelectric nematic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' References [1] Xi Chen, Eva Korblova, Dengpan Dong, Xiaoyu Wei, Renfan Shao, Leo Radzihovsky, Matthew A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfZPdk/content/2301.00219v1.pdf'} +page_content=' Glaser, Joseph E.' metadata={'source': 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our daily life. However, the unfair forecasting of +MTSs not only degrades their practical benefit but even brings about serious potential risks. Such unfair MTS forecasting may be +attributed to variable disparity leading to advantaged and disadvantaged variables. This issue has rarely been studied in the existing +MTS forecasting models. To address this significant gap, we formulate the MTS fairness modeling problem as learning informative +representations attending to both advantaged and disadvantaged variables. Accordingly, we propose a novel framework, named +FairFor, for fairness-aware MTS forecasting. FairFor is based on adversarial learning to generate both group-irrelevant and -relevant +representations for downstream forecasting. FairFor first adopts the recurrent graph convolution to capture spatio-temporal variable +correlations and to group variables by leveraging a spectral relaxation of the K-means objective. Then, it utilizes a novel filtering & +fusion module to filter the group-relevant information and generate group-irrelevant representations by orthogonality regularization. The +group-irrelevant and -relevant representations form highly informative representations, facilitating to share the knowledge from +advantaged variables to disadvantaged variables and guarantee fairness. Extensive experiments on four public datasets demonstrate +the FairFor effectiveness for fair forecasting and significant performance improvement. +Index Terms—Multivariate time series, forecasting, fairness, adversarial learning. +! +1 +INTRODUCTION +M +ULTIVARIATE time-series (MTS) forecasting, penetrat- +ing our daily living, studying, working, and en- +tertaining, has played a critical role in a wide range of +real-world applications. Examples include climate forecast- +ing [1], stock trend analysis [2], [3], road-use monitoring [4], +[5], and clinical risk forecasting [6]. For example, in quanti- +tative finance analysis, multiple financial factors co-involve +and their forecasts assist financial practitioners in optimiz- +ing investment portfolios and strategies. MTS forecasting +capably enhances investment opportunities, schedules suit- +able services, and adjusts optimal plans. It has been one of +the most fundamental yet beneficial real-world tasks. +However, unfair MTS forecasting results in the unequal +forecasting accuracy among variables. It may degrade the +practical benefits of MTS forecasting or even bring about +serious potential risks. For example, in finance, the trends of +high-frequency trading stocks are difficult to be predicted +owing to their highly volatile temporal patterns which +are easily overwhelmed by those of stable low-frequency +trading stocks. In the urban police deployment, the crimes +in regions with geographical neighborhoods are easier to +• +Hui He is with the School of Medical and Technology, Beijing Institute of +Technology, Beijing 100081, China. +E-mail: hehui617@bit.edu.cn +• +Qi Zhang, Shoujin Wang and Longbing Cao are with Data Science Lab, +University of Technology Sydney, Ultimo, NSW 2007, Australia. +E-mail: qi.zhang-13@students.uts.edu.au, +{shoujin.wang, longbing.cao}@uts.edu.au +• +Kun Yi and Zhendong Niu are with the School of Computer Science and +Technology, Beijing Institute of Technology, Beijing 100081, China. +E-mail: {yikun, zniu}@bit.edu.cn +Manuscript received April 19, 2005; revised August 26, 2015. +Advantaged Variable +Disadvantaged Variable +Methods +StemGNN +Informer +TS2Vec +MTGNN +AGCRN +VAR +97.0574 +46.9595 +119.3862 +144.0949 +122.7769 +Fig. 1. An example of MTS forecasting unfairness: the table at the upper +part shows the variance of the forecasting performance (MAE) over all +variables derived from five advanced MTS forecasting methods on real- +world traffic dataset; the curves below show the true traffic flow data +and its corresponding prediction from StemGNN model on four sensors +(variables) numbered 15, 16, 20 and 21. It is clear that StemGNN +achieves desirable performance on those advantaged variables (#20 +and #21 marked in dotted ellipse) while poor performance on those +disadvantaged variables (#15 and #16 marked in solid ellipse). +be predicted than the isolated regions due to the shared +socioeconomic factors. This results in the inherent variable +disparity, e.g., high vs low-frequency stocks and isolated vs +neighborhood regions, which may cause MTS forecasting +models to generate unequal forecasting accuracy over dif- +ferent variables, i.e., well-performed (advantaged) variables +and badly-performed (disadvantaged) variables. The unfair +arXiv:2301.11535v1 [cs.LG] 27 Jan 2023 + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +2 +results may deceive people by the overall performance into +trusting the results on disadvantaged variables, potentially +causing practical failures or risks, e.g., great investment +losses or fatal treatment plans. To this end, it is necessary +to develop fair MTS forecasting models to eliminate unfair +forecasting for improving both the overall performance and +specifically the performance on disadvantaged variables. +Previously, great efforts have been made to design com- +petent MTS forecasting models to explore temporal depen- +dencies (i.e., short- and long-term temporal patterns) [7], [8], +[9], [10], spatial dependencies (i.e. inter-series correlations +across multiple time series) [4], [11], [12], [13], [14], inter- +pretability of modeling [15], [16], [17], [18], or complicated +nonstationary [19], [20]. Although achieving great success, +those models focusing on accuracy improvement [21] may +be vulnerable to variable disparity [22], leading to serious +performance unfairness among variables. As illustrated in +Figure 1, all five advanced MTS forecasting models have +large accuracy variances (i.e., VAR) over different variables, +and one representative, StemGNN, attends to certain vari- +ables and accordingly leads obviously bad performance on +variables #15 and #16 (disadvantaged variables). The il- +lustration obviously reveals the significant (widely existing) +yet challenging (rarely considered) unfairness issue in MTS +forecasting models. +Although fair MTS forecasting has been less studied +in the literature, a variety of methods for fairness mod- +eling recently emerge in other learning tasks, such as +classification [23], [24], [25], [26], clustering [27], [28] and +ranking [29], [30], including by customizing fairness reg- +ularization [31] and adversarial learning [32], [33]. These +methods generally model fairness from two perspectives, +i.e., individual and group perspectives. Some methods +model individual fairness by defining a reasonable similar- +ity metric based on a fairness graph [34], [35], weighted +ℓp-metrics [27], [36] or the Wasserstein distance [37] at a +fine granularity. The other methods aim to eliminate group +unfairness on sensitive attributes (e.g., gender, age, or race +of users). Specifically, they learn to filter out the sensitive +information via adversarial learning to obtain insensitive +(group-irrelevant) representations [32], [33] or disentangle +data representations into sensitive and insensitive repre- +sentations by orthogonality regularization [29], [30]. How- +ever, individual fairness may be potentially harmful to +overall forecasting performance and is costly in calculating +the similarity measurement among all variables for high- +dimensional MTS data [38]. Group fairness generally de- +pends on natural groups pre-defined by sensitive attributes, +while the attributes are usually inaccessible in MTS data. +This also brings challenges for fairness modeling of MTS +scenarios. +Considering +the +inter-variable +correlations +in +MTS +data [11], correlated variables may have similar (advan- +taged/disadvantaged) performance and can be clustered +into one group to handle. To this end, we deliberatively +employ group fairness to achieve fair MTS forecasting, +simplifying fairness modeling from the individual (vari- +able) level to a group-wise manner and avoiding the high +computational cost on individual variable fairness. We ex- +pect to share the knowledge between advantaged variables +(groups) and disadvantaged variables (groups) to improve +the performance of disadvantaged variables, guaranteeing +performance fairness and overall improvement simultane- +ously. Accordingly, two main challenges (CH) are necessar- +ily addressed in our work: CH1, how can we adaptively +learn variable grouping? CH2, how can we effectively learn +group-relevant and group-irrelevant representations? In or- +der to address these critical challenges, we propose a novel +fair MTS forecasting model FairFor which consists of two +main modules: variable correlating & grouping and filtering +& fusion, which aims to address CH1 and CH2 respectively. +To be specific, in the variable correlating & grouping mod- +ule, we first adopt recurrent graph convolution to capture +spatio-temporal variable correlations and introduce cluster- +ing objectives to learn variable grouping. Inspired by [32], +[39], we design a novel filtering & fusion to generate group- +irrelevant representations via adversarial learning. Owing to +our novel and specific design, FairFor effectively improves +the overall MTS forecasting performance and achieves much +fairer performance among variables. +The contributions of our work are summarized below: +• +We study a significant and common problem in MTS +forecasting, i.e., the unfairness of forecasting perfor- +mance. Accordingly, we propose a novel fairness- +aware MTS forecasting framework to address this +problem. To the best of our knowledge, this is the +first effort on fair MTS forecasting. +• +We propose a novel variable correlating & grouping +module to learn spatio-temporal variable correla- +tions by a recurrent graph convolutional network +and adaptive variable grouping via the spectral re- +laxation of the K-means clustering objective. +• +We design a novel filtering & fusion module to learn +group-irrelevant representations by an orthogonality +regularization in an adversarial learning framework. +Extensive experiments on four public datasets demon- +strate the superior forecasting performance of our proposed +FairFor model compared with the state-of-the-art models. +We further verify the effectiveness and rationality of our +proposed model in enhancing the fairness of MTS forecast- +ing without performance loss. +2 +RELATED WORK +2.1 +Multivariate Time-series Forecasting +Statistical methods for MTS forecasting, such as Gaussian +process (GP) [40], vector auto-regressive model (VAR) [41] +and +autoregressive +integrated +moving +average +model +(ARIMA) [42], all rely on powerful assumptions regarding +a stationary process and can only learn linear relation- +ship among different time steps within time series data. +In contrast, deep learning based methods are immune to +stationary assumptions and have an inherent efficiency in +capturing non-linearity, complicated and hidden signals +existing in many real-world time series collections. The first +two deep learning based models created for MTS forecasting +are LSTNet [7] and TPA-LSTM [8]. They marry convolu- +tional neural network (CNN) with recurrent neural net- +work (RNN) to capture short-range temporal dependencies +and long-range temporal patterns respectively. Informer [9] + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +3 +embeds sparse self-attention mechanism into Transformer- +based architecture to enhance the latter’s capacity in ex- +tracting the long-range dependencies and alleviate mem- +ory burden. Pyraformer [10] explores the multi-resolution +representation of time series via introducing pyramidal +attention module. Although these models are cutting-edge +MTS forecasting algorithms, they only focus on exploiting +temporal dependencies that deflate the ability when dealing +with highly-correlated data. To further explicitly address +the inter-series correlations among multiple variables, recent +works extract the unidirected relations among variables and +captures shared patterns via priori graph [43], [44], [45] or +graph learning [4], [12], [46], [47]. Similarly, CATN [11] intro- +duces a tree (i.e., an ordered graph) to structure time-series +variables with a clear hierarchy. Although great achieve- +ment has been achieved in MTS modeling, all these existing +works overlook a critical and practical issue, the forecasting +performance bias over different time series variables. In +this work, we particularly focus on performing fair MTS +forecasting to avoid performance disparity on variables. +2.2 +Fairness-aware Algorithms +With the popularity of machine learning, the fairness issue +has received broad attention as algorithms are vulnerable to +data biases that render the decision skewed toward partic- +ular individuals. This data bias mostly relates to the issue +of data imbalance (e.g., historical user-item interactions of +active users are much more than those of inactive users in +recommendations) and then algorithms have been shown to +amplify biases in the raw data to some extent. Therefore, +researchers have proposed many debiasing algorithms [23], +[24], [34], [35], [37], [48] to mitigate inequity for each specific +domain. Specifically, some methods [34], [35], [37] model- +ing individual fairness mostly define reasonable similarity +metrics, but high-dimensional data make similarity measure +between individuals very costly [38]. Other methods fall +into group fairness [23], [24] or a combination [48] of both +group and individual fairness. Lin et al. [23] minimized +the disproportionate impacts by calculating group-wise im- +portance separately when pruning on different groups in +the process of face classification model compression. Fu +et al. [48] devised a fairness-constrained approach through +heuristic re-ranking to mitigate the unfair recommendation +issue where the user-item interactions of inactive users tend +to be neglected and are easily overwhelmed by active users. +In this work, we adopt group fairness to avoid high com- +putation cost and achieve performance fairness and overall +improvement simultaneously. +2.3 +Adversarial Learning for Fairness +Next, a brief review is provided of adversarial learning +algorithms applied in exploring fairness, which is the most +relevant to our work. Some studies [26], [29], [30], [32], +[33], [39] adversarially train models to discriminate sen- +sitive attributes. For example, Li et al. [39] learned a set +of filters for erasing the sensitive attributes from the user +representations and meanwhile use a set of classifiers as +discriminators to predict sensitive attributes. Wu et al. [32] +also tried to obfuscate all sensitive attributes of users and +items under a graph-based adversarial training process. +Inspired by [25], [49] disentangling data representation into +orthogonal subspaces including sensitive attributes or not, +Patro et al. [30] simultaneously learned a bias-aware user +representation and a bias-free user representation that only +carries insensitive user information for fair news recommen- +dation. Similarly, Qi et al. [29] designed an adversarial learn- +ing task to preclude encoding provider bias for provider- +fair news representation and further render the provider- +fair and provider-biased representations to be orthogonal by +an orthogonal regularization term. However, these methods +strongly depend on the pre-defined sensitive attributes that +are unavailable in MTS analysis scenarios. In this study, we +learn informative representations attending to each variable +to promote fairness by leveraging a group-based adversarial +learning framework. +3 +METHODOLOGY +This section presents the problem formulation, then de- +scribes the overview of the proposed framework, named +fairness-aware multivariate time-series forecasting (FairFor) +network, followed by the details of each module and the +learning objectives. +3.1 +Problem Formulation +We use X:,0:T −1 += +{X:,0, X:,1, ..., X:,t, ..., X:,T −1} +∈ +RT ×N to denote N univariate time series with T time +steps, where X:,t = {x0,t, x1,t, ..., xi,t, ..., xN−1,t}T ∈ RN×1 +records the observed values of N variables at time step t. +Herein, i ∈ {0, 1, ..., N − 1} and t ∈ {0, 1, ..., T − 1} is the +index of the variable and the time step respectively, and T +denotes transpose operation. The time interval between any +two time steps is fixed. +Under the sliding forecasting setting with a fixed win- +dow size of w ∈ N+ and a sliding step of 1, we have +the input, i.e., the observed values of all the N variables +in w successive steps till the tth time step, X:,t−w+1:t = +{X:,t−w+1, X:,t−w+2, ..., X:,t} ∈ Rw×N. We articulate the +research problem of MTS forecasting on a graph G = +(V, E, M) to emphasize the spatio-temporal correlations +simultaneously. The set of nodes V denotes input series +X:,t−w+1:t, where |V| = N and each series/variable Xi,: +corresponds to a node. E represents the set of edges, +and M ∈ RN×N is defined as the adjacent matrix. In +addition, we evaluate the forecasting unfairness with the +variance in forecasting errors of different variables, where +the larger the variance is, the more unfair the forecasts +are. Accordingly, the target is to accurately forecast based +on the graph G the future sequence of h ∈ N+ steps +Y:,t+1:t+h = {Y:,t+1, Y:,t+2, ..., Y:,t+h} successive to the tth +time step through one forward procedure and guarantee a +small forecasting error variance simultaneously. +3.2 +Framework Overview +Figure 2 illustrates an overview of FairFor which consists +of four modules: adaptive graph construction, variable cor- +relating & grouping, filtering & fusion and predictor. To +be specific, the adaptive graph construction is to learn an +adjacent matrix (the implicit graph G) to capture the inter- +series dependencies. Then, based on the learned adjacent + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +4 +Input 𝑿:,𝒕%𝒘'𝟏:𝒕 +Time Step +Variable +𝓕𝑲 +𝓕𝟐 +𝓕𝟏 +… +AGC +Filtering & Fusion +𝓩 +… +𝑯. +⨁ +𝑯∗ +FC Layer +Clustering +Layer +Clustering +Results 𝑪 +Discriminator +Adversarial Loss +𝓛𝑪 +Clustering Loss +Output 𝒀.:,𝒕'𝟏:𝒕'𝒉 +Predictor +𝓛𝑫 +Orthogonality +Regularization Loss +𝓛𝑭 +Forecasting +Loss +𝓛𝑨 +Variable Correlating & Grouping +… +… +… +Variable 0 +𝑬𝑴 +𝑴; +1 +2 +𝑁 − 1 +… +… +… +Node Embedding +… +… +… +Softmax +RGCU +… +𝑿:,𝒕%𝒘'𝟏 +… +𝑿:,𝒕%𝒘'𝟐 +… +RGCU +… +𝑿:,𝒕%𝒘'𝟑 +𝑯𝒕(𝑯) +𝑯𝒕%𝒘'𝟏 +… +RGCU +𝑯𝒕%𝒘'𝟐 +𝑯𝒕%𝟏 +… +… +𝑿:,𝒕 +… +… +… +… +Adaptive Graph Construction (AGC) +Spatio-temporal Correlation Learning (STCL) +STCL +𝑯 +Fig. 2. The framework of our FairFor network. AGC (the blue dotted line box) takes X:,t−w+1:t as input to generate the learnable node embedding +EM and adjacent matrix � +M; Then the variable correlating & grouping (the green dotted line box) outputs the last hidden state H through multiple +RGCUs and produces the clustering results C; Next the filtering & fusion (the yellow dotted line box) is applied to filter out the group-relevant +information from H and form the group-irrelevant representation � +H, which is further input to the discriminator D with C; The predictor integrates +the group-relevant representation H and group-irrelevant representation � +H to form more informative representation H⋆ for final prediction. +matrix, the variable correlating & grouping learns a spatio- +temporal representation for each time series with a recurrent +graph convolutional network and then clusters all time +series variables into multiple groups. Subsequently, the +filtering & fusion learns group-relevant variable represen- +tation and also group-irrelevant variable representation by +filtering out the group-relevant information. Both the group- +relevant representation and group-irrelevant representation +are beneficial for the forecasting task. Therefore, the pre- +dictor integrates the two kinds of representations to form +a more informative representation for producing the final +forecast of each time series variable at one forward step. +3.3 +Adaptive Graph Construction +Graph convolutional network (GCN) has been widely stud- +ied for accurate MTS forecasting due to its capability to +capture the spatial correlations of time series data. Let +X ∈ RN×w (i.e., w-dimensional feature vector for each +node) be the input time series matrix, W and b be the +learnable weight matrix and bias respectively, then the +layer-wise graph convolution operation of GCN can be +well-approximated by the Chebyshev polynomial expan- +sion form as: +G(X) = (I + B− 1 +2 MB− 1 +2 )XW + b +(1) +where I ∈ RN×N, B is the degree matrix, and G(X) ∈ +RN×q is the output of GCN layer. The GCN operation can +be viewed from the perspective of a node (e.g., node i) as +transforming the features of node Xi ∈ R1×w to G(Xi) ∈ +R1×q with the shared W and b. +The pre-defined adjacent matrix M embedded in the +graph convolution operation is usually explicitly con- +structed by defining the similarity function of the dataset +itself, or by defining the distance function according to the +geographic distance on urban maps. However, the fixed +explicit graph structure is not always available or complete. +The reason is that it is hard to manually capture latent rela- +tionships from substantial time-series data to construct the +graph structure, especially for non-traffic datasets. Besides, +the pre-defined graph is not directly related to the final fair +forecasting task, which may lead to sizable biases. Inspired +by [4], [12], [46], [47] precisely and automatically disclosing +the implicit inter-series dependencies at each time step from +MTSs without prior knowledge, in this paper, we propose +an adaptive graph construction module to randomly initial- +ize a learnable node embedding matrix EM ∈ RN×d for all +nodes. Herein, each row of EM represents the embedding +of a node, and d is the dimension of node embedding. Next, +the inter-series dependencies between each pair of nodes +can be inferred by multiplying EM and ET +M. Formally: +B− 1 +2 MB− 1 +2 = δ(ReLU(EM · ET +M)) +(2) +where δ(·) is the element-wise softmax function which is +used to normalize the adaptive adjacent matrix, T is the +transpose operation. Here, to eliminate needless and re- +peated calculations during the iterative training process, +we directly produce B− 1 +2 MB− 1 +2 in this case rather than +producing M and computing a Laplacian matrix. The EM +will be automatically updated throughout training to dis- +cover the hidden inter-series dependencies between various +series and obtain the adaptive adjacent matrix M for graph +convolutions. Therefore, the GCN enhanced by the adaptive +graph construction can be formulated as: +G(X) = (I + δ(ReLU(EM · ET +M)))XW + b +(3) +3.4 +Variable Correlating & Grouping +Accurate MTS forecasting relies on capturing two essential +properties, i.e., temporal correlations over a sequence of +time steps and spatial correlations over different time series +variables. Accordingly, we adopt a recurrent graph convolu- +tional network (RGCN) based on an implicit graph structure +to capture the complex temporal and spatial correlations + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +5 +among a set of time series variables. We further group the +learned spatio-temporal representation into several groups +and produce the corresponding clustering results. +Spatio-temporal Correlation Learning. We introduce a +recurrent graph convolutional unit (RGCU) via integrating +a gated recurrent unit (GRU) with a GCN layer to learn the +spatial and temporal inter-correlations between time series +variables (see Figure 2). A GRU takes X:,t as input and +updates the hidden state from the previous state Ht−1 to +Ht by employing the reset gate and update gate to govern +how much information from the history should be taken +into account. By doing this, the GRU remembers historical +hidden states that are relevant to future predictions and +forgets those that are irrelevant. RGCU replaces the MLP +layers in GRU with GCN and send the embedded graph +information to reset gate and update gate to update the +hidden state Ht collectively. On the one hand, the spatial +dependencies among different series, i.e., the inter-series +dependencies, can be well captured by the implicit graph +structure constructed based on the node embedding matrix +EM in RGCU, where the implicit graph is encoded into +graph nodes by GCN for time series variable interaction. +On the other hand, the GRU structure can well capture the +temporal dependencies over different time steps. As a result, +the RGCU unit can well capture both temporal and spatial +dependencies embedded in MTS data. Formally, we have +� +M = δ(ReLU(EM · ET +M)) +(4) +rt = σ( � +M[X:,t||Ht−1]Wr + br) +(5) +ut = σ( � +M[X:,t||Ht−1]Wu + bu) +(6) +ct = θ( � +M[X:,t||r ⊙ Ht−1]Wc + bc) +(7) +Ht = u ⊙ Ht−1 + (1 − u) ⊙ ct +(8) +where X:,t ∈ RN×1 and Ht ∈ RN×o refer to the input and +output at time step t, o is the hidden dimension, ut and +rt refer to the reset gate and update gate respectively, ct +is the memory state, [·||·] denotes a concatenation operator, +⊙ denotes en element-wise multiplication, σ(·) and θ(·) are +sigmoid and tanh activation function. EM, Wu, Wr, Wc, bu, +br and bc are learnable parameters. The last hidden state +of RGCU denoted by H is sent to the variable grouping +module and the filtering & fusion module respectively to +generate clustering results and group-irrelevant representa- +tion respectively as the inputs to the discriminator D. +Variable Grouping. Then, we propose to group the +learned hidden state H according to variable correlations +and regard clustering/grouping results as a counterweight +to the filtering & fusion module. Hence, we introduce a +clustering layer consisted of three-layer fully connected +(FC) neural network with LeakyReLU as the activation +and output the clustering results into discriminator D for +adversarial learning. The learned hidden state H may not +suitable for forming cluster structures. Hence, to stimulate +to group variables in H, we introduce the spectral relaxation +of the K-means objective [50], [51] as a loss: +LC = Tr(HTH) − Tr(F THTHF ) +s.t.F TF = I +(9) +where H ∈ Ro×a with a = b×N after reshaping, F ∈ Ra×K +is the cluster indicator matrix, K is the number of clusters +and b is the batch size. The optimization process of Eq. (9) +requires iteratively updating F and H due to the dynamic +learning of H. When F is fixed, updating H can follow the +SGD of the clustering layer that boosts the representation +to