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+{"text":"\\section{Introduction}\nThe ejection of material from the surfaces of atmosphereless bodies is a ubiquitous phenomenon in the Solar System. Prominent examples are comets, active asteroids, ejecta clouds from hypervelocity impacts, or plumes from active satellites. For the dynamics of the ejected material, it is in many cases possible to neglect any other forces than the mass point gravity of the source body  to a good degree of approximation. This is for instance the case for impact-generated dust clouds around planetary satellites as were detected around the Galilean moons \\citep{1999Natur.399..558K, 2003Icar..164..170K} or the Moon \\citep{2015Natur.522..324H}, or dust plumes ejected from cryovolcanically active satellites \\citep{2006Sci...311.1416S, 2006Sci...311.1393P, 2015GeoRL..4210541S}. We note that for higher-order gravity terms to be negligible, the source body does not necessarily need to be spherical. For instance, mass-point gravity can be a good approximation to describe dust ejection from a satellite with surface topography.\n\n\nIn this paper we derive a semianalytical model to assess the spatial configuration of the emitted dust.  The model relates the distribution of dust sources on the surface of the atmosphereless body and the parameters of ejection (e.g., source strength or directional and velocity distribution) to observable quantities such as number density, fluxes, or optical depth. The mathematical foundations are described in Sect. \\ref{sec:mathematics}. Expanding on work in the literature \\citep{2003P&SS...51..251K, Sremcevic:2003gf}, our model can handle emission through inclined jets, and we develop a method for carrying out two of three integrations over the velocity distribution analytically. The code that implements the new model, carrying out the one remaining integration numerically, is called DUDI (for \\textquotedblleft dust distribution\\textquotedblright) and is freely available under the GNU General Public License on https:\/\/github.com\/Veyza\/dudi. Aspects of the numerical algorithm for the integration are outlined in Sect. \\ref{sec:numerics}. Examples for an application of the model to current problems in planetary science are given in Sect. \\ref{sec:applications}, including cases of steady and nonsteady dust emission.\n\n\\section{Mathematical formulation}\n\\label{sec:mathematics}\n\\subsection{Phase-space density}\nWe followed the derivations by \\citet{2003P&SS...51..251K}, \\citet{Sremcevic:2003gf}, and \\citet{2011Natur.474..620P} to relate the phase-space density of dust in a certain point of interest in space to the distributions that describe the ejection of the dust from a source on the moon surface. We generalized the existing model to allow emission from a point source in a direction that is not normal to the surface, with an axisymmetric distribution of ejection angles around this direction. Moreover, we allowed for a general coupling of the distribution of ejection velocities and grain size.\n\n\nOur model was developed initially to fit in situ measurements by the Cassini Cosmic Dust Analyzer at the Saturn satellite Enceladus. For convenience, we use the words \\textquotedblleft spacecraft position\\textquotedblright \\ or \\textquotedblleft spacecraft coordinates\\textquotedblright\\   from now on to denote the point in space at which the dust density is calculated. We also use the term \\textquotedblleft density\\textquotedblright,  which at any point can be understood as the number density, mass density, the average radius of dust particles, or the cross section that is covered by the dust at the spacecraft position. The model allows us to obtain any of these quantities with a change of only one parameter.\n\nWe first consider a stationary process. We can equate the differential number of dust particles in a certain point of phase space \n        \\begin{equation}\n                \\label{dninspace}\n                \\mathrm{d}n = n(r, \\alpha, \\beta, v, \\theta, \\lambda,R)r^2\\sin\\alpha \\mathrm{d}r\\mathrm{d}\\alpha \\mathrm{d}\\beta\\  v^2\\sin\\theta \\mathrm{d}v\\mathrm{d}\\theta \\mathrm{d}\\lambda \\mathrm{d}R\n        \\end{equation}\nto the number of particles ejected from the satellite surface\n        \\begin{equation}\n                \\label{dnonsurface}\n                \\mathrm{d}n = \\gamma \\mathrm{d}t\\ f(\\alpha_M, \\beta_M, u, \\psi,\\lambda_M, R)\\sin\\alpha_M \\mathrm{d}\\alpha_M \\mathrm{d}\\beta_M \\mathrm{d}u \\sin\\psi \\mathrm{d}\\psi \\mathrm{d}\\lambda_M \\mathrm{d}R\n        .\\end{equation}\nThe variables used here are defined in Table \\ref{vars}, and Fig. \\ref{sptr} illustrates the geometry of the problem.\nFor the phase-space  density at the spacecraft, we obtain\n\\begin{multline}\n\\label{ndens0}\nn(r, \\alpha, \\beta, v, \\theta, \\lambda, R) v^2 \\sin \\theta = \\frac{\\gamma}{|\\mathrm{d}r\/\\mathrm{d}t|r^2}\\frac{\\sin\\alpha_M\\sin\\psi}{\\sin\\alpha}\\times\\\\\\times f(\\alpha_M, \\beta_M, u, \\psi, \\lambda_M, R) \\abs*{\\frac{\\partial (\\alpha_M, \\beta_M, u, \\psi, \\lambda_M)}{\\partial(\\alpha, \\beta, v, \\theta, \\lambda)}}.\n\\end{multline}\nFor the two-body problem, the Jacobian can be obtained analytically (see \\citet{Sremcevic:2003gf}),\n\\begin{equation}\n\\label{Jalpha}\n\\abs*{\\frac{\\partial (\\alpha_M, \\beta_M, u, \\psi, \\lambda_M)}{\\partial(\\alpha, \\beta, v, \\theta, \\lambda)}} = \\frac{r}{r_M}\\frac{v^2}{u^2}\\frac{|\\cos\\theta|}{\\cos\\psi}\\frac{\\sin\\alpha}{\\sin\\alpha_M}\n.\\end{equation}\n\n\\begin{figure}\n        \\centering\n        \\includegraphics[width=0.5\\textwidth]{sph_triangle.pdf}\n        \\caption{Directions to the north and to the positions of the source and the spacecraft form a spherical triangle on a sphere. This establishes the relations between the angles of the problem. Here $\\Delta\\beta$ is the angle between the projections of vectors $\\mathbf{r}$ and $\\mathbf{r}_M$ on the equatorial plane.}\n        \\label{sptr}\n\\end{figure}\n\n\\begin{table*}[t]\n        \\caption{Definition of variables}\n        \\vskip0.3cm\n        \\begin{tabular}{|c|p{6cm}|c|p{6cm}|}\n                \\hline\n                Variable & Definition & Variable & Definition \\\\\n                \\hline\n                $r$ & Radial distance from the moon center to a point in space where the density is to be calculated (spacecraft position) & $r_M$ & Radial distance of the dust source from the center of the moon (source position)\\\\\n                $\\alpha$ & Colatitude of the spacecraft (measured from the moon north pole) & $\\alpha_M$ & Colatitude of the source  (measured from the moon north pole) \\\\\n                $\\beta$ & Eastern longitude of the spacecraft & $\\beta_M$ & Eastern longitude of the source \\\\\n                $v$ & Particle speed at the spacecraft position & $u$ & Particle speed at the moment of ejection \\\\\n                $\\theta$  & Angle\n                between the particle velocity and position vectors & $\\psi$  & Initial angle\n                between the particle velocity and position vectors\\\\\n                $\\lambda$ & Azimuth angle of the particle velocity, measured clockwise from local north. & $\\lambda_M$ & Azimuth angle of the particle initial velocity, measured clockwise from local north.\\\\\n                $R$ & Radius of the particle & $\\gamma$ & Rate of dust particle production \\\\\n                $n$ & Phase-space density of particles with fixed radius & $f$ & Distribution describing the dust ejection process\\\\\n                $\\zeta$ & Zenith angle of the source symmetry axis & $\\eta$ & Azimuth of the source symmetry axis measured clockwise from local north \\\\\n                 & & $\\eta^*$ & Auxiliary angle used in the derivation of the expression for the ejection direction distribution in case of a tilted symmetry axis\\\\\n                \\hline\n        \\end{tabular}\n        \\label{vars}\n\\end{table*}\nWe assume that the distribution function $f$ factorizes, and that the distributions of the source position ($f_{\\alpha_M,\\beta_M}$), ejection speed ($f_u$), ejection direction ($f_{\\psi,\\lambda_M}$), and size of the ejected dust particles ($f_R$) can be defined separately,\n\\begin{equation}\n\\label{fff}\nf(\\alpha_M, \\beta_M, u, \\psi, \\lambda_M, R) = f_{\\alpha_M,\\beta_M} (\\alpha_M,\\beta_M) f_{\\psi, \\lambda_M}(\\psi, \\lambda_M) f_u(u, R)  f_R(R).\n\\end{equation}\n\nIt is physically plausible that the distribution of the ejection speed of the dust particles depends on the grain size \\citep{2008Natur.451..685S, 2011Natur.474..620P}, which we emphasize with the notation $f_u(u,R)$.  \n\nWe describe the position of the point source located at the coordinates $(\\alpha_M^0,\\beta_M^0)$ on the surface of the spherical moon as the product of two Dirac $\\delta$-functions,\n\\begin{equation}\n\\label{fpos}\nf_{\\alpha_M,\\beta_M} (\\alpha_M,\\beta_M) = \\frac{\\delta(\\alpha_M - \\alpha_M^0)\\,\\delta(\\beta_M - \\beta_M^0)}{\\sin\\alpha_M} \\equiv\\frac{\\delta\\left((\\alpha_M,\\beta_M)-(\\alpha_M^0,\\beta_M^0)\\right)}{\\sin\\alpha_M}\n.\\end{equation}\nThe term $\\sin\\alpha_M$ in the denominator comes from the normalization.\n\nTo formulate the directional distribution of $\\psi$ and $\\lambda_M$ so that it describes the ejection of dust that is axisymmetric around the axis of an inclined jet, we consider two coordinate systems centered at the location of the point source. The \\textit{Z}-axis of system $(X,Y,Z)$ points along the local normal to the surface, and the $X$-axis points toward the local north. The  $\\tilde{Z}$-axis\nof system $(\\tilde{X}, \\tilde{Y}, \\tilde{Z})$ is aligned with the axis of the jet. \n The axis $\\tilde{X}$ lies on the line of nodes, so that the angle $\\eta^*$ measured from $X$ to $\\tilde{X}$ is related to the jet azimuth $\\eta$ as $\\eta^* = \\eta - \\pi \/ 2$. We define azimuth angles always \\emph{\\textup{clockwise}} from the local north, allowing a direct comparison to the derivations in \\citet{2003P&SS...51..251K} and \\citet{Sremcevic:2003gf}. Then, the transformation of the $(X,Y,Z)$ coordinate system to the $(\\tilde{X}, \\tilde{Y}, \\tilde{Z})$ coordinate system may be performed as two subsequent rotations, as shown in Fig. \\ref{2sys}. The first rotation is clockwise around the $Z$-axis with angle $\\eta^*$. The second rotation is counterclockwise around the $\\tilde{X}$-axis with angle $\\zeta$.\n\n\n\\begin{figure}\n        \\centering\n        \\includegraphics[width=0.5\\textwidth]{twosystems_newnew}\n        \\caption{Two coordinate systems centered at the location of the dust source on the surface of a spherical body. The  $Z$-axis of system ($X,Y,Z$) is normal to the surface, and $X$ points to the local north. The $\\tilde{Z}$-axis of the coordinate system ($\\tilde{X},\\tilde{Y}, \\tilde{Z}$) is aligned with a jet that is tilted from the surface normal by an angle $\\zeta$.}\n        \\label{2sys}\n\\end{figure}\n\nLet $(\\psi,\\lambda_M)$ and $(\\tilde{\\psi}, \\tilde{\\lambda}_M)$ be the polar angle and azimuth in the two systems $(X,Y,Z)$ and $(\\tilde{X}, \\tilde{Y}, \\tilde{Z})$, respectively. The distribution of the ejection direction we wish to use can be formulated in a simple manner in terms of the variables $(\\tilde{\\psi}, \\tilde{\\lambda}_M)$ because the distribution is symmetrical with respect to the axis $\\tilde{Z}$. \nHowever, in Eq. (\\ref{ndens0}) and in the formulae that are to be derived later in the course of solving the two-body problem, we have to deal with the angles $(\\psi,\\lambda_M)$ that are defined in the local horizontal coordinate system. Therefore a replacement of the coordinates must be performed in the expression to obtain the distribution of $\\psi$ and $\\lambda_M$ , which can be used in further calculations. The desired function $f_{\\psi, \\lambda_M}(\\psi, \\lambda_M)$ can be obtained by multiplication with the Jacobian,\n\\begin{equation}\n\\label{transpsi}\nf_{\\psi, \\lambda_M}(\\psi, \\lambda_M)\\sin\\psi = f_{\\tilde{\\psi}, \\tilde{\\lambda}_M}(\\tilde{\\psi}, \\tilde{\\lambda}_M) \\sin\\tilde{\\psi} \\abs*{\\frac{\\partial (\\tilde{\\psi}, \\tilde{\\lambda}_M)}{\\partial (\\psi, \\lambda_M)}}\n.\\end{equation}\nTo express $\\tilde{\\psi}$ and $\\tilde{\\lambda}_M$ through $\\psi$ and $\\lambda_M$ , we consider a unit vector $\\mathbf{k}$ pointing in an arbitrary direction (Fig. \\ref{2sys}). In both coordinate systems the vector can be defined by Cartesian coordinates related to the corresponding polar coordinates as (we recall that $\\lambda_M$ and $\\tilde{\\lambda}_M$ are azimuthal angles counted \\emph{\\textup{clockwise}} from the $X$ and $\\tilde X$ axes, respectively)\n\n\\begin{equation}\n\\label{kvec}\n\\mathbf{k} = \\colvec{3}{k_1}{k_2}{k_3} = \\colvec{3}{\\sin\\psi\\cos\\lambda_M}{-\\sin\\psi\\sin\\lambda_M}{\\cos\\psi},\\ \n\\mathbf{k} = \\colvec{3}{\\tilde{k}_1}{\\tilde{k}_2}{\\tilde{k}_3} = \\colvec{3}{\\sin\\tilde{\\psi}\\cos\\tilde{\\lambda}_M}{-\\sin\\tilde{\\psi}\\sin\\tilde{\\lambda}_M}{\\cos\\tilde{\\psi}}\n.\\end{equation}\nThe Cartesian coordinates of $\\mathbf{k}$ in the two systems are related through the rotation matrix, which can be expressed in terms of  $\\eta$,\n\\begin{equation}\n\\label{rotmatr}\n\\colvec{3}{\\tilde{k}_1}{\\tilde{k}_2}{\\tilde{k}_3} = \\begin{pmatrix}\n\\sin\\eta & \\cos\\eta & 0\\\\\n-\\cos\\eta\\cos \\zeta &\\sin \\eta \\cos \\zeta & \\sin \\zeta\\\\\n\\cos\\eta\\sin \\zeta& -\\sin\\eta\\sin \\zeta & \\cos \\zeta\n\\end{pmatrix}\\colvec{3}{k_1}{k_2}{k_3}\n.\\end{equation}\nUsing Eqs. \\ref{kvec} and \\ref{rotmatr}, we obtain\n\\begin{equation}\n\\label{wpsi}\n\\tilde{\\psi} = \\arccos (\\cos \\zeta\\cos\\psi + \\cos(\\eta-\\lambda_M)\\sin \\zeta\\sin\\psi),\n\\end{equation}\n\\begin{equation}\n\\label{wlambdaM}\n \\tilde{\\lambda}_M =\\arctan \\left( \\frac{\\cos\\zeta \\sin\\psi \\cos(\\eta-\\lambda_M) - \\sin\\zeta \\sin\\psi)}{\\sin\\psi \\sin(\\eta-\\lambda_M)} \\right)\n.\\end{equation}\nThis gives the Jacobian\n\\begin{multline}\n\\label{Jpsi}\n\\abs*{\\frac{\\partial (\\tilde{\\psi}, \\tilde{\\lambda}_M)}{\\partial (\\psi, \\lambda_M)}} = 4\\sin\\psi\\ \/ [10-2\\cos 2\\psi-3\\cos 2(\\psi - \\zeta) - 2\\cos 2\\zeta -3\\cos 2(\\psi + \\zeta) \\\\- 8\\cos 2(\\lambda_M-\\eta)\\sin^2\\zeta\\sin^2\\psi -8\\cos(\\lambda_M-\\eta)\\sin 2\\zeta \\sin 2\\psi] ^{1\/2}.\n\\end{multline}\n\n\n\n\\subsection{Integration}\n\\label{integration}\nTo compute the density of dust at the point $(r, \\alpha,\\beta),$ we must integrate Eq. (\\ref{ndens0}) over all possible velocities and over all possible particle sizes,\n\\begin{multline}\n\\label{ndens}\nn(r,\\alpha,\\beta,R_{min} < R < R_{max}) = \\frac{\\gamma}{r r_M} \\int_{v_{min}}^{v_{max}}\\mathrm{d}v\\int_{0}^{\\pi}\\mathrm{d}\\theta\\int_{0}^{2\\pi}\\mathrm{d}\\lambda \\frac{v}{u^2}G^p_u(R_{min}, R_{max})\\times\\\\\\times\\frac{f_{\\psi, \\lambda_M}(\\psi, \\lambda_M)\\sin\\psi }{\\cos\\psi}\\,\\frac{\\delta\\left((\\alpha_M(\\theta,\\lambda),\\beta_M(\\theta,\\lambda))-(\\alpha_M^0,\\beta_M^0)\\right)}{\\sin\\alpha_M}\n.\\end{multline}\nHere,\n\\begin{equation}\n\\label{Gu}\nG^p_u(R_{min},R_{max}) \\equiv \\int_{R_{min}}^{R_{max}}\\mathrm{d}Rf_R(R)f_u(u,R)R^p\n\\end{equation}\nis defined in a similar way as in \\citet{2011Natur.474..620P}. The parameter $p$ defines the moment of the size distribution related to the quantity we are interested in. Using p=0, we obtain the number density of particles in the specified range of sizes. Setting p=1 gives the average radius of the grains per unit volume. For p=2 we obtain the average cross section of the dust particles per volume. This setting is used below to compute the geometrical optical depth of the dust population. Finally, p=3 gives the average volume occupied by dust grains per unit volume. This setting is used to compute the mass density of the dust. For more details of the evaluation of $G^p_u(R_{min},R_{max}),$ see Appendix B.\nWe replace variables in the argument of the $\\delta$ function in equation (\\ref{ndens}) as\n\\begin{equation}\n\\label{repl}\n\\int_{0}^{\\pi}\\mathrm{d}\\theta\\int_{0}^{2\\pi}\\mathrm{d}\\lambda\\, \\delta\\left((\\alpha_M(\\theta,\\lambda),\\beta_M(\\theta,\\lambda))-(\\alpha_M^0,\\beta_M^0)\\right)F(\\theta,\\lambda) =\\sum_{i} \\frac{F(\\theta_i^*,\\lambda_i^*)}{\\abs*{\\frac{\\partial(\\alpha_M,\\beta_M)}{\\partial(\\theta,\\lambda)}}}_{\\theta_i^*,\\lambda_i^*}\\,\n.\\end{equation}\nto integrate over $\\theta$ and $\\lambda$ analytically.  Eq. (\\ref{repl}) is derived in greater detail in Appendix A.\nHere, $F(\\theta,\\lambda)$ represents the integrand of equation \\ref{ndens}, while $\\theta_i$ and $\\lambda_i$ are the roots of the equation\n\\begin{equation}\n\\label{alphaMbetaM}\n\\alpha_M(\\theta_i^*,\\lambda_i^*) = \\alpha_M^0,\\quad \\beta_M(\\theta_i^*,\\lambda_i^*)=\\beta_M^0\n.\\end{equation}\nAll the necessary dependences between the variables in question can be found from spherical trigonometry  \\citep{2003P&SS...51..251K, Sremcevic:2003gf}, for instance,\n\\begin{equation}\n\\label{alphaM}\n\\alpha_M = \\arccos\\left(\\cos\\alpha\\cos\\Delta\\phi(\\theta) - \\sin\\alpha\\sin\\Delta\\phi(\\theta)\\cos\\lambda\\right)\n\\end{equation}\nand\n\\begin{equation}\n\\label{betaM}\n\\beta_M = \\beta \\pm \\arcsin\\left(\\frac{\\sin\\Delta\\phi(\\theta)\\sin\\lambda}{\\sin\\alpha_M(\\theta,\\lambda)}\\right)\n.\\end{equation}\n\nThe spherical triangle used to obtain these relations is shown in Fig. \\ref{sptr}. The angle $\\Delta\\phi$ is the angle between the position vectors of the spacecraft and the source location on the moon.\nBecause $\\theta$ enters expressions \\ref{alphaM} and \\ref{betaM} only through $\\Delta\\phi$, the partial derivatives of $\\alpha_M$ and $\\beta_M$ with respect to $\\theta$ can be computed as partial derivatives with respect to $\\Delta\\phi$ multiplied by $\\partial\\Delta\\phi \/ \\partial\\theta$. The Jacobian reads\n\\begin{equation}\n\\label{JalphaM}\n\\abs*{\\frac{\\partial(\\alpha_M,\\beta_M)}{\\partial(\\theta,\\lambda)}}=\\frac{\\sin\\Delta\\phi}{\\sin\\alpha_M}\\abs*{\\frac{\\partial\\Delta\\phi}{\\partial\\theta}},\n\\end{equation}\nand our final formula is\n\\begin{multline}\n\\label{workformula}\nn(r,\\alpha,\\beta,R_{min}<R<R_{max}) = \\frac{\\gamma}{rr_M\\sin\\Delta\\phi} \\int_{v_{min}}^{v_{max}}\\mathrm{d}v\\frac{v}{u^2}G^p_u(R_{min},R_{max})\\times\\\\ \\times \\sum_{i}\\frac{f_{\\psi,\\lambda_M}(\\psi_i,\\lambda_{Mi})\\sin\\psi_i}{\\cos\\psi_i}\\abs*{\\frac{\\partial\\Delta\\phi}{\\partial\\theta}}_{\\theta^*_i}^{-1}.\n\\end{multline}\n\nThe integration over $v$ must be carried out numerically. The lower integration limit is restricted by the minimum energy, or minimum semimajor axis, of the orbits that pass through the two points $(r_M,\\alpha_M,\\beta_M)$ and $(r,\\alpha,\\beta)$. It is obtained from\n\\begin{equation}\n        \\label{vmin}\n        v_{min} = \\sqrt{GM\\left(\\frac{2}{r}-\\frac{1}{a_{min}}\\right)}\n,\\end{equation}\nwhere\n \\begin{equation}\n\\label{amin}\na_{min} = \\frac{r + r_M}{4} + \\frac{1}{2} \\sqrt{\\frac{r^2 + r_M^2}{4} - \\frac{r r_M \\cos\\Delta\\phi}{2}}\n.\\end{equation}\nThe upper limit is constrained by the maximum ejection speed,\n\\begin{equation}\n        \\label{vmax}\n        v_{max} = \\sqrt{u_{max}^2+ 2GM \\left(\\frac{1}{r}-\\frac{1}{r_M}\\right)}\n.\\end{equation}\n\nThe angle $\\Delta\\phi$  in the expressions above is the angle between the position vector of the dust source, $\\mathbf{r}_M$ , and the position vector of the spacecraft, $\\mathbf{r}$. For a fixed position of the source and spacecraft, the angle $\\Delta\\phi$ is also fixed, but it formally depends on $v$ and $\\theta$. In the integrand of equation \\ref{workformula}, the value $v$ is given and $\\Delta\\phi(\\theta)$ is a function of the variable $\\theta$ alone. From the conservation equations of the two-body problem, we obtain \n\\begin{equation}\n\\label{wp}\n\\tilde{p} = 2\\tilde{r}^2\\tilde{v}^2\\sin^2\\theta,\n\\end{equation}\n\\begin{equation}\ne^2 = 1 + 4 \\tilde{r}^2\\tilde{v}^2\\sin^2\\theta\\left(\\tilde{v}^2-\\frac{1}{\\tilde{r}}\\right),\n\\end{equation}\n\\begin{equation}\n\\label{cosphiM}\n\\cos\\phi_M= \\frac{1}{e}\\left(\\frac{\\tilde{p}}{1}-1\\right),\n\\end{equation}\n\\begin{equation}\n\\label{cosphi}\n\\cos\\phi= \\frac{1}{e}\\left(\\frac{\\tilde{p}}{\\tilde{r}}-1\\right),\n\\end{equation}\n\\begin{equation}\n\\label{dphi}\n\\Delta\\phi = \\phi - \\phi_M,\n\\end{equation}\nwhich give the relation between $\\Delta\\phi$ and $\\theta$. Following the notational convention of \\citet{2003P&SS...51..251K}, we use dimensionless variables $\\tilde{r} = r\/r_M$ and $\\tilde{v} = v\/ v_{escape}$, where $v_{escape}$ is the escape velocity on the satellite surface. The angles $\\phi$ and $\\phi_M$ are the true anomalies at $\\mathbf{r}$ and $\\mathbf{r}_M$, respectively.\n\nEquations (\\ref{wp}) -- (\\ref{dphi}) can be used to calculate the partial derivative $\\partial\\Delta\\phi \/ \\partial\\theta$. However, it is not possible to reverse these expressions to obtain $\\theta$ from a given value of $\\Delta\\phi$ analytically. The desired $\\theta^*_i$ are the values of $\\theta$ that (for a given $v$) lead to a $\\Delta\\phi$ satisfying equations \\ref{alphaM} and \\ref{betaM}. We use a geometrical approach to calculate all possible $\\theta_i^*$. We consider the two-body problem, therefore the motion is restricted to a plane. We define a two-dimensional coordinate system in the plane containing the vectors $\\mathbf{r}$ and $\\mathbf{r_M}$. The origin is located at the moon center and $\\mathbf{r}$ points along the $x$-axis. We know the lengths of $\\mathbf{r}$ and $\\mathbf{r_M}$ as well as the angle $\\Delta\\phi$ between them. Then the coordinates of the vectors $\\mathbf{r}$ and $\\mathbf{r_M}$ in the plane are ($r,0$) and ($r_M\\cos\\Delta\\phi,r_M\\sin\\Delta\\phi$). \n\nWe wish to obtain $\\theta$ , which is the angle between $\\mathbf{r}$ and the grain velocity vector, which is tangential to the particle trajectory at the point $\\mathbf{r}$. This angle can be calculated when we know the equation of the trajectory, which is either an ellipse or a hyperbola. At this step (calculating the integrand of equation \\ref{workformula}), we know the value of the particle speed and the distance to the moon center, which determines the orbital energy, and thus, the semimajor axis $a$. We also know whether the particle moves along an ellipse (negative orbital engery) or along a hyperbola (positive orbital energy).\n\nWe first consider the case of an ellipse (Fig. \\ref{ellipse}). One of the focal points ($F_1$) is located in the origin, which is the center of the moon. To draw the ellipse, we must find the position of the second focus. At each point of the ellipse, the sum of the distances to the focal points is a constant equal to $2a$. Therefore the second focus must be removed from point $\\mathbf{r}$ by $2a-r$ and from point $\\mathbf{r_M}$ by $2a-r_M$. These conditions are met at the intersection points of two circles centered at $\\mathbf{r}$ and $\\mathbf{r_M}$ with radii $2a-r$ and $2a-r_M$, respectively.\n\\begin{figure\n        \\centering\n        \\begin{minipage}{0.45\\textwidth}\n                \\centering\n                \\includegraphics[width=0.9\\textwidth]{ellipse_new.pdf}\n                \\caption{Finding the second focus for an elliptic trajectory and the solutions for $\\theta^*_i$\\\\\n                }\n                \\label{ellipse}\n        \\end{minipage}\\hfill\n        \\begin{minipage}{0.45\\textwidth}\n                \\centering\n                \\includegraphics[width=0.9\\textwidth]{hyperbola_new.pdf}\n                \\caption{Finding the second focus for an hyperbolic trajectory and the solutions for $\\theta^*_i$. Only $\\theta^*_1$ corresponds to a physically possible trajectory.}\n                \\label{hyperbola}\n        \\end{minipage}\\hfill\n\\end{figure}\nThis leads to the conditions\n\\begin{equation}\n\\label{circleel}\n\\begin{array}{rl}\n(x-r_M\\cos\\Delta\\phi)^2 + (y-r_M\\sin\\Delta\\phi)^2 & = (2a-r_M)^2,\\\\\n(x-r)^2 + y^2 &= (2a-r)^2.\n\\end{array}\n\\end{equation}\nBy solving the system of equations \\ref{circleel}, we find two possible positions for the second focal point, $F_2^1$ and $F_2^2$. In either case, we can calculate the eccentricity $e$ of the ellipse, the true anomaly at point $\\mathbf{r,}$ and, finally, the two solutions for $\\theta$ using Eqs. (\\ref{eccentricity}) -- (\\ref{theta}). Knowing $\\cos\\phi_M$ is sufficient to obtain $\\phi_M$ because we know that the ejection point $\\mathbf{r_M}$ cannot have a true anomaly $\\phi_M > \\pi$. We have\n\\begin{equation}\n\\label{eccentricity}\ne = \\frac{|\\overline{F_2F_1}|}{2a},\n\\end{equation} \n\\begin{equation}\n\\label{cosf1}\n        \\cos\\phi_M = \\frac{\\overline{F_2F_1}\\cdot\\mathbf{r}_M}{|\\overline{F_2F_1}|r_M},\n\\end{equation}\n\\begin{equation}\n        \\label{difphi}\n        \\phi = \\phi_M + \\Delta\\phi,\n\\end{equation}\n\\begin{equation}\n\\label{theta}\n\\theta = \\frac{\\pi}{2} - \\arctan\\frac{e\\sin \\phi}{\\sqrt{1+e\\cos\\phi}},\n\\end{equation}\n\\begin{equation}\n        \\label{rad}\n        r = \\frac{a(1-e^2)}{1+e\\cos\\phi}\n.\\end{equation}\nThese equations determine the solutions $\\theta^*_i$ used in equation \\ref{workformula} if the particle travels from $\\mathbf{r_M}$ to $\\mathbf{r}$ along the shorter arc of the ellipse. However, there are cases when a particle reaches $\\mathbf{r}$ over an arc of $2\\pi - \\Delta\\phi$ (Fig. \\ref{bigdphi}), leaving $\\mathbf{r_M}$ in the opposite direction. To distinguish this case, we must recalculate $r$ from the obtained value for $\\phi$ using Eq. (\\ref{rad}) and verify that it matches the starting value for $r$ that we used to obtain $\\phi$. If this is not the case, then $\\Delta\\phi$ in Eq. (\\ref{difphi}) must be replaced by $2\\pi-\\Delta\\phi$, which corresponds to the motion along the same ellipse, but in the opposite direction. This case is relatively rare, and in the examples we explored was only encountered at large distances from the source.\n\n\\begin{figure}\n        \\centering\n        \\includegraphics[width=0.5\\textwidth]{bigdphi_new.pdf}\n        \\caption{The case when $\\Delta\\phi$ should be replaced by $2\\pi-\\Delta\\phi.$}\n        \\label{bigdphi}\n\\end{figure}\n\n\n\nIn case of hyperbolic motion ($a<0$, Fig. \\ref{hyperbola}), we follow almost the same line. For every point on a hyperbola, the difference between the distances to the focal points is the same. Because $r_M$ and $r$ must be distances to the nearest focus, the system of equations for the coordinates of the second focal point reads \n\\begin{equation}\n\\label{circlehy}\n\\begin{array}{rl}\n(x-r_M\\cos\\Delta\\phi)^2 + (y-r_M\\sin\\Delta\\phi)^2 & = (r_M+2|a|)^2,\\\\\n(x-r)^2 + y^2 &= (r+2|a|)^2.\n\\end{array}\n\\end{equation}\n\\vskip0.4cm\nEquations \\ref{eccentricity} -- \\ref{theta} remain the same for the hyperbolic case. However, when the coordinates of the second focus are found, we must make sure that the particle does not pass the pericenter on its way from $\\mathbf{r_M}$ to $\\mathbf{r}$. For this purpose, we verify that the points $\\mathbf{r_M}$ and $\\mathbf{r}$ lie on the same side of the line $F_1F_2$. If this condition is not satisfied, the solution is rejected. In Fig. \\ref{hyperbola} the hyperbola with second focus at the point $F_2^2$  does not meet this condition. Therefore only one hyperbolic trajectory is possible to get from $\\mathbf{r_M}$ to $\\mathbf{r}$. Furthermore, the motion along the hyperbola is possible only in one direction. The value for $\\sin\\phi$ is always positive and $\\Delta\\phi=\\phi-\\phi_M$.