mine variable correlations. When H is fixed, F can be +updated once by computing the K-truncated SVD of H +after several epochs (e.g., 3) to prevent instability. Since the +cluster-friendly representation H is not favorable for the +forecasting process, it is difficult to jointly optimize H in the +clustering and forecasting process. Hence, H is delivered to +a FC layer before to Eq. (9) to alleviate model instability, +referring to Figure 2. +3.5 +Filtering & Fusion +In our fairness-aware forecasting method, a key challenge +is to learn the group-relevant (specific to each group) and +group-irrelevant (shared by all groups) representations as il- +lustrated in the Introduction. Given a learning algorithm that +learns spatio-temporal hidden state H to directly generate +forecasts, we require the hidden state H to be independent +from the learned group information C to achieve forecasting +fairness. Therefore, we design a filter layer with a series +of filter functions {F1, F2, ..., FK}, which are used to filter +out the group-relevant information in the hidden state H +and K is the number of groups. The filter function is +represented as F : Ro �→ Ro, and the representation F(H) +preserves the features shared by all groups when specific +features to per group are filtered out. We use the three FC +layers followed by a batch normalization layer to represent +each filter function F. Finally, K filtered representations are +combined to generate the group-irrelevant representation: +� +H = Z(F1(H), F2(H), ...FK(H)) +(10) +where Z is a fusion function. The K filtered representations +are fed into Z together and the group-irrelevant represen- +tation unrelated to all group information is output without +the representation dimension altered, e.g., using the average +of the K filtered representations as Z. +Discriminator. To learn filter functions, we use the idea +of adversarial learning to train a group discriminator. Specif- +ically, we train two mappers M(� +H) : Ro �→ Ro and M(C) : +RK �→ Ro, which attempt to map the group-irrelevant +representation and corresponding clustering results into +the same space, more specifically, obtaining � +H ∈ Ro and +C ∈ Ro respectively, and calculate their Euclidean distance. +Similar to [52], we accordingly use the three FC layers with +LeakyReLu as the activation function to represent M(� +H) +and M(C) of discriminator D. The aim of filter functions is +to render it difficult to infer variable correlations and group +variables from the group-relevant representation H, while +that of discriminator D is to fail the filter functions. The +adversarial loss function is shown: +LA = 1 +N +N−1 +� +i=0 +||� +Hi − Ci||2 +2 +(11) +Unfortunately, some group information may still be in- +cluded in the group-irrelevant representation � +H. Because +the group-irrelevant representation � +H just needs to fool +the discriminator D, the filtering & fusion module does + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +6 +Algorithm 1 Adversarial Training for FairFor +Require: Hidden state H, group results C, network R, +filter and fusion functions F, Z ⊆ R, discriminator D +and cluster number K +1: Orthogonal initialize cluster indicator matrix F +2: for each training iteration i do +3: +� +H ← Z({Fk}k∈[K](H)) by Eq. (10) +4: +H⋆ ← H + � +H +5: +Optimize L w.r.t H, � +H, H⋆, C, R with D being fixed +6: +if i%3 == 0 then +7: +Update F by computing K-truncated SVD of H +8: +end if +9: +LA ← E[||� +H − C||2 +2] by Eq. (11) +10: +Optimize LA w.r.t � +H, C, D with R, H, H⋆ fixed +11: end for +not necessarily completely filter the group information so +that the discriminator D generally cannot perfectly assess +the group information. To address this issue, we design an +orthogonality regularization method [30] to further purge +the group-irrelevant representation. Specifically, the group- +relevant representation H and group-irrelevant representa- +tion � +H are regularized by boosting them to be orthogonal +to each other. The orthogonality regularization is calculated: +LD = 1 +N +N−1 +� +i=0 +| +� +Hi · Hi +∥� +Hi∥ · ∥Hi∥ +| +(12) +where � +Hi and Hi are the group-relevant and corresponding +group-irrelevant representations of the ith variable. +3.6 +Prediction +The group-relevant representation mainly contains informa- +tion on group attributes, and the group-irrelevant represen- +tation mainly encodes group-free time-series information. +Considering the information in both representations is rele- +vant to the forecasting task, we integrate the group-relevant +representation H and group-irrelevant representation � +H by +an addition operation to form the informative representa- +tion H⋆, i.e., H⋆ = H + � +H, which is then fed into a +predictor. Then the future time-series sequence is estimated +by a 2-D convolutional layer at one forward step style rather +than a step-by-step style: +�Y = Conv2D(H⋆) +(13) +where Conv2D is a convolutional operation to directly +map H⋆ to the predictions for all horizons. Finally, the +forecasting loss is denoted as: +LF = 1 +N +N−1 +� +i=0 +||Yi − �Yi||2 +2 +(14) +3.7 +Adversarial Training +Adversarial learning techniques encourage the deep repre- +sentation to be maximally informative to generate group- +irrelevant representation, and meanwhile to be minimally +TABLE 1 +Dataset statistics. +Tasks +#Time Step +#Variable +Interval +Start Time +Traffic +10,392 +963 +1hour +1/1/2008 +PeMSD7(M) +11,232 +228 +5min +7/1/2016 +Solar-Energy +52,560 +137 +10min +1/1/2016 +ECG5000 +5,000 +140 +− +− +discriminative in a group-relevant discriminator. There- +fore, adversarial learning has the potential to learn group- +irrelevant representations and treat each variable fairly +whether it is advantaged or disadvantaged. Note that we +are not trying to purely render the prediction accuracy of +advantaged variables and disadvantaged variables closer +like common fairness-aware algorithms [30], [32]. Instead, +we try to learn the group-relevant and group-irrelevant +representation by adversarial learning and then form in- +formative representations focusing on both advantaged and +disadvantaged variables for enhancing the performance on +disadvantaged variables. To the end, the FairFor optimiza- +tion involves playing a min-max game: +L = arg min +R (LF + LC + LD + arg max +D λaLA) +(15) +where R = FairFor − D denotes the remaining part after +removing discriminator D from FairFor. The adversarial +training algorithm is presented in Algorithm 1. Our pro- +posed FairFor is carried out via alternately optimizing the +subsequent processes. Concretely, we first feed input to the +model to obtain L, then fix the parameters in the discrimi- +nator D, and optimize R by minimizing L. Then, fixing the +parameters of R, D is optimized by minimizing LA. +4 +EXPERIMENT AND EVALUATION +4.1 +Datasets +To evaluate the models under different scales and appli- +cation scenarios, we employ four real-world datasets (see +dataset statistics in Table 1) for extensive evaluation. +PeMSD7(M) 1 records the traffic flow data of the detec- +tors in California. It includes 228 variables and 11,232 time +steps at a 5-minute interval; +Solar-Energy +2 is collected from National Renewable +Energy Laboratory (NREL) and records the solar power +production in 2006. It includes 137 variables and 52,560 time +steps at a 10-minute interval; +Traffic 3 is originally collected from the California De- +partment of Transportation and describes the occupancy +rate of different lanes in San Francisco highway. It contains +963 variables and 10,560 time steps with a 10-minute inter- +val, where each observation is between 0 and 1; +ECG5000 4 records 5,000 heartbeats randomly selected +from a 20-hour long ECG downloaded from Physionet. It +contains 140 variables and 5,000 time steps. +1. https://dot.ca.gov/programs/traffic-operations/mpr/ +pems-source +2. http://www.nrel.gov/grid/solar-power-data.html +3. https://archive.ics.uci.edu/ml/datasets/PEMS-SF +4. http://www.timeseriesclassification.com/description.php? +Dataset=ECG5000 + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +7 +4.2 +Experimental Setting and Metrics +Following many classical time-series data split practice [7], +[12], we split the data into training, validation, and test parts +with the ratio of 7:2:1. All data is normalized by min-max +method by following [4], [12]. In our experiments, the FC +layer, clustering layer, filters {F1, F2, ..., FK}, discriminator +D are all multi-layer FC networks with the number of layers +set to 3. The predictor is a 2-D convolutional layer with the +kennel size set to (1, o) and the number of features o in +the hidden state H is fixed to 64. The cluster number K +in the filtering & fusion module is set to 6 for all datasets. +We examine values of trade-off parameter λa in Eq. (15) +in the range of {0, 0.1, 0.5.0.7, 1} and choose {λa = 0.1} +for all datasets and the node embedding dimension d in +learnable node embedding matrix EM is set to 10. For +network R, we use Adam optimizer with a learning rate +of 3e-3. For discriminator D, we use Adam optimizer with +the learning rate of 5e-2. Model parameters are turned on a +Ubuntu 18.04.6 server with four NVIDIA GeForce 3090 GPU +cards for 50 training epochs, through which we save the +best-performing model based on values of accuracy metrics +MAE/RMSE/MAPE/VAR on the validation set and reload +it for the evaluation on the test set. The batch size is set as +64. Our code will be released at GitHub. +Four typical metrics, i.e., MAE, RMSE, MAPE, and VAR, +are employed for MTS forecasting evaluation. Their formal +definitions to evaluate all comparative methods are as fol- +lows: Mean Absolute Error MAE = 1 +N +N−1 +� +i=0 +|Yi − ˆYi|, Root +Mean Squared Error RMSE = +� +1 +N +N−1 +� +i=0 +(Yi − ˆYi)2, Mean +Absolute Percent Error MAPE = +1 +N +N−1 +� +i=0 +|Yi− ˆYi| +Yi +1{|Yi| > +0} and Variance (the variance of MAE over each variable) +V AR = +1 +N +N−1 +� +i=0 +[|Yi − ˆYi| − 1 +N +N−1 +� +i=0 +|Yi − ˆYi|]2, where N +denotes the number of variables, Yi and ˆYi denote the +ground truth and prediction respectively. +4.3 +Baselines +Because the fairness problem in MTS forecasting is rarely +considered in previous studies, eight representative and +SOTA MTS forecasting methods from different classes are +deliberately chosen as baselines to compare with our pro- +posed method. To be specific, two representative RNN- +based methods, i.e., LSTNet and TPA-LSTM, and two +Transformer-based methods, i.e., Informer and Pyraformer +are chosen since they are good at capturing temporal pat- +terns with different ranges such as short- and long-range. +TS2Vec is a very popular universal time-series represen- +tation framework. Furthermore, MTGNN, StemGNN and +AGCRN are based on graph neural networks, which are +selected to justify the effectiveness of incorporating both +intricate temporal and spatial dependencies among time- +series data in our method. Traditional methods such as +VAR and GP are not compared since the latest deep neural +network-based methods [46], [47], [53] have been verified to +outperform these methods. +For a fair comparison, the length of time-series input +(historical sequence), time-series output (future sequence), +and hardware technical indicators for all baselines are iden- +tical. Hyper-parameters of each baseline are consistent with +the corresponding paper settings. Some methods [7], [8] +were originally proposed for single-step output, then we +carefully modify them into sequence output. More details +about these methods are as follows: +LSTNet [7]: integrates RNNs and CNNs to capture the +short- and long-term temporal patterns respectively; +TPA-LSTM] [8]: embeds temporal pattern attention into +RNNs to discover both relevant time series and time steps; +TS2Vec [54]: is a dilated CNN based universal frame- +work to capture multi-scale contextual information in MTSs; +Informer [9]: is a Transformer-based model with Prob- +Sparse self-attention mechanism and generative style de- +coder to predict long time series at one forward step; +Pyraformer [10]: is a Transformer-based model to simul- +taneously extract multiple ranges of temporal dependencies +via the compact multi-resolution operation; +MTGNN [46]: combines graph learning, graph convolu- +tion and temporal convolution together to learn the spatial- +temporal correlations without pre-defined graph structure; +StemGNN [12]: uses a GCN-based spectral network that +can capture inter-series dependencies and temporal correla- +tions jointly in the spectral domain; +AGCRN [4]: employs GCNs embedded with an adap- +tive node-specific pattern learning module to capture fine- +grained inter-series relationships and uses RNNs to capture +temporal patterns. +4.4 +Experimental Results +4.4.1 +Overall Results +We compare FairFor with the baseline models on both MTS +forecasting and fairness performance. Table 2 and Table 3 +show the fairness and forecasting performance of different +methods respectively under the commonly-used setting of +w = 12, h = 12 following [4], [46]. +Fairness +Improvement +In +the +fairness +aspect, +we +find +that +our +FairFor +obviously +outperforms +all +baseline +MTS +forecasting +methods +on +four +datasets, +specifically +the +VAR +decrease +over +the +best +baseline +is +10.59%/1.74%/2.34%/2.37% +on +PeMSD7(M)/Solar- +Energy/Traffic/ECG5000 respectively. This is consistent +with previous analysis: 1) general forecasting models +are vulnerable to variable disparity and prone to focus +on certain variables (advantaged variables), leading to +generating unequal performance over different variables; +2) FairFor proposes to combine group-relevant and group- +irrelevant representation together to form informative +representation +focusing +on +both +advantaged +and +disadvantaged variables. Consequently, the performance +of disadvantaged variables is improved by enriching the +group-irrelevant representation and drawing support from +the knowledge of advantaged variables. +Forecasting Performance From Table 3, we can ob- +serve that FairFor can still achieve high multi-step forecast- +ing quality on PeMSD7(M) and ECG5000. FairFor makes +3.21%/3.32%/2.49% improvements in average w.r.t. MAE, +RMSE and MAPE over the best baseline on PeMSD7(M) and +ECG5000. This is explainable: different from the motivation +of sacrificing overall performance to guarantee the fairness + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +8 +TABLE 2 +The fairness performance of different methods evaluated on VAR under the setting of w = 12, h = 12, where the best results are highlighted in +bold (smaller values indicate better fairness). +Dataset +Metric +LSTNet +TPA-LSTM +TS2Vec +Informer +Pyraformer +MTGNN +StemGNN +AGCRN +FairFor +PeMSD7(M) +VAR +23.6003 +42.2336 +28.4647 +28.1218 +48.4494 +42.7716 +26.5534 +22.7220 +20.3152 +Solar-Energy +5.6880 +10.1448 +5.2330 +5.2096 +10.6149 +9.7419 +5.7216 +5.6543 +5.1188 +Traffic +9.51e-4 +8.07e-4 +5.30e-4 +4.27e-4 +7.63e-4 +6.43e-4 +6.71e-4 +5.59e-4 +4.17e-4 +ECG5000 +0.5668 +0.2577 +0.2360 +0.2257 +0.2618 +0.2255 +0.1774 +0.2467 +0.1732 +TABLE 3 +The prediction performance of different methods evaluated on four real-world datasets under the setting of w = 12, h = 12, where the best results +are highlighted in bold (smaller values indicate better results). +Dataset +Metric +LSTNet +TPA-LSTM +TS2Vec +Informer +Pyraformer +MTGNN +StemGNN +AGCRN +FairFor +PeMSD7(M) +MAE +3.2004 +4.7573 +4.6001 +3.9728 +4.2925 +4.4072 +3.3064 +2.7983 +2.7336 +RMSE +5.6566 +7.8292 +7.0445 +6.6654 +8.1646 +7.5602 +5.7927 +5.4950 +5.2798 +MAPE +0.0870 +0.1932 +0.1117 +0.0957 +0.1067 +0.1127 +0.0817 +0.0694 +0.0678 +Solar-Energy +MAE +1.3030 +2.0399 +1.3810 +1.1426 +2.2316 +1.5043 +1.1205 +0.9090 +1.0807 +RMSE +2.7057 +3.1682 +2.6721 +2.5702 +3.9558 +2.7029 +2.6720 +2.5457 +2.5608 +MAPE +3.4220 +3.4175 +3.3694 +3.3826 +3.4080 +3.3727 +3.4096 +3.2905 +3.3584 +Traffic +MAE +0.0220 +0.0158 +0.0163 +0.0095 +0.0445 +0.0117 +0.0169 +0.0123 +0.0108 +RMSE +0.0374 +0.0317 +0.0282 +0.0215 +0.0523 +0.0279 +0.0310 +0.0260 +0.0240 +MAPE +0.7736 +0.4924 +0.5630 +0.2482 +2.2674 +0.3429 +0.4768 +0.2676 +0.2669 +ECG5000 +MAE +0.4978 +0.4268 +0.3634 +0.3448 +0.3287 +0.3493 +0.3303 +0.3534 +0.3152 +RMSE +0.9022 +0.6874 +0.6067 +0.5921 +0.6110 +0.5769 +0.5352 +0.6096 +0.5207 +MAPE +0.9334 +0.9561 +0.9664 +0.9481 +1.0132 +0.951 +0.9176 +0.9854 +0.8931 +requirement in existing recommendation methods [30], [32], +FairFor approaches an unbiased process of enriching the +representation of per variable and thus can achieve supe- +rior performance of prediction and fairness simultaneously. +Although our model is surpassed on some specific exam- +ples, i.e. Solar-Energy and Traffic, the forecasting perfor- +mance is still sub-optimal. More specifically, we reveal the +cause of FairFor’s under-performance on Solar-Energy and +Traffic datasets. Our method has a performance decrease +over Informer of 13.68% (MAE), 11.63% (RMSE) and 7.53% +(MAPE). This is intuitively attributable to the fact that the +capture of mid-term (e.g., h ∈ {12, 24}) and long-term (e.g., +h ∈ {48, 72, 96}) dependency coupling between output and +input in Informer is more precise and contributes more to +improve overall regression indicators, which is also verified +in Figure 3(b), 3(c). Moreover, MAE, RMSE and MAPE +decrease of AGCRN over proposed FairFor is 15.89%, 0.59% +and 2.02% on Solar-Energy respectively. This is attributable +that fine-grained node-specific (e.g., similar, dissimilar, or +contradictory) patterns across different data sources may be +captured by node-specific parameter learning strategy in- +stead of parameter sharing strategy commonly used in MT- +GNN, StemGNN and FairFor (with GNN-based backbone). +However, these methods neglect to focus on disadvantaged +variables in design and cause poor VAR performance. +Next, to verify the ability of forecasting different +horizons, we compare FairFor with TS2Vec, Informer, +Pyraformer, MTGNN, StemGNN and AGCRN, which are +the top-6 baselines from the previous experiments. We fix +w = 12 and adjust h in the range of {3, 6, 12, 24, 48, 72, 96}. +Figure 3 further shows the MAE and VAR performance +comparison at different horizons. We can observe that: 1) the +FairFor shows significantly better VAR results at different +horizons across all datasets; 2) On the Solar-Energy and +Traffic dataset, Informer achieves better MAE results on +the long horizon (at {48, 72, 96}). We attribute this to the +potential value to capture the long-range dependencies. +4.4.2 +Ablation Study +Besides, we also perform an ablation study to better evalu- +ate the effectiveness of several core designs in our FairFor, +i.e., spatio-temporal correlation learning enhanced by AGC +(STCL+AGC) , adversarial learning - D (ALD), cluster- +ing loss (CL), and orthogonal regularization loss (ORL), +by removing them individually. We select the setting as +w = 12, h = 12, K = 6 on all datasets. +Results are summarized in Table 4. We can observe that: +1) In w/o STCL+AGC, the STCL module enhanced by AGC +module of FairFor is replaced by gated TCN previously used +in [45] to learn temporal dependencies. Table 4 shows our +method averagely outperforms its variant w/o STCL+AGC +with a significant margin (MAE: -30.14%, MAPE: -16.04%, +VAR: -40.55%) across all datasets, indicating that jointly +capturing the temporal patterns and inter-series correla- +tions can significantly improve the forecasting and fairness +performance. 2) Variate w/o ALD denotes FairFor without +discriminator D, and La is removed from training objective +L in Eq. (15). As verified in Table 4, the score w.r.t MAE, +MAPE, and VAR increases by 9.11%, 1.58%, and 1.74% on +average. This proves that group-based adversarial learning +is worth adopting, especially when balancing and optimiz- +ing the performance of each variable is required. 3) To study +the importance of variable correlating & grouping module +without affecting the normal work of discriminator D, we + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +9 +3 +6 +12 +24 +48 +72 +96 +Horizon +2 +3 +4 +5 +6 +7 +8 +MAE +TS2Vec +Informer +MTGNN +StemGNN +AGCRN +FairFor +3 +6 +12 +24 +48 +72 +96 +Horizon +10 +20 +30 +40 +50 +60 +70 +80 +VAR +TS2Vec +Informer +MTGNN +StemGNN +AGCRN +FairFor +(a) PeMSD7(M) +3 +6 +12 +24 +48 +72 +96 +Horizon +3 +4 +5 +6 +7 +8 +VAR ( + 10-4) +TS2Vec +Informer +MTGNN +StemGNN +AGCRN +FairFor +3 +6 +12 +24 +48 +72 +96 +Horizon +1 +1.2 +1.4 +1.6 +1.8 +2 +MAE ( + 10-2) +TS2Vec +Informer +MTGNN +StemGNN +AGCRN +FairFor +(b) Solar-Energy +3 +6 +12 +24 +48 +72 +96 +Horizon +1 +2 +3 +4 +5 +MAE +TS2Vec +Informer +MTGNN +StemGNN +AGCRN +FairFor +3 +6 +12 +24 +48 +72 +96 +Horizon +5 +10 +15 +20 +25 +30 +35 +40 +VAR +TS2Vec +Informer +MTGNN +StemGNN +AGCRN +FairFor +(c) Traffic +3 +6 +12 +24 +48 +72 +96 +Horizon +0.3 +0.4 +0.5 +0.6 +0.7 +MAE +TS2Vec +Informer +MTGNN +StemGNN +AGCRN +FairFor +3 +6 +12 +24 +48 +72 +96 +Horizon +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +VAR +TS2Vec +Informer +MTGNN +StemGNN +AGCRN +FairFor +(d) ECG5000 +Fig. 3. Performance comparison at different horizons. +TABLE 4 +Results of ablation study. +Models +PeMSD7(M) +Solar-Energy +Traffic +ECG5000 +MAE +MAPE +VAR +MAE +MAPE +VAR +MAE +MAPE +VAR +MAE +MAPE +VAR +w/o STCL+AGC +4.2412 +0.1018 +30.2543 +1.5797 +3.4438 +6.1499 +0.0164 +0.4758 +5.93e-4 +0.4500 +0.9024 +1.0522 +w/o ALD +2.8889 +0.0689 +20.0300 +1.3456 +3.4157 +5.3100 +0.0142 +0.3541 +4.38e-4 +0.3159 +0.8982 +0.1744 +w/o CL +2.8522 +0.0689 +20.3052 +1.2181 +3.3994 +5.2194 +0.0145 +0.3471 +4.47e-4 +0.3161 +0.8976 +0.1749 +w/o ORL +2.8497 +0.0690 +20.3152 +1.1736 +3.3865 +5.2112 +0.0138 +0.3784 +4.56e-4 +0.3159 +0.8960 +0.1747 +FairFor +2.7983 +0.0678 +19.9953 +1.0807 +3.3826 +5.1188 +0.0123 +0.3429 +4.27e-4 +0.3152 +0.8931 +0.1732 +only discard LC from L in Eq. (15) to form variate w/o +CL. It yields 7.16%/0.95%/2.23% rise in average across all +datasets, demonstrating the significance of setting a soft +K-means objective to auxiliarily infer variable correlations +and group variables in group-based adversarial training. +4) Variate w/o ORL represents discarding LD from L in +the training process. The results demonstrate that removing +ORL also hurts the fairness of FairFor. Because ORL pro- +motes the orthogonality of group-relevant and -independent +representation, which can better identify and remove group- +relevant information. +The STCL module enhanced by AGC exhibits a greater +impact on the performance because it has resorted to distill- +ing complex temporal patterns and inter-series correlations +as the basis of time-series representation. Besides, other +modules enhance TSRL from the perspective of learning +more informative representations attending to both advan- +taged and disadvantaged variables. The ablation variants +except for w/o STCL+AGC still outperform most baselines, +indicating the stable and complementary advantages of +different modules. +4.4.3 +Hyper-parameter Analysis +We explore the sensitivity analysis of several critical hyper- +parameters on PeMSD7(M) and ECG5000 datasets, includ- +ing cluster number K, window size w, embedding dimen- +sion d and trade-off coefficient λa. The results in terms of +MAE and VAR are demonstrated in Figure 4. When we +change a hyper-parameter, the other variables keep their +default values explained in Section 4.2. +First, we investigate the influence of variable correlating +& grouping by varying the value of K in a range of +{2, 3, 4, 5, 6, 7, 8, 9, 12, 15}, and plot the forecasting results +MAE and fairness results VAR under w = 12, h = 12 in +Figure 4(a). When K is less than 6, MAE and VAR display +a trend of violent fluctuation. A moderate value for K, +i.e., K = 6, makes MAE and VAR descend to the lowest +on PeMSD7(M). A similar trend exists as increasing cluster +number from 6 since unsuitable clustering objective possibly +cannot fully describe the complex group information inside +variables and confuse latent data regularities. Meanwhile, +the performance on ECG5000 is relatively stable with differ- +ent K. The possible reason is that the range of K we set is +too small to display the data regularities (K = 56) [55]. Note +that the increase of K will bring a burden to the filtering +& fusion module and hurt the performance. Therefore, the +setting of K is 6 for PeMSD7(M) and ECG5000. +Then, we turn to explore the impact of input win- +dow with different sizes by varying w in a range of +{1, 3, 6, 9, 12, 15, 18, 21, 24, 48}. As shown in Figure 4(b), +when predicting mid-term sequence (h = 12), initially +increasing window size degrades performance, but further +increasing causes the MAE and VAR to drop since it brings +sufficient temporal patterns and dependency information. +However, further increasing w makes the MAE and VAR + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +10 +(a) Cluster Number K +(b) Window Size w +(c) Embedding Dimension d +(d) Trade-off Coefficient λa +Fig. 4. Hyper-parameter analysis on PeMSD7(M) and ECG5000 +datasets. +get higher. This is intuitive due to the noise introduced +by redundant temporal patterns, dependencies and group +features that are not fit for the group-based adversarial +learning task. Hence, the input window size and output +window size need to be carefully weighed. +Moreover, we check a key parameter d of learnable +node embedding matrix EM, which not only influences the +quality of adjacent matrix � +M but also decides the parameter +diversity in STCL module (Eq. (5), (6) and (7)). Figure 4(c) +reports the effects taking different dimensions of EM. As +can be seen, FairFor achieves relatively good performance +with all tested dimensions, demonstrating that the model +capacity is fairly robust. Note that FairFor achieves the best +performance when the dimension is set to 10. As the dimen- +sion goes larger, the performance first increases since more +information is contained to thus help our STCL module to +deduce more accurate intra-series correlations, then starts +to be weaker probably due to over-fitting. Intuitively, an +appropriate embedding dimension should make the model +obtain sufficient correlation information meanwhile avoid- +ing introducing additional parameters causing over-fitting. +To learn the importance of adversarial losses, we control +La to change within a range {0, 0.1, 0.5, 0.7, 1}. The results +are exhibited in Figure 4(d). From Figure 4(d), we find that +the VAR drops steadily with the increase of λa, then rises +until λa = 1, and the MAE has a similar trend. When +λa = 0.1, forecasting performance and fairness achieve +the best together. Thus, a moderate value for λa (e.g., 0.1) +may be preferable to make adversaries achieve an appropri- +ate equilibrium and protect group information from being +leaked to the group-irrelevant representation. +4.4.4 +Visualization of Fairness +As previously discussed, the filtering & fusion module +filters the group-relevant information from H and obtains +the group-irrelevant representation � +H in the group-based +adversarial training. To further understand how the group- +relevant (specific to each group) and -independent (shared +by all groups) representation work, we respectively visu- +alize the feature distribution by t-SNE in 2-D space and +evaluate the informativeness of H and � +H. We randomly +choose a batch of the Traffic-train data. Figure 5(a)-5(d) and +Figure 5(e)-5(h) show the feature distributions of H and � +H, +respectively. Firstly, H and � +H all exhibit a good clustering +structure, which proves the clustering ability of our model. +In addition, H shows smaller inter-cluster distances than +� +H, indicating different variable representations learned in +H are close and concentrated, while those learned in � +H +after filtering are informative and discriminative and exhibit +less overlap. Furthermore, H with a compact clustering +structure may only contain the information of original ad- +vantaged groups, and � +H with a decentralized clustering +structure is enriched by drawing support from advantaged +groups and changes into informative representation shared +by all groups. Therefore, the FairFor model has good predic- +tion and fairness performance and is also interpretable. +5 +CONCLUSION +We argue that the performance unfairness between vari- +ables widely exists in MTSs due to the variable disparity, +equally attending to both advantaged and disadvantaged +variables can facilitate overall forecasting performance and +fairness simultaneously. In this paper, we study the unfair- +ness problem and propose a fairness-aware MTS forecasting +method (FairFor), which aims to (1) capture spatio-temporal +variable correlations via recurrent graph convolution, (2) +group variables by leveraging a spectral relaxation of the +K-means objective, (3) filter the group-relevant information +and generate group-irrelevant representation via adversar- +ial learning with an orthogonality regularization, and (4) +integrate the group-relevant and -irrelevant representations +to form a highly informative representation for the final +prediction. Experimental results on four real-world datasets +prove that FairFor can simultaneously and effectively im- +prove the performance of fairness and MTS forecasting. Cur- +rently, FairFor is an enhanced framework that sequentially +performs to capture spatio-temporal correlations and bal- +ance the forecasting performance of all variables. Intuitively, +the variable disparity and correlations dynamically evolve +over time in coordination. Therefore, we will embed the +balance operation into spatio-temporal correlation learning +at each time step in the future work. +ACKNOWLEDGMENTS +This work is supported by the National Key R&D Program +of China under Grant 2019YFB1406302 and National Natu- +ral Science Foundation of China under Grant 62272048. +REFERENCES +[1] +C. Shang, J. Chen, and J. Bi, “Discrete graph structure learning for +forecasting multiple time series,” in ICLR, 2021. +[2] +R. Sawhney, S. Agarwal, A. Wadhwa, T. Derr, and R. R. 