\n\nThe values of $\\lambda_i$ can be inferred from spherical trigonometry. The two solutions can be either identical or they differ by $180^\\circ$ because the motion is restricted to a plane,\n\\begin{equation}\n\\label{coslam}\n\\cos\\lambda= \\frac{\\cos\\alpha_M\\cos\\Delta\\phi-\\cos\\alpha}{\\sin\\alpha_M\\sin\\Delta\\phi},\n\\end{equation}\n\\begin{equation}\n\\label{sinlam}\n\\sin\\lambda=\\pm\\frac{\\sin\\alpha_M\\sin(\\beta-\\beta_M)}{\\sin\\Delta\\phi}\n.\\end{equation}\nThe sign in Eq. (\\ref{sinlam}) depends on the specific orientation of $\\mathbf{r}$ and $\\mathbf{r_M}$ relative to the direction of zero-longitude. When the particle travels from $\\mathbf{r_M}$ to $\\mathbf{r}$ over the angle of $2\\pi-\\Delta\\phi$ , the signs of $\\sin\\lambda$ and $\\cos\\lambda$ both change because the sign of the $\\sin\\Delta\\phi$ term in the denominator changes.\n\nAs soon as the values of $(v,\\theta^*_i,\\lambda_i)$ that satisfy the orbital geometry are known, they may be used to calculate the corresponding $(u,\\psi_i,\\lambda_{Mi})$ from Eqs. (\\ref{ufromv}) -- (\\ref{coslambdaM}) and the integrand in Eq. (\\ref{workformula}) is fully determined,\n\\begin{equation}\n\\label{ufromv}\nu = \\sqrt{v^2_{escape}+2\\left(\\frac{v^2}{2}-\\frac{GM}{r}\\right)},\n\\end{equation}\n\\begin{equation}\n\\label{psi}\n\\sin\\psi = \\frac{rv}{r_M u}\\sin\\theta,\n\\end{equation}\n\\begin{equation}\n\\label{sinlambdaM}\n\\sin\\lambda_M=\\frac{\\sin\\alpha\\sin\\lambda}{\\sin\\alpha_M},\n\\end{equation}\n\\begin{equation}\n\\label{coslambdaM}\n\\cos\\lambda_M = \\frac{\\cos\\alpha -\\cos\\alpha_M\\cos\\Delta\\phi}{\\sin\\alpha_M\\sin\\Delta\\phi}\n.\\end{equation}\n\n\\subsection{Nonstationary case}\nNonstationary dust ejection can be modeled by allowing a time-dependent production rate $\\gamma$  in Eq. (\\ref{workformula})\nthat will result in a time-dependent spatial distribution of the dust. When we determine the orbital geometry for a fixed velocity value at the given point in space (Sect. \\ref{integration}), the time $\\Delta t$ required for traveling from $\\mathbf{r_{M}}$ to $\\mathbf{r}$ along the Keplerian orbit can be calculated from Kepler's equation. Thus, we know that the properties of the dust configuration at location $\\mathbf{r}$ and time $t$ are caused by the production of dust at the source location $\\mathbf{r_{M}}$ with the rate $\\gamma(t-\\Delta t)$.\nIn this case, the production rate cannot be put outside the integral and Eq. (\\ref{workformula}) becomes\n\\begin{multline}\n\\label{timedep}\n        n(r,\\alpha,\\beta,R_{min}<R<R_{max},t) = \\frac{1}{rr_M\\sin\\Delta\\phi} \\int_{v_{min}}^{v_{max}}\\mathrm{d}v\\frac{v}{u^2}G^p_u(R_{min},R_{max})\\times\\\\\\times\\sum_{i}\\gamma(t-\\Delta t_i)\\frac{f_{\\psi,\\lambda_M}(\\psi_i,\\lambda_{Mi})}{\\cos\\psi_i}\\abs*{\\frac{\\partial\\Delta\\phi}{\\partial\\theta}}_{\\theta^*_i}^{-1}.\n\\end{multline}\nThe two solutions for $\\Delta t_i$ follow from Kepler's equation using the two solutions for eccentricity from equation \\ref{eccentricity} and computing the eccentric anomaly with the half-angle formula from the true anomaly given by equation \\ref{difphi}.\n\n\n\\subsection{Singularities of the coordinates}\n\\label{spepoints}\nThere are coordinates for which solutions of equations (\\ref{workformula}) or (\\ref{timedep}) cannot be obtained (see Table \\ref{sppoints}). The problem arises from the use of a spherical coordinate system. In practice, it is possible to avoid the singularities by carefully choosing the pole axis of the coordinate system after the source location and the detector position of interest are known. If coordinates close to the singularities need to be evaluated, then the stability and accuracy of the numerical integration becomes a challenge. However, double-precision calculations allow approaching the singular angles as close as $10^{-4}$ radians. This difference from the values listed in Table \\ref{sppoints} can be considered safe, and with a possible moderate loss of accuracy, the model can be applied within an even closer vicinity of the singularities.\n\\begin{table*}\n        \\caption{Coordinate singularities of the model}\n        \\begin{tabular}{|c|c|}\n                \\hline\n                Variable value    &  Physical and geometrical meaning   \\\\    \n                \\hline\n                $\\alpha_M = 0^\\circ$ & Point source of dust is located at the north or south pole,  \\\\\n                $\\alpha_M = 180^\\circ$ & its latitude is not defined, and no spherical triangle from Fig. \\ref{sptr} exists\\\\\n                \\hline\n                $\\alpha = 0^\\circ$ & Spacecraft is located directly above the north or south pole, \\\\\n                $\\alpha = 180^\\circ$ & its latitude  is not defined, no spherical triangle from Fig. \\ref{sptr} exists\\\\\n                \\hline\n                $\\beta = \\beta_M$ & Spacecraft has the same longitude \\\\\n                ($\\Delta\\beta = 0^\\circ$) &as the source. No spherical triangle from Fig. \\ref{sptr} exists\\\\\n                \\hline\n                $\\Delta\\phi = 0^\\circ$& Spacecraft position, source position, and moon center \\\\\n                $\\Delta\\phi = 180^\\circ$ & are on a straight line. No spherical triangle from Fig. \\ref{sptr} exists\\\\\n                &  and the orbit geometry (Figs. \\ref{ellipse} and \\ref{hyperbola}) is undefined\\\\\n                \\hline\n                $\\psi = z$, & A particle is ejected exactly along\\\\\n                $z \\neq 0^\\circ$ &  the jet axis, the azimuth of ejection is undefined, and the Jacobian\\\\\n                &  in Eq. (\\ref{Jpsi}) diverges\\\\\n                \\hline\n        \\end{tabular}\n        \\label{sppoints}\n\\end{table*}\n\n\\section{Numerical integration}\n\\label{sec:numerics}\nIn this section we present and discuss the algorithm for the numerical solution of equations \\ref{workformula} and \\ref{timedep}. We have implemented this algorithm in a code written in Fortran-95, which we call dust distribution (DUDI). The source code with technical documentation and instructions for usage and compilation is freely available under the GNU General Public License on https:\/\/github.com\/Veyza\/dudi. \nThe \\textit{makefile} provided for compilation uses the \\textit{gfortran} compiler. DUDI can be compiled and run without installing additional libraries. The library of \\textit{OpenMP}, which is used to speed the computation up, is included in the compiler. \n\nDUDI allows us to compute the number density of dust or related quantities at given points in space as a mean radius,  average cross section, or mass density of the dust grains ejected from the surface of a spherical body without an atmosphere. The input data are the spacecraft coordinates, the properties of the source (i.e., the\\ location and the distributions of the direction and speed of the ejection) and the three parameters of $G^p_u$. \n\nThe first preliminary step is to calculate $G^p_u$ on a grid of $u$-values. The array of pairs $(u, G^p_u)$ is later used to interpolate $G^p_u(R_{min}, R_{max})$ for the actually required value $u$ under the integrand of equations \\ref{workformula} or \\ref{timedep}. The second preliminary step is to compute the values of $\\Delta\\phi$ and $\\Delta\\beta$ for the given positions of the source and the spacecraft.\n\nThen we proceed directly to the numerical evaluation of the integral over velocity in Eq (\\ref{workformula}) in the stationary case, or Eq. (\\ref{timedep}) if there is a time-dependence. The ejection speed distribution implies a certain lower limit for the possible ejection speed $u_{min}$. At any point $\\mathbf{r,}$ this minimum ejection velocity restricts the corresponding minimum velocity at spacecraft position $v_{min}^0$ , which may be higher than the lower integration limit given by Eqs. (\\ref{amin}) -- (\\ref{vmin}). The actual numerical integration is performed over the interval where $(v_{min},v_{max})$ and $(v_{min}^0,v_{max})$ overlap. The minimum and maximum ejection speed $u_{min}$ and $u_{max}$ (in m\/s) must be explicitly specified as a property of the dust source, along with the expression for the ejection speed distribution.\n\nAt each step of the integration, we find for a given $v$, $r$, and $\\Delta\\phi$ the solutions for the angles $\\theta$ and $\\lambda$ as described in Sect. \\ref{integration}. We control the accuracy of the solution for $\\theta$ by recalculating the value of $\\Delta\\phi$ from Eqs. (\\ref{wp})--(\\ref{dphi}) and comparing it to the starting value of $\\Delta\\phi$ as given by the positions of the dust source and the spacecraft. We require the difference between the two values of $\\Delta\\phi$ to be smaller than $10^{-4}\\Delta\\phi$. In most cases the accuracy is much better, but it may degrade for certain values of $v$ near the singularities of the coordinates (see Sect. \\ref{spepoints}) or when $r \\approx r_M$. Even for poor accuracy for $\\theta$ , this means that the accuracy decreases only for one or two integration steps. The error in the final result is smaller. We consider the relative accuracy of $10^{-4}$ sufficient for the $\\theta$ solutions. If it is worse, a warning is produced by the program.\n\n We divide the integration domain into two regions as $(v_{min},v_{par})$ and $(v_{par},v_{max})$ to obtain the number density of the particles on elliptic and hyperbolic (escaping) trajectories separately.  Here $v_{par}$ (the subscript stands for \"\\emph{\\textup{parabolic\"}}) is the minimum escape velocity at radial distance $r$. \n\n When $v_{min} > v_{min}^0$ , the integrand in Eq. (\\ref{workformula}) has a pole at $v = v_{min}$. We replace $v_{min}$ from Eq. (\\ref{vmin}) with $v_{min} + \\Delta$, where $\\Delta = 10^{-10}$ has turned out to be a good choice to evaluate the integrand near the pole to reasonable accuracy in a stable manner. To better resolve the pole, we additionally subdivide the elliptic part of the integral into two parts that are treated separately. The first integration subinterval is $(v_{min},v_{1})$, where $v_{1} = v_{min} + 10^{-4}(v_{par} - v_{min})$ is the domain that contains the pole. Here integration is performed using the trapezoidal rule with a large number of supports that become denser toward $v_{min}$ as\n \\begin{equation}\n \\label{vsteps}\n v_i = v_{min} + \\left(\\frac{i}{N}\\right)^k (v_{1} - v_{min})\n ,\\end{equation}\n where $N$ is the number of supports. Then we use the Gauss-Legendre quadrature of a moderate order to compute the integral from $v_{1}$ to $v_{par}$. The integration over the hyperbolic velocities is also performed with a Gauss-Legendre quadrature. The choice of the quadrature order depends on the integration domain and ejection speed distribution. The nodes and weights of the Gauss-Legendre formula are tabulated in our code for the following orders: 5, 10, 20, 30, 40, and 50. \n \n Figures \\ref{integrandpole} -- \\ref{integrand3} show examples for the behavior of the integrand in the three domains. Depending on the choice of the ejection speed distribution $f_u(u,R),$ the integrand may decrease (Fig. \\ref{integrand1}) or increase (Fig. \\ref{integrand2}) toward higher velocities. Remarkably, the integrand can jump at $v = v_{par}$ (Fig. \\ref{integrand3}) if a significant part of the dust number density is due to the particles on their way back to the moon after passage of their apocenter,\n\n\n\n\\begin{figure\n        \\centering\n        \\begin{minipage}{0.45\\textwidth}\n                \\centering\n                \\includegraphics[width=0.9\\textwidth]{integrand_vs_velocity_pole.pdf}\n                \\caption{Pole at $v = v_{min.}$\\\\\n                }\n                \\label{integrandpole}\n        \\end{minipage}\\hfill\n        \\begin{minipage}{0.45\\textwidth}\n                \\centering\n                \\includegraphics[width=0.9\\textwidth]{integrand_vs_velocity1.pdf}\n                \\caption{Integrand from Eq. (\\ref{workformula}) for an ejection speed distribution that favors low velocities.}\n                \\label{integrand1}\n        \\end{minipage}\\hfill\\\\\n        \\begin{minipage}{0.45\\textwidth}\n                \\centering\n                \\includegraphics[width=0.9\\textwidth]{integrand_vs_velocity2.pdf} \n                \\caption{Integrand from Eq. (\\ref{workformula}) for an ejection speed distribution that favors high velocities.\\\\\n                }\n                \\label{integrand2}\n        \\end{minipage}\\hfill\n        \\begin{minipage}{0.45\\textwidth}\n                \\centering\n                \\includegraphics[width=0.9\\textwidth]{integrand_vs_velocity3.pdf} \n                \\caption{Integrand from Eq. (\\ref{workformula}). A high abundance of dust falling back causes a jump at the transition from the elliptic to the hyperbolic case.}\n                \\label{integrand3}\n        \\end{minipage}\\hfill\n        \\captionsetup{labelformat=empty}\n\\end{figure}\n\n The sharpness of the pole varies. At each integration step, the given velocity $v$ determines the values of $\\theta$ that enter the integrand of equation \\ref{workformula} through the derivative $\\partial\\Delta\\phi \/ \\partial\\theta$ and $\\psi(\\theta)$. The latter is needed to compute the ejection angle distribution $f_{\\psi,\\lambda_M}(\\psi,\\lambda_M)$. The factor $\\partial\\Delta\\phi \/ \\partial\\theta$ is the reason for the pole. Its value depends on $\\theta$ and on the spacecraft position relative to the source. The pole is less strongly peaked if the value of $\\psi(\\theta)$ corresponds to a very unlikely ejection direction. Thus, the sharpness of the pole depends on the spacecraft position relative to the source position and also on the ejection angle distribution, and so does the number of integration steps required to achieve a given accuracy goal.\n\nThe accuracy can be estimated from Eq. (\\ref{eps}), where $P$ is the value of the pole integral (between $v_{min}$ and $v_1$) obtained with $N$ steps for the integration with the trapezoidal rule, and $N_{max}$ is the maximum reasonable number of steps. $N_{max}$ is limited by accumulated rounding errors, and we determine its value in test integrations. $I(N_{max})$ is the sum of $P(N_{max})$ and the remaining part of the integral between $v_1$ and $v_{max}$. In this way, we can quantify the discrepancy induced by the pole integration in the final result,\n\n\\begin{equation}\n        \\label{eps}\n        \\epsilon(N) = |P(N) - P(N_{max})| \/ I(N_{max})\n.\\end{equation}\n\nWe require $\\epsilon \\le 10^{-3}$ and perform tests to determine the corresponding number of pole integration steps $N$ necessary to achieve this goal. This number we compute for different spacecraft positions relative to the source and to the axis of ejection symmetry (the polar angle in the coordinate system $\\tilde{X} \\tilde{Y} \\tilde{Z}$ in Fig. \\ref{2sys}, in the following denoted by $\\xi$). We adopt \n\\begin{equation}\n\\label{psidistr}\nf_{\\tilde{\\psi}, \\tilde{\\lambda}_M}(\\tilde{\\psi}, \\tilde{\\lambda}_M)\\sin\\tilde{\\psi} = e^{-(\\tilde{\\psi}-\\tilde{\\psi}_{max})^2\/2\\omega^2} \\frac{\\sin\\tilde{\\psi}}{2\\pi}\n.\\end{equation}\nfor the ejection direction distribution. Normalization in this expression does not matter for an evaluation of $\\epsilon$ from Eq. (\\ref{eps}). We vary the parameters $\\psi_{max}$ and $\\omega$, along with the polar angle of the ejection symmetry axis, to investigate the behavior of the pole for different ejection distributions, of which two main classes can be defined. The \\textquotedblleft jets\\textquotedblright\\ are the sources with a preferred direction of ejection, and the \\textquotedblleft diffuse sources\\textquotedblright \\ have no such direction. \n We find that for diffuse ejection ($\\omega = 45^\\circ$ and $\\psi_{max} = 45^\\circ$), N = 15 is a sufficient number of supports to achieve $\\epsilon \\le 10^{-3}$ at any spacecraft position. For a vertical jet ($\\psi_{max} = 0^\\circ$ and $\\omega$ in the range of $3^\\circ$ and $5^\\circ$), the value of $P$ can be neglected for all $\\xi > 40^\\circ$ and $N = 15$ is sufficient for $\\xi < 40^\\circ$.\n \n However, for a narrow and inclined jet, we find that there are points where a large number of supports is required to integrate the pole accurately. The narrower and the more inclined the jet, the greater the number of these points and the greater the required $N$. We focus on the worst-case scenarios generally to constrain an optimal number of steps required for the pole integration.\n\nEmpirically, we find that an accuracy of $10^{-3}$ can be achieved with a minimum number of steps when an exponent of $k = 4$ is used in Eq. (\\ref{vsteps}). Figs. \\ref{accumap1} and \\ref{accumap2} show examples of how the minimum number of steps required to integrate the pole with the given accuracy is distributed over $r$ and $\\xi$ values. $N=0$ means that the pole does not have to be integrated at all because its value is negligible.\n\n\\begin{figure}[H]\n        \\centering\n        \\begin{minipage}{0.45\\textwidth}\n                \\centering\n                \\includegraphics[scale=0.5]{omega3deg_inc20.pdf}\n                \\caption{Minimum number of supports necessary to achieve an accuracy of $10^{-3}$ (equation \\ref{eps}) in the integration of the pole of the integrand (Fig.\\ref{integrandpole})  for the case of a narrow jet ($\\omega = 3^\\circ,\\ \\psi_{max} = 0^\\circ$), tilted by $z = 20^\\circ.$}\n                \\label{accumap1}\n        \\end{minipage}\\hfill\n        \\begin{minipage}{0.45\\textwidth}\n                \\centering\n                \\includegraphics[scale=0.5]{omega3deg_inc30.pdf}\n                \\caption{Minimum number of supports necessary to achieve an accuracy of $10^{-3}$ (equation \\ref{eps}) in the integration of the pole of the integrand (Fig. \\ref{integrandpole})  for the case of a narrow jet ($\\omega = 3^\\circ,\\ \\psi_{max} = 0^\\circ$), tilted by $z = 30^\\circ.$}\n                \\label{accumap2}\n        \\end{minipage}\\hfill\n        \n\\end{figure}\n\nFor jets, we select the number of supports $N$ for the pole integration\nbased on the distributions shown in Figs. \\ref{accumap1} and \\ref{accumap2} as follows: \n\\begin{equation}\n\\label{Nprestep}\nN = \\left\\{ \\begin{array}{cl}\n0,& \\xi > 45^\\circ\\ \\mathrm{or}\\ \\xi < 10^\\circ,\\ r\/r_M > 1.05,\\\\\n80,&\\xi > 45^\\circ\\ \\mathrm{or}\\ \\xi < 10^\\circ,\\ r\/r_M < 1.05,\\\\\n15 + 10z[\\mathrm{deg}],& 10^\\circ < \\xi < 45^\\circ, \\ r\/r_M <2,\\\\\n10 + 5z[\\mathrm{deg}],& 10^\\circ < \\xi < 45^\\circ, \\ r\/r_M >2.\\\\\n\\end{array}\\right.\n\\end{equation}\n\n\\section{Applications}\n\\label{sec:applications}\nIn this section we present three applications of the model to phenomena of scientific interest in the Solar System. The purpose of these examples is to demonstrate the wide range of applicability of the model. We leave a rigorous scientific analysis of these problems with a comprehensive comparison to data for future work.\n\n\\subsection{Density profile of the Enceladus dust plume}\n\\label{enceladusex}\nOn July 14, 2005, the Cassini spacecraft performed a flyby at the Saturnian moon Enceladus (labeled E2). During the flyby, the High Rate Detector (HRD), a subsystem of the Cassini Cosmic Dust Analyzer instrument \\citep{Srama:2004uz}, measured the number density of dust particles in the vicinity of the satellite. The significant increase in dust density near Enceladus (see Fig. \\ref{hrdex}), about one minute prior to the closest approach of the spacecraft to the satellite, was the first in situ measurement of particles in the Enceladus dust plume \\citep{2006Sci...311.1416S}. Dust and vapor are emitted from four fissures called the tiger stripes in the anomalously warm south polar terrain of Enceladus \\citep{2006Sci...311.1401S, 2006Sci...311.1393P}.\nA part of this dust escapes the moon gravity and forms the dusty E ring of Saturn \\citep{2009sfch.book..511H, 2018eims.book..195K}.\n\nFrom an analysis of high phase-angle images, \\citet{2014AJ....148...45P} suggested a list of 100 jets of dust emission for which the coordinates and tilts were derived from images (see also \\citet{2015Natur.521...57S}). To demonstrate an application of our model to the dust emission from Enceladus, we selected one single jet from this list with coordinates ($-80.25^\\circ$\\ N, $ 55.23^\\circ$\\ E), which is tilted by $5^\\circ$ from the surface normal in an azimuthal direction $38^\\circ$ away from local north. The ejection is stationary, so that the production rate $\\gamma(t)$ is constant. The distributions we implemented for particle sizes, ejection speed, and direction are given by \n\n\\begin{equation}\n\\label{lognorm}\nf_R(R) = \\frac{1}{\\sigma \\sqrt{2\\pi}}\\frac{1}{R}\\exp\\left(-\\frac{(\\ln R - \\mu)^2}{2\\sigma ^2}\\right)\n,\\end{equation}\n\n\\begin{equation}\n\\label{fu}\nf_u(u,R) = \\frac{R}{R_c}\\left(1 + \\frac{R}{R_c}\\right)\\frac{u}{u_{gas}^2}\\left(1 - \\frac{u}{u_{gas}}\\right)^{\\frac{R}{R_c}-1}\n,\\end{equation}\nand\n\\begin{equation}\n\\label{unicon}\nf_{\\tilde{\\psi}, \\tilde{\\lambda_M}}(\\tilde{\\psi}, \\tilde{\\lambda}_M)\\sin\\tilde{\\psi} = \\left\\{ \\begin{array}{rl}\n\\frac{\\sin\\tilde{\\psi}}{1-\\cos\\omega}\\frac{1}{2\\pi},\\ \\tilde{\\psi} \\le \\omega,\\\\\n0,\\ \\tilde{\\psi} > \\omega.\n\\end{array}\\right.\n\\end{equation}\n Equation (\\ref{fu}) was derived by \\citet{2008Natur.451..685S} to describe the acceleration of dust grains in the gas flux in the vents that supply the sources. Particles smaller than $R_c$ (measured in the same units as $R$) tend to accelerate up to the gas velocity ($u_{gas}$), while particles larger than $R_c$ are significantly slower. For this distribution, we have $u_{min} = 0,$ while $u_{max}$ is equal to the gas velocity $u_{gas}$. Table \\ref{enceladus_params} lists the parameters of the distributions and other parameters that are necessary to set up the model.\nFigure \\ref{hrdex} shows the result for the number density of dust obtained from the model for the single jet, evaluated along the trajectory of Cassini during the E2 flyby. The model was evaluated for grains with a radius larger that 1.6 micron, which corresponds to the size threshold for the HRD data shown in the plot. We multiplied the model profile by a factor so that the peak matches the measured peak density. To match the HRD profile at a large distance from the plume, we added a constant background of 0.01 particles\/$m^3$ to the model number density. The selection of the grain size in the model was realized by adjusting the function $G^p_u$ appropriately (see Table \\ref{enceladus_params}). We obtained the position of the spacecraft from the reconstructed spice kernels of the mission (https:\/\/naif.jpl.nasa.gov\/pub\/naif\/CASSINI\/kernels\/), using the NAIF Spice toolkit (https:\/\/naif.jpl.nasa.gov\/naif\/toolkit.html). Using these parameters, we recovered the location of the maximum number density on the Cassini trajectory (Fig. \\ref{hrdex}). Our model could now be applied to all the jets identified by \\citep{2014AJ....148...45P}, and the results could be fit to in situ\\emph{} data. Similarly, we could calculate the geometrical optical depths of the dust emitted along a given line of sight and compare this to the brightness distribution in images (see Sect. \\ref{ioex} for an example). For a quantitative comparison to images, we can apply light scattering modeling to the dust configuration that is derived from the dust distribution model.\n\\begin{table*}\n        \\caption{Parameters used to model the number density profile of the E2 flyby}\n        \\begin{tabular}{|c|c|}\n                \\hline\n                Parameter & Comment \\\\\n                \\hline\n        $\\gamma(t) = 1.35 \\cdot 10^{14} \\ s^{-1}$ & The dust ejection is stationary, the rate \\\\\n        & was chosen to fit the data\\\\\n        \\hline\n        Eq. (\\ref{lognorm}) with $\\mu = -1.0$ and $\\sigma = 1.5$  & Small particles dominate the population,\\\\\n        is used as a size distribution & but the distribution is not too steep\\\\\n        \\hline\n         Eq. (\\ref{fu}) with $R_c=0.5\\ \\mu m$  & This expression describes  \\\\\n         and $u_{gas} = 1000$ m\/s is used  & the dust acceleration by the gas flux \\\\\n         as the ejection speed distribution&inside the channels as it is at Enceladus\\\\\n         \\hline\n         Eq. (\\ref{unicon}) with $\\omega=10^\\circ$ is used& The distribution describes uniform emission \\\\\n         as the distribution  & into a cone of a given width\\\\\n         of the ejection direction&\\\\\n         \\hline\n         $R_{min} = 1.6 \\ \\mu m$ & The lower sensitivity threshold of the HRD \\\\\n         \\hline\n          & An arbitrarily chosen but reasonably  \\\\\n           &high value, meaning that we can neglect\\\\\n           & particles with larger sizes.  With our\\\\\n         $R_{max} = 6 \\ \\mu m$ & choice of size distribution and the size  \\\\\n         & -dependent ejection speed distribution, it \\\\\n          &is highly unlikely that the HRD detects \\\\\n          & a particle of this size at the altitude\\\\\n          &  of E2 flyby\\\\\n          \\hline\n         $p=0$ & We are interested in the number density \\\\\n         \\hline\n        \\end{tabular}\n        \\label{enceladus_params}\n\\end{table*}\n\\begin{figure}[H]\n        \\centering\n        \\includegraphics[width=0.5\\textwidth]{hrd_profile}\n        \\caption{Dust number density profile observed by the HRD during the E2 flyby of the Cassini spacecraft at Enceladus and results from modeling the emission from one single jet in the south polar terrain (see text for details).}\n        \\label{hrdex}\n\\end{figure}\n\n\n\n\\subsection{Surface deposition of material from plumes on Europa}\n\\label{europaex}\nThere is evidence that cryovolcanic activity also generates plumes on the Jupiter satellite Europa \\citep{2000Icar..144...54F, 2000JGR...10522579P, 2014Sci...343..171R,Sparks:2016gl,Jia:2018ii}. Owing to the higher gravity of Europa, plumes would be more confined than those on Enceladus, and they may be harder to observe or sample directly \\citep{2015GeoRL..4210541S, Quick:2020kp}. However, surface features from plume deposits may provide evidence for past cryovolcanic activity, and if spectral features exist, allow the remote characterization of material from the interior \\citep{2000Icar..144...54F,2000JGR...10522579P,Quick:2020kp}. \n\n\nWe calculated the radial variation of the mass flux of dust grains falling back onto the surface, considering four different dust sources with identical characteristics except for the particle size distribution and the distribution of ejection directions. We considered two size distributions and two ejection modes: one describing a narrow jet, and the other describing a broader, more diffuse ejection. For the size distributions we employed a power law\n\n\\begin{equation}\n\\label{pow}\nf_R(R) = \\frac{1-q}{R_2^{1-q}-R_1^{1-q}}R^{-q}\n,\\end{equation}\nwith two different values of the exponent $q$. The greater $q$, the more abundant the small dust particles. For the ejection direction, we used a pseudo-Gaussian distribution of the polar angle \n\\begin{equation}\n\\label{Gauss}\nf_{\\tilde{\\psi}, \\tilde{\\lambda}_M}(\\tilde{\\psi}, \\tilde{\\lambda}_M)\\sin\\tilde{\\psi} = C_{norm}e^{-(\\tilde{\\psi}-\\tilde{\\psi}_{max})^2\/2\\omega^2} \\frac{\\sin\\tilde{\\psi}}{2\\pi}\n,\\end{equation}\ngiving a nonzero probability of ejection in any direction. The normalization constant $C_{norm}$ was found numerically for fixed values of $\\psi_{max}$ and $\\omega$. The dust production rate $\\gamma$ was identical for all the sources, meaning that they produced the same number of dust particles per unit time, but we then obtained different mass production rates for the different size distributions. For the ejection speed distribution we again used Eq. (\\ref{fu}). Table \\ref{europa_params} lists the distribution parameters.\n\nWe can compute the rate of dust mass produced per second from the size distribution as\n\\begin{equation}\n        \\label{totmass}\n        \\frac{\\mathrm{d}m}{\\mathrm{d}t} =\\gamma\\rho\\frac{4\\pi}{3}\\int_{R_{min}}^{R_{max}}f_R(R)R^3\\mathrm{d}R,\n\\end{equation}\nwhere $\\rho$ is the density of the ice grains, and the grains were assumed to be spherical.\n\nTo compute flux instead of density, we must modify Eq. (\\ref{workformula}) as\n\\begin{multline}\n        \\label{flux}\n        n(r,\\alpha,\\beta,R_{min}<R<R_{max}) = \\frac{\\gamma}{rr_M\\sin\\Delta\\phi} \\int_{v_{min}}^{v_{max}}\\frac{G^p_u(R_{min},R_{max})}{u^2}\\times\\\\\\times \\sum_{i}\\frac{f_{\\psi,\\lambda_M}(\\psi_i,\\lambda_{Mi})}{\\cos\\psi_i}\\abs*{\\frac{\\partial\\Delta\\phi}{\\partial\\theta}}_{\\theta^*_i}^{-1}v^2|\\cos\\theta_i^*|dv.  \n\\end{multline}\nAs  the vector $\\mathbf{r}$ is normal to the surface of the satellite, the factor $v|\\cos\\theta|$ turns the density into the flux of dust falling back to the moon. Multiplying equation \\ref{flux} by $4\\pi\\rho\/3 $ ($\\rho$ is the density of the particle material) and setting $p=3$ and $r=r_M$ gives the mass flux onto the surface. Because $r = r_M$ on the surface, it is geometrically impossible to obtain $\\theta^*_i < \\pi\/2$ or any particles moving upward.\n\nWith the parameters from Table \\ref{europa_params}, Eq (\\ref{totmass}) gives $0.6$ kg of dust produced each second for the source with the shallow size distribution ($q=3$) and $0.05$ kg for the source with the steep size distribution ($q=5$). Figure \\ref{deposits} shows the distribution of the dust deposition (mass flux onto the surface) with distance from the source on the moon surface.\n \\begin{table*}\n        \\caption{Parameters used to model the radial distribution of dust deposition on the surface of Europa}\n        \\begin{tabular}{|c|c|}\n                \\hline\n                Parameter & Comment \\\\\n                \\hline\n        $\\gamma(t) = 10^{14} \\ s^{-1}$ & The dust ejection is considered stationary\\\\\n                \\hline\n        Eq.( \\ref{pow}) with $q = 3$, $R_1 = 0.2\\ \\mu m$& We paired these size and ejection \\\\\n          and $R_2 = 20\\ \\mu m$ as   & direction distributions\\\\\n           the shallow\\ size distribution & to model four sources: a narrow jet\\\\\n         and with the same boundaries, but $q=5$ & with a shallow size distribution,\\\\\n         as the steep\\ size distribution & a narrow jet with a steep size distribution,\\\\\n         & a diffuse source  with a shallow size\\\\\n         Eq. (\\ref{Gauss}) with $\\omega=5^\\circ$ and $\\psi_{max} = 0^\\circ$&  distribution, and a diffuse source\\\\\n         for the  narrow jet\\ and with $\\omega=45^\\circ$ & with a steep size distribution \\\\\n         and $\\psi_{max}=45^\\circ$ for the  diffuse source& \\\\\n        \\hline\n                Eq. (\\ref{fu}) with $R_c=0.5\\ \\mu m$  & This expression describes  \\\\\n                and $u_{gas} = 700$ m\/s is used&the acceleration of the dust by the gas  \\\\\n                 as the ejection speed distribution &flux inside the vents as it is thought be at Europa \\\\\n                \\hline\n & We wish to compute the total mass\\\\\n                $R_{min} = 0.2 \\ \\mu m$ &  produced by each source, \\\\\n                $R_{max} = 20 \\ \\mu m$ & therefore the interval $(R_{min},R_{max})$ \\\\\n &coincides with $(R_1,R_2)$\\\\\n                \\hline\n                $p=3$ & We are interested in the mass \\\\\n                \\hline\n                $\\rho = 920\\ kg\/m^3$ & Water-ice density\\\\\n                \\hline\n        \\end{tabular}\n        \\label{europa_params}\n \\end{table*}\n \\begin{figure}[H]\n        \\centering\n        \\includegraphics[width=0.5\\textwidth]{mass_deposition}\n        \\caption{Dust flux on the surface of the moon at various distances from the dust source. The production rate of the sources with shallow dust size distribution is 0.6 kg\/s, and the production rate of the sources with steep size distribution is 0.05 kg\/s.}\n        \\label{deposits}\n \\end{figure}\n\nTaking the observational evidence together, we expect the outgassing activity on Europa to be intermittent or episodic, with a potentially complex time dependence. This case could be modeled using a time-dependent function for the dust production rate. We consider a time-dependent dust ejection in our final example.\n\\subsection{Images of a volcanic eruption on Io}\n\\label{ioex}\nThe innermost satellite of Jupiter, Io, is a geologically very active body. It has multiple volcanic centers scattered over its surface \\citep{1979Natur.280..733S,2001JGR...10633025K, 2004Icar..169...29G}. The volcanic plumes vary in many features as shape, size and period of activity and often they are asymmetric. Our model allows to investigate such plumes by using inclined jets with time-dependent eruptions, or even a simultaneous dust ejection from sources with different properties.\n\nThe main flaw of our model in application to the Io volcanic plumes is that the gas emerging from the vents is generally not collisionless. In the dense cores of the plumes, condensation occurs up to a height of several hundred kilometers \\citep{2002JGRE..107.5109C}. Moreover, the dynamics of the dust will be influenced by the surrounding gas. Condensation (i.e.,\\ dust production) in the plume might be taken into account in principle by modeling such a plume with additional point sources placed within the condensation column, with appropriately adjusted time-dependent dust production rates. For this purpose, we implemented in the code the possibility of computing the dust density also at locations below (closer to the moon center) the location of the source. In this case, we still use the assumption that the true anomaly at the location of the source is lower than $\\pi$, that is, downward ejection from a source located above the surface is not possible. \n\nIn our simple example presented here, we considered just one source located on the surface to represent a volcanic plume with a finite duration of activity. We took several images of the expanding dust plume from the same point in space with a favorable geometry when the volcano is located at the limb of the moon. We also assumed that the glow of the surface of Io was already subtracted from the images but a homogeneous background brightness was present, so that the moon disk looks dark against this background. We tilted the volcano slightly by $3^\\circ$ toward local south. The volcano was not exactly in the center of the image for the purpose of reducing the computational difficulties arising when $\\Delta\\phi$ or $\\Delta\\beta \\approx 0$ (see Sect. \\ref{spepoints}). \n\nSynthetic images of size 128x128 pixels were then constructed in the following way. Each pixel corresponds to a line of sight. We placed a grid of points on the line of sight for which we calculated the total cross section covered by the dust grains ($p=2$ in Eq. (\\ref{Gu})). Then we integrated over the lines of sight to obtain the geometrical optical depth (total particle cross section per area covered by the pixel). The color of a pixel then corresponded to the value determined for the geometric optical depth.\n\nWe employed Eq. (\\ref{pow}) for the size distribution, Eq. (\\ref{Gauss}) for the distribution of the ejection direction, and a simple size-independent expression (Eq. (\\ref{fuun})) for a uniform ejection speed distribution,\n\\begin{equation}\n\\label{fuun}\nf_u(u,R) = \\frac{1}{u_{max} - u_{min}}\n.\\end{equation}\n\nWhen the time interval in which the plume is observed is much shorter than the characteristic time of the plume variability, the ejection can be considered steady. However, here we modeled a nonstationary volcanic plume that was active for only 1000 seconds and constructed nine snapshots of this plume to show the evolution of the space distribution of dust with time \n\\begin{equation}\n\\label{gammat}\n\\gamma(t) = \\left\\{ \\begin{array}{rl}\n\\gamma_0\\frac{-t^2+2t_{max}t}{t_{max}^2} ,\\ 0 < t < 2t_{max},\\\\\n0,\\ t < 0\\ \\text{or}\\ t > 2t_{max},\n\\end{array}\\right\n.\\end{equation}\nwhere $t_{max} = 500$ s. Table \\ref{io_params} describes the model setup, and the images are shown in Fig. \\ref{9im}\n\\begin{table*}\n        \\caption{Parameters used to construct the volcanic plume images.}\n        \\begin{tabular}{|c|c|}\n                \\hline\n                Parameter & Comment \\\\\n                \\hline\n                Eq. (\\ref{gammat}) with $t_{max} = 500$ s  & The volcanic plume \\\\\n                and $\\gamma_0 = 10^{14} \\ s^{-1}$ as the time- & ejected dust for 1000 s\\\\\n                dependent dust production rate& \\\\\n                \\hline\n                Eq. (\\ref{pow}) with $q = 3.0$, & \\\\\n                 $R_1 = 0.2\\ \\mu m$, and $R_2 = 20\\ \\mu m$  & \\\\\n                is used as the size distribution & \\\\\n                \\hline\n                The ejection speed is uniformly  & The narrow range of initial  \\\\\n                distributed between &velocities allows us to obtain \\\\\n                700 m\/s and 750 m\/s & the umbrella-shaped plume\\\\\n                \\hline\n                Eq. (\\ref{Gauss}) with $\\psi_{max} = 0^\\circ$ & The jets are very narrow,   \\\\\n                and $\\omega=5^\\circ$ is used as the distribution  &but there is a nonzero probability \\\\\n                of the ejection direction & of ejection in any direction\\\\\n                \\hline\n                $R_{min} = 0.2 \\ \\mu m$ & We observe the particles with the sizes \\\\\n                $R_{max} = 0.4 \\ \\mu m$ &  close to the optical wavelength range, \\\\\n                 & so that the radii are twice smaller\\\\\n                \\hline\n                $p=2$ & We are interested in the area \\\\\n                & covered by the particles \\\\\n                \\hline\n        \\end{tabular}\n        \\label{io_params}\n\\end{table*}\n\n\\begin{figure*}[h!]\n        \\centering\n        \\begin{minipage}{0.33\\textwidth}\n                \\centering\n                \\includegraphics[width=0.95\\textwidth]{5}\n        \\end{minipage}\\hfill\n        \\begin{minipage}{0.33\\textwidth}\n                \\centering\n                \\includegraphics[width=0.95\\textwidth]{10}\n        \\end{minipage}\\hfill\n        \\begin{minipage}{0.33\\textwidth}\n                \\centering\n                \\includegraphics[width=0.95\\textwidth]{15}\n        \\end{minipage}\\hfill\\\\\n        \\vskip0.1cm\n        \\begin{minipage}{0.33\\textwidth}\n                \\centering\n                \\includegraphics[width=0.95\\textwidth]{20}\n        \\end{minipage}\\hfill\n        \\begin{minipage}{0.33\\textwidth}\n                \\centering\n                \\includegraphics[width=0.95\\textwidth]{25}\n        \\end{minipage}\\hfill\n        \\begin{minipage}{0.33\\textwidth}\n                \\centering\n                \\includegraphics[width=0.95\\textwidth]{30}\n        \\end{minipage}\\hfill\n        \\vskip0.1cm\n        \\begin{minipage}{0.33\\textwidth}\n                \\centering\n                \\includegraphics[width=0.95\\textwidth]{35}\n        \\end{minipage}\\hfill\n        \\begin{minipage}{0.33\\textwidth}\n                \\centering\n                \\includegraphics[width=0.95\\textwidth]{40}\n        \\end{minipage}\\hfill\n        \\begin{minipage}{0.33\\textwidth}\n                \\centering\n                \\includegraphics[width=0.95\\textwidth]{45}\n        \\end{minipage}\\hfill\n        \\caption{Images of a fictive volcanic plume on Io taken at different stages of eruption.}\n        \\label{9im}\n\\end{figure*}\n\nCompared to the examples from Sects \\ref{enceladusex} and \\ref{europaex}, the construction of images is computationally expensive. Therefore it makes sense to avoid calculations of the properties in those points where the value would not affect the final result. In the case considered here, the maximum ejection velocity was set to be lower than the escape velocity of Io, so that there is a maximum height that the particles can reach. Moreover, the disk of the moon covers part of the images. Taking these two facts into account, we excluded the points on the lines of sight from the calculations that crossed the moon disk and the points that lie at a greater distance from the surface than the maximum height. \n\n \n\\section{Discussion}\nWe developed a semianalytical model that allows the user to derive the average properties of dust (number or mass density, fluxes, and optical depths) ejected from an atmosphereless body. Physically, our approach is based on the two-body problem, that is,\\ it neglects any forces on the dust particles other than the point-mass gravity of the source body. We show that the model still has a wide range of applications. These are situations where the gravity of other perturbing bodies and higher-order gravity from a nonspherical mass distribution of the central body can be neglected along the entire path that a dust particle takes from its source to its sink. The nongravitational forces must also be negligibly small, for instance, electromagnetic forces acting on charged grains, radiation-induced forces (solar radiation pressure and Poynting-Robertson drag), and drag exerted by ambient gas or plasma. For instance, the model can be applied to the Enceladus dust plume in a region that is sufficiently close to the dust sources on the south polar terrain of this satellite. It cannot be applied to estimate the dust density at higher northern latitudes of Enceladus, however, because this region is too far away from the sources near the south pole, and three-body forces due to Saturn have already affected the shape of the dust configuration. This can be seen in a peculiar pattern of color variation \\citep{2011Icar..211..740S}, that can be matched with two wedge shaped regions extending deep into the northern hemisphere \\citep{2011epsc.conf.1358S,2017LPI....48.2601S}, for which three body models of the plume predict enhanced fall back rates of south polar dust \\citep{2010Icar..206..446K}. The model can be applied to investigate the dust clouds around the Galilean moons \\citep{1999Natur.399..558K} at any longitude and latitude, however, because in this case, the dust emission occurs (nearly) uniformly over the whole surface of the satellites as long as the point of interest is located deeply enough in the Hill sphere of the satellite. \n\nMathematically, our model relates the dust distribution at the site of ejection to the distribution at the point of interest (spacecraft position) by using the conservation laws of energy and momentum provided by the two-body problem \\citep{2003P&SS...51..251K, Sremcevic:2003gf}. We use the fact that the position of the spacecraft, the position of the source, and the center of the moon define the plane to which the movement is restricted. In the evaluation of the dust properties at a given point, this allows us to carry out two of the three integrations over velocity space analytically. Only one remaining integration must then be performed numerically. The distribution-based approach is very flexible, and because it allows employing asymmetric and nonstationary modes of dust emission, it allows modeling quite complex situations. A time dependence could relatively easily be introduced in the distributions of ejection speed, direction, and size as well. Furthermore, the model could also be extended to model dust emission from a comet or an active asteroid, making the Sun the central body and using a rescaled solar mass to account for the radiation pressure. \n\n\nRelying only on one numerical integration, the model becomes computationally very efficient, so that even image reconstruction becomes feasible, although it involves the evaluation of the dust properties in the three-dimensional region that is in the field of view. From the examples discussed in Sect. \\ref{sec:applications}, the image of the volcano (Sect. \\ref{ioex}) is the most computationally demanding. Nevertheless, on a usual four-core PC, each image in Fig. \\ref{9im} required 0.4 s of elapsed time to be obtained.\n\n The model is implemented in Fortran-95, and the package called DUDI is available at \\\\\n https:\/\/github.com\/Veyza\/dudi for free usage under the terms of GNU General Public License. A user can use the  probability density functions and choose from several variants that are already implemented in the described examples, or they can implement new variants.  \n\n\\section*{Acknowledgements}\nThis work was supported by the Academy of Finland.\n\n\n\\section*{Appendix A. Replacement of the variable in the argument of Dirac's $\\delta$-function}\n\\label{replsec}\n\nLet $x\\in S \\subset \\mathbb{R}^n$; $f,g : S \\rightarrow \\mathbb{R}^n$. Then we can perform a replacement of the variable under the integral based on Eqs. (\\ref{dg}) and (\\ref{dint}) from \\citet{GelfShil},\n\\begin{equation}\n\\label{dg}\n\\delta(g(x)) = \\sum_{i}\\frac{\\delta(x-x_i)}{|g'(x_i)|}\n.\\end{equation}\nHere, $g'$ is the Jacobian matrix of the function $g$, $|g'|$ denotes the Jacobi determinant, and $x_i$ are the zeros of $g$,\n\\begin{equation}\n\\label{dint}\n\\int_{\\mathbb{R}^n}\\delta(g(x))f(g(x))|g'(x)|dx = \\int_{g(S)}\\delta(h)f(h)dh\n\\end{equation}\n\n\\begin{eqnarray}\n\\nonumber\n\\int_S \\delta(g(x))f(x)dx &=& \\int_S \\delta(g(x))f(g^{-1}(g(x)))dx\\\\\n\\nonumber\n&=& \\int_S\\delta(g(x))f^*(g(x))dx \\\\\n\\nonumber\n&=&\\int_S \\delta(g(x))\\frac{f^*(g(x))}{| g'(x)|}|g'(x)|dx \\\\\n\\nonumber\n&=&\\int_S \\delta(g(x))\\frac{f^*(g(x))}{| g'(g^{-1}(g(x)))|}|g'(x)|dx \\\\\n\\nonumber\n&=&\\int_S \\delta(g(x))f^*_D(g(x))|g'(x)|dx \\\\\n\\nonumber\n&=& \\int_{g(S)}\\delta(h)f^*_D(h)dh,\n\\end{eqnarray}\nwhere\n\\begin{equation*}\nf^*_D(h) = \\frac{f(x)}{\\abs*{g'(x)}},\\quad h = g(x).\n\\end{equation*}\nUpon integration, we obtain\n\\begin{equation*}\n\\int_{g(S)}\\delta(h)f^*_D(h)dh = \\sum_{i}\\frac{f(x_i)}{\\abs*{g'(x_i)}},\\ \\ g(x_i) = \\textbf{0}.\n\\end{equation*}\n\nIn our case, $n=2,$ the function $g$ performs a transformation from $(\\alpha_M,\\beta_M)$ to $(\\theta,\\lambda),$ and the function $f$ represents all the dependences on $\\theta$ and $\\lambda$ in the integrand in Eq. (\\ref{ndens}).\n\n\\section*{Appendix B. Physical meaning of the function $G_u$}\nFor each source, the first step in the numerical calculations is to obtain values of $G^p_u(R_{min}, R_{max})$ (Eq. (\\ref{Gu})) on a dense grid of $u$. This means that we perform integrations over the particle size $R$ treating $u$ as a parameter. The size distribution $f_R(R)$ can be defined on an interval of grain sizes, its lower and upper boundaries being parameters of the distribution. \n\nThese boundaries need not necessarily be equal to $R_{min}$ and $R_{max}$. Instead, $R_{min}$ and $R_{max}$ should be understood as limits of an observable interval of particle radii. Particles smaller than $R_{min}$ or larger than $R_{max}$ may exist, and they contribute to the normalization of $f_R(R)$, but they would not contribute to the value of $n$ (Eq. (\\ref{workformula})). This concept is illustrated in Figs. \\ref{GRexlognorm} and \\ref{GRexpower}. The quantity plotted in the two figures is $f_R(R)f_u(u,R)$. In Fig. \\ref{GRexlognorm} ,$f_R(R)$ is a lognormal distribution that is formally defined in the interval $(0,+\\infty)$, or in other words, $\\int_{0}^{+\\infty}f_R(R)dR = 1$, and the interval $(R_{min}, R_{max})$ does not cover the whole domain of $f_R(R)$. In Fig. \\ref{GRexpower}, the particle sizes are distributed between certain $R_1$ and $R_2$ as Eq. (\\ref{pow}). In this example, $R_{min} = R_1$ , but $R_{max} < R_2$. If $R_{min}$ were lower than $R_1$ , it would not have changed the result because for $R < R_1$ , we have $f_R(R) = 0$.\n\\begin{figure}\n        \\centering\n        \\begin{minipage}{0.45\\textwidth}\n                \\centering\n                \\includegraphics[width=0.9\\textwidth]{GR_lognorm_example.pdf}\n                \\caption{Integrand in Eq (\\ref{Gu}) with a lognormal size distribution and a fixed value of $u.$}\n                \\label{GRexlognorm}\n        \\end{minipage}\\hfill\n        \\begin{minipage}{0.45\\textwidth}\n                \\centering\n                \\includegraphics[width=0.9\\textwidth]{GR_powerlow_example.pdf}\n                \\caption{Integrand in Eq. (\\ref{Gu}) with a power-law size distribution and a fixed value of $u.$}\n                \\label{GRexpower}\n        \\end{minipage}\\hfill\n        \\captionsetup{labelformat=empty}\n\\end{figure}\n\nAlthough we defined the size distribution outside the interval $(R_{min}, R_{max}),$ its shape there does not affect the final result as long as it remains the same inside $(R_{min}, R_{max})$. However, we can investigate the defined size distribution by applying different values of $R_{min}$ and $R_{max}$. This approach implies that the size distribution and the initial speed distribution are physical characteristics of the dust source, while $R_{min}$ and $R_{max}$ represent the sensitivity range of the instrument with which we perform observations. In the model implementation DUDI, the interval $(R_{min},R_{max})$ can coincide with the $f_R(R)$ domain or can be even larger.\n\nThe units of the particle radius $R$ matter only in the definition of $f_R$ and $f_u$, so that we suggest measuring $R$ in microns to have simpler numbers in the expressions.\nWith different formulae for $f_R$ and $f_u$ , shorter expressions for $G^p_u$ can be obtained that require less cumbersome computations (see, e.g., \\citet{2011Natur.474..620P}). However, we purposefully consider a set of $f_R$ and $f_u$ in our model that cannot be simplified to show the general form of the solution and to allow flexibility in applying the model.\n\\newpage\n\\bibliographystyle{apalike}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\t\nStudies on the topology of band structures in ${\\bf k}$-space reveal various nontrivial phases of insulators and semimetals~\\cite{Hasan10,Qi11,Hirayama18r}.\nIn recent years, higher-order topological phases have attracted high attention,\nwhere higher-order topological states appear due to nontrivial bulk topology.\nThere are two types of higher-order topological phases: those supporting currents~\\cite{Langbehn17,Song17,Schindler18,Schindler18b,Tang18} and those supporting charges~\\cite{Benalcazar17,Benalcazar17b,Peterson18,Benalcazar19,Schindler19}.\nOne of the realistic materials having chiral hinge states is bismuth.\nSuch chiral hinge states have been confirmed by the scanning tunneling microscope (STM)~\\cite{Schindler18b}.\nThe other example is Bi$_4$Br$_4$, where the bulk is completely insulating unlike bismuth~\\cite{Tang18}.\nOn the other hand, there has been no proposal for a realistic three-dimensional material having higher-order topological states supporting quantized charges.\n\nHere, we focus on electride as a system that can realize a higher-order topological phase.\nIn electrides, electrons enter the voids surrounded by cations\nand are responsible for stabilizing the structure as anions~\\cite{Ellaboudy83,Singh93,Sushko03,Matsuishi03}.\nInterstitial electrons are not strongly stabilized by the electric field from positive ions,\nas compared to electrons belonging to atomic orbitals stabilized by the strong electric field from the nuclei.\nTherefore, the work function of an electride is generally small~\\cite{Singh93,Toda07}, and electrides are actively studied in the chemistry field such as catalyst~\\cite{Kitano12}.\nCompared with covalent crystals originating from $p$ orbitals, there is less structural instability in electrides when making the cleavage plane~\\cite{Hirayama18}.\n\nIn recent years, it has been proposed that electrides have a high affinity with topological phases in physics~\\cite{Hirayama18}.\nSince the interstitial electrons lead to a small work function, the bands originating from interstitial electrons appear near the Fermi level.\nTherefore, electrides are likely to have band inversion.\nActually, various topological phases with\/without the spin-orbit interaction (SOI) are proposed in the electrides~\\cite{Hirayama18,Huang18,Zhang18,Park18}.\nFor example, a two-dimensional electride Sc$_2$C is a nontrivial insulator having a quantized Zak phase $\\pi$~\\cite{Hirayama18}.\nBecause anionic electrons exist in the two-dimensional regions between Sc$_2$C layers, floating topological charges protected by the bulk topology appear on the surface.\n\n\nIn this letter, we propose an apatite electride as a higher-order topological crystalline insulator (TCI).\nThe apatite electride is a one-dimensional electride, having anionic electrons in one-dimensional hollows surrounded by cations~\\cite{Zhang15}.\nWe show that quantized topological charges protected by the $C_6$ symmetry appear at the hinge.\n\n\\begin{figure}[htp]\n\t\\centering \n\t\\includegraphics[clip,width=0.45\\textwidth ]{fig1.pdf} \n\n\n\t\\caption{Bulk structure and symmetry of the topological apatite A$_6$B$_4$(SiO$_4$)$_6$.\n\t\t(a) Crystal structure. \n\t\tThe green, brown, black and gray balls represent A, B, Si and O atoms, respectively.\n\t\t(b) One-dimensional hollow in the crystal structure.\n\t\t(c) Symmetry of the unit cell.\n\t}\n\t\\label{str}\n\n\\end{figure} \n\n\n\\begin{figure*}[htp]\n\t\\includegraphics[width=15cm]{fig2.pdf}\n\t\\caption{\n\t\tElectronic band structure.\n\t\t(a) (b) Electronic band structure of the La, Y apatite, respectively. \n\t\tThe energy is measured from the Fermi level.\n\t\t(c) Brillouin zone for bulk and the ($01\\bar{1}0$) surface.\n\t\t(d-f) Eigenfunctions at the $\\Gamma$ point.\n\t\tThe blue and yellow represent the positive and negative components of the wave function, respectively.\n\t\tThese eigenstates are marked in the band structure in (a).\n\t\t(g) Electronic band structure of the Wannier functions in the La apatite.\n\t\tThe four Wannier functions are at Wyckoff position $4e$.\n\t\tThe blue solid line is the Wannier band and the black dotted line is the Kohn-Sham band for comparison.\n\t\tThe band in the inset originates from the twelve Wannier functions at Wyckoff position $12i$.\n\t\tThe energy is measured from the Fermi level.\n\t}\n\t\\label{bulk}\n\\end{figure*} \n\n\n\nWe calculate the band structures within the density functional theory.\nThe electronic structure is calculated using the generalized gradient approximation (GGA).\nWe use VASP for the lattice optimization. \nThe energy cutoff is 50 Ry for the calculation by VASP~\\cite{vasp}.