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Park, “Graph neural controlled +differential equations for traffic forecasting,” in AAAI, 2022, pp. +6367–6374. +[54] Z. Yue, Y. Wang, J. Duan, T. Yang, C. Huang, Y. Tong, and +B. Xu, “Ts2vec: Towards universal representation of time series,” +in AAAI, 2022, pp. 8980–8987. +[55] H. A. Dau, A. J. Bagnall, K. Kamgar, C. M. Yeh, Y. Zhu, +S. Gharghabi, C. A. Ratanamahatana, and E. J. Keogh, “The UCR +time series archive,” CoRR, vol. abs/1810.07758, 2018. +Hui He received the M.E. degree from Univer- +sity of Shanghai for Science and Technology, +Shanghai, China in 2020. She is currently pursu- +ing the Ph.D. degree at Institute of Engineering +Medicine, Beijing Institute of Technology, Beijing, +China. Her current research interests focus on +multivariate time-series analysis and knowledge +services. +Qi Zhang received his Ph.D. degrees from Bei- +jing Institute of Technology China under the dual +Ph.D. program of Beijing Institute of Technol- +ogy and University of Technology Sydney Aus- +tralia. He is currently an AI Scientist in Deep- +Blue Academy of Sciences. His current research +interests focus on Collaborative Filtering, learn- +ing to hash and sequential recommendation and +multivariate time-series analysis. +Shoujin Wang received a Ph.D. degree in Data +Science from the University of Technology Syd- +ney in 2019. He is currently a Lecturer in Data +Science at Data Science Institute, University of +Technology Sydney. His main research interests +include data mining, machine learning, recom- +mender systems and fake news mitigation. He is +the social media editor of J. Data Science and +Analytics. +Kun Yi is a Ph.D. candidate from Beijing Institute +of Technology, China. His current research inter- +ests include multivariate time-series forecasting, +data science and knowledge discovery. +Zhendong Niu received the Ph.D. degree in +computer science from the Beijing Institute of +Technology in 1995. He was a Post-Doctoral Re- +searcher with the University of Pittsburgh from +1996 to 1998, a Researcher & Adjunct from +1999 to 2004, and a Joint Research Professor +with Information School, University of Pittsburgh +since 2006. His research interests include infor- +mational retrieval, software architecture, digital +libraries, and web-based learning techniques. + +NNANJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +13 +Longbing Cao (@SM in 2006) received PhD in +pattern recognition and intelligent systems from +the Chinese Academy of Science, China, and +PhD in computing sciences from the University +of Technology Sydney, Australia. He is a profes- +sor at UTS, an ARC Future Fellow (professorial +level), and the EiCs of IEEE Intelligent Systems +and J. Data Science and Analytics. His research +interests include artificial intelligence, data sci- +ence, machine learning, behavior informatics, +and their enterprise applications. + diff --git a/qdFJT4oBgHgl3EQfaSwR/content/tmp_files/load_file.txt b/qdFJT4oBgHgl3EQfaSwR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9ed01fdbee09a14f1e9989bde0e99ebb786e82da --- /dev/null +++ b/qdFJT4oBgHgl3EQfaSwR/content/tmp_files/load_file.txt @@ -0,0 +1,1279 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf,len=1278 +page_content='JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 8, AUGUST 2015 1 Learning Informative Representation for Fairness-aware Multivariate Time-series Forecasting: A Group-based Perspective Hui He, Qi Zhang, Shoujin Wang, Kun Yi, Zhendong Niu, and Longbing Cao, Senior Member, IEEE Abstract—Multivariate time series (MTS) forecasting has penetrated and benefited our daily life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' However, the unfair forecasting of MTSs not only degrades their practical benefit but even brings about serious potential risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Such unfair MTS forecasting may be attributed to variable disparity leading to advantaged and disadvantaged variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' This issue has rarely been studied in the existing MTS forecasting models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' To address this significant gap, we formulate the MTS fairness modeling problem as learning informative representations attending to both advantaged and disadvantaged variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Accordingly, we propose a novel framework, named FairFor, for fairness-aware MTS forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' FairFor is based on adversarial learning to generate both group-irrelevant and -relevant representations for downstream forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' FairFor first adopts the recurrent graph convolution to capture spatio-temporal variable correlations and to group variables by leveraging a spectral relaxation of the K-means objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Then, it utilizes a novel filtering & fusion module to filter the group-relevant information and generate group-irrelevant representations by orthogonality regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The group-irrelevant and -relevant representations form highly informative representations, facilitating to share the knowledge from advantaged variables to disadvantaged variables and guarantee fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Extensive experiments on four public datasets demonstrate the FairFor effectiveness for fair forecasting and significant performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Index Terms—Multivariate time series, forecasting, fairness, adversarial learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 1 INTRODUCTION M ULTIVARIATE time-series (MTS) forecasting, penetrat- ing our daily living, studying, working, and en- tertaining, has played a critical role in a wide range of real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Examples include climate forecast- ing [1], stock trend analysis [2], [3], road-use monitoring [4], [5], and clinical risk forecasting [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' For example, in quanti- tative finance analysis, multiple financial factors co-involve and their forecasts assist financial practitioners in optimiz- ing investment portfolios and strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' MTS forecasting capably enhances investment opportunities, schedules suit- able services, and adjusts optimal plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' It has been one of the most fundamental yet beneficial real-world tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' However, unfair MTS forecasting results in the unequal forecasting accuracy among variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' It may degrade the practical benefits of MTS forecasting or even bring about serious potential risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' For example, in finance, the trends of high-frequency trading stocks are difficult to be predicted owing to their highly volatile temporal patterns which are easily overwhelmed by those of stable low-frequency trading stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' In the urban police deployment, the crimes in regions with geographical neighborhoods are easier to Hui He is with the School of Medical and Technology, Beijing Institute of Technology, Beijing 100081, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' E-mail: hehui617@bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='cn Qi Zhang, Shoujin Wang and Longbing Cao are with Data Science Lab, University of Technology Sydney, Ultimo, NSW 2007, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' E-mail: qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='zhang-13@students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='uts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='au, {shoujin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='wang, longbing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='cao}@uts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='au Kun Yi and Zhendong Niu are with the School of Computer Science and Technology, Beijing Institute of Technology, Beijing 100081, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' E-mail: {yikun, zniu}@bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='cn Manuscript received April 19, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' revised August 26, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Advantaged Variable Disadvantaged Variable Methods StemGNN Informer TS2Vec MTGNN AGCRN VAR 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='0574 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='9595 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='3862 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='0949 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='7769 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' An example of MTS forecasting unfairness: the table at the upper part shows the variance of the forecasting performance (MAE) over all variables derived from five advanced MTS forecasting methods on real- world traffic dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' the curves below show the true traffic flow data and its corresponding prediction from StemGNN model on four sensors (variables) numbered 15, 16, 20 and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' It is clear that StemGNN achieves desirable performance on those advantaged variables (#20 and #21 marked in dotted ellipse) while poor performance on those disadvantaged variables (#15 and #16 marked in solid ellipse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' be predicted than the isolated regions due to the shared socioeconomic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' This results in the inherent variable disparity, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', high vs low-frequency stocks and isolated vs neighborhood regions, which may cause MTS forecasting models to generate unequal forecasting accuracy over dif- ferent variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', well-performed (advantaged) variables and badly-performed (disadvantaged) variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The unfair arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='11535v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='LG] 27 Jan 2023 JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 8, AUGUST 2015 2 results may deceive people by the overall performance into trusting the results on disadvantaged variables, potentially causing practical failures or risks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', great investment losses or fatal treatment plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' To this end, it is necessary to develop fair MTS forecasting models to eliminate unfair forecasting for improving both the overall performance and specifically the performance on disadvantaged variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Previously, great efforts have been made to design com- petent MTS forecasting models to explore temporal depen- dencies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', short- and long-term temporal patterns) [7], [8], [9], [10], spatial dependencies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' inter-series correlations across multiple time series) [4], [11], [12], [13], [14], inter- pretability of modeling [15], [16], [17], [18], or complicated nonstationary [19], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Although achieving great success, those models focusing on accuracy improvement [21] may be vulnerable to variable disparity [22], leading to serious performance unfairness among variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' As illustrated in Figure 1, all five advanced MTS forecasting models have large accuracy variances (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', VAR) over different variables, and one representative, StemGNN, attends to certain vari- ables and accordingly leads obviously bad performance on variables #15 and #16 (disadvantaged variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The il- lustration obviously reveals the significant (widely existing) yet challenging (rarely considered) unfairness issue in MTS forecasting models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Although fair MTS forecasting has been less studied in the literature, a variety of methods for fairness mod- eling recently emerge in other learning tasks, such as classification [23], [24], [25], [26], clustering [27], [28] and ranking [29], [30], including by customizing fairness reg- ularization [31] and adversarial learning [32], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' These methods generally model fairness from two perspectives, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', individual and group perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Some methods model individual fairness by defining a reasonable similar- ity metric based on a fairness graph [34], [35], weighted ℓp-metrics [27], [36] or the Wasserstein distance [37] at a fine granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The other methods aim to eliminate group unfairness on sensitive attributes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', gender, age, or race of users).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Specifically, they learn to filter out the sensitive information via adversarial learning to obtain insensitive (group-irrelevant) representations [32], [33] or disentangle data representations into sensitive and insensitive repre- sentations by orthogonality regularization [29], [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' How- ever, individual fairness may be potentially harmful to overall forecasting performance and is costly in calculating the similarity measurement among all variables for high- dimensional MTS data [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Group fairness generally de- pends on natural groups pre-defined by sensitive attributes, while the attributes are usually inaccessible in MTS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' This also brings challenges for fairness modeling of MTS scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Considering the inter-variable correlations in MTS data [11], correlated variables may have similar (advan- taged/disadvantaged) performance and can be clustered into one group to handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' To this end, we deliberatively employ group fairness to achieve fair MTS forecasting, simplifying fairness modeling from the individual (vari- able) level to a group-wise manner and avoiding the high computational cost on individual variable fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' We ex- pect to share the knowledge between advantaged variables (groups) and disadvantaged variables (groups) to improve the performance of disadvantaged variables, guaranteeing performance fairness and overall improvement simultane- ously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Accordingly, two main challenges (CH) are necessar- ily addressed in our work: CH1, how can we adaptively learn variable grouping?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' CH2, how can we effectively learn group-relevant and group-irrelevant representations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' In or- der to address these critical challenges, we propose a novel fair MTS forecasting model FairFor which consists of two main modules: variable correlating & grouping and filtering & fusion, which aims to address CH1 and CH2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' To be specific, in the variable correlating & grouping mod- ule, we first adopt recurrent graph convolution to capture spatio-temporal variable correlations and introduce cluster- ing objectives to learn variable grouping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Inspired by [32], [39], we design a novel filtering & fusion to generate group- irrelevant representations via adversarial learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Owing to our novel and specific design, FairFor effectively improves the overall MTS forecasting performance and achieves much fairer performance among variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The contributions of our work are summarized below: We study a significant and common problem in MTS forecasting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', the unfairness of forecasting perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Accordingly, we propose a novel fairness- aware MTS forecasting framework to address this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' To the best of our knowledge, this is the first effort on fair MTS forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' We propose a novel variable correlating & grouping module to learn spatio-temporal variable correla- tions by a recurrent graph convolutional network and adaptive variable grouping via the spectral re- laxation of the K-means clustering objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' We design a novel filtering & fusion module to learn group-irrelevant representations by an orthogonality regularization in an adversarial learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Extensive experiments on four public datasets demon- strate the superior forecasting performance of our proposed FairFor model compared with the state-of-the-art models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' We further verify the effectiveness and rationality of our proposed model in enhancing the fairness of MTS forecast- ing without performance loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 2 RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='1 Multivariate Time-series Forecasting Statistical methods for MTS forecasting, such as Gaussian process (GP) [40], vector auto-regressive model (VAR) [41] and autoregressive integrated moving average model (ARIMA) [42], all rely on powerful assumptions regarding a stationary process and can only learn linear relation- ship among different time steps within time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' In contrast, deep learning based methods are immune to stationary assumptions and have an inherent efficiency in capturing non-linearity, complicated and hidden signals existing in many real-world time series collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The first two deep learning based models created for MTS forecasting are LSTNet [7] and TPA-LSTM [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' They marry convolu- tional neural network (CNN) with recurrent neural net- work (RNN) to capture short-range temporal dependencies and long-range temporal patterns respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Informer [9] JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 8, AUGUST 2015 3 embeds sparse self-attention mechanism into Transformer- based architecture to enhance the latter’s capacity in ex- tracting the long-range dependencies and alleviate mem- ory burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Pyraformer [10] explores the multi-resolution representation of time series via introducing pyramidal attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Although these models are cutting-edge MTS forecasting algorithms, they only focus on exploiting temporal dependencies that deflate the ability when dealing with highly-correlated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' To further explicitly address the inter-series correlations among multiple variables, recent works extract the unidirected relations among variables and captures shared patterns via priori graph [43], [44], [45] or graph learning [4], [12], [46], [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Similarly, CATN [11] intro- duces a tree (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', an ordered graph) to structure time-series variables with a clear hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Although great achieve- ment has been achieved in MTS modeling, all these existing works overlook a critical and practical issue, the forecasting performance bias over different time series variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' In this work, we particularly focus on performing fair MTS forecasting to avoid performance disparity on variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='2 Fairness-aware Algorithms With the popularity of machine learning, the fairness issue has received broad attention as algorithms are vulnerable to data biases that render the decision skewed toward partic- ular individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' This data bias mostly relates to the issue of data imbalance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', historical user-item interactions of active users are much more than those of inactive users in recommendations) and then algorithms have been shown to amplify biases in the raw data to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Therefore, researchers have proposed many debiasing algorithms [23], [24], [34], [35], [37], [48] to mitigate inequity for each specific domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Specifically, some methods [34], [35], [37] model- ing individual fairness mostly define reasonable similarity metrics, but high-dimensional data make similarity measure between individuals very costly [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Other methods fall into group fairness [23], [24] or a combination [48] of both group and individual fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' [23] minimized the disproportionate impacts by calculating group-wise im- portance separately when pruning on different groups in the process of face classification model compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' [48] devised a fairness-constrained approach through heuristic re-ranking to mitigate the unfair recommendation issue where the user-item interactions of inactive users tend to be neglected and are easily overwhelmed by active users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' In this work, we adopt group fairness to avoid high com- putation cost and achieve performance fairness and overall improvement simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='3 Adversarial Learning for Fairness Next, a brief review is provided of adversarial learning algorithms applied in exploring fairness, which is the most relevant to our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Some studies [26], [29], [30], [32], [33], [39] adversarially train models to discriminate sen- sitive attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' For example, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' [39] learned a set of filters for erasing the sensitive attributes from the user representations and meanwhile use a set of classifiers as discriminators to predict sensitive attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' [32] also tried to obfuscate all sensitive attributes of users and items under a graph-based adversarial training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Inspired by [25], [49] disentangling data representation into orthogonal subspaces including sensitive attributes or not, Patro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' [30] simultaneously learned a bias-aware user representation and a bias-free user representation that only carries insensitive user information for fair news