\nWe use the \\textit{ab initio} code OpenMX for the calculation of the electronic structure~\\cite{openmx}. \nand the energy cutoff for the numerical integrations is 150 Ry for the calculation by OpenMX.\nThe $6\\times 8\\times 6$ regular ${\\bf k}$-mesh is employed for the bulk.\nWe construct Wannier functions from the Kohn-Sham bands, using the maximally localized Wannier function~\\cite{Marzari97,Souza01}.\nThe density of states on the ($01\\bar{1}0$) surface is calculated by the recursive Green's function method~\\cite{Turek97}.\nIn the slab (wire) calculations, we tune the on-site potential of the Wannier function at the surface (hinge) to satisfy the charge neutrality condition, respectively.  \nWe confirm the effect of the charge neutrality and the long-ranged Coulomb interaction by first-principles calculations (see Supplementary Material~\\cite{SM6}).\n\t\n\t\n\t\n\t\n\t\n\n\nFigure~\\ref{str}(a) shows the crystal structure of the apatite electride A$_6$B$_4$(SiO$_4$)$_6$.\nCrystals with apatite structure are so stable that they appear in nature. The most well-known one is\nhydroxyapatite Ca$_{10}$(PO$_4$)$_6$(OH)$_2$, \nthat is the main constituent of bones and teeth.\nIt has a hexagonal crystal structure having a cleavage plane of hexagonal shape around the (0001) direction~\\cite{Aizawa05}.\nThere are no atoms on the boundary of the unit cell except for the top and bottom planes.\nSince the apatite has a one-dimensional structure, it is used in the study of ionic conductors.\nExperimentally, the apatite electride A$_6$B$_4$(SiO$_4$)$_6$ can be synthesized from A$_6$B$_4$(SiO$_4$)$_6$O$_2$ by releasing two oxygen atoms located in a region surrounded by cations.\nThe apatite electride has a one-dimensional hollow along the $z$-axis as shown in Fig.~\\ref{str}(b).\nThe positive ions in the hollows are located at $z=1\/4$ and $3\/4$, where the lattice constant \nalong the $z$-direction is set to be unity.\nSince the Wannier orbitals in the interstitial region are $s$-orbital-like and there is no nucleus inside the interstitial region,\nthe SOI of the interstitial electron is very weak.\nIn the following, the apatite refers to the apatite electride, unless otherwise specified.\n\n\n\nFigure~\\ref{str}(c) is the spatial symmetry of the apatite.\nThe space group of the apatite is No. 176 ($P6_3\/m$).\nOne of the space inversion centers lies at $(x, y, z) = (0, 0, 0)$ of the unit cell.\nIt also has $C_6$ screw rotation symmetry with $1\/2$ translation along the $c$-axis with its screw axis at the hollow $(x, y) = (0, 0)$.\nThe unit cell consists of two triangular prisms, which are transformed to each other by space inversion.\nEach triangular prism region has the $C_3$ rotational axis around the $c$-axis at the center.   \nThe system also has mirror symmetry perpendicular to the $c$-axis.\n\n\n\\begin{figure*}[htp]\n\t\\includegraphics[width=15cm]{fig3.pdf}\n\t\\caption{Surface state.\n\t\t(a) ($01\\bar{1}0$) slab. The purple and the red circles represent the hollows along the $c$-axis supporting the bulk and surface bands, respectively, and the numbers in the figure represent\n\t\tthe fillings (i.e. the number of filled bands) in the bulk and the surface. \n\t\t(b) Electronic band structure of the La apatite for the ($01\\bar{1}0$) surface.\n\t\tThe energy is measured from the Fermi level.\n\t\t(c)  Electronic band structure of the La apatite for the ($01\\bar{1}0$) surface subtracted the bulk contribution.\n\t}\n\t\\label{surface}\n\n\\end{figure*}\n\n\nFigures~\\ref{bulk}(a) and (b) show electronic band structures of La and Y apatites, respectively.\nThe corresponding Brillouin zone is shown in Fig.~\\ref{bulk}(c).\nWhile the A site is La$^{3+}$\/Y$^{3+}$,\nthe B site is completely dissolved with La$^{3+}$\/Y$^{3+}$ and another positive ion or void and has $+2.5e$ charge on average.\nIn our numerical calculation, such mixed states are handled by the virtual crystal approximation.\nThe La apatite is an insulator, while the Y apatite has a semimetallic band structure.\nReflecting a quasi one-dimensional crystal structure, they have a large band dispersion along the $k_z$ direction.\nBoth the valence band and the conduction band near the Fermi level originate from the interstitial electron as shown in Figs.~\\ref{bulk} (d)-(f).\nThe corresponding Wannier bands for the interstitial electron are shown in Fig.~\\ref{bulk}(g).\nAs shown in Supplementary Material~\\cite{SM1}, we find that many topological materials such as Na$_3$Bi~\\cite{Wang12} and Ca$_3$P$_2$~\\cite{Xie15,Chan16} are topological electrides in which either the bottom of the conduction band or the top of the valence band originates from the interstitial orbital.\nNo topological material has been known so far in which the band inversion occurs between the interstitial orbitals, and apatite is a first example of such a case.\nWe discuss the irreducible representation (irrep) for the occupied bands in Supplementary Material~\\cite{SM2}.\nThe detail of the Wannier function is also shown in Supplementary Material~\\cite{SM3}.\n\n\n\nWe show our results on surface states of the La apatite. \nWe consider the slab with the $(01\\bar{1}0)$ surfaces as shown in Fig~\\ref{surface}(a),\nwhich preserves space inversion and $C_2$ symmetry.\nFigure~\\ref{surface}(b) is the surface states calculated\nusing the recursive Green's function.\nThe effect of the lattice optimization is shown in Supplementary Material~\\cite{SM2}.\nThe surface states are gapped similarly to the bulk states.\nThis gapped surface state means that the Zak phase of the bulk is zero~\\cite{Hirayama18r,Vanderbilt93,Hirayama17}.\nThe detail of the Zak phase is discussed in Supplementary Material~\\cite{SM4}.\nTwo spinless states are occupied in the hollow in the bulk, and one spinless state is occupied at the surface (Fig.~\\ref{surface}(a)), excluding the double degeneracy due to the spins.\nBecause the filling of the surface bands is an integer, the surface bands are gapped, as seen from Fig.~\\ref{surface}(c).\n\n\n\nNext we show the hinge band structure of the La apatite.\nHere, we consider a hexagonal prism, which preserves the space inversion and the $C_6$ symmetry (Fig.~\\ref{hinge}(a)).\nFigures~\\ref{hinge}(b),(c) shows the one-dimensional band structure of the La apatite prism.\nWhile the band structure is gapped in the bulk and the surface, as is shown in Fig.~\\ref{surface}(b), topological gapless hinge states appear at the Fermi level.  \nSuch hinge states are protected by nontrivial topology of the bulk La apatite, \nand is characterized as a higher-order topological insulator protected by a rotational symmetry, \nproposed in Ref.~\\cite{Benalcazar19}. \nIn Ref.~\\cite{Benalcazar19}, to describe this higher-order topology, integer topological invariants for the high symmetry $\\Pi$ point is defined in terms of \nthe $C_{n'}$ eigenvalues at high-symmetry points $\\Pi$ compared to those at the $\\Gamma$ point\n\\begin{align}\n\t[\\Pi _p^{(n')}]=\\# \\Pi _p^{(n')} -\\# \\Gamma _p ^{(n')},\n\t\\label{eq:pi}\n\\end{align}\nwhere $C_{n'}$ is a local symmetry at the high-symmetry point $\\Pi$, $\\Pi _p^{(n')}$ ($=e^{2\\pi i(p-1)\/n'}$)  is the $C_{n'}$ eigenvalue labeled by $p$ at the high-symmetry point denoted by $\\Pi$, and $\\# \\Pi _p^{(n')}$ is the number of the occupied bands having the eigenvalue $\\Pi _p^{(n')}$.\nIn Ref.~\\cite{Benalcazar19}, it was proposed that \nthe topological class of the two-dimensional TCI having the $C_6$ symmetry is characterized by $\\chi ^{(6)}=([M_1^{(2)}],[K_1^{(3)}])$.\nBy using the connections between the positions of the Wannier orbitals and the irrep at high-symmetry points, the corner charge is given in terms of the topological numbers by \n\\cite{Benalcazar19}\n\\begin{align}\n\tQ_{\\rm corner}^{\\text{H}(6)}=-\\frac{|e|}{4}[M_1^{(2)}]-\\frac{|e|}{6}[K_1^{(3)}]\\ \\text{mod}\\ e,\n\t\\label{eq:Qcorner}\n\\end{align}\nwhere we adopt the convention $e=-|e|<0$, following Ref.~\\cite{Benalcazar19}. \nThe rotational symmetry along the $c$-axis in the apatite is $C_6$ screw rotation $6_3$.\nSince the apatite is insulating, the eigenvalues for the rotational symmetries are same between $k_z=0$ and $k_z=\\pi$ of the high-symmetry points,\nand we can limit our discussion to the irrpdf at $k_z=0$ for characterization of the hinge charge.\nTherefore, the $6_3$ helically symmetric system is essentially the same as the 6-fold symmetric system in terms of the topological charges.\n\n\n\n\n\\begin{figure*}[htp]\n\t\\includegraphics[width=15cm]{fig4.pdf}\n\t\\caption{Topological hinge states and higher-order TCI phase in the apatite electride.\n\t\t(a) Crystal with a shape of a hexagonal prism along the $c$ axis. Along the $a$,$b$ directions, it consists of triangular blocks\n\t\tand electrons are located at the Wyckoff position $a$. \n\t\tThe purple, red and orange circles represent the hollows along the\n\t\t$c$-axis supporting the bulk, surface, and hinge bands, respectively, and the numbers in the figure represent\n\t\tthe fillings (i.e. the number of filled bands) of the hollows in the bulk, surfaces and hinges. \n\t\t(b)(c) Electronic structure of the $120^\\circ $ hinge of the La apatite.\n\t\tThe energy is measured from the Fermi level. The high-symmetry wavenumbers are $\\tilde{\\Gamma}=0$ and $\\tilde{Z}=\\pi$.\t\t\n\t\tIn the magnified figure (c) the bands originate from the surface and hinge are colored in orange and yellow, respectively. \n\t\t(d) Maximal Wyckoff positions in the unit cell consisting of hexagonal blocks.\n\t\t(e) Maximal Wyckoff positions in the unit cell consisting of triangular blocks.\n\t\t(f) Relationship between the hexagonal and triangular blocks.\n\t\t(g) Crystal consisting of hexagonal blocks. In contrast with (a), the Wannier orbitals at the Wyckoff position $a$ do not lead to hinge states.\n\t}\n\t\\label{hinge}\n\\end{figure*}\n\n\nNonetheless, this formula (\\ref{eq:Qcorner}) does not apply to our system. Equation (\\ref{eq:Qcorner}) is derived for a crystal consisting of \nhexagonal blocks (Fig.~\\ref{hinge}(d)), with the sixfold rotation axis at the center of the hexagon, and Eq.~(\\ref{eq:Qcorner})\nrepresents a corner charge for a 2D crystal with a hexagonal shape, composed of the \nhexagonal blocks (Fig.~\\ref{hinge}(g)).\nMeanwhile, in the present case, if we focus on the crystal shape along the $ab$ plane, the fundamental unit is a regular triangle, rather than a hexagon.\nThe whole crystal forming a hexagonal prism is composed of the triangular blocks, and the unit cell consists of \ntwo triangular blocks (Fig.~\\ref{hinge}(e)).\nThis choice of the fundamental units of the crystal makes a\ndifference in the corner charge. In the present case, the Wannier orbital is at the hollow sites, located at the Wyckoff position $1a$.\nIn this case the representations are trivial, and the same between the high-symmetry points, and $[M_1^{(2)}]=0$ and $[K_1^{(3)}]=0$. This leads to an absence of the corner charge if we use Eq.~(\\ref{eq:Qcorner})\nwhich is natural because the Wyckoff position $1a$ is at the center of the\nhexagon. \nNonetheless,\nit is not valid,\nbecause the fundamental blocks are triangular and this Wyckoff position $1a$ is at the corners of the triangle (Fig.~\\ref{hinge}(f)), giving rise to a nonzero quantized corner charge as we show later. \nThus, even for trivial representations, one can have a nonvanishing quantized corner charge. \n\nFor a triangular block,\nwe show maximal Wyckoff positions in Fig.~\\ref{hinge}(e). For each Wyckoff position, \none can calculate the corner charge and irreducible representations at high-symmetry points\nsimilarly as in Ref.~\\cite{Benalcazar19}. \nThrough some calculations whose details are given in Supplementary Material~\\cite{SM5}, \nthe corner charge for $C_6$-symmetric 2D systems is given by  \n\\begin{align}\n\tQ^{\\text{T}(6)}_{\\text{corner}}\n\t=\\frac{|e|}{4}[M^{(2)}_1]\n\t+\\frac{|e|}{3}[K^{(3)}_1]\n\t+\\frac{|e|}{3}\\nu\\ \\text{mod}\\ e,\n\t\\label{eq:Qcorner2}\n\\end{align}\nwhere $\\nu$ is a number of electrons per unit cell, i.e. the number of occupied bands. \nIt means that the filling of the corner states is $Q_{\\text{corner}}\/|e|$, and it is well defined only when the \nbulk and the surface are gapped. In the present case, the surface is gapped only when $\\nu$ is even. \nThis equation is quite different from \n(\\ref{eq:Qcorner}) in that it depends on $\\nu$. \nWe now apply this theory to the apatite. We consider a hexagonal prism, having a structure within the $ab$ plane as shown in Fig.~\\ref{hinge}(a).\nLet us consider the whole system as a two-dimensional system, and consider only the interstitial electrons. \nThen we have\n$[M_1^{(2)}]=0$, $[K_1^{(3)}]=0$, and $\\nu=2$,\ngiving rise to \n$Q_{\\rm corner}^{\\text{T}(6)}=\\frac{2|e|}{3} \\ \\text{mod} \\ e$.\nFrom the charge neutrality condition, the filling of the topological hinge state is $\\frac{2}{3}$.\n\n\n\n\n\t\n\t\n\t\n\nTo summarize, \nwe show that the apatite electrides are higher-order topological insulators, having topological gapless hinge states, while the bulk and the \nsurface are gapped. The hinge states reside at interstitial sites forming  one-dimensional hollows in the electride. \nIn the bulk and the surface, these interstitial sites are integer-filled, but at the 120${}^\\circ$ hinge, they are $2\/3$ filled due to the \ntopological properties of the bulk occupied bands. \nRealization of the topological bands stemming from the band inversion, together with the floating hinge states at the hinges \noriginates from the characteristics as elelctrides.\n\n\t\n\n\n\t\n\t\n\\begin{acknowledgments}\n\t\n\nThis work was supported by JSPS KAKENHI Grant Number 18H03678,  and by the MEXT Elements Strategy Initiative to Form Core Research\nCenter (TIES).\n\\end{acknowledgments}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe primary target of objective image quality assessment (IQA) is to automatically predict the perceptual visual quality, providing a cost-effective alternative for the cumbersome subjective user study~\\cite{athar2019comprehensive}. Full-reference IQA (FR-IQA) feeds the pristine image ${\\bf x}$ and the counterpart distorted image ${\\bf y}$ into different perceptual distance measures. The predicted quality score is used to evaluate the image processing system and optimize various real-world applications, such as image compression, restoration, and rendering. The FR-IQA models can be summarized from a Bayesian perspective~\\cite{duanmu2021quantifying}:\n\\begin{equation}\\label{eq:gauss_assump_fr}\n    p(s | {\\bf x}, {\\bf y}, \\bm{\\theta}, \\beta) = \\mathcal{N}(s | d({\\bf x}, {\\bf y}; \\bm{\\theta}), \\beta),\n\\end{equation}\nwhere $s$ is the subjective quality rating which is assumed to follow a Gaussian distribution with the mean $d({\\bf x}, {\\bf y}; \\bm{\\theta})$ and variance $\\beta$. Herein, we denote the $d({\\bf x}, {\\bf y}; \\bm{\\theta})$ as the perceptual distance measured between ${\\bf x}$ and  ${\\bf y}$, with the $\\theta$ encoding the prior knowledge of  human vision system (HVS). In the last decades, efforts have been mainly devoted to exploring a meaningful and powerful parameter distribution $p\\bm{(\\theta)}$, aiming to achieve a fully perceptual consistent measure. Those models can be classified into knowledge-driven and data-driven approaches.\n\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[trim={0cm 0cm 0cm 0cm},clip, width=.99\\textwidth]{.\/figs\/fig_1.png\n \\caption{ Illustration of the contrastive preference by human and existing quality measures.  Left: Human and DID prefer the image (c) from generative adversarial network (GAN)~\\cite{navarrete2018multi} over the JPEG2000 distorted image (a), but the distance-based measure (PSNR, SSIM, LPIPS, and DISTS) prefer image (a) over (c); Right: Human, DISTS, and DID prefer the resampling texture image (f) over the pink noise image (d), but the PSNR, SSIM, and LPIPS prefer image (d) over (f). The better quality scores are highlighted in boldface.\n }\n \\label{fig:sameples}\n\\end{figure*}\n\nThe \\textit{knowledge-driven approaches} have dominated the  FR-IQA models for more than a  half-century. Mean squared error (MSE) is one of the most popular error visibility methods owing to its simplicity, clear physical meaning and desired properties for optimization, but it shows poor correlation with the HVS~\\cite{wang2009mean}. Afterward, methods that correlate better with HVS were developed, such as the structural similarity index (SSIM)~\\cite{wang2004image}, the visual information fidelity (VIF)~\\cite{sheikh2006image}, and the normalized Laplacian pyramid distance (NLPD)~\\cite{laparra2016perceptual}. Recently, the \\textit{data-driven approaches} are prevailing due to the perceptual meaningful characteristic of deep pre-trained convolution neural network (CNN), denoted as $\\tilde{\\bf x} = h(\\bf{x}, \\bm{\\theta}_c)$ and $\\bm{\\theta}_c$ is the parameters of network $h(\\cdot)$. In the deep learning feature domain, various distance measures have been developed, \\textit{i}.\\textit{e}., $d(\\tilde{\\bf x}, \\tilde{\\bf y}; \\bm{\\theta}_c)$. For the element-wise methods, Johnson~\\textit{et al}.~ constructs the perceptual loss by computing the weighted summation of $\\ell_2$-norm distance in both image and feature domains~\\cite{johnson2016perceptual}. The learned perceptual image patch similarity (LPIPS)~\\cite{zhang2018unreasonable} calculates the weighted MSE result with corresponding deep representations, attempting to account for the ``unreasonable'' effectiveness. Ding~\\textit{et al}.~ computes the global or local deep feature statistics~(\\textit{i}.\\textit{e}., mean and variance) to unify the structure and texture similarity~(DISTS)~\\cite{ding2020image,ADISTS}. In addition, the internal feature dependency (\\textit{i}.\\textit{e}., distribution-wise) comparisons also play an essential role in perceptual visual quality predictions. Representative examples include style loss~\\cite{gatys2016image}, deep self-dissimilarity~\\cite{kligvasser2021deep}, and deep Wasserstein distance (DeepWSD)~\\cite{liao2022deepwsd}. \n\n\n\n\n\n\n\n\nThe knowledge-driven and data-driven approaches primarily rely on the deterministic comparisons of the images in various domains. The deterministic comparisons based on distance metrics, though faithfully reflecting the fidelity, may fail when the test images are generated instead of physically acquired. This can be attributed to the rooted view that images with perfect quality can be feasibly modeled as the output of a stochastic source. One example can be observed in Fig.~\\ref{fig:sameples}, where the image (c) with pleasant texture  is disfavored by the distance-based measures, due to the large distance caused by the generated textures. In addition, as shown in the right case of Fig.~\\ref{fig:sameples}, human perception is usually invariant to the texture resampling even the resampling brings distance boosting in both signal space and feature space. \nThose phenomenons inspire us to bring the shift from the traditional distance-based FR-IQA paradigm, to the dependency modeling from statistical perspective. In particular, though statistical models dominate the no-reference IQA methods, much less work has been dedicated to characterizing the statistics for FR-IQA. More importantly, statistical and perceptual modeling of visual signals are broadly recognized as the dual problems~\\cite{sheikh2005information}.\nAs such, the shift is grounded on the mild assumption that the perception of texture variation can be well reflected by a reliable feature dependency measure.\n\nIn this work, we adopt the Brownian Distance Covariance (BDC) as the ideal feature dependency measure for FR-IQA, and propose the Deep Image Dependency (DID)\nbased FR-IQA model. The BDC is defined as the weighted Euclidean distance between the joint characteristic function and the product of the marginal characteristic functions~\\cite{szekely2009brownian}. The proposed DID model presents several desired advantages. First, the model is able to  naturally capture the feature dependency against both linear and non-linear transformations. Second, in the deep feature space, the model does not rely on any trainable parameters, demonstrating promising flexibility and generalization capability. \n{Third, the model presents superior performance on texture perception, no matter the texture is artificially generated, randomly reasmpled and geometrically transformed.}\nExtensive experiments based upon classical IQA datasets,  texture similarity datasets, and geometric transformation  dataset demonstrate that DID achieves state-of-the-art performance according to the correlation with mean opinion scores (MOSs). \nIt is also worth mentioning that though DID obtains competitive performance in quality evaluation tasks, it is independent of the training data (\\textit{i}.\\textit{e}., MOSs) and does not contain any controllable parameters. \n\n\n \n\n\n\n\n\\section{Related Work}\n\\subsection{Full-reference Image Quality Assessment}\nThe early works for FR-IQA capture the distortion relying on signal fidelity measure \\textit{e}.\\textit{g}., MSE, and peak signal-to-noise ratio (PSNR). Although those works enjoy the calculation simplicity and mathematical convenience, the low consistency with human perception has been widely criticized \\cite{lin2003discriminative}. In \\cite{wang2004image}, the SSIM was proposed by introducing three important components, including luminance, contrast, and structural similarity. This work was extended to several more advanced quality measures, such as MS-SSIM \\cite{wang2003multiscale}, IW-SSIM \\cite{wang2010information} and CW-SSIM \\cite{wang2005translation}. In the deep-learning era, pioneering works concentrate on image comparison in the deep learning feature space. For example, in  LPIPS \\cite{zhang2018unreasonable}, the multi-scale features were extracted from the pre-trained VGG  \\cite{simonyan2014very} network and the image quality is estimated by measuring the feature fidelity loss with Euclidean distance. Analogously, the combination of spatial averages and  correlations of the feature maps was adopted in DISTS \\cite{ding2020image}, aiming for the estimation of  texture similarity and structure similarity.  The work was further improved by processing the structure and texture information adaptively in A-DISTS \\cite{ADISTS}. Instead of measuring the feature distance point-by-point, the the Wasserstein distance was utilized in DeepWSD \\cite{liao2022deepwsd} to capture the  quality contamination. Driven by the quality annotated data, the human perception knowledge can also be learned by CNN, such as DeepQA \\cite{kim2017deep}, WaDIQaM \\cite{bosse2017deep}, PieAPP \\cite{prashnani2018pieapp} and JSPL \\cite{cao2022incorporating}. However, compared with the features extracted from pre-trained networks, the learned models usually suffer from the over-fitting problem due to the limited labeled data. \n\n\n\n\\subsection{Data Dependency Measure}\nClassical data dependency is measured only in the linear scenario. For example, the product-moment correlation and covariance are two widely used dependency measures between two random variables. In the case that the joint distribution of two vectors is multivariate normal distribution, the covariance matrix can be used to measure the dependence of different dimensions. This  measure which captures the high-order information is also utilized in computer vision tasks, \\textit{i}.\\textit{e}., the style transferring \\cite{gatys2016image} and deep self-similarity~\\cite{kligvasser2021deep}. For the image stylization, the Gram matrices of the neural activations of different CNN layers are extracted under the view that the channel dependency captured by Gram matrices well represents the artistic style of an image. However, in a more generic scene, the nonlinear or nonmonotone dependence is expected to be effectively captured. In \\cite{szekely2009brownian,szekely2007measuring}, the BDC was proposed which is able to measure the data dependency efficiently even in the nonlinear scenario. The BDC is designed based on the construction of joint characteristics of two variables, such that it could be more effective than only the marginal distribution involved. Benefiting from the properties, the BDC was also introduced for few-shot classification tasks \\cite{xie2022joint}, showing its robustness in different settings. \n\n\n\\section{Methodology}\n\\subsection{Preliminary of Brownian distance covariance}\nLet $X \\in \\mathbb{R}^p$, $Y \\in \\mathbb{R}^q$ be two random vectors, where $p$ and $q$ are their dimensions. The characteristic functions of $X$ and $Y$ are denoted as $f_{X}$ and $f_{Y}$ and their joint characteristic function is  $f_{XY}$. Assuming $X$ and $Y$ have finite first moments, analogous to classical covariance, the BDC measure is defined  as follows,\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{V}^2(X, Y ; w) &=\\left\\|f_{X, Y}(t, s)-f_X(t) f_Y(s)\\right\\|_w^2,\n\\end{aligned}\n\\end{equation}\n where $\\|\\cdot\\|_w^2$-norm is defined by,\n\\begin{equation}\n\\|\\gamma(t, s)\\|_w^2=\\int_{\\mathbb{R}^{p+q}}|\\gamma(t, s)|^2 w(t, s) d t  d s.\n\\end{equation}\nThe $w(t, s)$ is a positive weight function for which the integral above exists. As such,\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{V}^2(X, Y ; w) =\\int_{\\mathbb{R}^{p+q}}\\left|f_{X, Y}(t, s)-f_X(t) f_Y(s)\\right|^2 w(t, s) d t d s.\n\\label{v1}\n\\end{aligned}\n\\end{equation}\nTo endow the $\\mathcal{V}^2(X, Y; w)$ with the capability to capture  the dependence  between  $X$ and $Y$, a suitable  weight function can be found as follows \\cite{szekely2009brownian}, \n\\begin{equation}\nw(t, s)=\\left(c_p c_q|t|_p^{1+p}|s|_q^{1+q}\\right)^{-1},\n\\label{wts}\n\\end{equation}\nwhere\n\\begin{equation}\nc_p=\\frac{\\pi^{(1+p) \/ 2}}{\\Gamma((1+p) \/ 2)},  \\quad  c_q=\\frac{\\pi^{(1+q) \/ 2}}{\\Gamma((1+q) \/ 2)}, \n\\end{equation}\nand $\\Gamma (\\cdot)$ is the complete gamma function. From the Eq. (\\ref{v1}) and Eq. (\\ref{wts}),  the BDC measure can be formed by,\n\\begin{equation}\n \\begin{aligned}\n\\mathcal{V}^{2}(X, Y) &=\\left\\|f_{X, Y}(t, s)-f_{X}(t) f_{Y}(s)\\right\\|^{2} \\\\\n&=\\frac{1}{c_{p} c_{q}} \\int_{\\mathbb{R}^{p+q}} \\frac{\\left|f_{X, Y}(t, s)-f_{X}(t) f_{Y}(s)\\right|^{2}}{|t|_{p}^{1+p}|s|_{q}^{1+q}} d t d s.\n\\end{aligned}\n\\end{equation}\nHerein, we omit the $w$ in $\\mathcal{V}^2(X, Y; w)$ for simplification. In practice, the observations of $X$ and $Y$  are usually discrete. For $n$ i.i.d. observed  random vectors $(\\mathbf{X}, \\mathbf{Y})=\\left\\{\\left(X_k, Y_k\\right): k=1, \\ldots, n\\right\\}$, the  BDC measure $\\mathcal{V}^{2}(X, Y)$ can be efficiently acquired by \\cite{szekely2009brownian,szekely2007measuring},\n\\begin{equation}\n\\mathcal{V}^{2}(\\mathbf{X}, \\mathbf{Y}) = \\frac{1}{n^{2}} \\sum_{k, l = 1}^{n}{\\bf{A}}_{k l} {\\bf{B}}_{k l},\n\\label{pro}\n\\end{equation}\nwhere ${\\bf{A}}_{k l}=a_{k l}-\\bar{a}_{k \\cdot}-\\bar{a}_{\\cdot l}+\\bar{a}_{. .}$ and \n\n\\begin{equation}\na_{k l}=\\left\\|X_k-X_l\\right\\|_z, \\quad \\bar{a}_{k \\cdot}=\\frac{1}{n} \\sum_{l=1}^n a_{k l}, \\quad \\bar{a}_{\\cdot l},=\\frac{1}{n} \\sum_{k=1}^n a_{k l},  \\quad  \\bar{a}_{. .} =\\frac{1}{n^2} \\sum_{k, l=1}^n a_{k l},\n\\label{akl}\n\\end{equation}\nwhere $\\|\\cdot\\|_z$ means the z-norm.  Analogously,  we define ${\\bf{B}}_{k l}=b_{k l}-\\bar{b}_{k \\cdot}-\\bar{b}_{\\cdot l}+\\bar{b}_{. .}$, for $k,l =1, \\ldots, n$. The model enjoys several interesting properties which are summarized as follows,\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathcal{V}^{2}(\\mathbf{X}, \\mathbf{Y}) \\geq 0$.\n\\item $\\mathcal{V}^{2}(\\mathbf{X}, \\mathbf{Y})= 0$, if and only if $\\mathbf{X}$ and  $\\mathbf{Y}$ are independent.\n\\item For all constant vector $t_1$, $t_2$, nonzero real number $s_1$, $s_2$  and orthogonal matrix $\\mathbf{R_1}$,  $\\mathbf{R_2}$, $\\mathcal{V}^{2}(t_1+s_1\\mathbf{X}\\mathbf{R_1}, t_2+s_2\\mathbf{X}\\mathbf{R_2})= \\left|{s_1}{s_2}\\right|\\mathcal{V}^{2}(\\mathbf{X}, \\mathbf{Y})$.\n\\end{enumerate}\n{\nThe above properties reveal that the  BDC is  non-negative and able to  capture the dependency of the signals well under the translations and orthonormal transformations. Those properties exhibit surprising consistency with human  texture perception, which is usually not sensitive to texture resampling and geometric transformations. Incorporating the BDC in our method endows our method with a more powerful quality prediction capability and its effectiveness can be verified by the extensive experiments in Sec. \\ref{exp}.\n}\n\n\n\\subsection{BDC based FR-IQA model}\nGiven the reference image ${\\bf x  \\in \\mathbb{R}^3}$ and  the test image ${\\bf y  \\in \\mathbb{R}^3}$, the aim of FR-IQA is to predict the image quality  ${\\bf q  \\in \\mathbb{R}^1}$ of ${\\bf y}$. Herein, directly calculating the dependency between ${\\bf x}$ and  ${\\bf y}$ in the pixel space may not be adequate, as human perceptual sensitivity  is  usually non-uniform~\\cite{wang2008maximum,berardino2017eigen}. Recently, deep neural networks have shown a surprising power in capturing image distortions, popularly adopted as the quality-aware representation generator \\cite{zhang2018unreasonable,prashnani2018pieapp,ding2020image}. Following the vein, we first adopt the VGG16 network~\\cite{simonyan2014very} to nonlinearly  transform the images (${\\bf x}$ and  ${\\bf y}$) to the deep representations. Then, the BDC is calculated to evaluate the dependency between the representations of ${\\bf x}$ and  ${\\bf y}$, deemed that the higher dependency corresponds to the higher quality. The design details are shown in Fig.~\\ref{fig:frame}. In particular, the VGG16 network contains five stages in total and is pre-trained on the ImageNet~\\cite{deng2009imagenet} dataset. We empirically  abandon the layers after the fourth stage and use the rest layers as the deep-feature extractor. Supposing the extractor is represented by $\\phi(\\cdot)$, the deep-features of ${\\bf x}$ and  ${\\bf y}$ can be obtained by,\n\\begin{equation}\n\\mathbf{X}= \\phi(\\bf {x}),\n\\mathbf{Y}= \\phi (\\bf{y}),\n\\label{vgg}\n\\end{equation}\nwhere $\\mathbf{X,Y} \\in \\mathbb{R}^{h \\times w \\times d}$, $h$ and $w$ are the spatial dimensions and $d$ is the channel number. We reshape the $\\mathbf{X}$ and $\\mathbf{Y}$ into $hw \\times\\ d$, \\textit{i}.\\textit{e}., $\\mathbf{X,Y} \\in \\mathbb{R}^{hw \\times\\ d}$, then treat each column vector of $\\mathbf{X}$ and $\\mathbf{Y}$ (denoted as $X \\in \\mathbb{R}^{hw}$, $Y \\in \\mathbb{R}^{hw}$) as the observations of random vectors sampled from the marginal distributions of  $f_{X}$ and $f_{Y}$, respectively. However, the i.i.d. assumption may not hold for  $X$ and $Y$ due to distinct semantic information lying  in different channels. As such, instead of directly using the inner product in Eq. (\\ref{pro}), suggested by~\\cite{xie2022joint}, the cosine similarity is adopted for dependency estimation,\n\\begin{equation}\n{\\bf q}= \\mathcal{D}(\\mathbf{X,Y})= \\frac{\\bf A_u {\\bf B_u^T}}{\\left\\| \\bf A_u \\right\\|_2 \\left\\| \\bf B_u \\right\\|_2},\n\\label{qua}\n\\end{equation}\nwhere $\\bf  A_u$ and $\\bf  B_u$  are the flattened results of the upper triangular portions of $\\bf  A$ and $\\bf  B$ with ${\\bf{A}}_{k l}=a_{k l}-\\bar{a}_{k \\cdot}-\\bar{a}_{\\cdot l}+\\bar{a}_{. .}$ and ${\\bf{B}}_{k l}=b_{k l}-\\bar{b}_{k \\cdot}-\\bar{b}_{\\cdot l}+\\bar{b}_{. .}$. For $a_{k l}$ and $b_{k l}$, we adopt $z=2$ in Eq. (\\ref{akl}), \n\\begin{equation}\na_{k l}=\\left\\|X_k-X_l\\right\\|_2,  b_{k l}=\\left\\|Y_k-Y_l\\right\\|_2.\n\\label{akl2}\n\\end{equation}\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[ width=.95\\textwidth]{.\/figs\/fram.png}\n \\caption{Overall structure of DID. The tailored VGG16 (only the first four stages are reserved) network is adopted as the deep feature extractor. Then the features dependency is measured by the BDC and higher dependency indicates better quality. }\n \\label{fig:frame}\n\\end{figure*}\n\n   Algorithm 1 summarizes the framework of our method.  The $\\mathcal{D}(\\mathbf{X,Y})$ depicts the dependency level of $\\mathbf{Y}$ with $\\mathbf{X}$, ranging from (-1,1). Higher $\\mathcal{D}(\\mathbf{X,Y})$ corresponds to better quality of test image. It should be noted that although our DID-based IQA model is a deep learning-based quality measure, it enjoys the learning-free advantage, avoiding over-fitting to a specific dataset. \n\n\n\n\\begin{algorithm}[t]\n\t\\caption{DID based FR-IQA model.}\n\t\\label{alg:algorithm1}\n\t\\KwIn{ Reference image ${\\bf x}$; test image  ${\\bf y}$. }\n\t\\KwOut{The quality of test image  ${\\bf q}$.}  \n\t\\BlankLine\n        Step 1. Extract the deep representations $\\mathbf{X, Y}$ of ${\\bf x, y}$ by the tailored VGG16; \\\\[2pt]\n        Step 2. Obtain the matrices ${\\bf{A}}_{k l}=a_{k l}-\\bar{a}_{k \\cdot}-\\bar{a}_{\\cdot l}+\\bar{a}_{. .}$ and ${\\bf{B}}_{k l}=b_{k l}-\\bar{b}_{k \\cdot}-\\bar{b}_{\\cdot l}+\\bar{b}_{. .}$ by Eq.~(\\ref{akl}) and Eq. (\\ref{akl2}). \\\\[2pt]\n        Step 3. Select the upper triangular portions of $\\bf  A$ and $\\bf  B$  as $\\bf  A_u$ and $\\bf  B_u$. \\\\[2pt]\n        Step 4. Estimate the test image quality  ${\\bf q} = \\mathcal{V}^{2}(\\mathbf{X}, \\mathbf{Y}) = \\frac{1}{n^{2}} \\sum_{k, l = 1}^{n}{\\bf{A}}_{k l} {\\bf{B}}_{k l}$ by Eq. (\\ref{qua}).\n\\end{algorithm}\n\n\n\\begin{table}[t]\n  \\centering\n  \\small\n    \\begin{tabular}{l|cc|cc|cc|cc|cc}\n    \\toprule\n    \\multicolumn{1}{c|}{\\multirow{2}[4]{*}{Method}} & \\multicolumn{2}{c|}{LIVE} & \\multicolumn{2}{c|}{CSIQ} & \\multicolumn{2}{c|}{TID2013} & \\multicolumn{2}{c|}{KADID-10k} & \\multicolumn{2}{c}{PIPAL} \\\\\n\\cmidrule{2-11}          & SRCC  & PLCC  & SRCC  & PLCC  & SRCC  & PLCC  & SRCC  & PLCC  & SRCC  & PLCC \\\\\n    \\midrule\n    PSNR  & 0.873  & 0.868  & 0.809  & 0.815  & 0.688  & 0.679  & 0.676  & 0.680  & 0.407  & 0.415  \\\\\n    SSIM  & 0.931  & 0.928  & 0.872  & 0.868  & 0.720  & 0.745  & 0.724  & 0.723  & 0.498  & 0.505  \\\\\n    MS-SSIM & 0.931  & 0.931  & 0.908  & 0.896  & 0.798  & 0.810  & 0.802  & 0.801  & 0.552  & 0.590  \\\\\n    VIF   & 0.927  & 0.925  & 0.902  & 0.887  & 0.690  & 0.732  & 0.593  & 0.602  & 0.443  & 0.468  \\\\\n    FSIM  & \\textbf{0.965 } & \\textbf{0.961 } & 0.931  & 0.919  & \\textbf{0.851}  & \\textbf{0.877}  & 0.854  & 0.851  & 0.589  & 0.615  \\\\\n    NLPD  & 0.914  & 0.914  & 0.917  & 0.911  & 0.808  & 0.823  & 0.810  & 0.810  & 0.469  & 0.509  \\\\\n    \\midrule\n    Style & 0.898  & 0.882  & 0.853  & 0.837  & 0.675  & 0.681  & 0.701  & 0.707  & 0.339  & 0.337  \\\\\n    PieAPP & 0.908  & 0.919  & 0.877  & 0.892  & 0.850  & 0.848  & 0.836  & 0.836  & \\textbf{0.700 } & \\textbf{0.712 } \\\\\n    LPIPS & 0.939  & 0.945  & 0.883  & 0.906  & 0.695  & 0.759  & 0.720  & 0.729  & 0.573  & 0.618  \\\\\n    DISTS & \\textbf{0.955}  & \\textbf{0.954}  & 0.939  & 0.941  & 0.848  & 0.870  & \\textbf{0.890 } & \\textbf{0.889 } & 0.624  & 0.644  \\\\\n    DSD   & 0.577  & 0.552  & 0.603  & 0.700  & 0.548  & 0.657  & 0.439  & 0.527  & 0.274  & 0.350  \\\\\n    DeepWSD & 0.896  & 0.890  & \\textbf{0.963 } & \\textbf{0.953 } & \\textbf{0.874 } & \\textbf{0.896 } & 0.888  & 0.888  & 0.514  & 0.517  \\\\\n    \\midrule\n    DID   & 0.949  & 0.943  & \\textbf{0.945 } & \\textbf{0.942 } & 0.850  & 0.871  & \\textbf{0.913 } & \\textbf{0.911 } & \\textbf{0.677 } & \\textbf{0.697 } \\\\\n    \\bottomrule\n    \\end{tabular}%\n  \\caption{Performance comparison of DID against twelve existing FR-IQA models on five standard IQA datasets. The best two results are highlighted in boldface.}    \n  \\label{tab:mainresult}%\n\\end{table}%\n\n\n\\section{Experiment}\n\\label{exp}\nIn this section, we first describe the experimental setup. Next, we conduct comprehensive experiments to verify the effectiveness of the proposed model, including image quality prediction, texture quality assessment, and invariance of geometric transformation. Finally, the  ablation studies are performed. \n\\subsection{Experimental Setups}\nThe VGG16 is tailored at ``ReLU4$\\_$3\" layer and pre-trained on the ImageNet~\\cite{deng2009imagenet}. Inspired by the SSIM~\\cite{wang2004image} and DISTS~\\cite{ding2020image}, we resize the shorter side of the input images to $224$ while keeping the aspect ratio. We apply the Spearman's rank-order correlation coefficient (SRCC) and Pearson linear correlation coefficient (PLCC) to evaluate the monotonicity and linearity. The larger SRCC and PLCC values reflect better quality prediction results. In particular,  a five-parameter nonlinear logistic function is fitted to map the predicted scores to the same scale as MOSs when computing PLCC~\\cite{video2000final}. \n\n\\subsection{Performance on Image Quality Prediction}\nWe compare DID with $12$ FR-IQA models on five standard IQA datasets, including the LIVE~\\cite{sheikh2003image} CSIQ~\\cite{larson2010most}, TID2013~\\cite{ponomarenko2015image}, KADID-10k~\\cite{2019KADID}, and PIPAL~\\cite{jinjin2020pipal}. In particular, LIVE, CSIQ, and TID2013 contain limited image contents and distortion types, and they have been widely used for more than ten years. The KADID-10k and PIPAL are two large-scale IQA datasets with more than ten thousand distorted images. KADID-10k has $81$ pristine images, and $25$ distortion types with $5$ levels are adopted to generate $10,125$ distorted images. PIPAL is so far the largest human-rated IQA dataset with $23,200$ images, which are generated by $200$ reference images with $40$ distortions types. It is worth noting that PIPAL introduces $19$ GAN-based distortions, challenging the existing FR-IQA a lot. In addition, the $12$ FR-IQA models cover various design methodologies: the error visibility methods - PSNR, and NLPD~\\cite{laparra2016perceptual}, the structural similarity methods - SSIM~\\cite{wang2004image}, MS-SSIM~\\cite{wang2003multiscale}, and FSIM~\\cite{zhang2011fsim}; the information-theoretic methods - MAD~\\cite{larson2010most}, and VIF~\\cite{sheikh2006image}; the learning-based methods - PieAPP~\\cite{prashnani2018pieapp}, LPIPS~\\cite{zhang2018unreasonable}, and DISTS~\\cite{ding2020image}; the distribution-based methods - Style~\\cite{gatys2016image}, DSD~\\cite{kligvasser2021deep}, DeepWSD~\\cite{liao2022deepwsd}. \nThe experimental results are reported in the Table~\\ref{tab:mainresult}, from which we can find DID achieves superior performance on both classical (LIVE, CSIQ, and TID2013) and latest (KADID-10k and PIPAL) IQA datasets.\nIt demonstrates that the dependency-based model is well correlated with human ratings, and the learning-free advantage equips the DID with strong generalization capability. \nIn addition, the knowledge-driven methods (\\textit{e}.\\textit{g}., FSIM) generally perform better on small-scale IQA datasets, indicating the potential over-fitting problem because of the extensive parameter tuning. \nMoreover, DeepWSD outperforms most learning-based methods on synthetic distortions, which further reflects the success of the joint distribution based FR-IQA model. \nFinally, though PieAPP obtains the best performance on the PIPAL dataset, it requires plenty of the human-rated images to train the model~\\cite{prashnani2018pieapp}.\n\n\n\\begin{table}[t]\n\t\\begin{minipage}[b]{0.48\\linewidth}\n\t\t\\centering\n\t\t\\small\n\t\t\\resizebox{1.0\\textwidth}{!}{\n\t\t\n    \\begin{tabular}{l|cc}\n    \\toprule\n    \\multicolumn{1}{c|}{\\multirow{2}[4]{*}{Method}} & \\multicolumn{2}{c}{PIPAL (GAN distortion)} \\\\\n\\cmidrule{2-3}          & SRCC  & PLCC  \\\\\n    \\midrule\n    SSIM  & 0.322  & 0.472  \\\\\n    MS-SSIM & 0.387  & 0.615  \\\\\n    VIF   & 0.324  & 0.543  \\\\\n    FSIM  & 0.410  & 0.621  \\\\\n    NLPD  & 0.341  & 0.570  \\\\\n    \\midrule\n    LPIPS & 0.486  & 0.617  \\\\\n    PieAPP & \\textbf{0.553 } & \\textbf{0.632 } \\\\\n    DISTS & 0.549  & 0.607  \\\\\n    DeepWSD & 0.397  & 0.560  \\\\\n    \\midrule\n    \\textit{FID} & 0.413  & 0.496  \\\\\n    \\midrule\n    DID   & \\textbf{0.5742} & \\textbf{0.6403} \\\\\n    \\bottomrule\n    \\end{tabular}%\n\t\t}\n\t\t\\caption{Performance comparison of DID against state-of-the-art methods on the GAN distortion of PIPAL dataset. The measure specifically designed for GAN images is represented in italics.}\n\t\t\\label{tab:gan}\t\t\n\t\\end{minipage}\n\t\\hfill\n\t\\begin{minipage}[b]{0.48\\linewidth}  \n\t\t\\centering\n\t    \\small\n   \n\t\n              \\resizebox{200pt}{102pt}{\n\n    \\begin{tabular}{l|cc|cc}\n    \\toprule\n    \\multicolumn{1}{c|}{\\multirow{2}[4]{*}{Method }} & \\multicolumn{2}{c|}{SynTEX} & \\multicolumn{2}{c}{TQD} \\\\\n\\cmidrule{2-5}          & SRCC  & PLCC  & SRCC  & PLCC \\\\\n    \\midrule\n    SSIM  & 0.579  & 0.598  & 0.352  & 0.418  \\\\\n    VIF   & 0.606  & 0.697  & 0.549  & 0.614  \\\\\n    FSIM  & 0.081  & 0.115  & 0.386  & 0.272  \\\\\n    NLPD  & 0.606  & 0.607  & 0.409  & 0.457  \\\\\n    \\midrule\n    LPIPS & 0.788  & 0.788  & 0.203  & 0.188  \\\\\n    PieAPP & 0.715  & 0.719  & 0.718  & 0.721  \\\\\n    DISTS & \\textbf{0.923 } & \\textbf{0.901 } & \\textbf{0.910 } & \\textbf{0.903 } \\\\\n    DISTS$_\\text{s}$ & 0.877  & 0.868  & 0.795  & 0.780  \\\\\n    \\midrule\n   \\textit{STSIM} & 0.643  & 0.650  & 0.408  & 0.422  \\\\\n    \\textit{NPTSM} & 0.496  & 0.505  & 0.679  & 0.678  \\\\\n    \\textit{ISGTQA} & 0.820  & 0.816  & 0.802  & 0.804  \\\\\n    \\midrule\n    DID   & \\textbf{0.896 } & \\textbf{0.874 } & \\textbf{0.889 } & \\textbf{0.917 } \\\\\n    \\bottomrule\n    \\end{tabular}%\n\t\t}\n\t\t\\caption{Performance comparison of DID against state-of-the-art methods on two texture quality datasets. The texture similarity models are represented in italics.}\n\t\t\\label{tab:texutre}\t\n\t\\end{minipage}\n\\end{table}\n\n\n\\begin{figure*}\n    \\centering\n   \n    \\subfloat[MOS $\\uparrow$ \/ DID $\\uparrow$]{\\includegraphics[width=0.24\\textwidth]{figs\/textures\/13-0.png}}\\hskip.2em\n    \\subfloat[0.873 \/ 0.991 ]{\\includegraphics[width=0.24\\textwidth]{.\/figs\/textures\/13-4.png}}\\hskip.2em\n    \\subfloat[0.727 \/ 0.980]{\\includegraphics[width=0.24\\textwidth]{.\/figs\/textures\/13-2.png}}\\hskip.2em\n    \\subfloat[0.693 \/ 0.931]{\\includegraphics[width=0.24\\textwidth]{.\/figs\/textures\/13-15.png}}\n    \\vspace{-8pt}\n    \\addtocounter{subfigure}{0}\n    \\subfloat[0.528 \/ 0.821]{\\includegraphics[width=0.24\\textwidth]{.\/figs\/textures\/13-9.png}}\\hskip.2em\n    \\subfloat[0.481 \/ 0.758]{\\includegraphics[width=0.24\\textwidth]{.\/figs\/textures\/13-14.png}}\\hskip.2em\n    \\subfloat[0.437 \/ 0.506]{\\includegraphics[width=0.24\\textwidth]{.\/figs\/textures\/13-11.png}}\\hskip.2em\n    \\subfloat[0.401 \/ 0.210]{\\includegraphics[width=0.24\\textwidth]{.\/figs\/textures\/13-10.png}}\n    \\vspace{-5pt}\n    \\caption{Texture  images sampled from TQD~\\cite{ding2020image}. (a) Reference image. (b) Resampling image. (c) Resampling image. (d) Texture synthesis method~\\cite{snelgrove2017high}. (e) Pink noise. (f) Texture synthesis method~\\cite{gatys2015texture}. (g) Color\nquantization. (h) Chromatic aberration. }\\label{fig:texture}\n\n\\end{figure*}\n\nWe further compare DID with the FR-IQA models on the GAN-generated images of the PIPAL dataset. As shown in Table~\\ref{tab:gan}, most distance-based DR-IQA models present poor performance, and the underlying reason may lie in that synthesized textures which not appear in reference images are usually introduced by the generation networks.  Although the  Fr\u00e9chet Inception Distance (FID)~\\cite{heusel2017gans} is designed especially for the quality evaluation of  GAN models,  the poor performance reveals its limitations and the great challenges of GAN-generated IQA.  Compared with those methods, our model achieves the best result, the dependency rather than the distance mitigates the strict requirement of point-by-point alignment  during feature comparison. \n\n\n\n\\subsection{Performance on Texture Similarity}\n\\label{sec:texture}\nTo verify the effectiveness of our method on texture quality prediction,\nwe conduct  experiments on SynTEX~\\cite{golestaneh2015effect} and TQD~\\cite{ding2020image} datasets. In particular, SynTEX consists of $105$ synthesized texture images, which were generated by five texture synthesis methods for $21$ high-quality texture images. TQD contains ten reference texture images, and each of them is degraded by 15 distortion types, including seven synthetic distortions, four texture synthesis methods, and four randomly resampling versions. We show several sampled texture images in Fig.~\\ref{fig:sameples}. For performance comparison, four representative knowledge-driven methods: SSIM~\\cite{wang2004image}, VIF~\\cite{sheikh2006image}, FSIM~\\cite{zhang2011fsim}, and NLPD~\\cite{laparra2016perceptual} and three data-driven methods: LPIPS~\\cite{zhang2018unreasonable}, PieAPP~\\cite{prashnani2018pieapp}, and DISTS~\\cite{ding2020image} are selected. Furthermore, three models: STSIM~\\cite{zujovic2013structural}, NPTSM~\\cite{alfarraj2016content}, and IGSTOA~\\cite{golestaneh2018synthesized}, that designed especially for texture similarity are also included. \n\nThe SRCC and PLCC results are listed in Table~\\ref{tab:texutre}. It is not surprising that DISTS achieves the best performance on two texture quality datasets since a great number of texture images were used to train the weights for measuring texture similarity. However, when the training session for the texture similarity term is absent (denoted as DISTS$_{\\text{s}}$ in Table~\\ref{tab:texutre}), DID outperforms DISTS$_{\\text{s}}$ by a significant margin. Besides, ISGTQA exhibits noteworthy improvements for the texture similarity models but still falls behind DID. Furthermore, as shown in Fig.~\\ref{fig:texture}, we rank the sample texture images based on the DID score and MOS. We can observe that DID is consistent with human perception of texture quality. \\bl{In particular,  the images with visible artifacts have lower quality scores while the resampling images correspond to higher quality scores.} Thus, we may draw the conclusion that the proposed dependency-based model provides a promising texture perception in the scenarios of texture synthesis, resampling, and transformation.\n\n\n\\subsection{Performance on Geometric Transformations}\n\n\\begin{table}[t]\n  \\centering\n  \\small\n    \\begin{tabular}{l|cc|cc|cc|cc|cc}\n    \\toprule\n    \\multicolumn{1}{c|}{\\multirow{2}[4]{*}{Method}} & \\multicolumn{2}{c|}{Translation} & \\multicolumn{2}{c|}{Rotation} & \\multicolumn{2}{c|}{Scaling} & \\multicolumn{2}{c|}{Mixed} & \\multicolumn{2}{c}{Overall} \\\\\n\\cmidrule{2-11}          & SRCC  & PLCC  & SRCC  & PLCC  & SRCC  & PLCC  & SRCC  & PLCC  & SRCC  & PLCC \\\\\n    \\midrule\n    PSNR  & 0.104  & 0.365  & 0.088  & 0.365  & 0.088  & 0.366  & 0.093  & 0.373  & 0.102  & 0.320  \\\\\n    SSIM  & 0.194  & 0.388  & 0.199  & 0.390  & 0.196  & 0.394  & 0.207  & 0.393  & 0.211  & 0.232  \\\\\n    MS-SSIM & 0.183  & 0.381  & 0.191  & 0.373  & 0.202  & 0.387  & 0.213  & 0.389  & 0.191  & 0.202  \\\\\n    VIF   & 0.239  & 0.417  & 0.224  & 0.409  & 0.209  & 0.402  & 0.214  & 0.399  & 0.369  & 0.373  \\\\\n    FSIM  & 0.370  & 0.558  & 0.382  & 0.575  & 0.376  & 0.566  & 0.393  & 0.590  & 0.336  & 0.522  \\\\\n    NLPD  & 0.153  & 0.180  & 0.170  & 0.196  & 0.189  & 0.223  & 0.172  & 0.202  & 0.265  & 0.324  \\\\\n    \\midrule\n    Style & 0.740  & 0.758  & 0.735  & 0.747  & 0.729  & 0.738  & 0.740  & 0.752  & 0.744  & 0.758  \\\\\n   \n    LPIPS & 0.781  & 0.788  & 0.806  & 0.808  & 0.811  & 0.822  & 0.843  & 0.853  & 0.746  & 0.762  \\\\\n    DISTS & \\textbf{0.885 } & \\textbf{0.886 } & \\textbf{0.887 } & \\textbf{0.887 } & \\textbf{0.881 } & \\textbf{0.882 } & \\textbf{0.905 } & \\textbf{0.905 } & \\textbf{0.890 } & \\textbf{0.890 } \\\\\n    DSD   & 0.497  & 0.670  & 0.500  & 0.682  & 0.501  & 0.686  & 0.487  & 0.688  & 0.496  & 0.638  \\\\\n    DeepWSD & 0.163  & 0.189  & 0.187  & 0.207  & 0.215  & 0.194  & 0.280  & 0.302  & 0.234  & 0.288  \\\\\n    \\midrule\n    DID   & \\textbf{0.899 } & \\textbf{0.907 } & \\textbf{0.899 } & \\textbf{0.905 } & \\textbf{0.891 } & \\textbf{0.898 } & \\textbf{0.913 } & \\textbf{0.918 } & \\textbf{0.903 } & \\textbf{0.909 } \\\\\n    \\bottomrule\n    \\end{tabular}%\n        \\caption{Performance comparison of DID against the state-of-the-art FR-IQA models on LIVE-GT  dataset. The best two results are highlighted in boldface.}\n  \\label{tab:geo}%\n\\end{table}%\n\n\n\\bl{Texture image quality usually presents an invariance to mild geometric transformations~\\cite{ding2020image}. To study the performance of  existing FR-IQA models on such a prior, we construct a new IQA dataset denoted as LIVE-GT  with four geometric transformations (translation, rotation, scaling, and their mixed) involved. \n In particular, the four geometric transformations are implemented by randomly shifting $5\\%$ pixels in vertical or horizontal directions, randomly rotating $3^{\\circ}$ in clockwise or anticlockwise directions, scaling the image by a factor of $1.05$, and mixing the above-mentioned transformations.  We impose the four transformations to the reference images in the LIVE dataset and a total of $3,895$ distorted images are finally generated. Herein, we make a mild assumption  that the modest geometric transformations will not change the MOS of each image. Then, the performance comparison on the  LIVE-GT dataset can be conducted. \n\n\n \n\n\n\n\nWe list the SRCC and PLCC results for overall and individual geometric transformation types in Table~\\ref{tab:geo}. We can observe that the performance of knowledge-driven models drops dramatically on the LIVE-GT dataset while  our DID model achieves the best performance on all four transformations.  We believe the invariance property is brought by the dependency-based measure which avoids measuring the feature difference in a deterministic way.\nIn addition, it is worth noting that the DeepWSD  is also a joint distribution-based FR-IQA. However, it is vulnerable to transformations as the feature statistic comparison is performed locally. On the contrary, we construct the feature joint distribution in a global manner with both spatial and channel dimensions involved. In summary, we can conclude that the dependency derived from the global statistic of features contributes  to the robustness of our model.\n\n\n}\n\n\n\n\\subsection{Ablation studies}\n\\begin{table}[t]\n  \\centering\n  \\small \n\\setlength{\\tabcolsep}{4pt}\n    \\begin{tabular}{c|c|c|c|c|c|c|c|c}\n    \\toprule\n    \\multirow{2}[4]{*}{Backbone} & \\multirow{2}[4]{*}{Layer} & \\multicolumn{4}{c|}{Quality prediction} & \\multicolumn{2}{c|}{Texture similarity} & Geo invariance \\\\\n\\cmidrule{3-9}          &       & LIVE  & TID2013 & KADID-10k & PIPAL & SynTEX & TQD   & LIVE-GT  \\\\\n    \\midrule\n    \\multirow{5}[2]{*}{VGG16} & ReLU1\\_2 & 0.825  & 0.637  & 0.709  & 0.573  & 0.699  & 0.733  & 0.679  \\\\\n          & ReLU2\\_2 & 0.892  & 0.771  & 0.854  & 0.601  & 0.837  & 0.806  & 0.774  \\\\\n          & ReLU3\\_3 & 0.938  & 0.845  & \\textbf{0.899 } & 0.652  & 0.869  & 0.860  & 0.892  \\\\\n          & \\underline{ReLU4\\_3} & {0.948}  & {\\textbf{0.858 }} & {\\textbf{0.905 }} & {\\textbf{0.677 }} & {0.896}  & {0.889}  & {0.903}  \\\\\n          & ReLU5\\_3 & 0.946  & 0.855  & 0.881  & 0.656  & 0.893  & 0.867  & 0.893  \\\\\n    \\midrule\n    \\multirow{2}[2]{*}{ResNet50} & Layer\\_3 & \\textbf{0.953 } & \\textbf{0.865 } & 0.877  & 0.612  & 0.903  & 0.897  & 0.889  \\\\\n          & Layer\\_4 & \\textbf{0.950 } & 0.857  & 0.875  & 0.640  & \\textbf{0.925 } & 0.898  & \\textbf{0.905 } \\\\\n    \\midrule\n    \\multirow{2}[2]{*}{DenseNet121} & Block\\_3 & 0.937  & 0.856  & 0.850  & \\textbf{0.658 } & 0.916  & \\textbf{0.920 } & \\textbf{0.919 } \\\\\n          & Block\\_4 & 0.949  & \\textbf{0.865 } & 0.881  & 0.654  & \\textbf{0.918 } & \\textbf{0.921 } & 0.856  \\\\\n    \\bottomrule\n    \\end{tabular\n    \\caption{Ablation experiments of DID with different CNN backbones and tailored at different layers in terms of SRCC results. Geo invariance is the abbreviation of geometric invariance. The best two results are highlighted in boldface, and the default setting of DID is highlighted with an underline.}\n  \\label{tab:ablation}%\n\\end{table}%\n\nIn this subsection, we conduct ablation experiments to investigate the effect of the pre-trained CNN backbone and the tailored layer in the proposed model. Except for the default VGG16~\\cite{simonyan2014very} backbone, we applied the proposed BDC based FR-IQA method to other two widely used ImageNet pre-trained image classification networks - ResNet50~\\cite{he2016deep} and DenseNet121~\\cite{huang2017densely}. We tailored VGG16 at the last ReLU nonlinearity layer of each stage. We take the last two stages of ResNet50 (denoted as Layer$\\_$3 and Layer$\\_$4, respectively) and DenseNet121 (denote as Block$\\_$3 and Block$\\_$4, respectively) as the comparison variants. The results of three kinds of quality assessment tasks, \\textit{i}.\\textit{e}., quality prediction, texture similarity, and geometric invariance, are shown in Table~\\ref{tab:ablation}, \\bl{from which we can have the following observations. \nFirst, the proposed dependency model performs better in the deeper layers, as the deeper layers are able to capture the semantic information.\nSecond, the proposed BDC based model is quite robust to the CNN backbones according to the superior performance. Last, while the ResNet50 and DenseNe121 outperform the VGG16 in some small-scale datasets, we still choose VGG16 as the default backbone due to the satisfactory trade-off between model complexity and performance.}\n\n\n\n\\section{Conclusion}\nIn this paper, we have presented the new design philosophy for FR-IQA method and shown that the feature-dependency is particularly effective for generative images. The paradigm shift brings a fresh new perspective regarding how image quality shall be alternatively defined, given the available reference image. We obtain the conclusion that instead of gauging the feature distance, the dependency which is characterized by BDC could well reflect the image quality. In addition,  the proposed measure can be treated as a plug-and-play module and incorporated into  different backbones, dynamically adapting the application scenarios. We also believe the paradigm shifted from distance to dependency will shed light on more generalized quality measures and inspire more works on the exploration of feature dependency.\n\n\n\n\n\\section*{Acknowledgements} The authors would like to thank Keyan Ding for providing the TQD dataset and Haoming Cai for giving the rule of the name for the distortion images in PIPAL.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION.}\n\nEdge magnetoplasmons (EMP's) have atracted considerable interest in the past\nyears \\cite{1}-\\cite{11} and Ref. \\cite{1} contains a discussion of older\nstudies. In \\cite{11} we have treated EMPs, $\\propto A(\\omega,q_{x},y) \\exp[\n-i(\\omega t-q_{x} x)]$, for $\\nu=1,2$ and very low temperatures when the\nunperturbed density profile drops sharply at the edge on a length of the\norder of the magnetic length $\\ell_{0}$. Such a profile is valid for $\nk_{B}T\\ll \\hbar v_{g}\/\\ell_{0}$, where $v_{g}$ is the group velocity of the\nedge states. Then for $\\nu=1,2$ the unperturbed electron density $n_{0}(y)$,\nnormalized to the bulk value $n_{0}$, is calculated as $n_{0}(y)\/n_{0}=\\{1+\n\\Phi[(y_{re}-y)\/\\ell_{0}]\\}\/2$, where $y_{re}$ is the coordinate of the\nright edge and $\\Phi(y)$ the probability integral.\n\nThe main results of Ref. \\cite{11} were the existence of many {\\it symmetric}\nand {\\it antisymmetric} modes with respect to the edge, some of which can\npropagate nearly undamped even when the dissipation is strong and were,\ntherefore, termed {\\it edge helicons}. In this {\\it quasi-microscopic}\ntreatment, however, the contributions to the current density\n$j_{\\mu}(\\omega,q_{x},y)$ were obtained from microscopic expressions valid\nwhen the components of the electric field ${\\bf E}$, associated with the\nwave, are smooth on the $\\ell_{0}$ scale. This is not well justified.\nIn Ref. \\cite{11} we neglected possible nonlocal effects and approximated the\ncontributions to $j_{\\mu}(\\omega,q_{x},y)$ with those obtained when\n$E_{y}(y)$ is smooth on the scale of $\\ell_{0}$.\nThe model \\cite{11} also neglects\nthe screening of the two-dimensional electron gas (2DEG). Here, using a RPA\nframework we study the effect of nonlocality and of edge-states screening on\nEMPs for filling factors $6\\geq\\nu\\geq 1$. For even filling factors $\\nu$ we\nneglect the spin splitting. As we will show, the existence of additional\nmodes for $\\nu=1(2)$, predicted in Ref. \\cite{11}, is confirmed by the\npresent {\\it fully microscopic} treatment and the results for the\nfundamental mode remain valid for weak dissipation despite the neglect of\nscreening and nonlocality in \\cite{11}.\n\nIn Sec. II we present the channel-edge characteristics and derive an\nintegral equation for the inhomogeneous wave charge density within a RPA\nframework. In Sec. III we obtain the fundamental and dipole EMP's for\n$\\nu=1,2$ and in Sec. IV we consider fundamental EMPs for $\\nu=4$\nand $\\nu=6$. Finally, in Sec. V we discuss briefly the results and\nmake concluding remarks.\n\n\\section{BASIC RELATIONS}\n\n\\subsection{Channel-edge characteristics}\n\nWe consider a zero-thickness 2DEG of width $W$ and of length $L_x=L$ in the\npresence of a strong magnetic field $B$ along the $z$ axis. We take the\nconfining potential flat in the interior of the 2DEG ($V_{y}=0$) and parabolic\nat its edges, $V_{y} =m^*\\Omega^2 (y-y_r)^2\/2$, $y\\geq y_r$. $V_{y}$ is\nassumed smooth on the scale of $\\ell_{0}=(\\hbar\/m^{*} \\omega_{c})^{1\/2}$\nsuch that $\\Omega \\ll \\omega_{c}$, where $\\omega_{c}=|e|B\/m^* $ is the\ncyclotron frequency ($e<0$). In the Landau gauge for the vector potential\n${\\bf A}=(-By, 0, 0)$ the one-electron Hamiltonian $\\hat{h}^0$ is given by\n\n\\begin{equation}\n\\hat{h}^0=[(\\hat{p}_x +eBy)^2 +\\hat{p}_{y}^{2}]\/2m^* +V_{y} \\;\\; , \\label{1}\n\\end{equation}\nwhere $\\hat{{\\bf p}}$ is the momentum operator. The eigenvalues and\neigenfunctions corresponding to Eq. (\\ref{1}) near the right edge of the\nchannel, with $y_{0} \\equiv y_{0}(k_{x})=\\ell_{0}^{2} k_{x} \\geq y_{r}$, are\nwell approximated by\n\n\\begin{equation}  \nE_{\\alpha}\\equiv E_{n}(k_{x})= (n+1\/2)\\hbar \\omega_{c}+m^*\\Omega^2\n(y_{0}-y_{r})^2\/2 , \\\\    \\label{2}\n\\end{equation}\nand\n\\begin{equation}  \n|\\alpha> \\equiv \\psi_\\alpha({\\bf r}) =e^{ik_x x}\\Psi_n(y-y_{0})\/\\sqrt{L}, \\\\\n\\label{3}\n\\end{equation}\nrespectively. Here ${\\bf r}=\\{x,y\\}$, $\\alpha\\equiv \\{n,k_x\\}$, $\\Psi_n(y)$\nis a harmonic oscillator function. \nThe energy spectrum (\\ref{2}) of the n-th LL leads to the group velocity of\nthe edge states $v_{gn}=\\partial E_{n}(k_{r}+k_{e}^{(n)})\/\\hbar\\partial\nk_{x}=\\hbar \\Omega^{2}k_{e}^{(n)}\/m^{*}\\omega_{c}^{2}$ with characteristic\nwave vector $k_{e}^{(n)}=(\\omega_{c}\/ \\hbar\\Omega) \\sqrt{2m^{*}\\Delta_{Fn}}$,\n$\\Delta_{Fn}=E_{F}-(n+1\/2)\\hbar \\omega_{c}$, and $E_F$ is the Fermi\nenergy. The edge of the n-th LL is denoted by $y_{rn}=y_{r}+\n\\ell_{0}^{2}k_{e}^{(n)}=\\ell_{0}^{2}k_{rn}$,\nwhere $k_{rn}=k_{r}+k_{e}^{(n)}$,\nand $W=2y_{r0}$. We can also write $v_{gn}=E_{en}\/B$, where $E_{en}=\\Omega\n\\sqrt{2m^{*}\\Delta_{Fn}}\/|e|$ is the electric field associated with the\nconfining potential $V_{y}$ at $y_{rn}$. We also introduce the wave vector\n$k_{r}=y_{r}\/\\ell_{0}^{2}$.\n\nFor definiteness, we take the background dielectric constant $\\epsilon$ to\nbe spatially homogeneous. Assuming $|q_{x}| W \\gg 1$, we can consider an EMP\nalong the right edge of the channel of the form $A(\\omega, q_{x}, y) \\exp[\n-i(\\omega t-q_{x} x)]$ totally independent of the left edge. For $\\nu$ even\nthe spin-splitting is neglected.\n\n\\subsection{ Wave charge density and electric potential at the channel edge}\n\nAs in Refs. \\cite{11} and \\cite{12}, we assume that without interaction the\none-electron density matrix $\\hat{\\rho}^{(0)}$ is diagonal, i.e., $<\\alpha|\n\\hat{\\rho}^{(0)}|\\beta>=f_{\\alpha}\\delta_{\\alpha\\beta}$, where\n$f_{\\alpha}=1\/[1+exp((E_{\\alpha}-E_F)\/k_BT)]$ is the Fermi-Dirac function.\n\nFor the application of the RPA we follow the self-consistent field approach\nof Ref. \\cite{13} and references cited therein. The one-electron Hamiltonian\nin the presence of a self-consistent wave potential $V(x,y,t)=V(%\n\\omega_{0},q_{x},y) \\exp[-i(\\omega_{0} t-q_{x} x)]+ c.c.$, is $\\hat{H}(t)=%\n\\hat{h}^{0}+V(x,y,t)$. The corresponding equation of motion for the\none-electron density matrix $\\hat{\\rho}$ reads\n\n\\begin{equation}\ni\\hbar\\frac{\\partial \\hat{\\rho}}{\\partial t}=[\\hat{H}(t),\\hat{\\rho}]%\n-\\frac{i\\hbar}{\\tau}(\\hat{\\rho}-\\hat{\\rho}^{(0)}),   \\label{5}\n\\end{equation}\nwhere $[,]$ denotes the commutator. On the right hand side (RHS) of Eq.\n(\\ref{5}) we have introduced the phenomenological infinitesimal term\n$\\propto 1\/\\tau$ ($\\tau \\rightarrow \\infty$) that leads to the correct\nrules for contour integration around the singularities, cf. Refs. \\cite{13}\nand \\cite{14}. Notice that $\\tau \\rightarrow \\infty$ corresponds to the\ncollisionless case while using a finite $\\tau$ provides the possibility of\nestimating roughly the influence of collisions. Here the most effective\ncollisions are related to intra-LL and intra-edge transitions,\ncf. Refs. \\cite{11}-\\cite{12}.\n\nWe take the Laplace trasform, with respect to the time t, of Eq. (\\ref{5}),\nwrite $\\hat{R}(\\omega)=\\int_{0}^{\\infty}e^{i\\omega t} \\hat{\\rho} dt$ and\n$R_{\\alpha \\beta}(\\omega)=<\\alpha|\\hat{R}(\\omega)|\\beta>$, and look for\nits solution in a power series in $V$\n\n\\begin{equation}\nR_{\\alpha \\beta}(\\omega)=\\sum_{n=0}^{\\infty} R_{\\alpha \\beta}^{(n)}(\\omega),\n\\label{6}\n\\end{equation}\nwhere $R_{\\alpha \\beta}^{(0)}(\\omega)=(i f_{\\alpha}\/\\omega) \\delta_{\\alpha\n\\beta}$. Because we consider linear EMPs, it is sufficient  to take into\naccount only the first two terms $n=0$ and $n=1$ on the RHS of\nEq. (\\ref{6}).  Then in $V(x,y,t)$ we can consider only the term\n$V(\\omega_{0},q_{x},y) \\exp[-i(\\omega_{0} t-q_{x} x)]$, which leads to\n\n\\begin{equation}\nR_{\\alpha \\beta}^{(1)}(\\omega)=\\frac{i(f_{\\beta}-f_{\\alpha})\n<\\alpha|V(\\omega_{0},q_{x},y) e^{iq_{x} x}|\\beta>} {(\\omega-\\omega_{0})\n[E_{\\beta}-E_{\\alpha}+\\hbar \\omega+i\\hbar\/\\tau]} .  \\label{7}\n\\end{equation}\nTaking the trace of $\\hat{\\rho}$ with the electron charge density operator,\n$e \\delta({\\bf r}-\\hat{{\\bf r}})$, gives the wave charge density as\n\n\\begin{equation}\n\\delta \\rho(t,x,y) \\equiv \\rho(t,x,y)=\\frac{e}{2\\pi} \\int_{-\\infty+i\\eta}^{\n\\infty+i\\eta} d \\omega e^{-i\\omega t} \\sum_{\\alpha \\beta} R_{\\alpha\n\\beta}^{(1)}(\\omega) \\psi_{\\beta}^{*}({\\bf r}) \\psi_\\alpha({\\bf r}) ,\n\\label{8}\n\\end{equation}\nwhere $\\eta>0$. From Eqs. (\\ref{7}) and (\\ref{8}) it follows that\n$\\rho(t,x,y)=\\rho(t,q_{x},y) \\exp(iq_{x} x)$. Moreover, for $t\/\\tau \\gg 1$,\nthe contributions related to transitional processes are already negligible.\nIt follows that\n$\\rho(t,q_{x},y)=\\rho(\\omega_{0},q_{x},y) \\exp(-i\\omega_{0}t)$.\n\nFurther, from Poisson's equation the wave electric potential $\\phi(t,\nq_{x},y)$ induced by $\\rho(t,q_{x},y)$ is given as\n\n\\begin{equation}\n\\phi(t, q_{x},y)=\\frac{2}{\\epsilon}\\int_{-\\infty}^{\\infty}\ndy^{\\prime}K_{0}(|q_{x}||y-y^{\\prime}|)\\rho(t, q_{x},y^{\\prime}), \\label{9}\n\\end{equation}\nwhere $K_{0}(x)$ is the modified Bessel function; $\\phi$ and $\\rho$ pertain\nto the 2D plane.\n\n\\subsection{ Integral equation for EMP's}\n\nFor $t\/\\tau \\gg 1$ the relation $\\rho(t, q_{x},y^{\\prime})= \\rho(\\omega_{0},\nq_{x},y^{\\prime})\\exp(-i\\omega_0 t)$ holds; then from Eq. (\\ref{9}) it\nfollows that $\\phi(t, q_{x},y)=\\phi(\\omega_0, q_{x},y)\\exp(-i\\omega_0 t)$.\nIn the absence of an external potential we have $V(\\omega_{0},q_{x},y)=e\n\\phi(\\omega_0, q_{x},y)$. As a result,  from Eqs. (\\ref{7})-(\\ref{9}), for\n$t\/\\tau \\gg 1$ , we obtain the integral equation for $\\rho(\\omega, q_{x},y)$\nas\n\n\\begin{eqnarray}\n&&\\rho(\\omega, q_{x},y)=\\frac{2e^{2}}{\\epsilon L} \\sum_{n_{\\alpha},n_{\n\\beta}=0}^{\\infty} \\sum_{k_{x\\alpha}} \\frac{f_{\\beta}-f_{\\alpha}}{\nE_{\\beta}-E_{\\alpha}+\\hbar \\omega+ i \\hbar\/\\tau} \\\n\\Pi_{n_{\\alpha}n_{\\beta}}( y, k_{x\\alpha}, k_{x\\beta})  \\nonumber \\\\\n* \\   \\nonumber \\\\\n&&\\times\\int_{-\\infty}^{\\infty} d\\tilde{y} \\int_{-\\infty}^{\\infty}\ndy^{\\prime}\\ \\Pi_{n_{\\alpha}n_{\\beta}}( \\tilde{y}, k_{x\\alpha}, k_{x\\beta})\nK_{0}(|q_{x}||\\tilde{y}-y^{\\prime}|)\\ \\rho(\\omega, q_{x},y^{\\prime}) ,\n\\label{10}\n\\end{eqnarray}\nwhere $\\Pi_{n_{\\alpha}n_{\\beta}}( y, k_{x\\alpha}, k_{x\\beta})=\n\\Psi_{n_{\\alpha}}(y-y_0(k_{x\\alpha})) \\Psi_{n_{\\beta}}(y-y_0(k_{x\\beta}))$,\n$k_{x\\beta}=k_{x\\alpha}-q_{x}$.\nWe dropped the subscript $0$ from $\\omega_{0}$.\n\nFor flat LL's, i.e., for $\\Omega \\rightarrow 0$ and fixed width of the 2DEG,\nwe first apply the Fourier transformation along $y$ to Eq. (\\ref{10}) and\nthen carry out the sum over $k_{x\\alpha}$ as well as the integral over\n$y^{\\prime}$. This leads to $\\rho(\\omega, q_{x},q_{y}) \\epsilon_{l}(\\omega,\nq_{x},q_{y})\/\\epsilon=0$, where $\\epsilon_{l}(\\omega,\nq_{x},q_{y})=\\epsilon_{l}(\\omega, {\\bf q})  \\equiv \\epsilon_{l}(\\omega, q)$\nis the RPA longitudinal dielectric function (${\\bf q}=\\{q_{x},q_{y}\\}$).\nThat is, for $\\rho(\\omega, q_{x},q_{y}) \\neq 0$ we have\n\n\\begin{equation}\n\\frac{\\epsilon_{l}(\\omega, {\\bf q})}{\\epsilon}=1- \\frac{2 \\pi e^{2}}{%\n\\epsilon q L^{2}} \\sum_{\\alpha,\\beta} \\frac{f_{\\beta}-f_{\\alpha}}{%\nE_{\\beta}-E_{\\alpha}+\\hbar \\omega+ i \\hbar\/\\tau} |<\\alpha|e^{i\\vec q \\cdot\n\\vec r}|\\beta>|^{2}=0 .  \\label{12}\n\\end{equation}\nThis equation gives the RPA dispersion relation for \"bulk\" longitudinal wave\nexcitations of a 2DEG in a strong magnetic field. It also shows that the\n\"bulk\" excitations should have $\\omega>\\omega_{c}$. For definitness, we take\n$\\omega>0$.\n\nThe integral equation (\\ref{10}) can be considered as a generalization of\nEq. (10) of Ref. \\cite{11} since it takes into account nonlocal\ncontributions to the current density $\\propto \\int dy^{\\prime}\\sigma_{\\mu\n\\gamma}(y,y^{\\prime}) E_{\\gamma}(y^{\\prime})$ and the screening by the edge\nand bulk states of the 2DEG. As shown in Ref. \\cite{16}, the screening by\nedge states can be strong. We will obtain only solutions of Eq. (\\ref{10})\nthat are low in frequency, i.e., solutions for which $\\omega\/\\omega_{c} \\ll 1\n$. The appearence of any wave branch with $\\omega < \\omega_{c}$ is\nprincipally related to the existence of the edge of the 2DEG. Hence, any\nwave with $\\omega < \\omega_{c}$ should be localized at the edge and can be\ncalled edge magnetoplasmon.\n\nWe consider very low temperatures $T$ satisfying $\\hbar v_{gn}\\gg \\ell_{0}\nk_{B} T$. Further, we will assume the long-wavelength limit $q_{x} \\ell_{0}\n\\ll 1$, which is well satisfied, e.g., for fundamental EMP \\cite{11} in the\nlow-frequency region. Then if we compare the terms $\\propto f_{\\beta^{*}}$,\nfor given $n_{\\beta^{*}}$, on the RHS of Eq. (\\ref{10}), we obtain that the\ncontribution to the sum over $n_{\\alpha}$ with $n_{\\alpha}=n_{\\beta^{*}}$ is\nmuch larger than any other term of this sum or the sum of all terms with\n$n_{\\alpha} \\neq n_{\\beta^{*}}$. The small parameter is $|\\omega-q_{x}v_{g\nn_{\\beta^{*}}}(k_{x \\beta})|\/\\omega_{c} \\ll 1$, where $v_{g\nn_{\\beta^{*}}}(k_{x \\beta})=\\hbar^{-1} \\partial E_{n_{\\beta^{*}}}(k_{x\n\\beta})\/\\partial k_{x \\beta}$ is the group velocity of any occupied state\n$\\{ n_{\\beta^{*}},k_{x \\beta} \\}$ of the $n_{\\beta^{*}}$ LL. We assume that\neach occupied n-th LL has one intersection with the Fermi level at the edge\nof the channel and denote the group velocity of its edge states as $v_{gn}\n\\equiv v_{gn}(k_{rn})$. The small parameter given above implies $q_{x}\nv_{g0}\/\\omega_{c} \\ll 1$, as $v_{g0}$ is typically the largest among\n$v_{gn}$. Similar results follow from an analysis of the terms $\\propto\nf_{\\alpha^{*}}$ in the sum over $n_{\\beta}$ on the RHS of Eq. (\\ref{10}).\nHence, for $\\omega \\ll \\omega_{c}$ and $q_{x} v_{g 0} \\ll \\omega_{c}$ the\nterms with $n_{\\alpha} \\neq n_{\\beta}$ can be neglected. This leads to the\nintegral equation\n\n\\begin{eqnarray}\n&&\\rho(\\omega, q_{x},y)=\\frac{2e^{2}}{\\epsilon L}\n\\sum_{n_{\\alpha}=0}^{\\bar{n}}\n\\sum_{k_{x\\alpha}} \\frac{f_{n_{\\alpha},k_{x\\alpha}-q_x}\n-f_{n_{\\alpha},k_{x\\alpha}}} {E_{n_{\\alpha},k_{x\\alpha}-q_x}\n-E_{n_{\\alpha},k_{x\\alpha}}+ \\hbar \\omega+i \\hbar\/\\tau} \\\n\\Pi_{n_{\\alpha}n_{\\alpha}}( y, k_{x\\alpha}, k_{x\\alpha}-q_x)  \\nonumber \\\\\n* \\   \\nonumber \\\\\n&&\\times\\int_{-\\infty}^{\\infty} d\\tilde{y} \\int_{-\\infty}^{\\infty}\ndy^{\\prime}\\ \\Pi_{n_{\\alpha}n_{\\alpha}}( \\tilde{y}, k_{x\\alpha},\nk_{x\\alpha}-q_x) K_{0}(|q_{x}||\\tilde{y}-y^{\\prime}|)\\ \\rho(\\omega,\nq_{x},y^{\\prime}) ,  \\label{14}\n\\end{eqnarray}\nwhere $\\bar{n}$ denotes the highest occupied LL. For even $\\nu$ the RHS of\nEq. (\\ref{14}) should be multiplied by 2, the spin degeneracy factor. We now\nstudy EMPs following from Eq. (\\ref{14}).\n\n\\section{EMPs FOR $\\protect\\nu=1$($2$)}\n\nWe first consider the case $\\nu=1$ and then explain how the pertinent\nformulas should be  modified for $\\nu=2$. For $\\nu=1$ we have $\\bar{n}=0$\nand Eq. (\\ref{14}) takes the form\n\n\\begin{eqnarray}\n&&\\rho(\\omega, q_{x},y)=\\frac{e^{2}}{\\pi \\hbar \\epsilon} \\int_{-\\infty}^{%\n\\infty} dk_{x\\alpha} \\frac{f_{0,k_{x\\alpha}-q_{x}}-f_{0,k_{x\\alpha}}} {%\n\\tilde{\\omega}-v_{g0}(k_{x \\alpha}) q_{x}} \\ \\Pi_{00}( y, k_{x\\alpha},\nk_{x\\alpha}-q_{x})  \\nonumber \\\\\n* \\   \\nonumber \\\\\n&&\\times\\int_{-\\infty}^{\\infty} d\\tilde{y} \\int_{-\\infty}^{\\infty}\ndy^{\\prime}\\ \\Pi_{00}( \\tilde{y}, k_{x\\alpha}, k_{x\\alpha}-q_{x}) \\\nK_{0}(|q_{x}||\\tilde{y}-y^{\\prime}|)\\ \\rho(\\omega, q_{x},y^{\\prime}) ,\n\\label{15}\n\\end{eqnarray}\nwhere $\\tilde{\\omega}=\\omega+i\/\\tau$. As in Ref. \\cite{11}, we seek a\nsolution of Eq. (\\ref{15}) in the form\n\n\\begin{equation}\n\\rho_{0}(\\omega,q_{x},y)=\\Psi_{0}^{2}(\\bar{y}) \\sum_{n=0}^{\\infty}\n\\rho^{(n)}_{0}(\\omega,q_{x}) H_{n}(\\bar{y}\/\\ell_{0}) = \\sum_{n=0}^{\\infty}\n\\sqrt{2^{n}n!} \\ \\rho^{(n)}_{0}(\\omega,q_{x})\n\\Psi_{n}(\\bar{y})\\Psi_{0}(\\bar{y}).  \\label{16}\n\\end{equation}\nHere $\\bar{y}=y-y_{r0}=y-y_{0}(k_{r0})$ and Eq. (\\ref{16}) is the {\\it exact}\nsolution due to the orthogonality of the Hermite polynomials $H_{n}(x)$. The\nexpansion (\\ref{16}) is specific to $\\nu=1 (2)$ under the assumed\nconditions. This implies that the charge distortion far from the LL edge,\n$\\agt 10 \\ell_{0}$, is much smaller than near this LL edge, $\\alt \\ell_{0}$.\nPhysically this assumption can be well justified. We call the terms $n=0, 1,\n2,...$, the monopole, dipole, quadrupole, etc. terms in the expansion of\n$\\rho_{0}(\\omega,q_{x},y)$ relative to $y=y_{r0}$. Notice that expansion\n(\\ref{16}) can also be understood as an expansion in a complete set of\noscillatory wave functions corresponding to all, occupied ($n=0$) and\nempty ($n \\geq 1$) LLs.\n\nWe now multiply Eq. (\\ref{15}) by $H_{m}(\\bar{y}\/\\ell_{0})$ and integrate\nover $y$ from $y_{r0}-\\Delta y_{0}$ to $y_{r0}+\\Delta y_{0}$, where $\\Delta\ny_{0} \\agt 3 \\ell_{0}$. With the abbreviation $\\rho_{0}^{(m)}(\\omega,\nq_{x})\\equiv \\rho_{0}^{(m)}$, we obtain\n\n\\begin{equation}\n\\rho_{0}^{(m)}=\\frac{e^{2}}{\\pi \\hbar \\epsilon} \\int_{-\\infty}^{\\infty}\ndk_{x\\alpha} \\frac{f_{0,k_{x\\alpha}-q_{x}}-f_{0,k_{x\\alpha}}}{\\tilde{\\omega}\n-v_{g0}(k_{x \\alpha}) q_{x}} \\ d_{m0}(q_{x},\\delta k_{x\\alpha})\n\\sum_{n=0}^{\\infty} \\Big(\\frac{2^{n}n!}{2^{m}m!}\\Big)^{1\/2}\nb_{n0}^{(0)}(q_{x},|q_{x}|,\\delta k_{x\\alpha}) \\rho_{0}^{(n)},  \\label{17}\n\\end{equation}\nwhere $\\delta k_{x\\alpha}=k_{x\\alpha}-k_{r0}$, $\\delta\nk_{x\\alpha^{^{\\prime}}}=\\delta k_{x\\alpha}-q_{x}$,\n\n\\begin{equation}\nd_{m0}(q_{x},\\delta k_{x\\alpha})= \\frac{1}{\\sqrt{2^{m}m!}}\n\\int_{-\\infty}^{\\infty} d\\bar{y} \\ H_{m}(\\bar{y}\/\\ell_{0}) \\ \\Pi_{00}( \\bar{y%\n}, \\delta k_{x\\alpha}, \\delta k_{x\\alpha^{^{\\prime}}})   \\label{18}\n\\end{equation}\nand\n\n\\begin{equation}\nb_{n0}^{(0)}(q_{x},|q_{x}|,\\delta k_{x\\alpha})= \\int_{-\\infty}^{\\infty} d\n\\bar{y}\\ \\int_{-\\infty}^{\\infty} d\\bar{y}^{\\prime}\\ \\Pi_{00}( \\bar{y},\n\\delta k_{x\\alpha}, \\delta k_{x\\alpha^{^{\\prime}}}) K_{0}(|q_{x}||\\bar{y}-\n\\bar{y}^{\\prime}|) \\Psi_{n}(\\bar{y}^{\\prime})\\Psi_{0}(\\bar{y}^{\\prime}) \\ .\n\\label{19}\n\\end{equation}\nIn both $d_{m0}(q_{x},\\delta k_{x\\alpha})$ and\n$b_{n0}^{(0)}(q_{x},|q_{x}|,\\delta k_{x\\alpha})$ the first argument\nrepresents the $q_{x}$-dependent term of the argument of the wave function\n$\\Psi_{0}(\\bar{y}-\\delta k_{x\\alpha^{^{\\prime}}})$ and, if replaced by 0, it\nmeans this term can be neglected. We will also use the coefficients $a_{mn}(\nq_{x})$ given by \\cite{11}\n\n\\begin{equation}\na_{mn}( q_{x})=a_{nm} ( q_{x})=\\int_{-\\infty}^{\\infty} dx\\\n\\Psi_{m}(x)\\Psi_{0}(x) \\int_{-\\infty}^{\\infty} dx^{\\prime}\\\nK_{0}(|q_{x}||x-x^{\\prime}|)\\ \\Psi_{n}(x^{\\prime})\\Psi_{0} (x^{\\prime}),\n\\label{20}\n\\end{equation}\nand satisfying $b_{n0}^{(0)}(0,|q_{x}|,0)=a_{n0}(q_{x})$.\n\n\\subsection{Fundamental EMP}\n\nLet us approximate the numerator on the RHS of Eq. (\\ref{17}) by the first\nnonzero term in the expansion over $q_{x}$: $f_{0,k_{x\\alpha}-q_x\n}-f_{0,k_{x\\alpha}} \\approx -q_{x} (\\partial f_{0,k_{x\\alpha}}\/\\partial\nk_{x\\alpha})= \\delta(k_{x\\alpha}-k_{r0}) q_{x}$. Then Eq. (\\ref{17}), after\nintegration over $k_{x\\alpha}$, gives\n\n\\begin{equation}\n\\rho_{0}^{(m)}=\\frac{e^{2}}{\\pi \\hbar \\epsilon} \\ \\frac{q_{x}}{\\tilde{\\omega}%\n_{0}} \\ d_{m0}(q_{x},0) \\sum_{n=0}^{\\infty} \\Big(\\frac{2^{n}n!}{2^{m}m!}\\Big)%\n^{1\/2} \\ b_{n0}^{(0)}(q_{x},|q_{x}|, 0)\\ \\rho_{0}^{(n)},  \\label{21}\n\\end{equation}\nwhere $\\tilde{\\omega}_{0}=(\\tilde{\\omega}-v_{g0} q_{x})$. Further, if we\nneglect a small term $\\propto q_{x} \\ell_{0}$\nin the argument of the wave function $\\Psi_{0}(\\bar{y}+\\ell_{0}^{2}q_{x})$,\nwe obtain $d_{m0}(q_{x},0) \\approx d_{m0}(0,0)=\\delta_{m,0}$ and\n$b_{n0}^{(0)}(q_{x},|q_{x}|,0) \\approx b_{n0}^{(0)}(0,|q_{x}|,0)=\na_{n0}(q_{x})$. Substituting the former in Eq. (\\ref{21}), we obtain\n$\\rho_{0}^{(m)} \\equiv 0$ for $m \\geq 1$; thus, only $\\rho_{0}^{(0)}$ can be\nfinite. Then Eq. (\\ref{21}) can be rewritten as\n\n\\begin{equation}\n\\rho_{0}^{(0)}=\\frac{e^{2}}{\\pi \\hbar \\epsilon} \\ \\frac{q_{x}}{\\tilde{\\omega}%\n_{0}}\\ a_{00}(q_{x})\\ \\rho_{0}^{(0)}.  \\label{22}\n\\end{equation}\nFor $\\rho_{0}^{(0)} \\neq 0$ Eq. (\\ref{22}) gives the dispersion relation\n\n\\begin{equation}\n\\omega=q_{x} v_{g0}+\\frac{2}{\\epsilon} \\sigma_{yx}^{0} q_{x} [\\ln(\\frac{1}{%\n|q_{x}|\\ell_{0}})+\\frac{3}{4}]-\\frac{i}{\\tau} ,  \\label{23}\n\\end{equation}\nwhere $\\sigma_{yx}^{0}=e^{2}\/2 \\pi \\hbar$ for $\\nu=1$ and $%\n\\sigma_{yx}^{0}=e^{2}\/\\pi \\hbar$ for $\\nu=2$. We have used the result\n$a_{00}(q_{x}) \\approx [\\ln(1\/|q_{x}|\\ell_{0})+3\/4]$ \\cite{11} for\n$|q_{x}|\\ell_{0} \\ll 1$. If we neglect the dissipation, the dispersion\nrelation (\\ref{23}) and the corresponding charge density profile coincide\nwith the corresponding expressions for the fundamental mode obtained in Ref.\n\\cite{11}.\n\nUntill now nonlocal effects and screening by the 2DEG have not changed the\nresults for the fundamental EMP of Ref. \\cite{11}. In Ref. \\cite{11} it is\nshown that the next term that can affect the fundamental mode of the $n=0$\nLL is the quadrupole term. Then, if we neglect dissipation $\\omega-q_{x}\nv_{g0}$ becomes slightly larger, by a factor $[1+0.125\/a_{00}^{2}(q_{x})]\n\\approx 1$. Typically we have $a_{00}(q_{x}) \\agt 10$. In addition, the\nspatial structure of the fundamental EMP acquires a small quadrupole term\n$|\\rho_{0}^{(2)}\/\\rho_{0}^{(0)}| \\approx 1\/8 a_{00}(q_{x}) \\ll 1$ \\cite{11}.\nIt can be shown from Eqs. (\\ref{17})-(\\ref{19}) that here too the\ncorrections to the fundamental EMP due to the quadrupole term are very\nsmall; namely, $a_{00}(q_{x})$ on the RHS of Eq. (\\ref{23}) should be\nchanged to $a_{00}(q_{x})+a_{20}(q_{x})q_{x}^{2} \\ell_{0}^{2}\/2\\sqrt{2}\n\\approx a_{00}(q_{x})-q_{x}^{2} \\ell_{0}^{2}\/8$. Further, the spatial\nstructure of the fundamental EMP acquires a small quadrupole term:\n$\\rho_{0}^{(2)}\/\\rho_{0}^{(0)}=q_{x}^{2} \\ell_{0}^{2}\/8 \\ll 1$. Thus, for\nweak dissipation the results of Ref. \\cite{11} for the fundamental EMP are\nnearly the same as those of the present microscopic treatment.\n\n\\subsection{Dipole EMP}\n\nFor a more acurate calculation of the RHS of Eq. (\\ref{17}) we must take\ninto account higher order terms in the expansion over $q_{x}$. As shown\nbelow, when this is done it leads to additional branches. To obtain the\ndipole branch correctly we must keep the first three nonzero terms in the\nexpansion of the numerator of RHS of Eq. (\\ref{17}) over $q_{x}$ by writing\n\n\\begin{equation}\nf_{0,k_{x\\alpha}-q_x }-f_{0,k_{x\\alpha}} \\approx [q_{x}-\\frac{q_{x}^{2}}{2!}%\n\\frac{\\partial}{\\partial k_{x\\alpha}}+ \\frac{q_{x}^{3}}{3!}\\frac{\\partial^{2}%\n}{\\partial k_{x\\alpha}^{2}}] \\delta(k_{x\\alpha}-k_{r0}).  \\label{24}\n\\end{equation}\nFurther, we will consider only the first two terms, $n=0$ and $n= 1$ in the\nsum of Eq. (\\ref{16}) . Then from Eq. (\\ref{17})  for $m=0$ and $m=1$ we\nobtain the following system of linear equations for $\\rho_{0}^{(0)}$ and\n$\\rho_{0}^{(1)}$\n\n\\begin{equation}\n\\rho_{0}^{(0)}=\\frac{e^{2}}{\\pi \\hbar \\epsilon \\tilde{\\omega}_{0}}\n\\int_{-\\infty}^{\\infty} dk_{x\\alpha} (f_{0, k_{x\\alpha}-q_x\n}-f_{0,k_{x\\alpha}}) \\ d_{00}(q_{x},\\delta k_{x\\alpha}) \\sum_{n=0}^{1} \\Big(%\n2^{n}n!\\Big)^{1\/2} \\ b_{n0}^{(0)}(q_{x},|q_{x}|,\\delta k_{x\\alpha})\n\\rho_{0}^{(n)},  \\label{25}\n\\end{equation}\n\n\\begin{eqnarray}\n\\rho_{0}^{(1)}=\\frac{e^{2}}{\\pi \\hbar \\epsilon \\tilde{\\omega}_{0}}\n\\int_{-\\infty}^{\\infty} dk_{x\\alpha} (f_{0,k_{x\\alpha}-q_x\n}-f_{0,k_{x\\alpha}}) \\ d_{10}(q_{x},\\delta k_{x\\alpha}) \\sum_{n=0}^{1} \\Big(%\n\\frac{2^{n}n!}{2^{1}1!}\\Big)^{1\/2} \\ b_{n0}^{(0)}(q_{x},|q_{x}|,\\delta\nk_{x\\alpha}) \\rho_{0}^{(n)}.  \\label{26}\n\\end{eqnarray}\nWe assumed that, for $T \\rightarrow 0$, $v_{g 0}(k_{x \\alpha})$ is\nindependent of $k_{x \\alpha}$ in the vicinity of $k_{x \\alpha}=k_{r0}$ and\nreplaced it by $v_{g 0}$. This assumption does not impose any important\nrestriction on the analysis. Therefore, taking into account Eq. (\\ref{24})\nwe can replace\n$1\/(\\tilde{\\omega}- v_{g 0}(k_{x \\alpha}) q_{x})$ by\n$1\/(\\tilde{\\omega}-q_{x} v_{g 0})$ and place it in front of the\nintegrals in Eqs. (\\ref{25})-(\\ref{26}).\n\nIn Eq. (\\ref{25}) we can assume\n$d_{00}(q_{x},\\delta k_{x \\alpha})=1$ since corrections to it, varying as\n$\\sim (q_{x} \\ell_{0})^{2}$, are not essential. To evaluate the\ncontribution $\\propto \\rho_{0}^{(0)}$ on the RHS of Eq. (\\ref{25}) it is\nsufficient to keep only the first term, $\\propto q_{x}$, on the RHS\nof Eq. (\\ref{24}). This gives\n\n\\begin{equation}\n\\rho_{0}^{(0)}=\\frac{e^{2}}{\\pi \\hbar \\epsilon \\tilde{\\omega}_{0}} \\{q_{x}\nb_{00}^{(0)}(0,|q_{x}|,0) \\rho_{0}^{(0)}+ \\sqrt{2} I_{01}^{(1)}(q_{x})\n\\rho_{0}^{(1)} \\} ,  \\label{27}\n\\end{equation}\nwhere\n\n\\begin{equation}\nI_{01}^{(1)}(q_{x})=\\int_{-\\infty}^{\\infty} dk_{x\\alpha} (f_{0,\nk_{x\\alpha}-q_x }-f_{0,k_{x\\alpha}}) b_{10}^{(0)}(q_{x},|q_{x}|,\\delta k_{x\n\\alpha}) ,  \\label{28}\n\\end{equation}\nand where $b_{00}^{(0)}(q_{x},|q_{x}|,0)$ has justifiably been approximated\nby $b_{00}^{(0)}(0,|q_{x}|,0)$. Before calculating $I_{01}^{(1)}(q_{x})$ we\nnotice that the approximation for\n$d_{00}(q_{x},\\delta k_{x \\alpha})$ entails the neglect of small corrections\nto $I_{01}^{(1)}(q_{x})$ proportional to $(q_{x} \\ell_{0})^{k}$ if\n$k \\geq 4$. Then Eqs. (\\ref{24}) and Eq. (\\ref{28}) give\n\n\\begin{eqnarray}\nI_{01}^{(1)}(q_{x})&=& -\\frac{1}{\\sqrt{2}\\ell_{0}} (q_{x} \\ell_{0})^{2}\na_{11}(q_{x})+  \\nonumber \\\\\n* && +\\frac{q_{x}^{2}}{2} \\int_{-\\infty}^{\\infty} dk_{x\\alpha}\n\\delta(k_{x\\alpha}-k_{r0}) [1+ \\frac{q_{x}}{3}\\frac{\\partial}{\\partial\nk_{x\\alpha}}] \\frac{\\partial}{\\partial k_{x\\alpha}}\nb_{10}^{(0)}(q_{x},|q_{x}|,\\delta k_{x \\alpha}) .  \\label{29}\n\\end{eqnarray}\nThe corrections $\\propto (q_{x} \\ell_{0})^{k}$, $k \\geq 4$,  we neglected in\nthe first term on the RHS  are related to the expansion\n\n\\begin{equation}\n\\Psi_{0}(\\bar{y}+\\ell_{0}^{2}q_{x})=\\Psi_{0}(\\bar{y})- \\frac{1}{\\sqrt{2}}\n\\Psi_{1}(\\bar{y}) (q_{x} \\ell_{0})+ \\frac{1}{4}[\\sqrt{2} \\Psi_{2}(\\bar{y}%\n)-\\Psi_{0}(\\bar{y})] (q_{x} \\ell_{0})^{2}+...  \\label{30}\n\\end{equation}\nIt can be shown that the term $(q_{x}\/3)\\partial\/\\partial k_{x\\alpha}$ in\nthe square brackets \n[...] of Eq. (\\ref{29}) gives, upon integration over  $k_{x\\alpha}$, a\nvanishing contribution  since the  remaining double integral \nchanges sign upon making the changes $\\bar{y}  \\rightarrow -\\bar{y}$\nand $\\bar{y}^{\\prime}\\rightarrow -\\bar{y}^{\\prime}$. As for the term\ncorresponding to $1$ in [...], its evaluation gives $q_{x}^{2}\\ell_{0}\na_{11}(q_{x})\/\\sqrt{2}$. Hence, the two finite terms on the RHS of\nEq. (\\ref{29}) cancel each other. The final result is $I_{01}^{(1)}(q_{x})=0$\nand Eq. (\\ref{27}) takes the form of Eq. (\\ref{22}).\nThat is,  in this approximation the fundamental mode is purely {\\it monopole}\nand is totally independent of {\\it dipole} excitations.\n\nWe now consider Eq. (\\ref{26}). Here we should consider six different\ncontributions $I_{1n}^{(1)}(q_{x})$, $I_{1n}^{(2)}(q_{x})$, and\n$I_{1n}^{(3)}(q_{x})$ for $n=0,1$ related, respectively, to the first\n($\\propto q_{x}$), second ($\\propto q_{x}^{2}$), and third\n($\\propto q_{x}^{3}$) term on the RHS of Eq. (\\ref{24}).