recommen- dation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Similarly, Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' [29] designed an adversarial learn- ing task to preclude encoding provider bias for provider- fair news representation and further render the provider- fair and provider-biased representations to be orthogonal by an orthogonal regularization term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' However, these methods strongly depend on the pre-defined sensitive attributes that are unavailable in MTS analysis scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' In this study, we learn informative representations attending to each variable to promote fairness by leveraging a group-based adversarial learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 3 METHODOLOGY This section presents the problem formulation, then de- scribes the overview of the proposed framework, named fairness-aware multivariate time-series forecasting (FairFor) network, followed by the details of each module and the learning objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='1 Problem Formulation We use X:,0:T −1 = {X:,0, X:,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', X:,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', X:,T −1} ∈ RT ×N to denote N univariate time series with T time steps, where X:,t = {x0,t, x1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', xi,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', xN−1,t}T ∈ RN×1 records the observed values of N variables at time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Herein, i ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', N − 1} and t ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', T − 1} is the index of the variable and the time step respectively, and T denotes transpose operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The time interval between any two time steps is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Under the sliding forecasting setting with a fixed win- dow size of w ∈ N+ and a sliding step of 1, we have the input, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', the observed values of all the N variables in w successive steps till the tth time step, X:,t−w+1:t = {X:,t−w+1, X:,t−w+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', X:,t} ∈ Rw×N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' We articulate the research problem of MTS forecasting on a graph G = (V, E, M) to emphasize the spatio-temporal correlations simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The set of nodes V denotes input series X:,t−w+1:t, where |V| = N and each series/variable Xi,: corresponds to a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' E represents the set of edges, and M ∈ RN×N is defined as the adjacent matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' In addition, we evaluate the forecasting unfairness with the variance in forecasting errors of different variables, where the larger the variance is, the more unfair the forecasts are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Accordingly, the target is to accurately forecast based on the graph G the future sequence of h ∈ N+ steps Y:,t+1:t+h = {Y:,t+1, Y:,t+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', Y:,t+h} successive to the tth time step through one forward procedure and guarantee a small forecasting error variance simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='2 Framework Overview Figure 2 illustrates an overview of FairFor which consists of four modules: adaptive graph construction, variable cor- relating & grouping, filtering & fusion and predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' To be specific, the adaptive graph construction is to learn an adjacent matrix (the implicit graph G) to capture the inter- series dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Then, based on the learned adjacent JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=" 8, AUGUST 2015 4 Input 𝑿:,𝒕%𝒘'𝟏:𝒕 Time Step Variable 𝓕𝑲 𝓕𝟐 𝓕𝟏 … AGC Filtering & Fusion 𝓩 … 𝑯." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' ⨁ 𝑯∗ FC Layer Clustering Layer Clustering Results 𝑪 Discriminator Adversarial Loss 𝓛𝑪 Clustering Loss Output 𝒀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=":,𝒕'𝟏:𝒕'𝒉 Predictor 𝓛𝑫 Orthogonality Regularization Loss 𝓛𝑭 Forecasting Loss 𝓛𝑨 Variable Correlating & Grouping … … … Variable 0 𝑬𝑴 𝑴;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=" 1 2 𝑁 − 1 … … … Node Embedding … … … Softmax RGCU … 𝑿:,𝒕%𝒘'𝟏 … 𝑿:,𝒕%𝒘'𝟐 … RGCU … 𝑿:,𝒕%𝒘'𝟑 𝑯𝒕(𝑯) 𝑯𝒕%𝒘'𝟏 … RGCU 𝑯𝒕%𝒘'𝟐 𝑯𝒕%𝟏 … … 𝑿:,𝒕 … … … … Adaptive Graph Construction (AGC) Spatio-temporal Correlation Learning (STCL) STCL 𝑯 Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The framework of our FairFor network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' AGC (the blue dotted line box) takes X:,t−w+1:t as input to generate the learnable node embedding EM and adjacent matrix � M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Then the variable correlating & grouping (the green dotted line box) outputs the last hidden state H through multiple RGCUs and produces the clustering results C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Next the filtering & fusion (the yellow dotted line box) is applied to filter out the group-relevant information from H and form the group-irrelevant representation � H, which is further input to the discriminator D with C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The predictor integrates the group-relevant representation H and group-irrelevant representation � H to form more informative representation H⋆ for final prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' matrix, the variable correlating & grouping learns a spatio- temporal representation for each time series with a recurrent graph convolutional network and then clusters all time series variables into multiple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Subsequently, the filtering & fusion learns group-relevant variable represen- tation and also group-irrelevant variable representation by filtering out the group-relevant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Both the group- relevant representation and group-irrelevant representation are beneficial for the forecasting task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Therefore, the pre- dictor integrates the two kinds of representations to form a more informative representation for producing the final forecast of each time series variable at one forward step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='3 Adaptive Graph Construction Graph convolutional network (GCN) has been widely stud- ied for accurate MTS forecasting due to its capability to capture the spatial correlations of time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Let X ∈ RN×w (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', w-dimensional feature vector for each node) be the input time series matrix, W and b be the learnable weight matrix and bias respectively, then the layer-wise graph convolution operation of GCN can be well-approximated by the Chebyshev polynomial expan- sion form as: G(X) = (I + B− 1 2 MB− 1 2 )XW + b (1) where I ∈ RN×N, B is the degree matrix, and G(X) ∈ RN×q is the output of GCN layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The GCN operation can be viewed from the perspective of a node (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', node i) as transforming the features of node Xi ∈ R1×w to G(Xi) ∈ R1×q with the shared W and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The pre-defined adjacent matrix M embedded in the graph convolution operation is usually explicitly con- structed by defining the similarity function of the dataset itself, or by defining the distance function according to the geographic distance on urban maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' However, the fixed explicit graph structure is not always available or complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The reason is that it is hard to manually capture latent rela- tionships from substantial time-series data to construct the graph structure, especially for non-traffic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Besides, the pre-defined graph is not directly related to the final fair forecasting task, which may lead to sizable biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Inspired by [4], [12], [46], [47] precisely and automatically disclosing the implicit inter-series dependencies at each time step from MTSs without prior knowledge, in this paper, we propose an adaptive graph construction module to randomly initial- ize a learnable node embedding matrix EM ∈ RN×d for all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Herein, each row of EM represents the embedding of a node, and d is the dimension of node embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Next, the inter-series dependencies between each pair of nodes can be inferred by multiplying EM and ET M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Formally: B− 1 2 MB− 1 2 = δ(ReLU(EM · ET M)) (2) where δ(·) is the element-wise softmax function which is used to normalize the adaptive adjacent matrix, T is the transpose operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Here, to eliminate needless and re- peated calculations during the iterative training process, we directly produce B− 1 2 MB− 1 2 in this case rather than producing M and computing a Laplacian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The EM will be automatically updated throughout training to dis- cover the hidden inter-series dependencies between various series and obtain the adaptive adjacent matrix M for graph convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Therefore, the GCN enhanced by the adaptive graph construction can be formulated as: G(X) = (I + δ(ReLU(EM · ET M)))XW + b (3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='4 Variable Correlating & Grouping Accurate MTS forecasting relies on capturing two essential properties, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', temporal correlations over a sequence of time steps and spatial correlations over different time series variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Accordingly, we adopt a recurrent graph convolu- tional network (RGCN) based on an implicit graph structure to capture the complex temporal and spatial correlations JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 8, AUGUST 2015 5 among a set of time series variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' We further group the learned spatio-temporal representation into several groups and produce the corresponding clustering results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Spatio-temporal Correlation Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' We introduce a recurrent graph convolutional unit (RGCU) via integrating a gated recurrent unit (GRU) with a GCN layer to learn the spatial and temporal inter-correlations between time series variables (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' A GRU takes X:,t as input and updates the hidden state from the previous state Ht−1 to Ht by employing the reset gate and update gate to govern how much information from the history should be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' By doing this, the GRU remembers historical hidden states that are relevant to future predictions and forgets those that are irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' RGCU replaces the MLP layers in GRU with GCN and send the embedded graph information to reset gate and update gate to update the hidden state Ht collectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' On the one hand, the spatial dependencies among different series, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', the inter-series dependencies, can be well captured by the implicit graph structure constructed based on the node embedding matrix EM in RGCU, where the implicit graph is encoded into graph nodes by GCN for time series variable interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' On the other hand, the GRU structure can well capture the temporal dependencies over different time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' As a result, the RGCU unit can well capture both temporal and spatial dependencies embedded in MTS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Formally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' we have � M = δ(ReLU(EM · ET M)) (4) rt = σ( � M[X:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='t||Ht−1]Wr + br) (5) ut = σ( � M[X:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='t||Ht−1]Wu + bu) (6) ct = θ( � M[X:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='t||r ⊙ Ht−1]Wc + bc) (7) Ht = u ⊙ Ht−1 + (1 − u) ⊙ ct (8) where X:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='t ∈ RN×1 and Ht ∈ RN×o refer to the input and output at time step t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' o is the hidden dimension,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' ut and rt refer to the reset gate and update gate respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' ct is the memory state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' [·||·] denotes a concatenation operator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' ⊙ denotes en element-wise multiplication,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' σ(·) and θ(·) are sigmoid and tanh activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' EM, Wu, Wr, Wc, bu, br and bc are learnable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The last hidden state of RGCU denoted by H is sent to the variable grouping module and the filtering & fusion module respectively to generate clustering results and group-irrelevant representa- tion respectively as the inputs to the discriminator D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Variable Grouping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Then, we propose to group the learned hidden state H according to variable correlations and regard clustering/grouping results as a counterweight to the filtering & fusion module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Hence, we introduce a clustering layer consisted of three-layer fully connected (FC) neural network with LeakyReLU as the activation and output the clustering results into discriminator D for adversarial learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The learned hidden state H may not suitable for forming cluster structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Hence, to stimulate to group variables in H, we introduce the spectral relaxation of the K-means objective [50], [51] as a loss: LC = Tr(HTH) − Tr(F THTHF ) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='F TF = I (9) where H ∈ Ro×a with a = b×N after reshaping, F ∈ Ra×K is the cluster indicator matrix, K is the number of clusters and b is the batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The optimization process of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' (9) requires iteratively updating F and H due to the dynamic learning of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' When F is fixed, updating H can follow the SGD of the clustering layer that boosts the representation to mine variable correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' When H is fixed, F can be updated once by computing the K-truncated SVD of H after several epochs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', 3) to prevent instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Since the cluster-friendly representation H is not favorable for the forecasting process, it is difficult to jointly optimize H in the clustering and forecasting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Hence, H is delivered to a FC layer before to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' (9) to alleviate model instability, referring to Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='5 Filtering & Fusion In our fairness-aware forecasting method, a key challenge is to learn the group-relevant (specific to each group) and group-irrelevant (shared by all groups) representations as il- lustrated in the Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Given a learning algorithm that learns spatio-temporal hidden state H to directly generate forecasts, we require the hidden state H to be independent from the learned group information C to achieve forecasting fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Therefore, we design a filter layer with a series of filter functions {F1, F2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', FK}, which are used to filter out the group-relevant information in the hidden state H and K is the number of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The filter function is represented as F : Ro �→ Ro, and the representation F(H) preserves the features shared by all groups when specific features to per group are filtered out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' We use the three FC layers followed by a batch normalization layer to represent each filter function F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Finally, K filtered representations are combined to generate the group-irrelevant representation: � H = Z(F1(H), F2(H), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='FK(H)) (10) where Z is a fusion function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The K filtered representations are fed into Z together and the group-irrelevant represen- tation unrelated to all group information is output without the representation dimension altered, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', using the average of the K filtered representations as Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' To learn filter functions, we use the idea of adversarial learning to train a group discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Specif- ically, we train two mappers M(� H) : Ro �→ Ro and M(C) : RK �→ Ro, which attempt to map the group-irrelevant representation and corresponding clustering results into the same space, more specifically, obtaining � H ∈ Ro and C ∈ Ro respectively, and calculate their Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Similar to [52], we accordingly use the three FC layers with LeakyReLu as the activation function to represent M(� H) and M(C) of discriminator D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The aim of filter functions is to render it difficult to infer variable correlations and group variables from the group-relevant representation H, while that of discriminator D is to fail the filter functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The adversarial loss function is shown: LA = 1 N N−1 � i=0 ||� Hi − Ci||2 2 (11) Unfortunately, some group information may still be in- cluded in the group-irrelevant representation � H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Because the group-irrelevant representation � H just needs to fool the discriminator D, the filtering & fusion module does JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 8, AUGUST 2015 6 Algorithm 1 Adversarial Training for FairFor Require: Hidden state H, group results C, network R, filter and fusion functions F, Z ⊆ R, discriminator D and cluster number K 1: Orthogonal initialize cluster indicator matrix F 2: for each training iteration i do 3: � H ← Z({Fk}k∈[K](H)) by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' (10) 4: H⋆ ← H + � H 5: Optimize L w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='t H, � H, H⋆, C, R with D being fixed 6: if i%3 == 0 then 7: Update F by computing K-truncated SVD of H 8: end if 9: LA ← E[||� H − C||2 2] by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' (11) 10: Optimize LA w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='t � H, C, D with R, H, H⋆ fixed 11: end for not necessarily completely filter the group information so that the discriminator D generally cannot perfectly assess the group information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' To address this issue, we design an orthogonality regularization method [30] to further purge the group-irrelevant representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Specifically, the group- relevant representation H and group-irrelevant representa- tion � H are regularized by boosting them to be orthogonal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The orthogonality regularization is calculated: LD = 1 N N−1 � i=0 | � Hi · Hi ∥� Hi∥ · ∥Hi∥ | (12) where � Hi and Hi are the group-relevant and corresponding group-irrelevant representations of the ith variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='6 Prediction The group-relevant representation mainly contains informa- tion on group attributes, and the group-irrelevant represen- tation mainly encodes group-free time-series information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Considering the information in both representations is rele- vant to the forecasting task, we integrate the group-relevant representation H and group-irrelevant representation � H by an addition operation to form the informative representa- tion H⋆, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', H⋆ = H + � H, which is then fed into a predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Then the future time-series sequence is estimated by a 2-D convolutional layer at one forward step style rather than a step-by-step style: �Y = Conv2D(H⋆) (13) where Conv2D is a convolutional operation to directly map H⋆ to the predictions for all horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Finally, the forecasting loss is denoted as: LF = 1 N N−1 � i=0 ||Yi − �Yi||2 2 (14) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='7 Adversarial Training Adversarial learning techniques encourage the deep repre- sentation to be maximally informative to generate group- irrelevant representation, and meanwhile to be minimally TABLE 1 Dataset statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Tasks #Time Step #Variable Interval Start Time Traffic 10,392 963 1hour 1/1/2008 PeMSD7(M) 11,232 228 5min 7/1/2016 Solar-Energy 52,560 137 10min 1/1/2016 ECG5000 5,000 140 − − discriminative in a group-relevant discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' There- fore, adversarial learning has the potential to learn group- irrelevant representations and treat each variable fairly whether it is advantaged or disadvantaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Note that we are not trying to purely render the prediction accuracy of advantaged variables and disadvantaged variables closer like common fairness-aware algorithms [30], [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Instead, we try to learn the group-relevant and group-irrelevant representation by adversarial learning and then form in- formative representations focusing on both advantaged and disadvantaged variables for enhancing the performance on disadvantaged variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' To the end, the FairFor optimiza- tion involves playing a min-max game: L = arg min R (LF + LC + LD + arg max D λaLA) (15) where R = FairFor − D denotes the remaining part after removing discriminator D from FairFor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The adversarial training algorithm is presented in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Our pro- posed FairFor is carried out via alternately optimizing the subsequent processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Concretely, we first feed input to the model to obtain L, then fix the parameters in the discrimi- nator D, and optimize R by minimizing L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Then, fixing the parameters of R, D is optimized by minimizing LA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 4 EXPERIMENT AND EVALUATION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='1 Datasets To evaluate the models under different scales and appli- cation scenarios, we employ four real-world datasets (see dataset statistics in Table 1) for extensive evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' PeMSD7(M) 1 records the traffic flow data of the detec- tors in California.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' It includes 228 variables and 11,232 time steps at a 5-minute interval;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Solar-Energy 2 is collected from National Renewable Energy Laboratory (NREL) and records the solar power production in 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' It includes 137 variables and 52,560 time steps at a 10-minute interval;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Traffic 3 is originally collected from the California De- partment of Transportation and describes the occupancy rate of different lanes in San Francisco highway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' It contains 963 variables and 10,560 time steps with a 10-minute inter- val, where each observation is between 0 and 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' ECG5000 4 records 5,000 heartbeats randomly selected from a 20-hour long ECG downloaded from Physionet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' It contains 140 variables and 5,000 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='html 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='uci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='edu/ml/datasets/PEMS-SF 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='timeseriesclassification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='com/description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Dataset=ECG5000 JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 8, AUGUST 2015 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='2 Experimental Setting and Metrics Following many classical time-series data split practice [7], [12], we split the data into training, validation, and test parts with the ratio of 7:2:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' All data is normalized by min-max method by following [4], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' In our experiments, the FC layer, clustering layer, filters {F1, F2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', FK}, discriminator D are all multi-layer FC networks with the number of layers set to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The predictor is a 2-D convolutional layer with the kennel size set to (1, o) and the number of features o in the hidden state H is fixed to 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The cluster number K in the filtering & fusion module is set to 6 for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' We examine values of trade-off parameter λa in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' (15) in the range of {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='7, 1} and choose {λa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='1} for all datasets and the node embedding dimension d in learnable node embedding matrix EM is set to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' For network R, we use Adam optimizer with a learning rate of 3e-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' For discriminator D, we use Adam optimizer with the learning rate of 5e-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Model parameters are turned on a Ubuntu 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='6 server with four NVIDIA GeForce 3090 GPU cards for 50 training epochs, through which we save the best-performing model based on values of accuracy metrics MAE/RMSE/MAPE/VAR on the validation set and reload it for the evaluation on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The batch size is set as 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Our code will be released at GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Four typical metrics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', MAE, RMSE, MAPE, and VAR, are employed for MTS forecasting evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Their formal definitions to evaluate all comparative methods are as fol- lows: Mean Absolute Error MAE = 1 N N−1 � i=0 |Yi − ˆYi|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Root Mean Squared Error RMSE = � 1 N N−1 � i=0 (Yi − ˆYi)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Mean Absolute Percent Error MAPE = 1 N N−1 � i=0 |Yi− ˆYi| Yi 1{|Yi| > 0} and Variance (the variance of MAE over each variable) V AR = 1 N N−1 � i=0 [|Yi − ˆYi| − 1 N N−1 � i=0 |Yi − ˆYi|]2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' where N denotes the number of variables,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Yi and ˆYi denote the ground truth and prediction respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='3 Baselines Because the fairness problem in MTS forecasting is rarely considered in previous studies, eight representative and SOTA MTS forecasting methods from different classes are deliberately chosen as baselines to compare with our pro- posed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' To be specific, two representative RNN- based methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', LSTNet and TPA-LSTM, and two Transformer-based methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', Informer and Pyraformer are chosen since they are good at capturing temporal pat- terns with different ranges such as short- and long-range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' TS2Vec is a very popular universal time-series represen- tation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Furthermore, MTGNN, StemGNN and AGCRN are based on graph neural networks, which are selected to justify the effectiveness of incorporating both intricate temporal and spatial dependencies among time- series data in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Traditional methods such as VAR and GP are not compared since the latest deep neural network-based methods [46], [47], [53] have been verified to outperform these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' For a fair comparison, the length of time-series input (historical sequence), time-series output (future sequence), and hardware technical indicators for all baselines are iden- tical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Hyper-parameters of each baseline are consistent with the corresponding paper settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Some methods [7], [8] were originally proposed for single-step output, then we carefully modify them into sequence output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' More details about these methods are as follows: LSTNet [7]: integrates RNNs and CNNs to capture the short- and long-term temporal patterns respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' TPA-LSTM] [8]: embeds temporal pattern attention into RNNs to discover both relevant time series and time steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' TS2Vec [54]: is a dilated CNN based universal frame- work to capture multi-scale contextual information in MTSs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Informer [9]: is a Transformer-based model with Prob- Sparse self-attention mechanism and generative style de- coder to predict long time series at one forward step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Pyraformer [10]: is a Transformer-based model to simul- taneously extract multiple ranges of temporal dependencies via the compact multi-resolution operation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' MTGNN [46]: combines graph learning, graph convolu- tion and temporal convolution together to learn the spatial- temporal correlations without pre-defined graph structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' StemGNN [12]: uses a GCN-based spectral network that can capture inter-series dependencies and temporal correla- tions jointly in the spectral domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' AGCRN [4]: employs GCNs embedded with an adap- tive node-specific pattern learning module to capture fine- grained inter-series relationships and uses RNNs to capture temporal patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='4 Experimental Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='1 Overall Results We compare FairFor with the baseline models on both MTS forecasting and fairness performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Table 2 and Table 3 show the fairness and forecasting performance of different methods respectively under the commonly-used setting of w = 12, h = 12 following [4], [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Fairness Improvement In the fairness aspect, we find that our FairFor obviously outperforms all baseline MTS forecasting methods on four datasets, specifically the VAR decrease over the best baseline is 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='59%/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='74%/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='34%/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='37% on PeMSD7(M)/Solar- Energy/Traffic/ECG5000 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' This is consistent with previous analysis: 1) general forecasting models are vulnerable to variable disparity and prone to focus on certain variables (advantaged variables), leading to generating unequal performance over different variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 2) FairFor proposes to combine group-relevant and group- irrelevant representation together to form informative representation focusing on both advantaged and disadvantaged variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Consequently, the performance of disadvantaged variables is improved by enriching the group-irrelevant representation and drawing support from the knowledge of advantaged variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Forecasting Performance From Table 3, we can ob- serve that FairFor can still achieve high multi-step forecast- ing quality on PeMSD7(M) and ECG5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' FairFor makes 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='21%/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='32%/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='49% improvements in average w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' MAE, RMSE and MAPE over the best baseline on PeMSD7(M) and ECG5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' This is explainable: different from the motivation of sacrificing overall performance to guarantee the fairness JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 8, AUGUST 2015 8 TABLE 2 The fairness performance of different methods evaluated on VAR under the setting of w = 12, h = 12, where the best results are highlighted in bold (smaller values indicate better fairness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Dataset Metric LSTNet TPA-LSTM TS2Vec Informer Pyraformer MTGNN StemGNN AGCRN FairFor PeMSD7(M) VAR 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='6003 42.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='63e-4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='43e-4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='71e-4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='59e-4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='17e-4 ECG5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='5668 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='2577 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='2360 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='2257 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='2618 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='2255 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='1774 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='2467 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='1732 TABLE 3 The prediction performance of different methods evaluated on four real-world datasets under the setting of w = 12, h = 12, where the best results are highlighted in bold (smaller values indicate better results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Dataset Metric LSTNet TPA-LSTM TS2Vec Informer Pyraformer MTGNN StemGNN AGCRN FairFor PeMSD7(M) MAE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='2004 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='7573 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='6001 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='6110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='5769 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='5352 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='6096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='5207 MAPE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='9334 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='9561 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='9664 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='9481 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='0132 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='951 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='9176 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='9854 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='8931 requirement in existing recommendation methods [30], [32], FairFor approaches an unbiased process of enriching the representation of per variable and thus can achieve supe- rior performance of prediction and fairness simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Although our model is surpassed on some specific exam- ples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Solar-Energy and Traffic, the forecasting perfor- mance is still sub-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' More specifically, we reveal the cause of FairFor’s under-performance on Solar-Energy and Traffic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Our method has a performance decrease over Informer of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='68% (MAE), 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='63% (RMSE) and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='53% (MAPE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' This is intuitively attributable to the fact that the capture of mid-term (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', h ∈ {12, 24}) and long-term (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', h ∈ {48, 72, 96}) dependency coupling between output and input in Informer is more precise and contributes more to improve overall regression indicators, which is also verified in Figure 3(b), 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Moreover, MAE, RMSE and MAPE decrease of AGCRN over proposed FairFor is 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='89%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='59% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='02% on Solar-Energy respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' This is attributable that fine-grained node-specific (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', similar, dissimilar, or contradictory) patterns across different data sources may be captured by node-specific parameter learning strategy in- stead of parameter sharing strategy commonly used in MT- GNN, StemGNN and FairFor (with GNN-based backbone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' However, these methods neglect to focus on disadvantaged variables in design and cause poor VAR performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Next, to verify the ability of forecasting different horizons, we compare FairFor with TS2Vec, Informer, Pyraformer, MTGNN, StemGNN and AGCRN, which are the top-6 baselines from the previous experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' We fix w = 12 and adjust h in the range of {3, 6, 12, 24, 48, 72, 96}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Figure 3 further shows the MAE and VAR performance comparison at different horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' We can observe that: 1) the FairFor shows significantly better VAR results at different horizons across all datasets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 2) On the Solar-Energy and Traffic dataset, Informer achieves better MAE results on the long horizon (at {48, 72, 96}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' We attribute this to the potential value to capture the long-range dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='2 Ablation Study Besides, we also perform an ablation study to better evalu- ate the effectiveness of several core designs in our FairFor, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', spatio-temporal correlation learning enhanced by AGC (STCL+AGC) , adversarial learning - D (ALD), cluster- ing loss (CL), and orthogonal regularization loss (ORL), by removing them individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' We select the setting as w = 12, h = 12, K = 6 on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Results are summarized in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' We can observe that: 1) In w/o STCL+AGC, the STCL module enhanced by AGC module of FairFor is replaced by gated TCN previously used in [45] to learn temporal dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Table 4 shows our method averagely outperforms its variant w/o STCL+AGC with a significant margin (MAE: -30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='14%, MAPE: -16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='04%, VAR: -40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='55%) across all datasets, indicating that jointly capturing the temporal patterns and inter-series correla- tions can significantly improve the forecasting and fairness performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 2) Variate w/o ALD denotes FairFor without discriminator D, and La is removed from training objective L in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' As verified in Table 4, the score w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='t MAE, MAPE, and VAR increases by 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='11%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='58%, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='74% on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' This proves that group-based adversarial learning is worth adopting, especially when balancing and optimiz- ing the performance of each variable is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 3) To study the importance of variable correlating & grouping module without affecting the normal work of discriminator D, we JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 8, AUGUST 2015 9 3 6 12 24 48 72 96 Horizon 2 3 4 5 6 7 8 MAE TS2Vec Informer MTGNN StemGNN AGCRN FairFor 3 6 12 24 48 72 96 Horizon 10 20 30 40 50 60 70 80 VAR TS2Vec Informer MTGNN StemGNN AGCRN FairFor (a) PeMSD7(M) 3 6 12 24 48 72 96 Horizon 3 4 5 6 7 8 VAR ( 10-4) TS2Vec Informer MTGNN StemGNN AGCRN FairFor 3 6 12 24 48 72 96 Horizon 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='8 2 MAE ( 10-2) TS2Vec Informer MTGNN StemGNN AGCRN FairFor (b) Solar-Energy 3 6 12 24 48 72 96 Horizon 1 2 3 4 5 MAE TS2Vec Informer MTGNN StemGNN AGCRN FairFor 3 6 12 24 48 72 96 Horizon 5 10 15 20 25 30 35 40 VAR TS2Vec Informer MTGNN StemGNN AGCRN FairFor (c) Traffic 3 6 12 24 48 72 96 Horizon 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='7 MAE TS2Vec Informer MTGNN StemGNN AGCRN FairFor 3 6 12 24 48 72 96 Horizon 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='8 VAR TS2Vec Informer MTGNN StemGNN AGCRN FairFor (d) ECG5000 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Performance comparison at different horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' TABLE 4 Results of ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Models PeMSD7(M) Solar-Energy Traffic ECG5000 MAE MAPE VAR MAE MAPE VAR MAE MAPE VAR MAE MAPE VAR w/o STCL+AGC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='2412 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='1018 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='2543 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='5797 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='8960 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='1747 FairFor 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='7983 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='0678 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='9953 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='0807 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='3826 5.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' (15) to form variate w/o CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' It yields 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='16%/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='95%/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='23% rise in average across all datasets, demonstrating the significance of setting a soft K-means objective to auxiliarily infer variable correlations and group variables in group-based adversarial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 4) Variate w/o ORL represents discarding LD from L in the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The results demonstrate that removing ORL also hurts the fairness of FairFor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Because ORL pro- motes the orthogonality of group-relevant and -independent representation, which can better identify and remove group- relevant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The STCL module enhanced by AGC exhibits a greater impact on the performance because it has resorted to distill- ing complex temporal patterns and inter-series correlations as the basis of time-series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Besides, other modules enhance TSRL from the perspective of learning more informative representations attending to both advan- taged and disadvantaged variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The ablation variants except for w/o STCL+AGC still outperform most baselines, indicating the stable and complementary advantages of different modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='3 Hyper-parameter Analysis We explore the sensitivity analysis of several critical hyper- parameters on PeMSD7(M) and ECG5000 datasets, includ- ing cluster number K, window size w, embedding dimen- sion d and trade-off coefficient λa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The results in terms of MAE and VAR are demonstrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' When we change a hyper-parameter, the other variables keep their default values explained in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' First, we investigate the influence of variable correlating & grouping by varying the value of K in a range of {2, 3, 4, 5, 6, 7, 8, 9, 12, 15}, and plot the forecasting results MAE and fairness results VAR under w = 12, h = 12 in Figure 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' When K is less than 6, MAE and VAR display a trend of violent fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' A moderate value for K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', K = 6, makes MAE and VAR descend to the lowest on PeMSD7(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' A similar trend exists as increasing cluster number from 6 since unsuitable clustering objective possibly cannot fully describe the complex group information inside variables and confuse latent data regularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Meanwhile, the performance on ECG5000 is relatively stable with differ- ent K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The possible reason is that the range of K we set is too small to display the data regularities (K = 56) [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Note that the increase of K will bring a burden to the filtering & fusion module and hurt the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Therefore, the setting of K is 6 for PeMSD7(M) and ECG5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Then, we turn to explore the impact of input win- dow with different sizes by varying w in a range of {1, 3, 6, 9, 12, 15, 18, 21, 24, 48}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' As shown in Figure 4(b), when predicting mid-term sequence (h = 12), initially increasing window size degrades performance, but further increasing causes the MAE and VAR to drop since it brings sufficient temporal patterns and dependency information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' However, further increasing w makes the MAE and VAR JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 8, AUGUST 2015 10 (a) Cluster Number K (b) Window Size w (c) Embedding Dimension d (d) Trade-off Coefficient λa Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Hyper-parameter analysis on PeMSD7(M) and ECG5000 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' get higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' This is intuitive due to the noise introduced by redundant temporal patterns, dependencies and group features that are not fit for the group-based adversarial learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Hence, the input window size and output window size need to be carefully weighed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Moreover, we check a key parameter d of learnable node embedding matrix EM, which not only influences the quality of adjacent matrix � M but also decides the parameter diversity in STCL module (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' (5), (6) and (7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Figure 4(c) reports the effects taking different dimensions of EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' As can be seen, FairFor achieves relatively good performance with all tested dimensions, demonstrating that the model capacity is fairly robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Note that FairFor achieves the best performance when the dimension is set to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' As the dimen- sion goes larger, the performance first increases since more information is contained to thus help our STCL module to deduce more accurate intra-series correlations, then starts to be weaker probably due to over-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Intuitively, an appropriate embedding dimension should make the model obtain sufficient correlation information meanwhile avoid- ing introducing additional parameters causing over-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' To learn the importance of adversarial losses, we control La to change within a range {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='7, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' The results are exhibited in Figure 4(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' From Figure 4(d), we find that the VAR drops steadily with the increase of λa, then rises until λa = 1, and the MAE has a similar trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' When λa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='1, forecasting performance and fairness achieve the best together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Thus, a moderate value for λa (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='1) may be preferable to make adversaries achieve an appropri- ate equilibrium and protect group information from being leaked to the group-irrelevant representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='4 Visualization of Fairness As previously discussed, the filtering & fusion module filters the group-relevant information from H and obtains the group-irrelevant representation � H in the group-based adversarial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' To further understand how the group- relevant (specific to each group) and -independent (shared by all groups) representation work, we respectively visu- alize the feature distribution by t-SNE in 2-D space and evaluate the informativeness of H and � H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' We randomly choose a batch of the Traffic-train data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Figure 5(a)-5(d) and Figure 5(e)-5(h) show the feature distributions of H and � H, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Firstly, H and � H all exhibit a good clustering structure, which proves the clustering ability of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' In addition, H shows smaller inter-cluster distances than � H, indicating different variable representations learned in H are close and concentrated, while those learned in � H after filtering are informative and discriminative and exhibit less overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Furthermore, H with a compact clustering structure may only contain the information of original ad- vantaged groups, and � H with a decentralized clustering structure is enriched by drawing support from advantaged groups and changes into informative representation shared by all groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Therefore, the FairFor model has good predic- tion and fairness performance and is also interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 5 CONCLUSION We argue that the performance unfairness between vari- ables widely exists in MTSs due to the variable disparity, equally attending to both advantaged and disadvantaged variables can facilitate overall forecasting performance and fairness simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' In this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' we study the unfair- ness problem and propose a fairness-aware MTS forecasting method (FairFor),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' which aims to (1) capture spatio-temporal variable correlations via recurrent graph convolution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' (2) group variables by leveraging a spectral relaxation of the K-means objective,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' (3) filter the group-relevant information and generate group-irrelevant representation via adversar- ial learning with an orthogonality regularization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' and (4) integrate the group-relevant and -irrelevant representations to form a highly informative representation for the final prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Experimental results on four real-world datasets prove that FairFor can simultaneously and effectively im- prove the performance of fairness and MTS forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Cur- rently, FairFor is an enhanced framework that sequentially performs to capture spatio-temporal correlations and bal- ance the forecasting performance of all variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Intuitively, the variable disparity and correlations dynamically evolve over time in coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Therefore, we will embed the balance operation into spatio-temporal correlation learning at each time step in the future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of China under Grant 2019YFB1406302 and National Natu- ral Science Foundation of China under Grant 62272048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' REFERENCES 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=', pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 1–14, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' e-MAEPeMSD7(M) MAE(×10~1)ECG5000 e-VAR PeMSD7(M) 30 VAR(× 10*2)ECG5000 E 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='2 25 VAR MA 3 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='8 15 2 3 4 5 6 7 8 9 12 15 Ke-MAE PeMSD7(M) 9-MAE(×10-1)ECG5000 e-VAR PeMSD7(M) VAR(×102) ECG5000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='3 28 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='2 26 VAR 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='1 24 22 3 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='9 18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='8 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='7 1 3 6 6 12 15 18 21 24 48 we-MAE PeMSD7(M) MAE(×10-1)ECG5000 VAR PeMSD7(M) VAR(×102) ECG5000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='3 28 E 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='2 26 VAR MA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='1 24 22 3 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='9 ： 18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='8 e 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='7 1 2 5 7 9 10 11 13 15 20θMAE PeMSD7(M) MAE(×10-1)ECG5000 e-VARPeMSD7(M) e-VAR(×102)ECG5000 C VAR 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='1 222987 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='7 1JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 14, NO.' 