\nThen Eq. (\\ref{26}) can be rewritten as\n\n\\begin{equation}\n\\rho_{0}^{(1)}=\\frac{e^{2}}{\\pi \\hbar \\epsilon \\tilde{\\omega}_{0}}\n\\sum_{n=0}^{1} \\sum_{l=1}^{3} I_{1n}^{(l)}(q_{x}) \\rho_{0}^{(n)} ,\n\\label{31}\n\\end{equation}\nwhere\n\n\\begin{equation}\nI_{1n}^{(1)}(q_{x})=\\Big(\\frac{2^{n}n!}{2^{1}1!}\\Big)^{1\/2} q_{x}\nd_{10}(q_{x},0) b_{n0}^{(0)}(q_{x},|q_{x}|,0) ,  \\label{32}\n\\end{equation}\n\n\\begin{equation}\nI_{1n}^{(2)}(q_{x})=-\\frac{\\sqrt{2^{n-1}n! }}{2!} \\ q_{x}^{2}\n\\int_{-\\infty}^{\\infty} dk_{x\\alpha} d_{10}(q_{x},\\delta k_{x\\alpha})\nb_{n0}^{(0)}(q_{x},|q_{x}|,\\delta k_{x\\alpha}) \\frac{\\partial}{\\partial\nk_{x\\alpha}}\\delta(k_{x\\alpha}-k_{r0}) ,  \\label{33}\n\\end{equation}\n\n\\begin{equation}\nI_{1n}^{(3)}(q_{x})=\\frac{\\sqrt{2^{n-1}n!}}{3!} \\ q_{x}^{3}\n\\int_{-\\infty}^{\\infty} dk_{x\\alpha} d_{10}(q_{x},\\delta k_{x\\alpha})\nb_{n0}^{(0)}(q_{x},|q_{x}|,\\delta k_{x\\alpha}) \\frac{\\partial^{2}}{\\partial\nk_{x\\alpha}^{2}} \\delta(k_{x\\alpha}-k_{r0}).  \\label{34}\n\\end{equation}\n\nAgain we will neglect contributions $\\propto (q_{x} \\ell_{0})^{m}$ for $m\n\\geq 4$. In Eq. (\\ref{32}) we have $d_{10}(q_{x},0)=-(q_{x} \\ell_{0})\/\\sqrt{2%\n}+O((q_{x} \\ell_{0})^{3})$ and $%\nb_{00}^{(0)}(q_{x},|q_{x}|,0)=a_{00}(q_{x})+O((q_{x} \\ell_{0})^{2})$, $%\nb_{10}^{(0)}(q_{x},|q_{x}|,0)=-(q_{x} \\ell_{0}\/\\sqrt{2}) \\\na_{11}(q_{x})+O((q_{x} \\ell_{0})^{3})$. It follows\n\n\\begin{equation}\nI_{10}^{(1)}(q_{x})=-\\frac{(q_{x} \\ell_{0})^{2}}{2 \\ell_{0}} a_{00}(q_{x}) ,\n\\;\\;\\;\\;\\; I_{11}^{(1)}(q_{x})=\\frac{(q_{x} \\ell_{0})^{3}}{2 \\ell_{0}} \\\na_{11}(q_{x}) .  \\label{35}\n\\end{equation}\nEvaluating the integral in Eq. (\\ref{33}) by parts we obtain\n\n\\begin{eqnarray}\nI_{1n}^{(2)}(q_{x})&=&\\frac{(2^{n-1}n!)^{1\/2}}{2!} \\ q_{x}^{2}\\ \\Big\\{ \\frac{%\nd}{d \\delta k_{x\\alpha}} \\ [d_{10}(q_{x},\\delta k_{x\\alpha})]_{|\\delta\nk_{x\\alpha}=0} \\ b_{n0}^{(0)}(q_{x},|q_{x}|,0)+  \\nonumber \\\\\n* &&d_{10}(q_{x},0)\\ \\frac{d}{d \\delta k_{x\\alpha}} \\\n[b_{n0}^{(0)}(q_{x},|q_{x}|,\\delta k_{x\\alpha})]_{|\\delta k_{x\\alpha}=0} %\n\\Big\\}.  \\label{36}\n\\end{eqnarray}\nThis gives the contributions\n\n\\begin{equation}\nI_{10}^{(2)}(q_{x})=\\frac{(q_{x} \\ell_{0})^{2}}{2 \\ell_{0}} \\ a_{00}(q_{x})\n, \\;\\;\\;\\;\\; I_{11}^{(2)}(q_{x})=-\\frac{(q_{x} \\ell_{0})^{3}}{\\ell_{0}} \\\na_{11}(q_{x}) .  \\label{37}\n\\end{equation}\nEvaluating the integral in Eq. (\\ref{34}) by parts we obtain\n\n\\begin{equation}\nI_{10}^{(3)}(q_{x})=0 , \\;\\;\\;\\;\\; I_{11}^{(3)}(q_{x})=\\frac{2(q_{x}\n\\ell_{0})^{3}}{3\\ell_{0}} \\ a_{11}(q_{x}) .  \\label{38}\n\\end{equation}\nUsing Eqs. (\\ref{35}),(\\ref{37}) and (\\ref{38}) we can rewrite\nEq. (\\ref{31}) as\n\n\\begin{equation}\n\\rho_{0}^{(1)}=\\frac{e^{2} \\ell_{0}^{2} a_{11}(q_{x})} {6\\pi \\hbar \\epsilon\n\\tilde{\\omega}_{0}} q_{x}^{3} \\rho_{0}^{(1)} .  \\label{39}\n\\end{equation}\nEq. (\\ref{39}) shows that the {\\it dipole} branch, similar to that in Ref.\n\\cite{11}, is not coupled to {\\it monopole} excitations of the charge\ndensity. Further, Eq. (\\ref{39}), with $a_{11}(q_{x})=1\/2$ for\n$|q_{x}|\\ell_{0} \\ll 1$ and precision\n$\\alt 0.2 \\% $ \\cite{11}, gives the dispersion relation of the dipole\nbranch, $\\rho_{0}^{(1)}(\\omega,q_{x}) \\neq 0$,\n\n\\begin{equation}\n\\omega=v_{g 0}q_{x}+\\frac{\\ell_{0}^{2}}{6 \\epsilon} \\sigma_{yx}^{0}\nq_{x}^{3} -\\frac{i}{\\tau} ,  \\label{40}\n\\end{equation}\nwhere $\\sigma_{yx}^{0}=e^{2}\/2 \\pi \\hbar$ for $\\nu=1$ and\n$\\sigma_{yx}^{0}=e^{2}\/\\pi \\hbar$ for $\\nu=2$.\n\nAn important difference of Eq. (\\ref{40}) from the corresponding result, Eq.\n(40), of Ref. \\cite{11} for the pure dipole EMP is that here we have\n$\\omega-v_{g 0}q_{x} \\propto q_{x}^{3}$ whereas in Ref. \\cite{11} this\nbecomes $\\omega-v_{g 0}q_{x} \\propto q_{x}$. That is, the term caused by the\nCoulomb interaction has a {\\it nonacoustic} behavior if we neglect\ndissipation. The calculations demonstrate clearly that the spatial\ndispersion effects, although they depend on $q_{x}$, they are essentially\nrelated to the wave structure along the $y$-direction. It is known that\nspatial dispersion is directly related to the nonlocality of responses, as\nexpressed by the dielectric function, etc. \\cite{151}. In Eq. (\\ref{40})\nboth nonlocal effects and edge states screening are taken into account.\nWe point out that in Ref. \\cite {17}\na similar dispersion law, $\\omega \\propto a q_{x}^{3}$, was obtained for\nshort-range forces, i.e., for an interaction that is essentially not of the\nCoulomb type.\n\n\\section{Fundamental EMPs for $\\protect\\nu=4$ and $6$}\n\nFor $\\nu=2(\\bar{n}+1)$ the $n=0$,...,$n=\\bar{n}$ LLs have intersections with\nthe Fermi level at $y_{r0}$,..., $y_{r\\bar{n}}$, respectively, and\nEq. (\\ref{14}) can be written as\n\n\\begin{eqnarray}\n\\rho(\\omega, q_{x},y&)&=\\frac{2e^{2}}{\\pi \\hbar \\epsilon} \\sum_{n=0}^{\\bar{n}%\n} \\int_{-\\infty}^{\\infty} d{k_{x\\alpha}} \\frac{f_{n,k_{x\\alpha}-q_x}\n-f_{n,k_{x\\alpha}}} {\\tilde{\\omega}-v_{gn}(k_{x \\alpha}) q_{x}} \\ \\Pi_{nn}(\ny, k_{x\\alpha}, k_{x\\alpha}-q_{x})  \\nonumber \\\\\n* \\   \\nonumber \\\\\n&&\\times\\int_{-\\infty}^{\\infty} d\\tilde{y} \\int_{-\\infty}^{\\infty}\ndy^{\\prime}\\ \\Pi_{nn}( \\tilde{y}, k_{x\\alpha}, k_{x\\alpha}-q_{x})\nK_{0}(|q_{x}||\\tilde{y}-y^{\\prime}|)] \\rho(\\omega, q_{x},y^{\\prime}) .\n\\label{41}\n\\end{eqnarray}\nFor confining potentials smooth on the $\\ell_{0}$ scale we have $\\Delta\ny_{m-1,m}=y_{rm-1}-y_{rm} \\gg \\ell_{0}$, where $m \\leq \\bar{n}$. For $\\nu=4$\nthere is only one inter-LL length $\\Delta y_{0,1}=y_{r0}-y_{r1}$. Further,\nmaking the same  approximations, in the long-wavelength limit $q_{x}\\ell_{0}\n\\ll 1$, as in Sec. IIIA and integrating over $k_{x\\alpha}$ in\nEq. (\\ref{41}), we obtain\n\n\\begin{eqnarray}\n\\rho(\\omega, q_{x},y)&=&\\frac{2e^{2}}{\\pi \\hbar \\epsilon} \\sum_{n=0}^{\\bar{n}%\n} \\frac{q_{x}}{\\tilde{\\omega}_{n}} \\Psi_{n}^{2}(y-y_{rn})  \\nonumber \\\\\n* \\   \\nonumber \\\\\n&&\\times\\int_{-\\infty}^{\\infty} d\\tilde{y} \\int_{-\\infty}^{\\infty}\ndy^{\\prime}[\\Psi_{n}^{2}(\\tilde{y}-y_{rn}) K_{0}(|q_{x}||\\tilde{y}%\n-y^{\\prime}|)] \\rho(\\omega, q_{x},y^{\\prime}) ,  \\label{42}\n\\end{eqnarray}\nwhere $\\tilde{\\omega}_{n} \\equiv \\tilde{\\omega}-v_{gn} q_{x}$. It follows\nthat $\\rho(\\omega, q_{x},y)$ can be represented by a sum of charges\n$\\rho_{n}(\\omega, q_{x},y)= \\rho_{n}^{(0)}(\\omega, q_{x}) \\Psi_{n}^2(y-y_{rn})\n$, localized at the edge of the $n$-th LL, within a region of extent $\\sim\n\\sqrt{2n+1} \\ell_{0}$, $n=0,...,\\bar{n}$. The result is\n\n\\begin{equation}\n\\rho(\\omega, q_{x},y)=\\sum_{n=0}^{\\bar{n}} \\rho_{n}^{(0)}(\\omega, q_{x})\n\\Psi_{n}^{2}(y-y_{rn}) .  \\label{43}\n\\end{equation}\nWe substitute Eq. (\\ref{43}) into Eq. (\\ref{42}) and demand that the\ncoefficients of $\\Psi_{n}^{2}(y-y_{rn})$ on both sides of\nEq. (\\ref{42})  be equal.\nThis leads to $\\bar{n}+1$ linear homogeneous\nequations for $\\rho_{n}^{(0)}(\\omega, q_{x})$.\n\n\\subsection{$\\protect\\nu=4$}\n\nDue to $\\Delta y_{0,1} \\gg \\ell_{0}$ we can neglect the exponentially small\noverlap between $\\rho_{0}(\\omega, q_{x},y)$ and $\\rho_{1}(\\omega, q_{x},y)$.\nWe will assume $q_{x} \\Delta y_{01} \\ll 1$.\nAs shown above, from Eqs. (\\ref{42}) and (\\ref{43}), with $\\bar{n}=1$, we\nobtain the system\n\n\\begin{equation}\n\\rho_{0}^{(0)}=\\frac{2e^{2}}{\\pi\\hbar\\epsilon} \\frac{q_x}{\\tilde{\\omega}_{0}}\n[a_{00}(q_{x})\\rho_0^{(0)}+ a_{00}^{11}( q_{x}, \\Delta y_{01})\\rho_{1}^{(0)}\n] ,  \\label{44}\n\\end{equation}\n\n\\begin{equation}\n\\rho_{1}^{(0)}=\\frac{2e^{2}}{\\pi\\hbar\\epsilon} \\frac{q_x}{\\tilde{\\omega}_{1}}\n[a_{00}^{11}( q_{x}, \\Delta y_{01}) \\rho_{0}^{(0)}+ a_{11}^{11}( q_{x},\n0)\\rho_{1}^{(0)} ] ,  \\label{45}\n\\end{equation}\nwhere $\\rho_{n}^{(0)}(\\omega, q_{x}) \\equiv \\rho_{n}^{(0)}$ and\n\n\\begin{equation}\na_{nn}^{mm}( q_{x}, \\Delta y)=\n\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty} dx\\\ndx^{\\prime}\\Psi_{n}^2(x)\\Psi_{m}^2(x^{^{\\prime}})\nK_{0}(|q_{x}||x-x^{\\prime}+\\Delta y|).  \\label{46}\n\\end{equation}\nHere $a_{nn}^{mm}( q_{x}, \\Delta y)=a_{mm}^{nn}( q_{x}, \\Delta y)$,\n$a_{nn}^{mm}( q_{x}, \\Delta y)=a_{nn}^{mm}( q_{x}, -\\Delta y)$ and\n$a_{00}^{00}( q_{x},0)=a_{00}(q_{x})$. For a nontrivial solution of the\nsystem of Eqs. (\\ref{44}) and (\\ref{45}) the determinant of the coefficients\nmust vanish. This gives the dispersion relation of the renormalized\nfundamental EMP of the $n=0$ LL as\n\n\\begin{equation}\n\\omega=q_{x} v_{01} +\\frac{\\sigma_{yx}^{0}}{\\epsilon} \\ q_{x} [2 \\ln(\\frac{1%\n}{|q_{x}|\\ell_{0}})-\\ln(\\frac{\\Delta y_{01}}{\\ell_{0}})+ \\frac{3}{5}]-\\frac{i%\n}{\\tau} ,  \\label{47}\n\\end{equation}\nand that of the $n=1$ LL as\n\n\\begin{equation}\n\\omega=q_{x} v_{01} +\\frac{\\sigma_{yx}^{0}}{\\epsilon} \\ q_{x} [\\ln(\\frac{%\n\\Delta y_{01}}{\\ell_{0}})+\\frac{2}{5}]-\\frac{i}{\\tau} ,  \\label{48}\n\\end{equation}\nwhere $\\sigma_{yx}^{0}=2e^{2}\/\\pi \\hbar$ and $v_{01} =(v_{g0}+v_{g1})\/2$.\nAll double integrals involved in the coefficients of Eqs. (\\ref{44}) and\n(\\ref{45}) can be evaluated within the approximation $K_{0}(x) \\approx\n\\ln(2\/|x|)-\\gamma$, where $\\gamma$ is the Euler constant. We obtain\n$a_{11}^{11}( q_{x}, 0) \\approx \\ln(1\/|q_{x}|\\ell_{0})+1\/4$ and\n$a_{00}^{11}(q_{x}, \\Delta y_{01}) \\approx\n(\\ln(2)-\\gamma)- \\ln(|q_{x}|\\Delta y_{01}) \\approx \\ln(1\/|q_{x}\\Delta\ny_{01}|)+0.1$. Notice that in the absence of dissipation Eqs. (\\ref{47}) and\n(\\ref{48}) coincide with the quasi-microscopic results of Ref. \\cite{18}.\n>From Eqs. (\\ref{44}) and (\\ref{45}) it is easy to see that, if the inter-LL\nCoulomb coupling is neglected by setting $a_{00}^{11}( q_{x}, \\Delta\ny_{01})\\equiv 0$, the dispersion laws of the {\\it decoupled} fundamental\nEMPs of the $n=0$ and $n=1$ LLs are given, respectively, by Eq. (\\ref{23})\nand by\n\n\\begin{equation}\n\\omega=q_{x} v_{g1}+ \\frac{2e^{2}}{\\pi \\hbar \\epsilon} q_{x} [\\ln(\\frac{1}{%\n|q_{x}|\\ell_{0}})+\\frac{1}{4}]-\\frac{i}{\\tau} .  \\label{49}\n\\end{equation}\nSubstituting Eq. (\\ref{47}) in Eq. (\\ref{44}) and Eq. (\\ref{48}) in Eq. (\\ref\n{45}) we obtain, respectively, $\\rho_{1}^{(0)}\/\\rho_{0}^{(0)} \\approx 1$\nand $\\rho_{0}^{(0)}\/\\rho_{1}^{(0)} \\approx -1$. This means that in the\nformer case the wave charges localized at the edges of the $n=0$ LL and $n=1$\nLLs are in phase whereas in the latter they are out of phase. Therefore, the\nformer EMP has an {\\it acoustic} spatial structure along the $y$ axis while\nthe latter EMP has an {\\it optical} spatial structure though its dispersion\nlaw is purely acoustic.\n\n\\subsection{$\\protect\\nu=6$}\n\nFor $\\nu=6$ we have the intersection of the $n=2$ LL with the Fermi level,\nat $y_{r2}$, in addition those of the $n=0$ and $n=1$ LLs; $\\bar{n}=2$.\nCorresponding to Eqs. (\\ref{42}) and Eq. (\\ref{43}) we now obtain\n\n\\begin{equation}\n\\rho_{0}^{(0)}=\\frac{2e^{2}}{\\pi\\hbar\\epsilon} \\frac{q_x}{\\tilde{\\omega}_{0}}\n[a_{00}(q_{x})\\rho_{0}^{(0)}+ a_{00}^{11}( q_{x}, \\Delta\ny_{01})\\rho_{1}^{(0)}+ a_{00}^{22}( q_{x}, \\Delta y_{02})\\rho_{2}^{(0)} ] ,\n\\label{50}\n\\end{equation}\n\n\\begin{equation}\n\\rho_{1}^{(0)}=\\frac{2e^{2}}{\\pi\\hbar\\epsilon} \\frac{q_x}{\\tilde{\\omega}_{1}}\n[a_{00}^{11}( q_{x}, \\Delta y_{01}) \\rho_{0}^{(0)}+ a_{11}^{11}( q_{x}, 0)\n\\rho_{1}^{(0)}+ a_{11}^{22}( q_{x}, \\Delta y_{12}) \\rho_{2}^{(0)} ] ,\n\\label{51}\n\\end{equation}\n\n\\begin{equation}\n\\rho_{2}^{(0)}=\\frac{2e^{2}}{\\pi\\hbar\\epsilon} \\frac{q_x}{\\tilde{\\omega}_{2}}\n[a_{00}^{22}( q_{x}, \\Delta y_{02}) \\rho_{0}^{(0)}+ a_{11}^{22}( q_{x},\n\\Delta y_{12}) \\rho_{1}^{(0)}+ a_{22}^{22}( q_{x}, 0) \\rho_{2}^{(0)} ] .\n\\label{52}\n\\end{equation}\nThe vanishing of the determinant of the coefficients leads to the cubic\nequation\n\n\\begin{equation}\n\\omega^{` 3}+a_{2}(q_{x}) \\omega^{` 2}+ a_{1}(q_{x})\n\\omega^{`}+a_{0}(q_{x})=0 ,  \\label{53}\n\\end{equation}\nwhere $\\omega^{` }=\\tilde{\\omega}\/ (2e^{2}q_{x}\/\\pi \\hbar \\epsilon)$. The\nexpressions for the coefficients $a_{k}(q_{x}), \\ k=0,1,2$, are given in the\nappendix. We will assume that $q_{x} \\Delta y_{ij} \\ll 1$, $i \\neq j \\leq 2$.\n\nIt can be shown that all three roots of Eq. (\\ref{53}) are real and\ndifferent as they correspond to the irreducible case. Assuming $\\Delta\ny_{02}, \\Delta y_{12}, \\Delta y_{01} \\gg \\ell_{0}$,  these roots give the\ndispersion law for the renormalized fundamental EMP of the $n=0$ LL as\n\n\\begin{equation}\n\\omega=q_{x} v_{012} + \\frac{2\\sigma_{yx}^{0}}{3\\epsilon} q_{x} \\{3 \\ln(%\n\\frac{1}{|q_{x}|\\ell_{0}})- \\frac{2}{3}[\\ln(\\frac{\\Delta y_{02}}{\\ell_{0}})+\n\\ln(\\frac{\\Delta y_{01}}{\\ell_{0}})+ \\ln(\\frac{\\Delta y_{12}}{\\ell_{0}})]+\n\\frac{8}{15} \\}-\\frac{i}{\\tau} ,  \\label{57}\n\\end{equation}\nand that for the renormalized fundamental EMP of the $n=1$ LL, $\\omega_{+}$,\nand $n=2$ LL, $\\omega_{-}$, as\n\n\\begin{eqnarray}\n\\omega_{\\pm}=&&q_{x} v_{012} + \\frac{2\\sigma_{yx}^{0}}{9\\epsilon} q_{x}\n\\{\\ln(\\frac{\\Delta y_{02}}{\\ell_{0}})+ \\ln(\\frac{\\Delta y_{01}}{\\ell_{0}})+\n\\ln(\\frac{\\Delta y_{12}}{\\ell_{0}})+\\frac{7}{10} \\} \\pm  \\nonumber \\\\\n* \\   \\nonumber \\\\\n&&\\frac{2 \\sqrt{2}\\sigma_{yx}^{0}}{9\\epsilon} q_{x} \\{\\ln^{2}(\\frac{\\Delta\ny_{02}}{\\Delta y_{01}})+ \\ln^{2}(\\frac{\\Delta y_{02}}{\\Delta y_{12}})+\n\\ln^{2}(\\frac{\\Delta y_{01}}{\\Delta y_{12}})+A \\}^{1\/2}-\\frac{i}{\\tau} .\n\\label{58}\n\\end{eqnarray}\nHere $\\sigma_{yx}^{0}=3e^{2}\/\\pi \\hbar$,\n$v_{012} =(v_{g0}+v_{g1}+v_{g2})\/3$, and\n\n\\begin{eqnarray}\nA=&&(\\tilde{v}_{g0}+3\/4)[ (\\tilde{v}_{g0}- \\tilde{v}_{g2}+ 3\/4)\/2+\\ln(\\Delta\ny_{02}\\Delta y_{01}\/ {\\Delta y_{12}}^{2})]  \\nonumber \\\\\n* &&+(\\tilde{v}_{g1}+1\/4)[(\\tilde{v}_{g1}- \\tilde{v}_{g0}- 1\/2)\/2+\n\\ln(\\Delta y_{01}\\Delta y_{12}\/ {\\Delta y_{02}}^{2})]  \\nonumber \\\\\n* &&+\\tilde{v}_{g2} [ (\\tilde{v}_{g2}- \\tilde{v}_{g1}-1\/4)\/2+\\ln(\\Delta\ny_{02}\\Delta y_{12}\/ {\\Delta y_{01}}^{2})] ,  \\label{59}\n\\end{eqnarray}\nwhere $\\tilde{v}_{gi}=v_{gi}\/(2e^{2}\/\\pi \\hbar \\epsilon)$, $i=0,1,2$. Notice\nthat the fundamental EMPs of the $n=1$ and $n=2$ LLs have purely acoustic\ndispersion laws, cf. Eq. (\\ref{58}) . Only the fundamental EMP of the $n=0$\nLL, Eq. (\\ref{57}), behaves in the previously obtained manner $\\propto q_{x}\n\\ln(q_{x})$, cf. Refs. \\cite{1}-\\cite{10}), for a fundamental EMP. It is\ndifficult to clearly disentangle the contributions of the $n=2$ LL in\nEq. (\\ref{58}). It can be shown though that, if they are neglected by\nsetting $\\tilde{a}_{22}^{22}$, $a_{11}^{22}$, $a_{00}^{22}$, $v_{g2}$\nto zero, then we have $\\omega^{^{\\prime}}_{2}=0$ and Re$\\omega_{-}=0$ as\nwe should in this limit.\n\nThe solutions of the cubic equation, Eqs. (\\ref{57}) and  (\\ref{58}), are\nnot obtained using the standard expressions,  reproduced in the appendix,\nwhich become very unwieldy for the  present case. Instead, the following\nreasoning is used. If $\\omega$ has an acoustic character for some branches,\nthen $\\omega^{`}$ in Eq. (\\ref{53}) is independent of $q_{x}$ and should\nsatisfy the quadratic equation obtained by differentiating Eq. (\\ref{53})\nwith respect to $q_{x}$\n\n\\begin{equation}\n\\frac{\\partial a_{2}(q_{x})}{\\partial q_{x}} \\omega^{`2}+ \\frac{\\partial\na_{1}(q_{x})}{\\partial q_{x}} \\omega^{`}+ \\frac{\\partial a_{0}(q_{x})}{%\n\\partial q_{x}}=0 .  \\label{60}\n\\end{equation}\nThe two roots $\\omega_{2}^{`}$ and $\\omega^{`}_{3}$ of Eq. (\\ref{60}) are\nindeed independent of $q_{x}$ and lead to the solutions $\\omega_{-}$ and\n$\\omega_{+}$, respectively, given by Eq. (\\ref{58}). The other root is given\nby $\\omega^{`}_{1}=-a_{2}(q_{x})-(\\omega^{^{\\prime}}_{2}+\n\\omega^{^{\\prime}}_{3})$ and results in Eq. (\\ref{57}). Of course both\nmethods give the same solutions $\\omega^{^{\\prime}}_{j}$, $j=1,2,3$.\nNotice that on the RHS of Eqs. (\\ref{57}) and (\\ref{58}) we  used\n{\\it approximate} expressions of $a_{ii}^{jj}(q_{x},\\Delta y_{ij})$ which ,\nassuming $\\Delta y_{ij}\/\\ell_{0} \\gg 1$, $i \\neq j$, are obtained with the\napproximation\n\n\\begin{equation}\n\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty} dx\\\ndx^{\\prime}\\Psi_{i}^2(x)\\Psi_{j}^2(x^{^{\\prime}}) \\ln(|x-x^{\\prime}+\\Delta\ny_{ij}|\/\\ell_{0}) \\approx \\ln(\\Delta y_{ij}\/\\ell_{0}) .  \\label{61}\n\\end{equation}\nBelow we will demonstrate that the approximation (\\ref{61}) is well\njustified even for not-too-large values $\\Delta y_{ij}\/\\ell_{0}$. For that\nwe will calculate the dispersion laws of the renormalized fundamental EMPs\nusing the {\\it exact} expressions of $a_{ii}^{jj}(q_{x},\\Delta y_{ij})$,\ni.e., with the double integral on the LHS of Eq. (\\ref{61}) evaluated\nnumerically. In the latter case we will calculate $\\omega^{^{\\prime}}_{j}$\nfrom the standard expressions (\\ref{A1}) -(\\ref{A3}) of the appendix.\n\nFor a $GaAs$-based 2DEG and negligible dissipation the dispersion laws\n(\\ref{57}) and (\\ref{58}) for the renormalized, by the inter-LL Coulomb\ncoupling, fundamental EMPs are shown in Fig. 1 by the solid curves. The\nparameters are $m^{*} \\approx 6.1 \\times 10^{-29} g$, $\\epsilon \\approx 12.5$,\nand $\\Omega \\approx 7.8 \\times 10^{11} sec^{-1}$ \\cite{19}. For $\\nu=6$\nand $B=3 $Tesla these parameters lead to $\\hbar \\omega_{c} \\approx 5 meV$,\n$\\omega_{c}\/\\Omega \\approx 10$. Here $\\omega_{*}=2e^{2}\/\\pi \\hbar\n\\epsilon \\ell_{0}$ is a characteristic frequency almost equal, for the\nconditions stated,  to $\\omega_{c}$. We have also assumed\n$\\Delta_{F2}=\\hbar \\omega_{c}\/2$. This gives $v_{g0} \\approx \\sqrt{5}\n\\Omega \\ell_{0}$, $v_{g1} \\approx \\sqrt{3} \\Omega \\ell_{0}$, $v_{g2} \\approx\n\\Omega \\ell_{0}$, $\\Delta y_{02}\/\\ell_{0} \\approx (\\omega_{c}\/\\Omega)\n[\\sqrt{5}-1]$, $\\Delta y_{01}\/\\ell_{0} \\approx (\\omega_{c}\/\\Omega)\n[\\sqrt{5}-\\sqrt{3}]$, and $\\Delta y_{12}\/\\ell_{0} \\approx (\\omega_{c}\/\n\\Omega)[\\sqrt{3}-1]$. The dashed curves show the fundamental EMPs of\nthe totally decoupled $n=0$ (Eq. (\\ref{23}), topmost dashed curve),\n$n=1$, and $n=2$ (lowest dashed curve) LLs, i.e., the modes obtained\nby neglecting the inter-LL Coulomb coupling. In this case all coefficients\n$a_{ii}^{jj}$ on the RHS of Eqs. (\\ref{50})-(\\ref{52}) vanish for $i \\neq j$.\nThis leads directly to the dispersion laws represented by the dashed curves\nin Fig. 1. The dotted curves show the dispersion laws of the renormalized\nfundamental EMPs using the exact expressions for $a_{ii}^{jj}(q_{x},\\Delta\ny_{ij})$, $i \\neq j$. It is seen that each dotted curve is very close to the\ncorresponding solid curve.\n\nFor the same parameters and conditions as in Fig. 1  we show, in Fig. 2 ,\n$\\rho(\\omega,q_{x},y)\/\\rho_{*}$ (solid and dotted curves) and\n$\\rho(\\omega,q_{x},y)\/2\\rho_{*}$ (dashed curve) as a function of\n$Y=(y-y_{r2})\/\\ell_{0}$ with\n$\\rho_{*}=\\rho_{0}^{(0)}(\\omega,q_{x})\/\\sqrt{\\pi} \\ell_{0}$. The solid curve\ncorresponds to the topmost solid curve in Fig. 1, Eq. (\\ref{57}), with\n$\\rho_{1}^{(0)}\/\\rho_{0}^{(0)} \\approx 1.0$ and\n$\\rho_{2}^{(0)}\/\\rho_{0}^{(0)} \\approx 1$.\nAs can be seen the renormalized fundamental EMP of the $n=0$ LL has an\n{\\it acoustic} spatial structure along the $y$ axis. The dashed curve\ncorresponds to the lowest solid curve in Fig. 1, $\\omega_{-}$ of Eq. (\\ref\n{58}), with $\\rho_{1}^{(0)}\/\\rho_{0}^{(0)} \\approx -2.0$ and\n$\\rho_{2}^{(0)}\/\\rho_{0}^{(0)} \\approx 1$. Hence the renormalized fundamental\nEMP of the $n=2$ LL has an {\\it optical} spatial structure though its\ndispersion law is purely acoustic. The dotted curve corresponds to the\nmiddle solid curve in Fig. 1, $\\omega_{+}$ of Eq. (\\ref{58}), with\n$\\rho_{1}^{(0)}\/\\rho_{0}^{(0)} \\approx -0.1$ and $\\rho_{2}^{(0)}\/\n\\rho_{0}^{(0)} \\approx -1.0$. Thus, the renormalized fundamental EMP\nof the $n=1$ LL has an {\\it optical} spatial structure though its\ndispersion law is purely acoustic. Notice that the dependence of\n$\\rho_{j}^{(0)}\/\\rho_{0}^{(0)}$, $j=1,2$, on $q_{x}$ is typically weak.\n\nIn Fig. 3 we use the same values for $m^{*}$, $\\epsilon$, $\\Omega$, and\n$\\Delta_{F2}=\\hbar \\omega_{c}\/2$, $\\nu=6$ as in Fig. 1. However, we take\n$B=1.5$Tesla and this leads to $\\omega_{c}\/\\Omega \\approx 5$. This gives\n$\\sqrt{2}$ times larger values for $v_{g0} \\approx \\sqrt{5} \\Omega \\ell_{0}$, $v_{g1} \\approx\n\\sqrt{3} \\Omega \\ell_{0}$, and $v_{g2} \\approx \\Omega \\ell_{0}$, and twice\nsmaller values for $\\Delta y_{02}\/\\ell_{0} \\approx (\\omega_{c}\/\\Omega)[\n\\sqrt{5}-1]$, $\\Delta y_{01}\/\\ell_{0} \\approx (\\omega_{c}\/\\Omega)[\\sqrt{5}-\n\\sqrt{3}]$, and $\\Delta y_{12}\/\\ell_{0} \\approx (\\omega_{c}\/\\Omega)[\\sqrt{3}\n-1]$ as compared with those in Fig. 1. All  curves are marked as in Fig. 1\nand show the corresponding  dispersion laws.\n\nIn Fig. 4 we plot $\\rho(\\omega,q_{x},y)\/\\rho_{*}$ as a function of\n$Y=(y-y_{r2})\/\\ell_{0}$ for the conditions corresponding to the solid curves\nin Fig. 3. All curves are marked as in Fig. 2. We have, respectively,\n$\\rho_{1}^{(0)}\/\\rho_{0}^{(0)} \\approx 1.0$ and\n$\\rho_{2}^{(0)}\/\\rho_{0}^{(0)} \\approx 1$ (solid  curve),\n$\\rho_{1}^{(0)}\/\\rho_{0}^{(0)} \\approx -2.1$ and\n$\\rho_{2}^{(0)}\/\\rho_{0}^{(0)} \\approx 1.2$ (dashed curve), and\n$\\rho_{1}^{(0)}\/\\rho_{0}^{(0)} \\approx -0.1$ and\n$\\rho_{2}^{(0)}\/\\rho_{0}^{(0)} \\approx -1.0$ (dotted  curve).\nAs in Fg. 2, the renormalized fundamental EMP of the $n=0$ LL has an\n{\\it acoustic} spatial structure along the $y$ axis while those of the\n$n=1$ and $n=2$ LL have an {\\it optical} spatial structure though their\ndispersion laws are purely acoustic. As noted above, the dependence of\n$\\rho_{j}^{(0)}\/\\rho_{0}^{(0)}$, $j=1,2$, on $q_{x}$ is typically weak.\n\n\\section{discussion and concluding remarks}\n\nWe presented a fully {\\it microscopic} model of EMPs in a RPA framework\nvalid for integer $\\nu\\geq 1$ and confining potentials that are smooth on\nthe $\\ell_{0}$ scale but still sufficiently steep that LL flattening can be\nneglected \\cite{20}. The model takes into account LL coupling and treats\nonly very weak dissipation. The main results of the present work are as\nfollows.\n\ni) For moderately steep confining potential we presented a microscopic model\nthat improves the quasi-microscopic approach of Refs. \\cite{11} and\n\\cite{18}. In particular, the model combines features of the following\ndistinct edge wave mechanisms. In mostly classical models, e.g., \\cite{1},\n\\cite{7}, the position of the edge does not vary but the charge density\nprofile at the edge does. In a sence this edge wave mechanism is the analog\nof that for the Kelvin wave \\cite{21} at the edge of a rotating \"shallow\"\nsea with chirality determined by the Coriolis parameter which corresponds to\nthe cyclotron frequency $\\omega_{c}$. Another edge-wave mechanism,\nfully quantum mechanical, is that of Refs. \\cite{10}, \\cite{17}, and\n\\cite{22}-\\cite{23}, in which, for $\\nu=1$, only the edge position of an\nincompressible 2DEG of the lowest LL varies and with respect to which the\ndensity profile is that of the undisturbed 2DEG. The approach of these\nworks is limited to the subspace of the lowest LL wave functions,\nneglects LL mixing and dissipation, and results in a single EMP with\ndispersion law similar to that in Ref. \\cite{1}.\n\nii) We confirmed, for $\\nu=1 (2)$, the existence of EMP modes in addition to\nthe fundamental EMP and single one obtained for vertically steep unperturbed\nelectron density profile \\cite{1}. This is in line with our earlier\nquasi-microscopic results \\cite{11} for moderately steep \\cite{11} confining\npotentials. The additional modes result from an {\\bf exact} solution of\nEq. (\\ref{16}) in terms of the complete set of the Hermite polynomials.\nWith this expansion we can also make contact, for $\\nu=1 (2)$, with other\nmicroscopic theories \\cite{10}, \\cite{17}, and \\cite{22}-\\cite{23} that are\nlimited to the subspace of the lowest LL wave functions. If we retain only\nthe $n=0$ term in Eq. (\\ref{16}), we obtain only the $n=0$ LL fundamental\nmode with the same dispersion relation $\\omega\\sim q_{x}\\ln q_{x}$. The\nadditional modes result from retaining ( correspond to) higher-order terms\nin the expansion Eq. (\\ref{16}). This shows that the limitation to this\nLL subspace is too restrictive and indicates the importance of LL\ncoupling even for $\\nu=1 (2)$.\n\niii) An important additional mode is the {\\it dipole} EMP, presented in Sec.\nIII B, with dispersion relation $\\omega\\sim q_{x}^{3} $. This differs\nmarkedly from the corresponding result $\\omega\\sim q_{x}$ of Ref. \\cite{11}\nand signals {\\it nonlocal} responses that were previously neglected. This\nand other differences between the present results and those of Ref. \\cite{11}\nfurther indicate the importance of having a fully {\\it microscopic} theory\nfor EMPs.\n\niv) As we showed, taking LL coupling into account is an essential ingredient\nespecially for $\\nu>2$. As Fig. 1 and Fig. 3 demonstrate, the coupling\nstrongly renormalizes the {\\it uncoupled} fundamental LL modes and results\nin one mode behaving as the usual one $\\omega\\sim q_{x}\\ln q_{x}$ and the\nothers behaving in an {\\it acoustic} manner $\\omega\\sim q_{x}$. This partly\nreminds the results of Ref. \\cite{7} but the resemblance should not be\noverestimated because the two models are drastically different.\n\nv) We have not treated spin effects beyond the HA or RPA. Our study is\nfocused on important wave effects of non-spin nature and spin splitting is\nassumed negligible for even $\\nu$. Thus, skyrmions and spin textures are not\ndealt with in this work. Though neglecting the spin splitting is a\nreasonable approximation, for even $\\nu \\geq 2$\nin the bulk of the channel, its validity near the edges remains uncertain\nin view of the work of Refs. \\cite{16} and \\cite{24}.\n\n   (vi) Finally, a  few remarks are in order about  the studies\nof Refs.  \\cite{25} and \\cite{26}  that used the RPA. \nThe study of Ref. \\cite{26} is too simplified and in essence  \nrepeats the results of Ref. \\cite{25} and of works cited therein\nfor the \"optical\" EMP. As for the results of Ref. \\cite{25}, \ndue to omitted important contributions,\nthey are essentially different from ours for both the acoustic\nand the optical modes. As Sec. IV A shows, an important logarithmic term,\ncaused by the Coulomb interaction, is missed\nin Eqs.  (54a) and (54b) of Ref. \\cite{25}. \nIn addition,  Refs. \\cite{25} and \\cite{26} did not study a\ndipole mode or   multipole modes pertaining to any occupied LL.\nThese modes of ours are totally different than those of\nSecs. 3.1.6 and  3.1.7 of Ref. \\cite{25}.\n\n\\acknowledgements\nThis work was supported by NSERC Grant No. OGP0121756. In addition, O G B\nacknowledges partial support by the Ukrainian SFFI Grant No. 2.4\/665.