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+page_content=' abs/1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='07758, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Hui He received the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' degree from Univer- sity of Shanghai for Science and Technology, Shanghai, China in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' She is currently pursu- ing the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' degree at Institute of Engineering Medicine, Beijing Institute of Technology, Beijing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Her current research interests focus on multivariate time-series analysis and knowledge services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Qi Zhang received his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' degrees from Bei- jing Institute of Technology China under the dual Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' program of Beijing Institute of Technol- ogy and University of Technology Sydney Aus- tralia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' He is currently an AI Scientist in Deep- Blue Academy of Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' His current research interests focus on Collaborative Filtering, learn- ing to hash and sequential recommendation and multivariate time-series analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Shoujin Wang received a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' degree in Data Science from the University of Technology Syd- ney in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' He is currently a Lecturer in Data Science at Data Science Institute, University of Technology Sydney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' His main research interests include data mining, machine learning, recom- mender systems and fake news mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' He is the social media editor of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Data Science and Analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Kun Yi is a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' candidate from Beijing Institute of Technology, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' His current research inter- ests include multivariate time-series forecasting, data science and knowledge discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Zhendong Niu received the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' degree in computer science from the Beijing Institute of Technology in 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' He was a Post-Doctoral Re- searcher with the University of Pittsburgh from 1996 to 1998, a Researcher & Adjunct from 1999 to 2004, and a Joint Research Professor with Information School, University of Pittsburgh since 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' His research interests include infor- mational retrieval, software architecture, digital libraries, and web-based learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' NNANJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' 8, AUGUST 2015 13 Longbing Cao (@SM in 2006) received PhD in pattern recognition and intelligent systems from the Chinese Academy of Science, China, and PhD in computing sciences from the University of Technology Sydney, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' He is a profes- sor at UTS, an ARC Future Fellow (professorial level), and the EiCs of IEEE Intelligent Systems and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' Data Science and Analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} +page_content=' His research interests include artificial intelligence, data sci- ence, machine learning, behavior informatics, and their enterprise applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFJT4oBgHgl3EQfaSwR/content/2301.11535v1.pdf'} diff --git a/sNE0T4oBgHgl3EQfrgFB/content/tmp_files/2301.02566v1.pdf.txt b/sNE0T4oBgHgl3EQfrgFB/content/tmp_files/2301.02566v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..099fb8678828ee01ab4081da16cad3985e796759 --- /dev/null +++ b/sNE0T4oBgHgl3EQfrgFB/content/tmp_files/2301.02566v1.pdf.txt @@ -0,0 +1,4115 @@ +arXiv:2301.02566v1 [math.RA] 6 Jan 2023 +COCHARACTERS OF UTn(E) +LUCIO CENTRONE, VESSELIN DRENSKY, AND DANIELA MARTINEZ CORREA +Abstract. Let F be a field of characteristic 0 and let E be the infinite dimensional Grassmann algebra over +F . In the first part of this paper we give an algorithm calculating the generating function of the cocharacter +sequence of the n × n upper triangular matrix algebra UTn(E) with entries in E, lying in a strip of a fixed +size. In the second part we compute the double Hilbert series H(E; Tk, Yl) of E, then we define the (k, l)- +multiplicity series of any PI-algebra. +As an application, we derive from H(E; Tk, Yl) an easy algorithm +determining the (k, l)-multiplicity series of UTn(E). +1. Introduction +We fix a field F of characteristic 0 and any algebra over F is considered associative with unit. +Let +X = {x1, x2, . . .} be a countable set of indeterminates. We denote by F⟨X⟩ the free algebra freely generated +by X over F. Let A be an algebra over F satisfying a polynomial identity, i.e., a PI-algebra. It is well +known that its set of polynomial identities T (A) is a T -ideal of F⟨X⟩, i.e., an ideal that is invariant under +all endomorphisms of F⟨X⟩. +Since F is a field of characteristic 0, all the polynomial identities follow from the multilinear ones. A +famous theorem by Kemer [32] says that if A is a PI-algebra, then its T -ideal is finitely generated, but it is +important to recall that the complete set of finite generators of T -ideals is well known only for few algebras. +By a result of Regev [44], it seems to be more efficient to study the set of multilinear polynomials which +(in a certain sense) are not polynomial identities for a given algebra. More precisely, if Pn is the vector space +of multilinear polynomials in the variables {x1, . . . , xn}, we study the factor space Pn(A) := Pn/(Pn ∩ T (A)) +for each n. We recall that Pn is also a left Sn-module under the canonical left action of the symmetric group +Sn. Since Pn(A) inherits the Sn-action on Pn, it affords an Sn-character χn(A) called the n-th cocharacter +of A. The sequence (χn(A))n∈N is called the sequence of cocharacters of A. We also observe that Pn(A) is +a finite dimensional vector space which dimension is called the n-th codimension of A (or in symbol cn(A)) +and the sequence (cn(A))n∈N is called the sequence of codimensions of A. +In [28], [29], see also [30], Giambruno and Zaicev proved that there always exists the limit +exp(A) = lim +n→∞ +n� +cn(A) +and it is a nonnegative integer called the PI-exponent of A. If we use the language of varieties, we say that +the variety generated by the algebra A is the class +V = V(A) = {B associative algebra | T (A) ⊆ T (B)}. +We say that the variety of algebra V is minimal with respect to its exponent if and only if for any proper +subvariety U of V we have that exp(U) < exp(V). We say that a PI-algebra is minimal if it generates a +minimal variety. +If S is any commutative ring with 1, we denote by UTn(S) the ring of upper triangular matrices with +entries in S. Let E be the infinite dimensional Grassmann algebra over F. Drensky [19] proved that the +T -ideals of the algebras UTn(F) and UTn(E) are examples of maximal T -ideals of a given exponent of the +codimension sequences (and the corresponding varieties of algebras are minimal varieties of this exponent). +Some years before Kemer’s works, Genov in [25] and [26] Genov and Latyshev in [36] proved that every +algebra belonging to V(UTn(F)) has a finite basis of its polynomial identities. Latyshev in [37] and Popov +2020 Mathematics Subject Classification. 16R10; 05A15; 05E05; 05E10; 15A75; 16R40; 20C30. +Key words and phrases. Algebras with polynomial identity; block triangular matrices; Grassmann algebra; cocharacter +sequence; multiplicities; multiplicity series. +D.M. Correa was partially supported by CAPES-Brazil. Financial Code 001. +1 + +2 +LUCIO CENTRONE, VESSELIN DRENSKY, AND DANIELA MARTINEZ CORREA +in [42] generalized the previous result for PI-algebras satisfying the polynomial identity +[x1, x2, x3] · · · [x3n−2, x3n−1, x3n] +which generates the T -ideal T (UTn(E)) = T (E)n of the algebra UTn(E). For a long time, until Kemer +developed his structure theory, the results of Genov, Latyshev and Popov covered all known examples of +classes of PI-algebras with the finite basis property. +The T-ideals T (UTn(F)) and T (UTn(E)) have another interesting property established by Volichenko and +Zalesskii in [46]. Let the algebra A satisfy a multilinear polynomial identity f(x1, . . . , xm) which generates +an irreducible Sm-module with character χλ, where χλ is the irreducible Sm-character associated with the +partition λ. Then the Young diagram of λ contains less than n boxes below of the first row if and only if +f(x1, . . . , xm) does not hold for UTn(F). Similarly, the Young diagram of λ contains less than n boxes to the +right of the first column if and only if f(x1, . . . , xm) does not hold for UTn(E). For the Grassmann algebra +this means that the algebra A satisfies a standard identity if and only if T (A) is not contained in T (E). A +proof can be found for example in the book by Giambruno and Zaicev [30, Theorem 7.2.1]. +Let +χn(A) = +� +λ⊢n +mλ(A)χλ, +n ∈ N, +be the cocharacter sequence of A. Let us set Xd := {x1, . . . , xd} and let us consider +Fd(A) := F⟨Xd⟩/(F⟨Xd⟩ ∩ T (A)). +Moreover, if T = {t1, . . . , td} is a set of commutative variables, then the Hilbert series H(Fd(A); Td) of Fd(A) +may be decomposed as +H(Fd(A); Td) = +� +λ +mλ(A)Sλ(Td), +where λ is a partition in no more than d parts and Sλ(Td) is the Schur function associated to λ in the +variables from Td. We shall refer to H(Fd(A); Td) as the Hilbert series of A and we shall write H(A, Td) +instead of H(Fd(A); Td). By a result of Berele and Drensky, (see [4] and [17]), the mλ(A)’s are the same as +in the cocharacter sequence of A. Hence, in principle, the knowledge of the Hilbert series of A will give us +the multiplicities mλ(A) of the cocharacter sequence of A, when λ is a partition in no more than d parts. +So if A is finite dimensional, working with a sufficiently large set of variables will be enough to capture all +the multiplicities. This is no longer true for infinite dimensional algebras. It is also important to recall that +Belov proved in [3] that the Hilbert series of the relatively free algebra of a PI-algebra A in d variables is a +rational function. +The explicit form of the multiplicities in the cocharacter sequence of a PI-algebra is known for few cases. +Among them are the infinite dimensional Grassmann algebra E (Olsson and Regev [41]), the 2 × 2 matrix +algebra M2(F) (Formanek [23] and Drensky [18]), the algebra UT2(F) of 2 × 2 upper triangular matrices +(Mishchenko et al [40], based on the approach of Berele and Regev [10], see also [20]), the tensor square +E ⊗ E of the Grassmann algebra (Popov [43], Carini and Di Vincenzo [13]), the algebra UT2(E) of 2 × 2 +upper triangular matrices with entries from the Grassmann algebra E (Centrone [14]), the algebra UTn(F) of +n × n upper triangular matrices (Boumova and Drensky [12]), the algebra Rp,q(F) of upper block triangular +(p + 2q) × (p + 2q) when p and q are small values (Drensky and Kostadinov [22]). +In [21] Drensky and Genov define the multiplicity series of a PI-algebra A, that is the generating function +of the cocharacter sequence of A which corresponds to the multiplicities mλ(A) when λ is a partition in no +more than d parts. Then, coming back to upper triangular matrices and their central role in PI-theory, in [12] +Boumova and Drensky found an easy algorithm with input the multiplicity series of a symmetric function, +and output the multiplicity series of its Young-derived. Applying it, they found the explicit form of the +multiplicity series of the Hilbert series of UTn(F). Following this line of research, in the first part of the +paper we work with UTn(E) and calculate its multiplicity series in d variables. +Due to the fact that E is infinite dimensional, we need more tools than the ones used by Boumova and +Drensky in order to know all multiplicities of UTn(E). Using the idea of Berele (see [7]), we work with +double Hilbert series instead of with Hilbert series of PI-algebras. Due to the analogue of the result of Berele +and Drensky for double Hilbert series, it suffices to study the decomposition of the double Hilbert series of +UTn(E) in order to achieve the explicit form of the cocharacter sequence of UTn(E). In the second part of the +present paper, we generalize the definition of multiplicity series of a PI-algebra defining a (k, l)-multiplicity +series which controls three sets of disjoint variables, where (k, l) means that the partitions λ = (λ1, . . . , λm) + +COCHARACTERS OF UTn(E) +3 +satisfy the condition λk+1 ≤ l. In other words, their young diagrams Dλ are in a hook of height k of the +arm and wide l of the leg. By a result of Amitsur and Regev [2] all nonzero multiplicities mλ(A) for a +PI-algebra A are concentrated for Young diagrams in a sufficiently large hook. Hence the information about +the multiplicities of A is contained in the related with the hook (k, l)-multiplicity series. +Then we compute the double Hilbert series of E and, as a consequence, we build up an algorithm with +output the (k, l)-multiplicity series of UTn(E). In the spirit of [14] we compute the (2, 3)-multiplicity series +of UT2(E), which contains all multiplicities of the cocharacter sequence of UT2(E) and finally we compute +the (1, 1)-multiplicity series of UT3(E). +2. Preliminaries +2.1. Symmetric functions. We fix a positive integer d and consider the algebra +C[[Td]] = C[[t1, . . . , td]] +of formal power series in d commutative variables. Let C[[Td]]Sd ⊆ C[[Td]] be the subalgebra of symmetric +functions. Every symmetric function g(Td) can be represented in the form +g(Td) = +� +λ +mλSλ(Td), mλ ∈ C, λ = (λ1, . . . , λd), +where Sλ(Td) is the Schur function related to the partition λ which has at most d parts. For details on the +theory of Schur functions see [39]. +There are several ways to define Schur functions. The most convenient for our purpose is to define them +as fractions of Vandermonde-type determinants: +Sλ(Td) = V (λ + δ, Td) +V (δ, Td) +, +where δ = (d − 1, . . . , 2, 1) and for µ = (µ1, . . . , µd) +V (µ, Td) = +����������� +tµ1 +1 +tµ1 +2 +. . . +tµ1 +m−1 +tµ1 +m +tµ2 +1 +tµ2 +2 +. . . +tµ2 +m−1 +tµ2 +m +... +... +... +... +... +tµm−1 +1 +tµm−1 +2 +. . . +tµm−1 +m−1 +tµm−1 +m +tµm +1 +tµm +2 +. . . +tµm +m−1 +tµm +m +����������� +. +Let λ = (λ1, . . . , λd) be a partition of a natural number. The Young diagram Dλ associated to λ is the subset +of Z × Z defined as Dλ = {(i, j) | i = 1, . . . , d, j = 1, . . . , λi}. Graphically we draw the diagrams replacing +the knots by square boxes, adopting the convention, as with matrices, that the first coordinate i (the row +index) increases as one goes downwards, and the second coordinate j (the column index) increases as one +goes from left to right. The first boxes from the left of each row are one above another and the i-th row +contains λi boxes. We denote by λ′ +j the length of the j-th column of Dλ. The partition λ′ = (λ′ +1, . . . , λ′ +m) +and its diagram Dλ′ are called conjugate respectively to λ and Dλ. +For the partition λ, we define a λ-tableau Tλ of content α = α(Tλ) = (α1, . . . , αd) if each integer i = 1, . . . , d +appears in the tableau exactly αi times. Recall that the λ-tableau Tλ is semistandard if its entries do not +decrease in rows reading from left to right, and increase strictly in columns reading from top to bottom. +Another definition of Schur functions is given in terms of semistandard Young tableaux: +Sλ(Td) = +� +Tα(Tλ) +d +, +where the summations runs on all semistandard λ-tableaux. +We recall the definition of elementary symmetric polynomials. Given 0 ≤ m ≤ d, the m-th elementary +symmetric polynomial in d variables t1, . . . , td is defined by +em(Td) = +� +1≤i1<···pi+1 +b(p1, . . . , pd)tp1 +1 · · · tpd +d + +COCHARACTERS OF UTn(E) +5 +where the summations in the latter equation runs on all p = (p1, . . . , pd) such that p1 > p2 · · · > pd. +In the general case, it is difficult to find M(g; Td) even if we know g(Td). But it is very easy to check +whether the formal power series +h(Td) = +� +h(q1, . . . , qd)tq1 +1 · · · tqd +d , +q1 ≥ · · · ≥ qd, +is equal to the multiplicity series M(f; Td) of f(Td) because h(Td) = M(f; Td) if and only if +f(Td) +� +i 0}, +E(1) := spanF {ei1 · · · ei2k+1 | 1 ≤ i1 < · · · < i2k+1 k ≥ 0}. +It is easily checked that E(0)E(0)+E(1)E(1) ⊆ E(0) and E(0)E(1)+E(1)E(0) ⊆ E(1). Hence the decomposition +E = E(0) ⊕ E(1) is a Z2-grading of E. Notice that E(0) coincides with the center of E. +The next fact is well known. +Proposition 2.5. The Grassmann algebra E satisfies the polynomial identity +[[x1, x2], x3], +where [·, ·] is the Lie commutator, i.e. [w, y] := wy − yw for any w, y ∈ F⟨X⟩. +The triple commutator is the only generator of T (E) when the field is infinite of characteristic different +from 2. This fact in characteristic zero was proved by Krakowski and Regev in [33]. It follows also from the +results of Latyshev in [34] and [35] (but not stated explicitly there). In positive characteristic, the reader can +find the proof in [27]. +Theorem 2.6. The T -ideal of E is generated by the polynomial +[x1, x2, x3]. + +COCHARACTERS OF UTn(E) +7 +The following theorem of Olsson and Regev gives the cocharacter sequence of E. +Theorem 2.7 (Olsson and Regev [41]). Let E be the infinite dimensional Grassmann algebra over a field of +characteristic zero. Then the cocharacter sequence of E for any n ≥ 1 is given by +χn(E) = +n +� +p=1 +χ(p,1n−p). +The following result gives us an expression for the Hilbert series of Fd(E). The proof of this can be found +in [20]. +Proposition 2.8. Let E the infinite dimensional Grassmann algebra over a field of characteristic zero. The +Hilbert series of Fd(E) in d variables is given by +H(E; Td) = 1 +2 + 1 +2 +d +� +i=1 +1 + ti +1 − ti +. +The next theorem talks about the polynomial identities of the F-algebra UTn(E) of n × n upper trian- +gular matrices with entries in the Grassmann algebra E. See also [15] for the case of UT2(E) in positive +characteristic. +Theorem 2.9 (Abakarov [1]). The T -ideal of UTn(E) is generated by the polynomial +[x1, x2, x3] · · · [x3n−2, x3n−1, x3n]. +3. The operator �Y +In this section, we shall talk about the tools used in the development of an algorithm to calculate the +multiplicities in the cocharacter sequence of the algebra UTn(E) of n × n upper triangular matrices with +entries in the Grassmann algebra E over a field F of characteristic zero. +We follow the ideas developed in [12]. We start studying the action of two basic operators (Y and �Y in +the text) in the language of multiplicity series. Then, we compute the Hilbert series of UTn(E) and find an +expression for its multiplicity series. We shall give a description of the partitions λ such that the multiplicities +mλ are nonzero in the cocharacter sequence of UTn(E). Finally, we compute the multiplicity mλ for UTn(E), +where 1 ≤ n ≤ 3 and λ is a partition in no more than 2 parts. +Definition 3.1. Let Y be the linear operator in C[[Vd]] which sends the multiplicity series of a symmetric +function to the multiplicity series of its Young-derived. That is, if g(Td) is a symmetric function, then +Y (M(g); Td) = M +�� d +� +i=1 +1 +(1 − ti) +� +g(Td); Td +� +. +The operator Y is called the Young-derived operator. +Definition 3.2. If g(Td) ∈ C[[Td]]Sd, then we define the linear operator �Y in C[[Vd]] ⊆ C[[Td]] as +�Y (M(g); Td) := M +� +g(Td) +� +1 +2 +d +� +i=1 +(1 − ti) + 1 +2 +d +� +i=1 +(1 + ti) +� +; Td +� +. +The following proposition is well known. It describes the multiple action of Y on 1. A proof can be found +in [12]. +Proposition 3.3. For d ≥ k ≥ 1 the following decomposition holds +d +� +i=1 +1 +(1 − ti)k = +� +µ +ηµSµ(Td), +where the summation is on all partitions µ = (µ1, . . . , µk) and +ηµ = Sµ (1, . . . , 1) +� +�� +� +k times += dim Wk(µ). +Equivalently, for k ≥ 1 + +8 +LUCIO CENTRONE, VESSELIN DRENSKY, AND DANIELA MARTINEZ CORREA +In the general case there is an easy formula which translates the action of Y on g(Td) in the language of +its multiplicity series. +Proposition 3.4 (Drensky and Genov [21]). Let g(Td) ∈ C[[Td]]Sd. Then +Y (M(g; Td)) = +d +� +i=1 +1 +1 − ti +� +(−t2)ε2 · · · (−td)εdM(g; t1tε2 +2 , t1−ε2 +2 +tε3 +3 , . . . , t1−εd−1 +d−1 +tεd +d , t1−εd +d +), +where the summation runs on all ε2, . . . , εd ∈ {0, 1}. +Consider the operator �Y and notice that +�Y j(M(1); Td) = +� +1 +2 +d +� +i=1 +(1 − ti) + 1 +2 +d +� +i=1 +(1 + ti) +�j +. +Now, we want to know which Schur functions participate in the decomposition of �Y j(M(1); Td), that is we +want to express �Y j(M(1); Td) as a linear combination of Schur functions. The next is a direct consequence +of the Young rule. +Proposition 3.5. Let j ≥ 1 and +� +1 +2 +d +� +i=1 +(1 − ti) + 1 +2 +d +� +i=1 +(1 + ti) +�j += +� +ρ +mρSρ(Td). +If mρ ̸= 0, then ρ = (js1, (j − 1)s2, . . . , 1sj) ⊢ 2(n1 + · · · + nj) for some n1, . . . , nj integers and (s1, . . . , sj) ∈ +Zj +≥0. Equivalently, if mρ ̸= 0, then ρ has at most j columns. +The following lemma allows us to describe M ′(f(Td)S(12)(Td); Vd) in terms of the multiplicity series +M ′(f(Td); Vd), where f(Td) is a symmetric function. +Lemma 3.6. Let f(Td) ∈ C[[Td]]Sd. Then +M ′(f(Td)S(12)(Td); Vd) =v2M ′(f(Td; Vd) + v1 +d−1 +� +j=2 +vj+1 +vj +gj((M ′(f(Td); v1, . . . , vd)) ++ +d−2 +� +i=1 +vi+2 +vi +gi((M ′(f(Td); v1, . . . , vd)) ++ +� +1≤i,j≤d−1 +i+1 µj+1 and j ≥ 2; +• λ = (µ1, . . . , µi, µi+1 + 1, µi+2 + 1, . . . , µd), if µi > µi+1; +• λ = (µ1 + 1, . . . , µi, µi+1 + 1, . . . , µj, µj+1 + 1, . . . , µd), if µi > µi+1, µj > µj+1 and i + 1 < j. + +COCHARACTERS OF UTn(E) +9 +In the language of multiplicity series, this means that M ′(Sµ(Td)S(1,1)(Td)) is a linear combination of the +following terms +• v2(vp1 +1 · · · vpd +d ) = v2M ′(Sµ(Td; Vd)); +• v1 +vj+1 +vj +M ′(Sµ(Td; Vd)), if µj > µj+1 and j ≥ 2; +• vi+2 +vi +M ′(Sµ(Td, Vd)), if µi > µi+1; +• vi+1 +vi +vj+1 +vj +M ′(Sµ(Td, Vd)), if µi > µi+1, µj > µj+1 and i + 1 < j. +Now, observe that +gj((M ′(f(Td); Vd)) = + + + +M ′(f(Td); Vd), +if +pj > 0, +0, +if +pj = 0; +gij((M ′(f(Td); Vd)) = + + + +M ′(f(Td); Vd), +if +pi > 0, pj > 0, +0, +for +all other cases. +Then the result follows easily. +□ +By Lemma 3.6, we have the following corollary. +Corollary 3.7. If f(T2) ∈ C[[T2]]S2, then +�Y (M(f; T2)) = (1 + t1t2)M(f; T2) = +� +1 +2 +2 +� +i=1 +(1 − ti) + 1 +2 +2 +� +i=1 +(1 + ti)] +� +M(f; T2). +Note that if we want to describe M ′(Sλ(Td)S(1k)(Td); Vd) in terms of the multiplicity series M ′(Sλ(Td) +where λ is a partition in no more than d parts and k ≤ d a positive integer, we need expressions obtained +from M ′(Sλ((T )d); Vd) making vj1 = vj2 = · · · = vjr = 0 where 1 ≤ r ≤ s and j1 ≤ · · · ≤ jr. +4. Hilbert series and multiplicity series of UTn(E) +In this section we give an algorithm to calculate the multiplicities in the cocharacter sequence of UTn(E). +We follow the ideas developed in [12]. +The next result follows from Corollary 2.3, Theorem 2.9 and Proposition 2.8. +Theorem 4.1. The Hilbert series H(UTn(E); Td) of the algebra Fd(UTn(E)) is +H(UTn(E)); Td) += +n +� +j=1 +�n +j +�� +1 +2 + 1 +2 +d +� +i=1 +1 + ti +1 − ti +�j +(t1 + · · · td − 1)j−1. +From Definitions 3.1 and 3.2 we derive +Y (�Y (M(g); Td)) += +Y +� +M +� +g(Td) +� +1 +2 +d +� +i=1 +(1 − ti) + 1 +2 +d +� +i=1 +(1 + ti) +�� +; Td +� += +M +� d +� +i=1 +1 +1 − ti +g(Td) +� +1 +2 +d +� +i=1 +(1 − ti) + 1 +2 +d +� +i=1 +(1 + ti) +� +; Td +� += +M +� +g(Td) +� +1 +2 + 1 +2 +d +� +i=1 +1 + ti +1 − ti +� +; Td +� +. +Notice that Y (�Y (M(g); Td)) is well-defined because g(Td) and +� +1 +2 + 1 +2 +d +� +i=1 +1 + ti +1 − ti +� +are symmetric functions. +Hence, g(Td) +� +1 +2 + 1 +2 +d +� +i=1 +1 + ti +1 − ti +� +is symmetric. Moreover we have Y ◦ �Y = �Y ◦ Y too. +Using the previous result, we can obtain an expression for the multiplicity series of UTn(E). This is the +content of the following corollary. + +10 +LUCIO CENTRONE, VESSELIN DRENSKY, AND DANIELA MARTINEZ CORREA +Corollary 4.2. The multiplicity series of UTn(E) is +M(UTn(E); Td)) = +n +� +j=1 +j−1 +� +q=0 +� +λ⊢q +(−1)j−1−q +�n +j +��j − 1 +q +� +dλZj(Tλ +d), +where dλ is the degree of the irreducible Sn-character χλ, Td = tλ1 +1 · · · tλd +d +and Z = Y ◦ �Y . +Proof. Notice that +(t1 + · · · + td − 1)j−1 = +j−1 +� +q=0 +(−1)j−1−q +�j − 1 +q +� +(t1 + · · · + td)q +and expanding the expression of H(Uk(E); Td) from Proposition 4.1, we get +H(UTn(E); Td) += +n +� +j=1 +�n +j +�� +1 +2 + 1 +2 +d +� +i=1 +1 + ti +1 − ti +�j j−1 +� +q=0 +(−1)j−1−q +�j − 1 +q +� +(t1 + · · · + td)q. +Using the well-known equality +(t1 + · · · + td)q = Sq +(1)(Td) = +� +λ⊢q +dλSλ(Td), +where dλ is the degree of the irreducible Sq-character χλ, we have +H(UTn(E); Td) += +n +� +j=1 +�n +j +�� +1 +2 + 1 +2 +d +� +i=1 +1 + ti +1 − ti +�j j−1 +� +q=0 +(−1)j−1−q +�j − 1 +q +� � +λ⊢q +dλSλ(Td) += +n +� +j=1 +j−1 +� +q=0 +� +λ⊢q +(−1)j−1−q +�n +j +��j − 1 +q +� +dλ +� +1 +2 + 1 +2 +d +� +i=1 +1 + ti +1 − ti +�j +Sλ(Td). +Indeed, the multiplicity series of Sλ(Td) is +M(Sλ(Td); Td) = tλ1 +1 · · · tλd +d += Tλ +d. +Then +M + + +� +1 +2 + 1 +2 +d +� +i=1 +1 + ti +1 − ti +�j +Sλ(Td); Td + + = Zj(M(Sλ(Td); Td)) = Zj(Tλ +d). +Hence, the multiplicity series of UTn(E) is +M(H(UTn(E); Td)) = +n +� +j=1 +j−1 +� +q=0 +� +λ⊢q +(−1)j−1−q +�n +j +��j − 1 +q +� +dλZj(Tλ +d). +□ +We want to describe those partitions λ such that mλ(Uk(E)) ̸= 0. Hence we get a sharpening result on +the height of λ. +Theorem 4.3. If mλ(UTn(E)) ̸= 0 and λ = (λ1, . . . , λd), then λn+1 ≤ 2n − 1. +Proof. By Theorem 4.1 and in the spirit of Corollary 4.2 the nonzero multiplicities mλ(UTn(E)) in the +cocharacter sequence of UTn(E) come from the decomposition +� +1 +2 + 1 +2 +d +� +i=1 +1 + ti +1 − ti +�j +(t1 + · · · + td)q = +� d +� +i=1 +1 +1 − ti +�j� +1 +2 +d +� +i=1 +(1 − ti) + 1 +2 +d +� +i=1 +(1 + ti) +�j +S(1)(Td)q, +j ≤ n and q ≤ n − 1, as linear combination of Schur functions. +By Proposition 3.5 the Schur functions Sπ(Td) participating in the product +� +1 +2 +d +� +i=1 +(1 − ti) + 1 +2 +d +� +i=1 +(1 + ti) +�j + +COCHARACTERS OF UTn(E) +11 +are indexed by partitions π having at most j ≤ n columns. By the Branching rule, the multiplication of +Sπ(Td) by S(1)(Td) is a linear combination of Sρ(Td) where the diagrams of ρ are obtained from the diagrams +of π by adding a box. Multiplying q times by S(1)(Td) we add to the diagram of π no more than q ≤ n − 1 +boxes in the first row. +� +1 +2 +d +� +i=1 +(1 − ti) + 1 +2 +d +� +i=1 +(1 + ti) +�j +S(1)(Td)q, +have at most j + q ≤ 2n − 1 boxes in the first row. +Due to the fact that +� d +� +i=1 +1 +1 − ti +�j += +� +ni≥0 +S(m1)(Td) · · · S(mj)(Td), +applying Young rule iteratively, it follows that if the partition λ appears in the decomposition of +� d +� +i=1 +1 +1 − ti +�j +Sρ(Td) +then λ is of the type (m1, . . . , mj, (j + q)s1, . . . , 1sj+q) with m1, . . . , mj, s1, . . . , sj+q nonnegative. +Hence +λj+1 ≤ j + q. +Therefore, if mλ(UTn(E)) ̸= 0, then λk+1 ≤ 2n − 1. +□ +It is worth mentioning that results about the cocharacters sequence of E and U2(E) satisfy the bound +given in the previous theorem. Olsson and Regev proved in [41] that +χn(E) = +n−1 +� +k=0 +χ(n−k,1k). +Note that the diagrams of the partition λ = (n − k, 1k) have at most one box in the second column, that is, +λ2 ≤ 1. Therefore, it agrees with Theorem 4.3. +In [14] Centrone proved that +H(UT2(E); Td) = +� +mλ(U2(E))Sλ(Td), +where λ = (m1, m2, 3, 2m, 1l) or λ = (m1, m2, 2m, 1l). Hence the diagrams of partition λ have at most 3 +boxes in the third row, that is, λ3 ≤ 3. +5. Some applications of the multiplicity series of UTn(E) +In this section, we shall compute the multiplicity series of UTn(E) in two variables for n ∈ {1, 2, 3}. As a +consequence, we get the multiplicities mλ in the cocharacter sequences of UTn(E) where 1 ≤ n ≤ 3 and λ is +a partition in no more than 2 parts. +Proposition 5.1. Consider the algebras E and UT2(E) then +(i) the multiplicity series of E in two the variables is +M ′(E; V2) += +1 + v2 +1 − v1 +. +(1) +(ii) the multiplicity series of UT2(E) in two variables is +M ′(UT2(E); V2) += +2(1 + v2) +1 − v1 ++ (1 + v2)2(−1 + v1 + 2v2 − v1v2) +(1 − v1)2(1 − v2) +. +(2) +Proof. (i) This case follows immediately from the result of Olsson and Regev [41] because +mλ = +� +1 for λ = (k, 1n−k), +0 otherwise +and +M(E; T2) = +� +n≥0 +tn +1 + +� +n≥2 +tn−1 +1 +t2. + +12 +LUCIO CENTRONE, VESSELIN DRENSKY, AND DANIELA MARTINEZ CORREA +We shall restate this proof in terms of Corollary 4.2, +By Corollary 4.2, we get +M(E; T2) += +Z(1; T2). +Since Z = Y ◦ �Y , applying Proposition 3.4 and Corollary 3.7 we have +M(E; T2) = 1 + t1t2 +1 − t1 +. +Recall that v1 = t1 and v2 = t1t2. Hence +M ′(E; V2) += +1 + v2 +1 − v1 +. +(ii) By Corollary 4.2, we have +M(UT2(E); T2) += +2Z(1) − Z2(1) + Z2(t1). +Now, we shall compute 2Z(1), Z2(1) and Z2(t1) using Proposition 3.4 and Corollary 3.7 +• 2Z(1) = 2(1 + t1t2) +1 − t1 +; +• Z2(1) = +(1 + t1t2)2 +(1 − t1)2(1 − t1t2); +• Z2(t1) = (1 + t1t2)2(t1 + 2t1t2 − t2 +1t2) +(1 − t1)2(1 − t1t2) +. +Hence +M(UT2(E); T2) += +2(1 + t1t2) +1 − t1 +− +(1 + t1t2)2 +(1 − t1)2(1 − t1t2) + (1 + t1t2)2(t1 + 2t1t2 − t2 +1t2) +(1 − t1)2(1 − t1t2) += +2(1 + t1t2) +1 − t1 ++ (1 + t1t2)2(−1 + t1 + 2t1t2 − t2 +1t2) +(1 − t1)2(1 − t1t2) +. +Finally, we have +M ′(UT2(E); V2) += +2(1 + v2) +1 − v1 ++ (1 + v2)2(−1 + v1 + 2v2 − v1v2) +(1 − v1)2(1 − v2) +and we are done. +□ +Now we are able to compute the multiplicities mλ in the cocharacter sequences of E and UT2(E) when λ +is a partition in no more than 2 parts. The goal of this result is to show how to find out the multiplicities +using Corollary 4.2 and Proposition 5.1. +Corollary 5.2. Let λ be a partition in no more than 2 parts. Then: +(i) the multiplicity mλ in the cocharacter sequences of E is given by +mλ = + + + + + +1 if λ = (n), +1 if λ = (λ1, 1), λ1 ≥ 1, +0 for all other λ. +(ii) the multiplicity mλ in the cocharacter sequence of UT2(E) is given by +mλ = + + + + + + + + + + + + + + + + + + + +1 +if +λ = (n), +λ1 +if +λ = (λ1, 1), λ1 ≥ 1, +3λ1 − 4 +if +λ = (λ1, 2), λ1 ≥ 2, +4(λ1 − λ2 + 1) +if +λ = (λ1, λ2), λ1 ≥ λ2 ≥ 3. + +COCHARACTERS OF UTn(E) +13 +Proof. (i) By Proposition 5.1, we get +M ′(E; V2) = +� +n≥0 +vn +1 + +� +n≥0 +vn +1 v2. +From the first summand of the above equality, we have that if λ = (n) with n ≥ 0, then mλ = 1. Observe +that vn +1 v2 with n ≥ 0 corresponds to the partition λ = (n + 1, 1). It follows that if λ = (λ1, 1), where λ1 ≥ 1, +then mλ = 1. +(ii) By Proposition 5.1, it follows that +M ′(UT2(E); V2) += +� +n≥0 +2vn +1 + +� +n≥0 +2vn +1 v2 − +� +m,n≥0 +(n + 1)vn +1 vm +2 − +� +n≥0,m≥1 +2(n + 1)vn +1 vm +2 +− +� +m≥2,n≥0 +(n + 1)vn +1 vm +2 + +� +m≥1,n≥0 +2(n + 1)vn +1 vm +2 ++ +� +m≥2,n≥0 +4(n + 1)vn +1 vm +2 + +� +m≥3,n≥0 +2(n + 1)vn +1 vm +2 ++ +� +n≥1 +nvn +1 + +� +n≥1 +2nvn +1 v2 + +� +n≥1 +nvn +1 v2 +2. +Therefore +M ′(UT2(E); V2) = +� +n≥0 +vn +1 + +� +n≥0 +(n + 1)vn +1 v2 + +� +n≥0 +(3n + 2)vn +1 v2 +2 ++ +� +n≥0,m≥3 +4(n + 1)vn +1 vm +2 . +First, consider the first summand of the above equality and observe that vn +1 corresponds to the partition +λ = (n). So, if λ = (n), then mλ = 1. +Now, notice that there is a one-to-one correspondence between monomials vn +1 v2 with n ≥ 0 and partitions +λ = (n + 1, 1). Hence the equality gives that if λ = (n, 1) where n ≥ 1 then mλ = n. +Finally, we have that vn +1 vm +2 +corresponds to the partition λ = (n + m, m). +Then mλ = 4(n + 1) = +4((n + m) − m + 1) and it follows that if λ = (λ1, λ2) with λ1 ≥ λ2 ≥ 3, then mλ = 4(λ1 − λ2 + 1). +The case n ≥ 0, m ≥ 3 is treated similarly. +□ +We highlight that Corollary 5.2 agrees with the results presented in [41] and [14] when the partitions have +no more than two parts. +Now, we shall compute the multiplicity series of UT3(E) in two variables. +Theorem 5.3. +(i) The multiplicity series of UT3(E) in two variables is +M ′(UT3(E); V2) =3(1 + v2) +1 − v1 +− +3(1 + v2)2 +(1 − v1)2(1 − v2) + 3 +� +2(1 + v2)2 +(1 − v1)2(1 − v2) + v1(1 + v2)2 +(1 − v1)2 +� ++ +1 +(1 − v1)3(1 − v2)3 +� +1 − 2v1 + v2 +1 − 2v2 + 2v1v2 − 4v2 +2 + 8v1v2 +2 +− 3v2 +1v2 +2 + 7v3 +2 − 5v1v3 +2 + 10v4 +2 − 13v1v4 +2 + 3v2 +1v4 +2 − v5 +2 − v1v5 +2 +−3v6 +2 + 3v1v6 +2 − v2 +1v6 +2 +� +. +(3) + +14 +LUCIO CENTRONE, VESSELIN DRENSKY, AND DANIELA MARTINEZ CORREA +(ii) Let λ be a partition in not more than 2 parts. Then the multiplicities mλ in the cocharacter sequence +of UT3(E) are given by +mλ = + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +1 +if +λ = (n), +λ1 +if +λ = (λ1, 1), λ1 ≥ 1, +1 +2(λ1 + 2)(λ1 − 1) +if +λ = (λ1, 2), λ1 ≥ 2, +1 +2(16 − 17λ1 + 5λ2 +1) +if +λ = (λ1, 3), λ1 ≥ 3, +14 − 16λ2 + 4λ2 +2 + 2(λ1 − λ2)(2 − 5(λ1 − λ2)) ++4λ2(λ1 − λ2)(−3 + λ2 + (λ1 − λ2)) +if +λ = (λ1, λ2), λ1 ≥ λ2 ≥ 4. +Proof. (i) By Corollary 4.2, we have +M(UT3(E); T2) = 3Z(1) − 3Z2(1) + 3Z2(t1) + Z3(1) − 2Z3(t1) + Z3(t2 +1) + Z3(t1t2). +Due to Corollary 3.7 and Proposition 3.4, we get +3Z(1) − 3Z2(1) + 3Z2(t1) =3(1 + v2) +1 − v1 +− +3(1 + v2)2 +(1 − v1)2(1 − v2) ++ 3 +� +2(1 + v2)2 +(1 − v1)2(1 − v2) + v1(1 + v2)2 +(1 − v1)2 +� +, +(4) +Z3(1) − 2Z3(t1) + Z3(t2 +1) + Z3(t1t2) = +1 +(1 − v1)3(1 − v2)3 +� +1 − 2v1 + v2 +1 − 2v2 + 2v1v2 +− 4v2 +2 + 8v1v2 +2 − 3v2 +1v2 +2 + 7v3 +2 − 5v1v3 +2 + 10v4 +2 +−13v1v4 +2 + 3v2 +1v4 +2 − v5 +2 − v1v5 +2 − 3v6 +2 + 3v1v6 +2 − v2 +1v6 +2 +� +. +(5) +Now, the result follows. +(ii) We have expanded the expression of M ′(UT3(E); V2) given in the part (i) into a power series using +the following well known equalities: +va1 +1 va2 +2 +1 − v1 += +� +n≥a1 +vn +1 va2 +2 , +va1 +1 va2 +2 +(1 − v1)2 = +� +n≥a1 +(n − a1 + 1)vn +1 va2 +2 , +va1 +1 va2 +2 +(1 − v1)2(1 − v2) = +� +n≥a1 +� +m≥a2 +(n − a1 + 1)vn +1 vm +2 , +va1 +1 va2 +2 +(1 − v1)3(1 − v2)3 = +� +n≥a1 +� +m≥a2 +�n − a1 + 2 +2 +��m − a2 + 2 +2 +� +vn +1 vm +2 . +Easy manipulations give the explicit expression for mλ where λ is a partition in no more than two parts. In +particular, if we want to compute the multiplicity of λ = (λ1, 1), we need to study the terms of type vn +1 v2 in +M ′(UT3; V2). Hence we shall study the following expression +3v2 +1 − v1 +− +� +3 +(1 − v1)2(1 − v2) + +6v2 +(1 − v1)2(1 − v2) +� ++ +6v1v2 +(1 − v1)2 ++ +6v2 +(1 − v1)2(1 − v2) + +1 +(1 − v1)3(1 − v2)3 +� +1 − 2v1 + v2 +1 − 2v2 + 2v1v2 +� +. + +COCHARACTERS OF UTn(E) +15 +Notice that if n ≥ 2 and m = 1, then λ = (n + 1, 1). By the last expression, we get +mλ =3 − (3(n + 1) + 6(n + 1)) + 6n + 6(n + 1) + 6(n + 1)(n + 2) +4 +− 6n(n + 1) +2 ++ 6(n − 1)n +4 +− 2(n + 2)(n + 1) +2 ++ 2n(n + 1) +2 +=n + 1. +Equivalently, if λ = (n, 1), with n ≥ 3, then mλ = n. Observe that, if m = 0 and n = 1, then λ = (1, 1) and +mλ = 1. Finally, if n = 1 and m = 1, then λ = (2, 1). By the last expression, mλ = 2. +We conclude that, if λ = (λ1, 1), with λ1 ≥ 1, then mλ = λ1. The other cases are treated similarly and +the proof follows. +□ +6. Double Hilbert series and hook-Schur functions +Notice that the multiplicity series of UTn(E) given in Corollary 4.2 gives us only the multiplicities mλ +when λ has no more than d parts. Our next goal is to find an algorithm that allows us to calculate the +multiplicities in the cocharacter sequence of UTn(E) having more “freedom” in the partition λ, and to find +mλ without any restriction of the height of λ. For this purpose we shall introduce some new concepts. +Consider the infinite dimensional Grassmann algebra E, as pointed out above. Then +B = {1, ei1 · · · eim | i1 < · · · < im, m = 1, 2, . . .} +is a basis of E. We recall the action ∗ of Sn introduced by Olsson and Regev [41] and used by Berele and +Regev in [9]. Given 1 ̸= a = ei1 · · · eim ∈ B, we write l(a) = m. Let (a) = (a1, . . . , an), where a1, . . . , an ∈ B, +and define +I = Odd(a) = {i | l(ai) ≡ 1 (mod 2)}. +Remark 6.1. Let I ⊆ {1, . . ., n} (I is possibly empty), σ ∈ Sn. Choose any (a) = (a1, . . . , an), ai ∈ B, such +that a1 · · · an ̸= 1 and Odd(a) = I. Then in E +aσ(1) · · · aσ(n) = ±a1 · · · an +Note that the sign ± depends on I and σ but does not depend on the concrete choice of a. +Definition 6.2. Let I ⊆ {1, . . . , n} (I is possibly empty), σ ∈ Sn. Choose any n-tuple (a) = (a1, . . . , an), +ai ∈ B, such that a1 · · · an ̸= 0 and Odd(a) = I. We define fI(σ) = ±1 by the equality +aσ(1) · · · aσ(n) = fI(σ)a1 · · · an. +Definition 6.3. Fixing two noncommuting sets of variables X = {x1, . . . , xk} and Z = {z1, . . . , zl} and a +vector space V with basis X ∪ Z = {x1, . . . , xk, z1, . . . , zl}, the tensors v1 ⊗ · · · ⊗ vn, vi ∈ X ∪ Z, form a basis +of V ⊗n. Given such (v) = v1 ⊗ · · · ⊗ vn, we define the Z-indices of (v) by IZ(v) = {i | vi ∈ Z}. Let σ ∈ Sn +and let us define the right action ∗ of σ by +(v1 ⊗ · · · ⊗ vn) ∗ σ = fIZ(v)(σ)vσ(1) ⊗ · · · vσ(n). +Finally, extend the action ∗ of σ to the whole V ⊗n by linearity. As usual, we may identify the vector space +Pn of multilinear polynomials of degree n with the group algebra FSn and define an action ∗ of Pn on V ⊗n. +Let us consider now the double Hilbert series (or the double Poincar´e series) related to the polynomial +identities of a PI-algebra A. In the above setup we identify the tensor algebra of V by TV and the free algebra +F⟨X, Z⟩ = F⟨x1, . . . , xk, z1, . . . , zl⟩. +The latter algebra is a free superalgebra assuming, as usual, that x1, . . . , xk and z1, . . . , zl are, respectively, +the even and the odd free generators. Let +⟨a; b⟩ = ⟨a1, . . . , ak; b1, . . . , bl⟩, +where a1 + · · · + ak + b1 + · · · + bl = n, and let V ⟨a; b⟩ ⊆ V ⊗n being the subspace of all polynomials which +are homogeneous in each of x1, . . . , xk, z1, . . . , zl of degree ai in yi and bj in zj. + +16 +LUCIO CENTRONE, VESSELIN DRENSKY, AND DANIELA MARTINEZ CORREA +Definition 6.4. Let A be a PI-algebra and consider the following set of commutative variables Tk = +{t1, . . . , tk}, Yl = {y1, . . . , yl}. The double Hilbert series of A is defined to be +H(A; Tk, Yl) = H(A; t1, . . . , tk; y1, . . . , yl) := +� +⟨a;b⟩ +dimF (V ⟨a; b⟩/V ⟨a; b⟩ ∗ Qn)ta1 +1 · · · tak +k yb1 +1 · · · ybl +l , +where Qn = T (A) ∩ Pn. +Note that the variables t’s and y’s count, respectively, the degrees of the x’s and z’s. +There is another way to define double Hilbert series which is an exact analog of the definition of Hilbert +series of relatively free algebras. We recall that if A is a PI-algebra, then AM := A ⊗F E inherits the super- +algebra structure from the natural Z2-grading of E, i.e, A(0) = A ⊗ E(0) and A(1) = A ⊗ E(1). +If T2(AM) ⊆ F⟨x1, x2, . . . , z1, z2, . . .⟩ is the T2-ideal of the Z2-graded polynomial identities of AM, then +the relatively free Z2-graded algebra +F⟨x1, . . . , xk, z1, . . . , zl⟩/(T2(AM) ∩ F⟨x1, . . . , xk, z1, . . . , zl⟩) +is called the magnum of A. For more details on the magnum of a PI-algebra see [5]. The following result is +well known (see [5]) and gives that the double Hilbert series related to the PI-algebra A coincides with the +Hilbert series of the magnum of A. +Proposition 6.5. Let A be a PI-algebra. If ⟨a; b⟩ = ⟨a1, . . . , ak; b1, . . . , bl⟩ be such that a1 + · · · + ak + b1 + +· · · + bl = n, then V ⟨a; b⟩ ∗ Qn = V ⟨a; b⟩ ∩ T2(AM). +Now, we are going to talk about hook Schur functions and its relations with the double Hilbert series of +a PI-algebra. We shall begin with a definition that generalizes the concept of a semistandard tableau. +Definition 6.6. Fix integers k, l ≥ 0 such that k + l > 0 and k + l variables t1, . . . , tk, y1, . . . yl, so that +t1 < · · · < tk < y1 < · · · yl. Let λ be a partition with Young diagram Dλ. Fill Dλ with elements from +{t1, . . . , tk, y1, . . . yl}, allowing repetitions, to get a (k, l)-tableau Tλ. Such Tλ is said to be (k, l)-semistandard +if +a) The “t part” (i.e., the cells filled with ti’s) of Tλ is a tableau. (Thus the “y part” is a skew tableau); +b) The ti’s are nondecreasing in rows, strictly increasing in columns; +c) The yj’s are nondecreasing in columns, strictly increasing in rows. +Definition 6.7. For a (k, l)-semistandard tableau Tλ we define wTλ = ta1 +1 · · · tak +k yb1 +1 · · · ybl +l , where each ai +counts the number of entries of ti in Tλ and each bj counts the number of entries of yj in Tλ. So the hook +Schur function is defined by +HSλ(Tk, Yl) = HSλ(t1, . . . , tk, y1, . . . , yl) = +� +{wTλ | Tλ is a (k, l)-semistandard}. +Let H(k, l; n) = {λ = (λ1, λ2, · · · ) ⊢ n | λk+1 ≤ l} and +H(k, l) = +� +n≥0 +H(k, l; n). +Note that if λ ∈ H(k, l), then the Young diagram Dλ lies in the hook of width k of the arm and width l of +the leg. It is not hard to see from the definition that HSλ(Tk, Yl) ̸= 0 if and only if λ ∈ H(k, l). +The following theorem of Amitsur and Regev shows that by taking k, l large enough we can capture all +partitions that have nonzero multiplicities in the cocharacter sequence of a PI-algebra. +Theorem 6.8 (Amitsur and Regev [2]). If A is a PI-algebra over a field of characteristic zero, then there exist +k and l such that the cocharacter sequence of A lies in the k by l hook, i.e., if mλ(A) ̸= 0, then λ ∈ H(k, l). +Let A be a PI-algebra. We shall write χ(A) ⊆ H(k, l) when the nonzero multiplicities mλ(A) in the +cocharacter sequence χn(A), n = 0, 1, 2 . . ., appear only for those λ ∈ H(k, l). By Theorems 6 and 11 of [8] +we have the following. + +COCHARACTERS OF UTn(E) +17 +Theorem 6.9 (Berele and Regev [8]). Let A be a PI-algebra with cocharacter sequence +χn(A) = +� +λ⊢n +mλ(A)χλ, n ≥ 0. +Then there exist nonnegative integers k and l such that +H(A; Tk, Yl) = +∞ +� +n=0 +� +λ∈H(k,l;n) +mλ(A)HSλ(Tk, Yl). +The following result is a generalization of Theorem 2.2. +Proposition 6.10. Let A1, A2 and A be PI-algebras such that T (A) = T (A1)T (A2). Then +H(A; Tk, Yl) = H(A1; Tk, Yl) + H(A2; Tk, Yl) + (HS(1)(Tk, Yl) − 1)H(A1; Tk, Yl)H(A2; Tk, Yl). +Proof. Berele and Regev proved in [10] that if T (A) = T (A1)T (A2) then +(6) +χn(A) = χn(A1) + χn(A2) + χ(1) �⊗ +n−1 +� +j=0 +χj(A1)�⊗χn−j−1(A2) − +n +� +j=0 +χj(A1)�⊗χn−j(A2) +where �⊗ denotes the “outer” tensor product of characters. Recall that for irreducible characters �⊗ behaves +as in the Littlewood-Richardson rule. +In virtue of Theorem 6.9 we have +H(A; Tk, Yl) = +∞ +� +n=0 +� +λ∈H(k,l;n) +mλ(A)HSλ(Tk, Yl), +H(A1; Tk; Yl) = +∞ +� +n=0 +� +α∈H(k,l;n) +mα(A1)HSα(Tk, Yl), +H(A2; Tk, Yl) = +∞ +� +n=0 +� +β∈H(k,l;n) +mβ(A2)HSβ(Tk, Yl). +Multiplying the hook Schur functions with the Littlewood-Richardson rule (see [9], section 6), the equality +(6) implies +H(A; Tk, Yl) = H(A1; Tk, Yl) + H(A2; Tk, Yl) + (HS(1)(Tk, Yl) − 1)H(A1; Tk, Yl)H(A2; Tk, Yl), +as desired. +□ +Corollary 6.11. Let A and C be PI-algebras such that T (C) = T (A)m. Then +H(C; Tk, Yl) = +m +� +j=1 +�m +j +� +H(A; Tk, Yl)j(S(1)(Tk; Yl) − 1)j−1. +7. The double Hilbert series of UTn(E) +We know that the cocharacter sequence of E lies in the hook H(1, 1). The double Hilbert series H(E; t1, y1) +was computed in [8]. We present once again the computation of H(E; t1, y1) as a direct application of the +definition of hook Schur functions. +We shall also compute H(E; Tk, Tl) for k, l nonnegative integers. Moreover, we shall find an expression +for the double Hilbert series of UTn(E). Finally, using H(UTn(E); Tk, Yl) we shall give a description of the +nonzero multiplicities mλ in the cocharacter sequence of UTn(E). +Proposition 7.1. Let E be the infinite dimensional Grassmann algebra. Then +H(E; t1, y1) = +1 + t1y1 +(1 − t1)(1 − y1). + +18 +LUCIO CENTRONE, VESSELIN DRENSKY, AND DANIELA MARTINEZ CORREA +Proof. By Theorem 2.7 we know that for any n ≥ 1, if λ = (p, 1n − p), we have mλ(E) = 1. +In light +of Theorem 6.9, we have to compute HSλ(t1, y1) to determine H(E; t1, y1). +Note that the only (1, 1)- +semistandard tableaux of shape λ are +t1 +t1 +y1 +y1 +t1 +t1 y1 +y1 +y1 +corresponding to the monomials tp +1yn−p +1 +and tp−1 +1 +yn−p+1 +1 +, respectively. Hence +H(E; t1, y1) = 1 + +∞ +� +n=1 +n +� +p=1 +(tp +1yn−p +1 ++ tp−1 +1 +yn−p+1 +1 +). +Note that +1 + +∞ +� +n=1 +n +� +p=1 +(tp +1yn−p +1 ++ tp−1 +1 +yn−p+1 +1 +) += +1 + (t1 + y1) +∞ +� +n=1 +n +� +p=1 +tp−1 +1 +yn−p +1 += +1 + (t1 + y1) +∞ +� +k=0 +� +n+p=k +tp +1yn +1 += +1 + (t1 + y1) +∞ +� +p=0 +tp +1 +∞ +� +n=0 +yn +1 += +1 + +t1 + y1 +(1 − t1)(1 − y1) += +1 + t1y1 +(1 − t1)(1 − y1). +It follows that +H(E; t1, y1) = +1 + t1y1 +(1 − t1)(1 − y1). +□ +In a similar way, we can calculate H(E; Tk, Yl) for any k, l ∈ N using Definition 6.7 with Theorems 6.9 +and 2.7. The results is the following. +Proposition 7.2. Let k, l ∈ N. Then +H(E; Tk, Yl) = 1 +2 + +1 + +k +� +i=1 +l� +j=1 +(1 + ti)(1 + yj) +(1 − ti)(1 − yj) + + . +Proof. By the definition of (k, l)-semistandard tableau, we have only two types of tableaux for λ = (p, 1n−p): +T +T Y +Y +T +T +Y +Y +Y +Y Y +Y +Y +Y +Y +where with the symbol T , we mean “elements lying in Tk” and with the symbol Y “elements lying in Yl”. +Remember that +• The elements in Tk are nondecreasing in rows and strictly increasing in columns. +• The elements in Yl are nondecreasing in column and strictly increasing in rows. + +COCHARACTERS OF UTn(E) +19 +Hence the tableaux of the first type have n ≥ 1 boxes and contain at least one symbol T . The tableaux of +the second type do not contain symbols T and have n ≥ 0 boxes. +Consider a tableau Tλ of the first type. The T -parts of Tλ forms a semistandard Tµ filled with elements +from Tk where µ = (q, qm−q) for some m ≤ n and q ≤ p. Hence the T -parts of such tableaux are in one-to-one +correspondence with the semistandard µ-tableaux filled with elements from Tk where µ = (q, qm−q), m ≤ n +and q ≤ p. The sum on all µ of the products of the entries of Tµ is equal to the sum of the Schur functions +Sµ(Tk). If the Y -part of the arm of the tableau Tλ consists of yj1, . . . , yjr, then 1 ≤ j1 · · · < jr ≤ l. Similarly, +if the Y -part of the leg of the tableau of Tλ consists of ym1, . . . , yms then 1 ≤ m1 · · · ≤ ms ≤ l. Hence the +sum of all monomials wTλ, when Tλ runs on all (k − l)-semistandard tableaux of type 1, is +� +m≥1 +m +� +q=1 +S(q,1m−q)(Tk) +� +ci≥0 +yc1 +1 · · · ycl +l +� +j1<··· 1, +m + 1 +if +λ = (2, 1m), m ≥ 1, +3m + 2 +if +λ = (2, 2, 1m), m ≥ 0, +4(m + 1) +if +λ = (2, 2, 2s, 1m), m ≥ 0, s > 0, +2nm − 3m − n + 3 +if +λ = (n, 1m), n ≥ 3, m ≥ 1, +6m(n − 3) + 9m + 3(n − 3) + 5 +if +λ = (n, 2, 1m), n ≥ 3, m ≥ 0, +(8(n − 3) + 12)(m + 1) +if +λ = (n, 2, 2s, 1m), n ≥ 3, s ≥ 1, m ≥ 0, +4(n1 − n2 + 1)(2m + 1) +if +λ = (n1, n2, 1m), n1 ≥ n2 ≥ 3, m ≥ 0, +12(n1 − n2 + 1)(m + 1) +if +λ = (n1, n2, 2s, 1m), n1 ≥ n2 ≥ 3, s ≥ 1, m ≥ 0, +4(n1 − n2 + 1)(m + 1) +if +λ = (n1, n2, 3, 2s, 1m), n1 ≥ n2 ≥ 3, s ≥ 0, m ≥ 0, +0 +for +all other λ. +Proof. Given a partition λ, by Theorem 7.4 we know that mλ = 0 if λ /∈ H(2, 3). Hence let λ ∈ H(2, 3). +In order to compute the multiplicity mλ, it is necessary to write the hook multiplicity series of UT2(E) as a +power series. +Let λ ∈ H(2, 3) and consider the triple (λ0, µ, ν). Notice that Dλ0 ⊆ D(3,3). It follows that +λ0 ∈ {(1), (2), (3), (1, 1), (2, 1), (3, 1), (2, 2), (3, 2), (3, 3)}. +First, let λ be a partition such that λ0 ∈ {(1), (2), (3)}. Using Theorem 9.3, we obtain mλ = 1. + +32 +LUCIO CENTRONE, VESSELIN DRENSKY, AND DANIELA MARTINEZ CORREA +Consider now λ such that λ0 = (1, 1). By Theorem 9.3, we have that λ0 corresponds to the summand +v1v2 +1 − y1 += v1v2 +� +m≥0 +ym +1 . +It follows that the partitions of type λ = (1, 1, 1m) with m ≥ 0 have multiplicity 1 or, equivalently, if λ = (1m), +with m > 1 then mλ = 1. +Now, let λ be such that λ0 = (2, 1). Observe that +v2 +1v2(2 − y1) +(1 − y1)2 += v2 +1v2 + +2 + +� +n≥1 +(m + 2)ym +1 + + . +Hence, if λ = (2, 1), then mλ = 2. Moreover, if λ = (2, 1, 1m), with m ≥ 1, then mλ = m + 2 or, equivalently, +if λ = (2, 1m), with m ≥ 2, then mλ = m + 1. +Let λ be a partition such that λ0 = (2, 2). Then +v2 +1v2 +2 +� 2 + y1 +(1 − y1)2 + +4y1y2 +(1 − y1)2(1 − y1y2) +� += v2 +1v2 + +2 + +� +m≥1 +(3m + 2)ym +1 + +� +m≥1 +� +s≥1 +4mym+s−1 +1 +ys +2 + + . +Hence if λ = (2, 2), then mλ = 2. If λ = (2, 2, 1m), then mλ = 3m + 2. +Observe that ym+s−1 +1 +ys +2 is in one-to-one correspondence with the partition ν = (m + s − 1, s), hence +ν′ = (2s, 1m−1). So, if λ = (2, 2, 2s, 1m−1), with m, s ≥ 1, then mλ = 4m or, equivalently, if λ = (2, 2, 2s, 1m), +with s ≥ 1 and m ≥ 0, then mλ = 4(m + 1). +The other cases are treated similarly. +□ +Now, we are going to calculate the multiplicities mλ in the cocharacter sequence of UT3(E) when λ ∈ +H(1, 1). +Theorem 9.5. Let λ be a partition such that λ ∈ H(1, 1). The multiplicity mλ in the cocharacter sequence +of UT3(E) is given by the following expressions: +mλ = + + + + + + + + + + + + + + + + + +1 +if +λ = (n), n ≥ 0, +1 +if +λ = (1m), m > 1, +n +if +λ = (n, 1), n ≥ 2, +m + 1 +if +λ = (2, 1m), m ≥ 2, +1 +4(76 − 90m + 26m2 − 54n + 68mn − 20m2n ++10n2 − 12mn2 + 4m2n2) +if +λ = (n, 1m), n ≥ 3, m ≥ 2. +Proof. By Theorem 8.12, we have +(15) +� +M(UT3(E); t, y, v) = 3G(1) − 3G2(1) + 3G2(v) + G3(1) − 2G3(v) + G3(vt) + G3(vy) +By Theorem 8.10 and Corollary 8.10, we obtain +• G(1) = 1 + +v +(1 − t)(1 − y); +• G2(1) = 1 + +v +(1 − t)(1 − y) + +v(1 + ty) +(1 − t)2(1 − y)2 ; +• G2(v) = v(1 + 2ty + t2y2) +(1 − t)2(1 − y)2 ; +• G3(1) = 1 + +v +(1 − t)(1 − y) + +v(1 + ty) +(1 − t)2(1 − y)2 + v(1 + 2ty + t2y2) +(1 − t)3(1 − y)3 ; +• G3(v) = v(1 + 3ty + 3t2y2 + t3y3) +(1 − t)3(1 − y)3 +; +• G3(vt) = v(t + 3t2y + 3t3y2 + t4y3) +(1 − t)3(1 − y)3 +; +• G3(vy) = v(y + 3ty2 + 3t2y2 + t3y4) +(1 − t)3(1 − y)3 +. + +COCHARACTERS OF UTn(E) +33 +By the equation (15) we obtain that the (1, 1)-multiplicity series of UT3(E) in the variables v, t, y is +� +M(UT3(E); v, t, y) =1 + +v +(1 − t)(1 − y) + v(1 + 4ty + 3t2y2) +(1 − t)2(1 − y)2 ++ +v +(1 − t)3(1 − y)3 +� +−1 + t + y − 4ty − 5t2y2 − 2t3y3 + 3t2y + 3t3y2 + t4y3 + 3ty2 + 3t2y3 + t3y4� +. +(16) +Note that to calculate the multiplicity mλ where λ ∈ H(1, 1), it is necessary to write (16) as a power series. +Recall that +ta1ya2 +(1 − t)(1 − y) = +� +n≥a1 +� +m≥a2 +tnym, +ta1ya2 +(1 − t)2(1 − y)2 = +� +n≥a1 +� +m≥a2 +(n − a1 + 1)(m − a2 + 1)tnym, +ta1ya2 +(1 − t)3(1 − y)3 = +� +n≥a1 +� +m≥a2 +�n − a1 + 2 +2 +��m − a2 + 2 +2 +� +tnym. +Using the previous equations and making some algebraic manipulations, we obtain the following expression +� +M(UT3(E); v, t, y) =1 + v + +� +n≥0 +tn + +� +m≥1 +ym + +� +n≥1 +(n + 1)tny + +� +m≥2 +(m + 1)tym ++ +� +n≥2 +� +m≥2 +(32 − 34(m + n) + 10(n2 + m2) − 12(m2n + n2m) + 44mn + 4m2n2) +4 +tnym + + . +(17) +By the equation (17), it follows that if λ = (n) or λ = (1m) then mλ = 1. Now, if λ = (n + 1, 1) and n ≥ 1 +then mλ = n + 1, which means that if λ = (n, 1) with n ≥ 2 then mλ = n. Observe that if λ = (2, 1m) with +m ≥ 2, its multiplicity is m + 1. +Finally if λ = (n + 1, 1m) with n, m ≥ 2, then we have that +mλ = 32 − 34(m + n) + 10(n2 + m2) − 12(mn2 + n2m) + 44mn + 4m2n2 +4 +, +or equivalently if λ = (n, 1m) with n ≥ 3 and m ≥ 2, we have that +mλ = 76 − 90m + 26m2 − 54n + 68mn − 20m2n + 10n2 − 12mn2 + 4m2n2 +4 +. +□ +Recall that if we want to know all multiplicities of cocharacter sequences of UT3(E), we have to work with +the hook H(3, 5) because by Theorem 7.4 we know that χ(UT3(E)) ⊆ H(3, 5). Hence the (3, 5)-multiplicity +series � +M(UT3(E); V3, T3, Y5) has 11 variables and the computations are very technical. +References +[1] A. Sh. Abakarov, Identities of the algebra of triangular matrices (Russian), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. +Inst. Steklov (LOMI) 114 (1982), 7-27, 217. Translation: J. Sov. Math. 27(4) (1984), 2831-2848. +[2] S. A. Amitsur and A. Regev, PI-algebras and their cocharacters, J. Algebra 78(1) (1982), 248-254. +[3] A. Ya. Belov, Rationality of Hilbert series with respect to free algebras, (Russian) Uspekhi Mat. Nauk 52 (1997),no. 2(314), +153–154; translation in Russian Math. Surveys 52 (1997), no. 2, 394–395. +[4] A. Berele, Homogeneous polynomial identities, Israel J. Math. 42(3) (1982), 258-272. +[5] A. Berele, Magnum P.I., Israel J. Math. 51(1-2) (1985), 13-19. +[6] A. Berele, Applications of Belov’s theorem to the cocharacter sequence of p.i. algebras, J. Algebra 298(1) (2006), 208-214. +[7] A. Berele, Properties of hook Schur functions with applications to p.i. algebras, Adv. Appl. Math. 41(1) (2008), 52-75. +[8] A. Berele and A. Regev, Applications of hook Young diagrams to P.I. algebras, J. Algebra 82(2) (1983), 559-567. +[9] A. Berele and A. Regev, Hook Young diagrams with applications to combinatorics and to representations of Lie superalge- +bras, Adv. Math. 64(2) (1987), 118-175. +[10] A. Berele and A. Regev, Codimensions of products and of intersections of verbally prime T-ideals, Israel J. Math. 103 +(1998), 17-28. +[11] A. Berele and A. Regev, Exponential growth for codimensions of some p.i. algebras, J. Algebra 241(1) (2001), 118-145. + +34 +LUCIO CENTRONE, VESSELIN DRENSKY, AND DANIELA MARTINEZ CORREA +[12] S. Boumova and V. Drensky, Cocharacters of polynomial identitities of upper triangular matrices, J. Algebra Appl. 11(1) +(2012), 1250018, 24 pp. +[13] L. Carini and O. M. Di Vincenzo, On the multiplicities of the cocharacters of the tensor square of the Grassmann algebra, +Atti Accad. Peloritana Pericolanti Cl. Sci. Fis. Mat. Natur. 69 (1991), 237-246. +[14] L. Centrone, Ordynary and Z2-graded cocharacters of UT2(E), Comm. Algebra 39(7) (2011), 2554-2572. +[15] L. Centrone, V. R. T. da Silva, On Z2-graded identities of UT2(E) and their growth, Linear Algebra Appl. 471 (2015), +469-499. +[16] O. M. Di Vincenzo and V. R. T. da Silva, On Z2-graded polynomial identities of the Grassmann algebra, Linear Algebra +Appl. 431(1-2) (2009), 56-72. +[17] V. Drensky, Representations of the symmetric group and varieties of linear algebras (Russian), Mat. Sb. 115(157)(1) +(1981), 98-115. Translation: Math. USSR Sb. 43(1) (1981), 85-101. +[18] V. Drensky, Codimension of T-ideals and Hilbert series of relatively free algebras, J. Algebra, 91(1) (1984), 1-17. +[19] V. Drensky, Extremal varieties of algebras. I, II (Russian), Serdica 13(4) (1987), 320-332; 14(1) (1988), 20-27. +[20] V. Drensky, Free Algebras and PI-Algebras. Graduate Course in Algebra, Springer-Verlag Singapore, 2000. +[21] V. Drensky and G. K. Genov, Multiplicities of Schur functions in invariants of two 3 × 3 matrices, J. Algebra 264(2) +(2003), 496-519. +[22] V. Drensky and B. Kostadinov, Cocharacters of polynomial identities of block triangular matrices, Comm. Algebra 45(5) +(2017), 2127-2141. +[23] E. Formanek, Invariants and the ring of generic matrices, J. Algebra 89(1) (1984), 178-223. +[24] E. Formanek, Noncommutative invariant theory, Contemp. Math. 43 (1985), 87-119. +[25] G. K. Genov, The Spechtness of certain varieties of associative algebras over a field of zero characteristic (Russian), C. R. +Acad. Bulgare Sci. 29 (1976), 939-941. +[26] G. K. Genov, Some Specht varieties of associative algebras (Russian), Pliska Stud. Math. Bulgar. 2 (1981), 30-40. +[27] A. Giambruno and P. Koshlukov, P. On the identities of the Grassmann algebras in characteristic p > 0, Israel J. Math. +122 (2001), 305-316. +[28] A. Giambruno and M. V. Zaicev,On codimension growth of finitely generated associative algbras, Adv. Math. 140(2) (1998), +145-155. +[29] A. Giambruno and M. V. Zaicev, Exponential codimension growth of PI algebras: an exact estimate, Adv. Math. 142(2) +(1999), 221-243. +[30] A. Giambruno and M. Zaicev, Polynomial Identities and Asymptotic Methods, Math. Surveys Monogr. 122. AMS, Provi- +dence, RI, 2005. +[31] P. Halpin, Some Poincar´e series related to identities of 2 × 2 matrices, Pacific J. Math. 107(1) (1983), 107-115. +[32] A. R. Kemer, Finite basis property of identities of associative algebras (Russian), Algebra Logika 26(5) (1987), 597-641. +Translation: Algebra Logic 26(5) (197), 362-397. +[33] D. Krakowski and A. Regev, The polynomial identities of the Grassmann algebra, Trans. Amer. Math Soc. 181 (1973), +429-438. +[34] V. N. Latyshev, On algebras with identity relations (Russian), Dokl. Akad. Nauk SSSR 146(5) (1962), 1003-1006. Trans- +lation: Sov. Math., Dokl. 3 (1962), 1423-1427. +[35] V. N. Latyshev, On the choice of basis in a T-ideal (Russian), Sibirsk. Mat. Zh. 4(5) (1963), 1122-1127. +[36] V. N. Latyshev, Partially ordered sets and nonmatrix identities of associative algebras (Russian), Algebra Logika 15(1) +(1976), 53-70. Translation: Algebra Logic 15(1) (1976), 34-45. +[37] V. N. Latyshev, Finite basis property of identities of certain rings (Russian), Usp. Mat. Nauk 32(4)(196) (1977), 259-260. +[38] J. Lewin, A matrix representation for associative algebras. I, Trans. Amer. Math. 188 (1974), 293-308. +[39] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Second edition, 1995. +[40] S. P. Mishchenko, A. Revev and M. V. Zaicev, A characterization of P.I. algebras with bounded multiplicities of the +cocharacters, J. Algebra 219(1) (1999), 356-368. +[41] J. B. Olsson and A. Regev, Colength sequence of some T-ideals, J. Algebra 38(1) (1976), 100-111. +[42] A. P. Popov, On the Specht property of some varieties of associative algebras (Russian), Pliska Stud. Math. Bulgar. 2 +(1981), 41-53. +[43] A. P. Popov, Identities of the tensor square of a Grassmann algebra (Russian), Algebra Logika 21(4) (1982), 442-471. +Translation: Algebra Logic 21 (1982), 296-316. +[44] A. Regev, Existence of identities in A ⊗ B, Israel J. Math. 11 (1972), 131-152. +[45] A. Regev, The representations of Sn and explicit identities of P.I. algebras, J. Algebra 51(1) (1978), 25-40. +[46] I.B. Volichenko, A.E. Zalesskii, Characterization of certain T-ideals from the view point of representation theory of the +symmetric groups, Serdica Math. J. 38 (2012), 211-236. + +COCHARACTERS OF UTn(E) +35 +Dipartimento di Matematica, Universit`a degli Studi di Bari Aldo Moro, Via Edoardo Orabona, 4, 70125 Bari, +Italy +Email address: lucio.centrone@uniba.it, centrone@unicamp.br +Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria +Email address: drensky@math.bas.bg +IMECC, Universidade Estadual de Campinas, Rua S´ergio Buarque de Holanda, 651 Cidade Universit´aria “Ze- +ferino Vaz” Distr. Bar˜ao Geraldo Campinas, S˜ao Paulo, Brasil, CEP 13083-859 +Email address: d190688@dac.unicamp.br + diff --git a/sNE0T4oBgHgl3EQfrgFB/content/tmp_files/load_file.txt b/sNE0T4oBgHgl3EQfrgFB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..33e7ff3cab83b14620bd219ed57ad2ebe2c09fbc --- /dev/null +++ b/sNE0T4oBgHgl3EQfrgFB/content/tmp_files/load_file.txt @@ -0,0 +1,2116 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf,len=2115 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content='02566v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content='RA] 6 Jan 2023 COCHARACTERS OF UTn(E) LUCIO CENTRONE, VESSELIN DRENSKY, AND DANIELA MARTINEZ CORREA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Let F be a field of characteristic 0 and let E be the infinite dimensional Grassmann algebra over F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' In the first part of this paper we give an algorithm calculating the generating function of the cocharacter sequence of the n × n upper triangular matrix algebra UTn(E) with entries in E, lying in a strip of a fixed size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' In the second part we compute the double Hilbert series H(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Tk, Yl) of E, then we define the (k, l)- multiplicity series of any PI-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' As an application, we derive from H(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Tk, Yl) an easy algorithm determining the (k, l)-multiplicity series of UTn(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Introduction We fix a field F of characteristic 0 and any algebra over F is considered associative with unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Let X = {x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content='} be a countable set of indeterminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' We denote by F⟨X⟩ the free algebra freely generated by X over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Let A be an algebra over F satisfying a polynomial identity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=', a PI-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' It is well known that its set of polynomial identities T (A) is a T -ideal of F⟨X⟩, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=', an ideal that is invariant under all endomorphisms of F⟨X⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Since F is a field of characteristic 0, all the polynomial identities follow from the multilinear ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' A famous theorem by Kemer [32] says that if A is a PI-algebra, then its T -ideal is finitely generated, but it is important to recall that the complete set of finite generators of T -ideals is well known only for few algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' By a result of Regev [44], it seems to be more efficient to study the set of multilinear polynomials which (in a certain sense) are not polynomial identities for a given algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' More precisely, if Pn is the vector space of multilinear polynomials in the variables {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' , xn}, we study the factor space Pn(A) := Pn/(Pn ∩ T (A)) for each n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' We recall that Pn is also a left Sn-module under the canonical left action of the symmetric group Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Since Pn(A) inherits the Sn-action on Pn, it affords an Sn-character χn(A) called the n-th cocharacter of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' The sequence (χn(A))n∈N is called the sequence of cocharacters of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' We also observe that Pn(A) is a finite dimensional vector space which dimension is called the n-th codimension of A (or in symbol cn(A)) and the sequence (cn(A))n∈N is called the sequence of codimensions of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' In [28], [29], see also [30], Giambruno and Zaicev proved that there always exists the limit exp(A) = lim n→∞ n� cn(A) and it is a nonnegative integer called the PI-exponent of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' If we use the language of varieties, we say that the variety generated by the algebra A is the class V = V(A) = {B associative algebra | T (A) ⊆ T (B)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' We say that the variety of algebra V is minimal with respect to its exponent if and only if for any proper subvariety U of V we have that exp(U) < exp(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' We say that a PI-algebra is minimal if it generates a minimal variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' If S is any commutative ring with 1, we denote by UTn(S) the ring of upper triangular matrices with entries in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Let E be the infinite dimensional Grassmann algebra over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Drensky [19] proved that the T -ideals of the algebras UTn(F) and UTn(E) are examples of maximal T -ideals of a given exponent of the codimension sequences (and the corresponding varieties of algebras are minimal varieties of this exponent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Some years before Kemer’s works, Genov in [25] and [26] Genov and Latyshev in [36] proved that every algebra belonging to V(UTn(F)) has a finite basis of its polynomial identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Latyshev in [37] and Popov 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' 16R10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' 05A15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' 05E05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' 05E10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' 15A75;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' 16R40;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' 20C30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Algebras with polynomial identity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' block triangular matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Grassmann algebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' cocharacter sequence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' multiplicities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' multiplicity series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Correa was partially supported by CAPES-Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Financial Code 001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' 1 2 LUCIO CENTRONE, VESSELIN DRENSKY, AND DANIELA MARTINEZ CORREA in [42] generalized the previous result for PI-algebras satisfying the polynomial identity [x1, x2, x3] · · · [x3n−2, x3n−1, x3n] which generates the T -ideal T (UTn(E)) = T (E)n of the algebra UTn(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' For a long time, until Kemer developed his structure theory, the results of Genov, Latyshev and Popov covered all known examples of classes of PI-algebras with the finite basis property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' The T-ideals T (UTn(F)) and T (UTn(E)) have another interesting property established by Volichenko and Zalesskii in [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Let the algebra A satisfy a multilinear polynomial identity f(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' , xm) which generates an irreducible Sm-module with character χλ, where χλ is the irreducible Sm-character associated with the partition λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Then the Young diagram of λ contains less than n boxes below of the first row if and only if f(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' , xm) does not hold for UTn(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Similarly, the Young diagram of λ contains less than n boxes to the right of the first column if and only if f(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' , xm) does not hold for UTn(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' For the Grassmann algebra this means that the algebra A satisfies a standard identity if and only if T (A) is not contained in T (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' A proof can be found for example in the book by Giambruno and Zaicev [30, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Let χn(A) = � λ⊢n mλ(A)χλ, n ∈ N, be the cocharacter sequence of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Let us set Xd := {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' , xd} and let us consider Fd(A) := F⟨Xd⟩/(F⟨Xd⟩ ∩ T (A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Moreover, if T = {t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' , td} is a set of commutative variables, then the Hilbert series H(Fd(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Td) of Fd(A) may be decomposed as H(Fd(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Td) = � λ mλ(A)Sλ(Td), where λ is a partition in no more than d parts and Sλ(Td) is the Schur function associated to λ in the variables from Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' We shall refer to H(Fd(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Td) as the Hilbert series of A and we shall write H(A, Td) instead of H(Fd(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Td).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' By a result of Berele and Drensky, (see [4] and [17]), the mλ(A)’s are the same as in the cocharacter sequence of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Hence, in principle, the knowledge of the Hilbert series of A will give us the multiplicities mλ(A) of the cocharacter sequence of A, when λ is a partition in no more than d parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' So if A is finite dimensional, working with a sufficiently large set of variables will be enough to capture all the multiplicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' This is no longer true for infinite dimensional algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' It is also important to recall that Belov proved in [3] that the Hilbert series of the relatively free algebra of a PI-algebra A in d variables is a rational function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' The explicit form of the multiplicities in the cocharacter sequence of a PI-algebra is known for few cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Among them are the infinite dimensional Grassmann algebra E (Olsson and Regev [41]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' the 2 × 2 matrix algebra M2(F) (Formanek [23] and Drensky [18]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' the algebra UT2(F) of 2 × 2 upper triangular matrices (Mishchenko et al [40],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' based on the approach of Berele and Regev [10],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' see also [20]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' the tensor square E ⊗ E of the Grassmann algebra (Popov [43],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Carini and Di Vincenzo [13]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' the algebra UT2(E) of 2 × 2 upper triangular matrices with entries from the Grassmann algebra E (Centrone [14]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' the algebra UTn(F) of n × n upper triangular matrices (Boumova and Drensky [12]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' the algebra Rp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content='q(F) of upper block triangular (p + 2q) × (p + 2q) when p and q are small values (Drensky and Kostadinov [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' In [21] Drensky and Genov define the multiplicity series of a PI-algebra A, that is the generating function of the cocharacter sequence of A which corresponds to the multiplicities mλ(A) when λ is a partition in no more than d parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Then, coming back to upper triangular matrices and their central role in PI-theory, in [12] Boumova and Drensky found an easy algorithm with input the multiplicity series of a symmetric function, and output the multiplicity series of its Young-derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Applying it, they found the explicit form of the multiplicity series of the Hilbert series of UTn(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Following this line of research, in the first part of the paper we work with UTn(E) and calculate its multiplicity series in d variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Due to the fact that E is infinite dimensional, we need more tools than the ones used by Boumova and Drensky in order to know all multiplicities of UTn(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Using the idea of Berele (see [7]), we work with double Hilbert series instead of with Hilbert series of PI-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Due to the analogue of the result of Berele and Drensky for double Hilbert series, it suffices to study the decomposition of the double Hilbert series of UTn(E) in order to achieve the explicit form of the cocharacter sequence of UTn(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' In the second part of the present paper, we generalize the definition of multiplicity series of a PI-algebra defining a (k, l)-multiplicity series which controls three sets of disjoint variables, where (k, l) means that the partitions λ = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' , λm) COCHARACTERS OF UTn(E) 3 satisfy the condition λk+1 ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' In other words, their young diagrams Dλ are in a hook of height k of the arm and wide l of the leg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' By a result of Amitsur and Regev [2] all nonzero multiplicities mλ(A) for a PI-algebra A are concentrated for Young diagrams in a sufficiently large hook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Hence the information about the multiplicities of A is contained in the related with the hook (k, l)-multiplicity series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Then we compute the double Hilbert series of E and, as a consequence, we build up an algorithm with output the (k, l)-multiplicity series of UTn(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' In the spirit of [14] we compute the (2, 3)-multiplicity series of UT2(E), which contains all multiplicities of the cocharacter sequence of UT2(E) and finally we compute the (1, 1)-multiplicity series of UT3(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Symmetric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' We fix a positive integer d and consider the algebra C[[Td]] = C[[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' , td]] of formal power series in d commutative variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Let C[[Td]]Sd ⊆ C[[Td]] be the subalgebra of symmetric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Every symmetric function g(Td) can be represented in the form g(Td) = � λ mλSλ(Td), mλ ∈ C, λ = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' , λd), where Sλ(Td) is the Schur function related to the partition λ which has at most d parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' For details on the theory of Schur functions see [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' There are several ways to define Schur functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' The most convenient for our purpose is to define them as fractions of Vandermonde-type determinants: Sλ(Td) = V (λ + δ, Td) V (δ, Td) , where δ = (d − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' , 2, 1) and for µ = (µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' , µd) V (µ, Td) = ����������� tµ1 1 tµ1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' tµ1 m−1 tµ1 m tµ2 1 tµ2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' tµ2 m−1 tµ2 m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' tµm−1 1 tµm−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' tµm−1 m−1 tµm−1 m tµm 1 tµm 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' tµm m−1 tµm m ����������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Let λ = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' , λd) be a partition of a natural number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' The Young diagram Dλ associated to λ is the subset of Z × Z defined as Dλ = {(i, j) | i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' , d, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' , λi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Graphically we draw the diagrams replacing the knots by square boxes, adopting the convention, as with matrices, that the first coordinate i (the row index) increases as one goes downwards, and the second coordinate j (the column index) increases as one goes from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' The first boxes from the left of each row are one above another and the i-th row contains λi boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' We denote by λ′ j the length of the j-th column of Dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' The partition λ′ = (λ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' , λ′ m) and its diagram Dλ′ are called conjugate respectively to λ and Dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' For the partition λ, we define a λ-tableau Tλ of content α = α(Tλ) = (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' , αd) if each integer i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' , d appears in the tableau exactly αi times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Recall that the λ-tableau Tλ is semistandard if its entries do not decrease in rows reading from left to right, and increase strictly in columns reading from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Another definition of Schur functions is given in terms of semistandard Young tableaux: Sλ(Td) = � Tα(Tλ) d , where the summations runs on all semistandard λ-tableaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' We recall the definition of elementary symmetric polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' Given 0 ≤ m ≤ d, the m-th elementary symmetric polynomial in d variables t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE0T4oBgHgl3EQfrgFB/content/2301.02566v1.pdf'} +page_content=' , td is defined by em(Td) = � 1≤i1<···