\n\\ \\newline\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA tensor is a multidimensional array of numbers. Formally, an order-$k$ tensor is an element of the tensor product of $k$ vector spaces. A first order tensor is a vector, and a second order tensor is a matrix. The rank of a tensor is the minimal number of terms in its expression as a sum of simple tensors. See Cichocki et al. \\cite{Cichocki} for a recent survey of matrix and tensor decomposition algorithms, as well as applications in areas such as data mining, email surveillance, gene expression classification, and signal processing. See also Kroonenberg \\cite{Kroonenberg}, and Smilde et al. \\cite{Smilde}, and references therein.\n\nA tensor is called symmetric if its elements remain fixed under any permutation of the indices. Then we can define the symmetric rank of a symmetric tensor to be the minimal number of terms in its expression as a sum of simple symmetric tensors. Further, we can associate a homogeneous polynomial in $\\mathbb{F}[x_0, \\dots, x_n]_d$ to any symmetric tensor $X \\in S^d V$, and the determination of the symmetric rank of $X$ is equivalent to the Big Waring Problem: determining the minimum integer $r$ such that a generic form of degree $d$ in $n+1$ variables can be written as a sum of $r$ $d$th powers of linear forms \\cite{Comon1}. See \\cite{B1, B2, Bernardi, B3, B4} and references therein for more information on symmetric tensors, symmetric rank, and decompositions of homogeneous polynomials.\n\nThe determination of canonical forms of tensors is a generalization of the matrix case. For any $m \\times n$ matrix $A$ (with $m \\leq n$) there exist invertible matrices $P \\in GL_m(\\mathbb{F})$ and $Q \\in GL_n(\\mathbb{F})$ such that \n\\[\nPAQ = \n\\begin{bmatrix} I_r & 0 \\\\ 0 & 0 \\end{bmatrix},\n\\]\nwhere $I_r$ is the $r \\times r$ identity and $r \\leq m$ is the rank. The matrix $PAQ$ is called the Smith normal form (canonical form), and is the orbit representative for the action of $GL_m(\\mathbb{F}) \\times GL_n(\\mathbb{F})$, the direct product of general linear groups. When $m, n = 2$, there are three canonical forms,\n\\[\n\\begin{bmatrix} 0 & 0 \\\\ 0 & 0 \\end{bmatrix}, \\qquad\n\\begin{bmatrix} 1 & 0 \\\\ 0 & 0 \\end{bmatrix}, \\qquad\n\\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix}.\n\\]\nDetermining the canonical forms of order-$k$ tensors for $k>2$ is a more complicated problem. For the next simplest case, consider $2 \\times 2 \\times 2$ tensors. Le Paige (1881) determined there are 7 canonical forms over $\\mathbb{C}$ \\cite{LePaige}, and later Oldenburger determined there are 8 over $\\mathbb{R}$ \\cite{Oldenburger1}. In previous work, we considered the same problem over the prime fields $\\mathbb{F}_2$ and $\\mathbb{F}_3$ \\cite{BremnerStavrou}. The maximum rank of these tensors over $\\mathbb{R}$, $\\mathbb{C}$, and the prime fields we considered is 3. For larger tensor formats over $\\mathbb{R}$ and $\\mathbb{C}$, the problem is more complicated and, in general, there is not a finite classification \\cite{deSilvaLim}. \n\nIn previous work, we considered the same problem of determining canonical forms over prime fields, but restricted our attention to symmetric tensors. For the $2 \\times 2 \\times 2$ and $2 \\times 2 \\times 2 \\times 2$ symmetric formats over $\\mathbb{R}$ and $\\mathbb{C}$, this problem was analyzed independently by Gurevich \\cite{Gurevich} first, then by Weinberg \\cite{Weinberg}.\nIn \\cite{Stavrou}, we extended these results by determining the canonical forms of $2 \\times 2 \\times 2$ symmetric tensors over the prime fields $\\mathbb{F}_p$ for $p = 2, 3, 5, 7, 11, 13, 17$, as well as $2 \\times 2 \\times 2 \\times 2$ symmetric tensors over $\\mathbb{F}_p$ for $p = 2, 3, 5, 7$. \n\nWe continue in this direction by considering order-$k$ (for $k = 2,3,4$) symmetric tensors of format $3 \\times \\dots \\times 3$ over $\\mathbb{F}_p$, for $p \\in \\{2,3,5\\}$. For $3 \\times 3$ symmetric tensors (i.e. symmetric matrices), we work over $\\mathbb{F}_p$ for $p = 2, 3, 5$. The maximum symmetric rank is 4, which is larger than the maximum rank of 3.  For $3 \\times 3 \\times 3$ symmetric tensors, we work over $\\mathbb{F}_p$ for $p=2,3$. The maximum symmetric rank is 7, which is larger than the maximum rank of 5 over $\\mathbb{C}$ \\cite{PierreComon}. Lastly, for $3 \\times 3 \\times 3 \\times 3$ symmetric tensors, we work over $\\mathbb{F}_p$ for $p=2, 3$. Over $\\mathbb{F}_3$, the number of symmetric tensors was too large to determine, and so the procedure had to be terminated. At termination, there were symmetric tensors with symmetric rank 13, thus we have a lower bound on the symmetric rank.\n\nIn each case, we compute the symmetric tensors of each rank. The general linear group GL$_3(\\mathbb{F}_p)$ partitions the tensors of each rank into a disjoint union of orbits, where the tensors in each orbit are equivalent. The elements are arranged in lexical order and the minimal element is the canonical form of its orbit under this group action. We comment here that due to limited computer memory, we are not able to consider larger prime fields, or other tensor formats. \n\n\n\\section{Preliminaries}\n\nAn order-$k$ {\\bf tensor} $X$ is an element of the {\\bf tensor product} of $k$ vector spaces $V_1 \\otimes \\dots \\otimes V_k$, where the {\\bf order} refers to the number of dimensions. Once we fix a basis in each vector space, we can associate to $X$ a $k$-dimensional array. Then an order-1 tensor is a vector, an order-2 tensor is a matrix, and we denote an order-$k$ tensor of format $d_1 \\times \\dots \\times d_k$ as $[x_{i_1 \\dots i_k}] \\in \\mathbb{F}^{d_1 \\times \\dots \\times d_k}$. We can represent the tensor $X$ in {\\bf vectorized (flattened)} form by writing the columns of $X$ in a vector format, where the entries are in lexical order of the $k$-tuples of the subscripts. \n\nAn order-$k$ tensor $[x_{i_1 \\dots i_k}] \\in \\mathbb{F}^{n \\times \\dots \\times n}$ is called {\\bf symmetric} if \n\\[\nx_{i_{\\pi(1)} \\dots i_{\\pi(k)}} = x_{i_1 \\dots i_k}, \\qquad i_1, \\dots, i_k \\in \\{1, \\dots, n\\},\n\\]\nfor all permutations $\\pi \\in S_k$  \\cite{Comon1}. \n\n\\begin{proposition} [Proposition 3.7 \\cite{Comon1}]\nLet $X = [x_{i_1 \\dots i_k}] \\in \\mathbb{F}^{n \\times \\dots \\times n}$ be an order-$k$ tensor. Then \n\\[\n\\pi(X) = X\n\\]\nfor all permutations $\\pi \\in S_k$ if and only if \n\\[\nx_{i_{\\pi(1)} \\dots i_{\\pi(k)}} = x_{i_1 \\dots i_k}, \\qquad i_1, \\dots, i_k \\in \\{1, \\dots, n\\}\n\\]\nfor all permutations $\\pi \\in S_k$.\n\\end{proposition}\n\nIn this paper we consider $3 \\times \\dots \\times 3$ symmetric tensors of order-$k$ for $k=2,3,4$. For $k = 2$, a symmetric matrix has the form\n\\[\n\\begin{bmatrix}\na & b & c \\\\\nb & d & e \\\\\nc & e & f\n\\end{bmatrix},\n\\]\nwhere the entries are scalars in $\\mathbb{F}_p$. For $k = 3$, \na $3 \\times 3 \\times 3$ symmetric tensor, in terms of its $3 \\times 3$ frontal slices, has the form \n\\[\n\\left[\n\\begin{array}{ccc|ccc|ccc}\na & b & c & b & d & e & c & e & f \\\\\nb & d & e & d & j & g & e & g & h \\\\\nc & e & f & e & g & h & f & h & k\n\\end{array}\n\\right].\n\\]\nLastly, for $k = 4$, a $3 \\times 3 \\times 3 \\times 3$ symmetric tensor, in terms of its $3 \\times 3$ frontal slices, has the form\n\\[\n\\left[\n\\begin{array}{ccc|ccc|ccc}\na & b & f & b & c & g & f & g & j \\\\\nb & c & g & c & d & i & g & i & h \\\\\nf & g & j & g & i & h & j & h & k \\\\ \\hline\nb & c & g & c & d & i & g & i & h \\\\\nc & d & i & d & e & l & i & l & m \\\\\ng & i & h & i & l & m & h & m & n \\\\ \\hline\nf & g & j & g & i & h & j & h & k \\\\\ng & i & h & i & l & m & h & m & n \\\\\nj & h & k & h & m & n & k & n & p\n\\end{array}\n\\right].\n\\]\n\nWe will denote the set of all order-$k$, $n$-dimensional symmetric tensors over the field $\\mathbb{F}$ by $\\mathcal{S}^k(\\mathbb{F}^n) \\subset \\mathbb{F}^{n \\times \\dots \\times n}$. The set of such tensors satisfies the property $\\pi(X) = X$ for all $\\pi \\in S_k$ and $X \\in \\mathcal{S}^k(\\mathbb{F}^n)$. \n\n\\begin{definition}\\cite{Hitchcock1} \nA tensor $X \\in \\mathbb{F}^{d_1 \\times \\dots \\times d_k}$ is {\\bf simple} if it can be written as\n\\[\nX = u^{(1)} \\otimes u^{(2)} \\otimes \\cdots \\otimes u^{(k)},\n\\]\nwith non-zero $u^{(i)} \\in \\mathbb{F}^{d_i}$ for $i = 1, \\dots, k$. The $(i_1, i_2, \\dots, i_k)$th entry of $X$ is \n\\[\nx_{ i_1 i_2 \\cdots i_k } = u^{(1)}_{i_1} u^{(2)}_{i_2} \\cdots u^{(k)}_{i_k}.\n\\]\n\\end{definition}\n\n\\begin{definition}\nA tensor $X \\in \\mathcal{S}^k(\\mathbb{F}^n)$ is called {\\bf simple symmetric} if it can be written as \n\\[\nX = u \\otimes \\dots \\otimes u\n\\]\nwith $k$-many non-zero vectors $u \\in \\mathbb{F}^n$. The $(i_1, \\dots, i_k)$th entry of $X$ is \n\\[\nx_{i_1 \\dots i_k} = u_{i_1} \\dots u_{i_k}.\n\\]\n\\end{definition}\n\n\\begin{definition}\\cite{Hitchcock1}\nA tensor has {\\bf outer product rank} $r$ if it can be written as a sum of $r$ (and no fewer) decomposable tensors, \n\\[\nX = \\sum_{i = 1}^r  u_i^{(1)} \\otimes \\dots \\otimes u_i^{(k)} = u_1^{(1)} \\otimes \\dots \\otimes u_1^{(k)} + \\dots + u_r^{(1)} \\otimes \\dots \\otimes u_r^{(k)}\n\\]\nwhere $u_i^{(1)} \\in \\mathbb{F}^{d_1}, \\dots, u_i^{(k)} \\in \\mathbb{F}^{d_k}, i = 1, \\dots, r$.\nWe write rank$(X)$ to denote the outer product rank of $X$.\n\\end{definition}\n\n\\begin{definition}\nA tensor has {\\bf symmetric outer product rank} $s$ if it can be written as a sum of $s$ (and no fewer) symmetric simple tensors,\n\\[\nX = \\sum_{i=1}^s u_i^{\\otimes k}.\n\\]\nWe write rank$_S(X)$ to denote the symmetric outer product rank of $X$. \n\\end{definition}\nThe only rank-0 symmetric tensor is the zero tensor. We will drop the words outer product and simply say symmetric rank. \n\n\\begin{definition}\nThe {\\bf maximum rank} is defined to be  \n\\[\n\\text{max}\\{ \\text{rank}(X) \\mid X \\in \\mathbb{F}^{d_1 \\times \\dots \\times d_k} \\}. \n\\]\nIf we replace rank$(X)$ with rank$_S(X)$ we get the analogous definition of maximum symmetric rank.\n\n\\end{definition}\n\nIn order to compute the canonical forms of symmetric tensors we use a tensor-matrix multiplication. {\\bf Multilinear matrix multiplication} is a tensor-matrix multiplication that allows us to multiply matrices on each of the modes of the tensor. If $X = [x_{i_1 \\dots i_k}] \\in \\mathbb{F}^{d_1 \\times \\dots \\times d_k}$ and \n\\[\nA_1 = [a^{(1)}_{u_1 i_1}] \\in \\mathbb{F}^{c_1 \\times d_1}, \\dots, A_k = [a^{(k)}_{u_k i_k}] \\in \\mathbb{F}^{c_k \\times d_k},\n\\]\nthen $Y= (A_1, \\dots, A_k)\\cdot X = [y_{u_1 \\dots u_k}] \\in \\mathbb{F}^{c_1 \\times \\dots \\times c_k}$ is the new tensor defined by\n\\[\ny_{u_1 \\dots u_k} = \\sum_{i_1, \\dots, i_k = 1}^{d_1, \\dots, d_k} a^{(1)}_{u_1 i_1}  \\dots  a^{(k)}_{u_k i_k} x_{i_1 \\dots i_k}.\n\\]\nSince we are restricting ourselves to symmetric tensors, we impose the condition that $A_1 = \\dots = A_k$, otherwise, multilinear matrix multiplication can transform a symmetric tensor into a non-symmetric tensor.\n\n\nWe consider the action of the symmetry group GL$_3(\\mathbb{F}_p)$, which does not change the rank of a tensor \\cite{deSilvaLim}. The action of this group decomposes the set of order-$k$ symmetric tensors into a disjoint union of orbits, where the tensors in each orbit are equivalent under the group action. The orbit of $X$ is the set \n\\[\n\\mathcal{O}_X := \\{ (g, \\dots, g) \\cdot X \\mid g \\in \\text{GL}_3(\\mathbb{F}_p)\\}.\n\\] \nWe define the {\\bf canonical form} of $X$ to be the minimal element in its orbit with respect to the lexical ordering. \n\n\n\n\\section{Algorithms}\n\nUsing computer algebra, we generate all the symmetric tensors in each symmetric rank. Then we apply the group action which decomposes the set of tensors within each rank into a disjoint union of orbits. The minimal element of each orbit with respect to the lexical ordering is the canonical form.  \n\nWe determine the symmetric tensors in each symmetric rank by first generating the set $R_1$ of simple symmetric tensors, which is achieved by computing all possible products $v \\otimes \\dots \\otimes v$ of non-zero vectors $v \\in \\mathbb{F}^3_p$, for each prime. Then to determine the elements in $R_i$ we compute all possible sums $(T + S)\\ \\textrm{mod} \\ p$ for $T \\in R_{i-1}$ and $S \\in R_1$, and then subtract the tensors already computed in the previous ranks: $R_i = R_i \\setminus \\cup_{k=1}^{i-1} R_k$. The procedure terminates once no new symmetric tensors are generated when taking sums.  In our previous work \\cite{Stavrou}, when $T+S$ was computed, the procedure searched to see if $T+S$ already existed in the current rank and the lower ranks. If it did not, then it was added to the current rank set. In this paper, we made the necessary modification of subtracting the sets so that the ranks were computed more efficiently. This is what allowed us to consider larger formats than what we considered in \\cite{Stavrou}.\n\n\nNext we apply the action of the general linear group to decompose the tensors in each rank into a disjoint union of orbits. The pseudocode is provided in Table \\ref{groupalgorithmtable}. This algorithm was used in our previous paper \\cite{Stavrou}. In this paper, we made two major modifications to the program. The first modification was removing an index (mode) from the procedures in order to consider second order $3 \\times 3$ symmetric tensors.  The second modification was to index the elements to 3, since each mode has length 3. The pseudocode in Table \\ref{groupalgorithmtable} displays the order-4 case. For the order-3 case, remove the $\\ell$ index, and for the order-2 case, remove the $\\ell$ and $k$ indices.\n\n\n\n\\begin{table}\n\\begin{itemize}\n\\item[]\n\\texttt{unflatten}$( x )$\n\\item[] \\quad\nset $t \\leftarrow 0$\n\\item[] \\quad\nfor $i, j, k, \\ell = 1, 2, 3$ do:\nset $t \\leftarrow t + 1$;\nset $y_{ijk\\ell} \\leftarrow x_t$\n\\item[] \\quad\n$\\texttt{return}( y )$\n\\end{itemize}\n\\medskip\n\\begin{itemize}\n\\item[]\n\\texttt{groupaction}$( g, x, m )$\n\\item[] \\quad\nset $y \\leftarrow \\texttt{unflatten}( x )$\n\\item[] \\quad\nif $m = 1$ then\nfor $j,k,\\ell = 1,2,3$ do:\n\\item[] \\quad \\quad\nset $v \\leftarrow [ \\, y_{1jk\\ell}, \\, y_{2jk\\ell} \\, ]$;\n\\item[] \\quad \\quad\nset $w \\leftarrow [ \\, g_{11} v_1 {+} g_{12} v_2 {+} g_{13}v_3 \\, \\mathrm{mod}\\,p,\n\\dots,\n\\, g_{31} v_1 {+} g_{32} v_2{+} g_{33} v_3 \\, \\mathrm{mod}\\,p \\,]$\n\\item[] \\quad \\quad\nfor $i = 1, 2, 3$ do: set $y_{ijk\\ell} \\leftarrow w_i$\n\\item[] \\quad\nif $m = 2$ then \\dots \\emph{(similar for second subscript)}\n\\item[] \\quad\nif $m = 3$ then \\dots \\emph{(similar for third subscript)}\n\\item[] \\quad\nif $m = 4$ then \\dots \\emph{(similar for fourth subscript)}\n\\item[] \\quad\n\\texttt{return}( \\texttt{flatten}( $y$ ) )\n\\end{itemize}\n\\medskip\n\\begin{itemize}\n\\item[]\n\\texttt{smallorbit}$( x )$\n\\item[] \\quad\nset $\\texttt{result} \\leftarrow \\{\\,\\}$\n\\item[] \\quad\nfor $a \\in GL_3(\\mathbb{F}_p)$ do:\n\\item[] \\quad \\quad\nset $y \\leftarrow \\texttt{groupaction}( a, x, 1 )$\n\\item[] \\quad \\quad\nset $z \\leftarrow \\texttt{groupaction}( a, y, 2 )$\n\\item[] \\quad \\quad \nset $w \\leftarrow \\texttt{groupaction}( a, z, 3 )$\n\\item[] \\quad \\quad \nset $u \\leftarrow \\texttt{groupaction}( a, w, 4 )$\n\\item[] \\quad\nset $\\texttt{result} \\leftarrow \\texttt{result} \\cup \\{ u \\}$\n\\item[] \\quad\n\\texttt{return}( \\texttt{result} )\n\\end{itemize}\n\\medskip\n\\begin{itemize}\n\\item\nfor $r = 0,\\dots,\\texttt{maximumrank}$ do:\n\\item[] \\quad\nset $\\texttt{representatives}[r] \\leftarrow \\{\\,\\}$;\nset $\\texttt{remaining} \\leftarrow \\texttt{arrayset}[r]$\n\\item[] \\quad\nwhile $\\texttt{remaining} \\ne \\{\\,\\}$ do:\n\\item[] \\quad \\quad\nset $x \\leftarrow \\texttt{remaining}[1]$;\nset $\\texttt{xorbit} \\leftarrow \\texttt{largeorbit}( x )$\n\\item[] \\quad \\quad\nappend $\\texttt{xorbit}[1]$ to $\\texttt{representatives}[r]$\n\\item[] \\quad \\quad\nset $\\texttt{remaining} \\leftarrow \\texttt{remaining} \\setminus \\texttt{xorbit}$\n\\end{itemize}\n\\medskip\n\\caption{Algorithm for group action (pseudocode)}\n\\label{groupalgorithmtable}\n\\end{table}\n\n\n\n\n\n\n\\section{Symmetric Tensors of Format $3 \\times 3$}\n\nEvery order-$k$ symmetric tensor of dimension $n$ may be uniquely associated with a homogeneous polynomial (also called a quantic) of degree $k$ in $n$ variables \\cite{Comon1}. The problem of determining symmetric rank is equivalent to the Big Waring Problem: determining the minimal number of $p$th powers of linear terms \\cite{Ehrenborg2} \\cite{Ehrenborg3}. The Alexander-Hirschowitz (AH) Theorem \\cite{AH} completely describes the calculation of the generic rank of symmetric tensors. \n\n\n\n\\subsection{Canonical Forms of $3 \\times 3$ Symmetric Tensors over $F_2$.}\n\nThere are 512 $3 \\times 3$ tensors over $\\mathbb{F}_2$, where 64 are symmetric. Every symmetric tensor has a symmetric decomposition. The maximum symmetric rank is 3, which is equal to the maximum rank. The symmetric ranks, orders of each orbit, and the minimal representatives of each orbit are given in Table \\ref{table33symmetricmod2}. The number of tensors in each symmetric rank is listed below.\n\\[\n\\begin{array}{lrrrrrrrr}\n\\text{rank} & 0 & 1 & 2 & 3 \\\\\n\\text{number} & 1 & 7 & 21 & 35 \\\\\n\\text{$\\approx$ $\\%$} & 1.5625\\% & 10.9375\\% & 32.8125\\% & 54.6875\\%\n\\end{array}\n\\]\n\n  \\begin{table}\n  \\[\n  \\begin{array}{ccc}\n  \\text{symmetric rank} & \\text{orbit size} &  \\text{canonical form} \n    \\\\\n  \\toprule \n0 & 1 & \\begin{bmatrix}0&0&0\\\\0&0&0\\\\0&0&0\\end{bmatrix}\n\\\\\n\\midrule \n1 & 7 & \\begin{bmatrix}0&0&0\\\\0&0&0\\\\0&0&1\\end{bmatrix}\n\\\\\n\\midrule\n2 & 21 &  \\begin{bmatrix}0&0&0\\\\0&0&1\\\\0&1&1\\end{bmatrix}\n\\\\\n\\midrule\n3 & 7 & \\begin{bmatrix}0&0&0\\\\0&0&1\\\\0&1&0\\end{bmatrix}\n\\\\\n\\midrule\n3 & 28 & \\begin{bmatrix}0&0&1\\\\0&1&0\\\\1&0&0\\end{bmatrix}\n  \\\\\n  \\bottomrule\n  \\end{array}\n  \\]\n  \\medskip\n  \\caption{Canonical forms of $3 \\times 3$ symmetric tensors over $\\mathbb{F}_2$}\n  \\label{table33symmetricmod2}\n  \\end{table}\n\n\n\\subsection{Canonical Forms of $3 \\times 3$ Symmetric Tensors over $F_3$.}\n\nThere are 19,683 $3 \\times 3$ tensors over $\\mathbb{F}_3$, where 729 are symmetric. Every symmetric tensor has a symmetric decomposition. The maximum symmetric rank in this case is 4, which is larger than the maximum rank of 3. The symmetric ranks, orders of each orbit, and the minimal representatives of each orbit are given in Table \\ref{table33symmetricmod3}. The number of tensors in each symmetric rank is listed below.\n\\[\n\\begin{array}{lrrrrrrrr}\n\\text{rank} & 0 & 1 & 2 & 3 & 4 \\\\\n\\text{number} & 1 & 13 & 91 & 390 & 234 \\\\\n\\text{$\\approx$ $\\%$} & 0.1372\\% & 1.7833\\% & 12.4829\\% & 53.4979\\% & 32.0988\n\\end{array}\n\\]\n\n  \\begin{table}\n  \\[\n  \\begin{array}{ccc}\n  \\text{symmetric rank} & \\text{orbit size} &  \\text{canonical form} \n    \\\\\n  \\toprule \n0 & 1 & \\begin{bmatrix}0&0&0\\\\0&0&0\\\\0&0&0\\end{bmatrix}\n\\\\\n\\midrule \n1 & 13 & \\begin{bmatrix}0&0&0\\\\0&0&0\\\\0&0&1\\end{bmatrix}\n\\\\\n\\midrule\n2 & 13 &  \\begin{bmatrix}0&0&0\\\\0&0&0\\\\0&0&2\\end{bmatrix}\n\\\\\n\\midrule\n2 & 78 &  \\begin{bmatrix}0&0&0\\\\0&1&0\\\\0&0&1\\end{bmatrix}\n\\\\\n\\midrule\n3 & 156 & \\begin{bmatrix}0&0&0\\\\0&0&1\\\\0&1&0\\end{bmatrix}\n\\\\\n\\midrule\n3 & 234 &  \\begin{bmatrix}0&0&1\\\\0&2&0\\\\1&0&0\\end{bmatrix}\n\\\\\n\\midrule\n4 & 234 & \\begin{bmatrix}0&0&1\\\\0&1&0\\\\1&0&0\\end{bmatrix}\n  \\\\\n  \\bottomrule\n  \\end{array}\n  \\]\n  \\medskip\n  \\caption{Canonical forms of $3 \\times 3$ symmetric tensors over $\\mathbb{F}_3$}\n  \\label{table33symmetricmod3}\n  \\end{table}\n\n\n\n\n\n\n\\subsection{Canonical Forms of $3 \\times 3$ Symmetric Tensors over $F_5$.}\n\nThere are 1,953,125 $3 \\times 3$ tensors over $\\mathbb{F}_5$, where 15,625 are symmetric.  The maximum symmetric rank is again 4. The order of the group GL$(\\mathbb{F}_5)$ is 427,307 which our program cannot handle, and thus we do not determine the orbits under the group action. The symmetric ranks, number of tensors in each rank, and the minimal representative of each rank are given in Table \\ref{table33symmetricmod5}. \n\\[\n\\begin{array}{lrrrrrrrr}\n\\text{rank} & 0 & 1 & 2 & 3 & 4 \\\\\n\\text{number} & 1 & 62 & 1922 & 7440 & 6200 \\\\\n\\text{$\\approx$ $\\%$} & 0.0064\\% & 0.3968\\% & 12.3008\\% & 47.6160\\% & 39.6800\n\\end{array}\n\\]\n\n  \\begin{table}\n  \\[\n  \\begin{array}{ccc}\n  \\text{symmetric rank} & \\text{rank size} &  \\text{minimal element} \n    \\\\\n  \\toprule \n0 & 1 & \\begin{bmatrix}0&0&0\\\\0&0&0\\\\0&0&0\\end{bmatrix}\n\\\\\n\\midrule \n1 & 62 & \\begin{bmatrix}0&0&0\\\\0&0&0\\\\0&0&1\\end{bmatrix}\n\\\\\n\\midrule\n2 & 1922 &  \\begin{bmatrix}0&0&0\\\\0&0&0\\\\0&0&2\\end{bmatrix}\n\\\\\n\\midrule\n3 & 7440 &  \\begin{bmatrix}0&0&0\\\\0&1&0\\\\0&0&2\\end{bmatrix}\n\\\\\n\\midrule\n4 & 6200 & \\begin{bmatrix}0&0&1\\\\0&2&0\\\\1&0&0\\end{bmatrix}\n  \\\\\n  \\bottomrule\n  \\end{array}\n  \\]\n  \\medskip\n  \\caption{Minimal elements of $3 \\times 3$ symmetric tensors over $\\mathbb{F}_5$}\n  \\label{table33symmetricmod5}\n  \\end{table}\n  \n  \n  \n  \n  \n  \n  \n\n\n\n\n\\section{Symmetric Tensors of Format $3 \\times 3 \\times 3$}\n\nIn Dickson's 1908 paper \\cite{Dickson}, he cites that a complete set of canonical forms of ternary cubics was determined over $\\mathbb{C}$. He considers canonical forms by determining the algebraic irrationalities occurring in the reducing linear transformations over $\\mathbb{F}_3$, and finds 11 distinct forms. \n\n\nMore recently, Comon \\cite{PierreComon} summarize the generic ranks of symmetric tensors of some specific formats. In particular, they tabulate the equivalence classes for ternary cubics (i.e. symmetric $3 \\times 3 \\times 3$ tensors). Below are the orbits under the action of the group of invertible three-dimensional changes of coordinates.\n\\[\n\\begin{array}{c|c}\n\\text{Orbit} & \\text{rank} \\\\ \\hline\nx^3 & 1 \\\\\nx^3+y^3 & 2 \\\\\nx^2y & 3 \\\\\nx^3 + 3y^2z & 4 \\\\\nx^3 + y^3 + 6xyz & 4 \\\\\nx^3 + 6xyz & 4 \\\\\na(x^3 + y^3 + z^3) + 6bxyz & 4 \\\\\nx^2y + xz^2 & 5\n\\end{array}\n\\]\n\n\nKogan and Maza \\cite{Kogan} determined the equivalence classes of ternary cubics under general complex linear changes of variables using a computational approach. See in particular Theorem 5 of \\cite{Kogan}. Their contribution provided a method of computing the signature manifolds for each of the equivalence classes, and their algorithm matches a cubic with its canonical form while producing the required linear transformation explicitly.\n\n\n\n\n\\subsection{Canonical Forms of $3 \\times 3 \\times 3$ Symmetric Tensors over $\\mathbb{F}_2$}\n\n\nThere are 1024 $3 \\times 3 \\times 3$ symmetric tensors over $\\mathbb{F}_2$. Only 128 (12.5\\%) of these have a symmetric decomposition. The maximum symmetric rank is 7. The symmetric ranks, orders of each orbit, and the minimal representatives (in flattened form) of each orbit are given in Table \\ref{table333symmetricmod2}. To make the canonical forms more legible we replace the zero entries with $\\cdot$. The number of tensors in each symmetric rank is listed below (percentages add to 12.5\\%).\n\\[\n\\begin{array}{lrrrrrrrr}\n\\text{rank} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\\n\\text{number} & 1 & 7 & 21 & 35 & 35 & 21 & 7 & 1 \\\\\n\\text{$\\approx$ $\\%$} & 0.10\\% & 0.68\\% & 2.05\\% & 3.42\\% & 3.42\\% & 2.05\\% & 0.68\\% & 0.10 \\%\n\\end{array}\n\\]\n\n  \\begin{table}\n  \\[\n  \\begin{array}{ccc}\n  \\text{symmetric rank} & \\text{orbit size} &  \\text{canonical form (flattened)} \n    \\\\\n  \\toprule \n0 & 1 & [ \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot ]\n\\\\\n\\midrule \n1 & 7 & [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1] \n\\\\\n\\midrule\n2 & 21 &  [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 1 \\cdot \\cdot \\cdot \\cdot 1 1 \\cdot 1 1]\n\\\\\n\\midrule\n3 & 7 & [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 1 \\cdot \\cdot \\cdot \\cdot 1 1 \\cdot 1 \\cdot]\n\\\\\n3 & 18 &  [\\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1]\n\\\\\n\\midrule\n4 & 7 &  [\\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot 1 1 1 1 \\cdot 1 \\cdot 1 1 1 \\cdot 1 \\cdot]  \n\\\\  \n4 & 18 & [\\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot]\n\\\\   \n\\midrule\n5 & 21 & [\\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot 1 1 1 1 \\cdot 1 \\cdot 1 1 1 \\cdot 1 1]    \n\\\\\n\\midrule \n6 & 7 & [\\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot \\cdot 1]\n\\\\  \n\\midrule\n7 & 1 & [\\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot]   \n  \\\\\n  \\bottomrule\n  \\end{array}\n  \\]\n  \\medskip\n  \\caption{Canonical forms of $3 \\times 3 \\times 3$ symmetric tensors over $\\mathbb{F}_2$}\n  \\label{table333symmetricmod2}\n  \\end{table}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Canonical Forms of $3 \\times 3 \\times 3$ Symmetric Tensors over $\\mathbb{F}_3$}\n\nThere are 59,049 $3 \\times 3 \\times 3$ symmetric tensors over $\\mathbb{F}_3$, and unlike in the previous case, every symmetric tensor has a symmetric decomposition. The maximum symmetric rank is 7. The symmetric ranks, orders of each orbit, and the minimal representatives of each orbit are given in Table \\ref{table333symmetricmod3}. The number of tensors in each rank is listed in the table below. We mention now that we cannot consider larger primes because of insufficient computer memory. There are over 9.7 million symmetric $3 \\times 3 \\times 3$ tensors over $\\mathbb{F}_5$.\n\n\\[\n\\begin{array}{lrrrrrrrr}\n\\text{rank} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\\n\\text{number} & 1 & 26 & 312 & 2288 & 11440 & 30342 & 14352 & 288 \\\\\n\\text{$\\approx$ $\\%$} & 0.00\\% & 0.05\\% & 0.53\\% & 3.87\\% & 19.37\\% & 51.38\\% & 24.31\\% & 0.49 \\%\n\\end{array}\n\\]\n\n\\begin{table}\n \\[\n  \\begin{array}{cccc}\n  \\text{symmetric rank} & \\text{orbit size} &  \\text{canonical form (flattened)} \n    \\\\\n  \\toprule\n0 & 1 &  [\\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot]   \n\\\\\n\\midrule  \n1 & 26 & [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1]  \n\\\\\n\\midrule\n2 & 312 &  [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1]  \n\\\\\n\\midrule  \n3 & 104 &  [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot]  \n\\\\ \n3 & 312 &  [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 2]    \n\\\\\n3 & 1872 & [\\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1]     \n\\\\\n\\midrule  \n4 & 208 & [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 1]  \n\\\\\n4 & 1872 & [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot ]    \n\\\\\n4 & 468 & [\\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot]    \n\\\\\n4 & 1404 & [\\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1]    \n\\\\\n4 & 5616 & [\\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 2]    \n\\\\\n4 & 1872 & [\\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 1 \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 1]   \n\\\\\n\\midrule  \n5 & 624 & [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot 1 \\cdot 1]     \n\\\\\n5 & 3744 & [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1]    \n\\\\\n5 & 2808  & [\\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot \\cdot 1]    \n\\\\\n5 & 5616  &   [\\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot \\cdot 1]    \n\\\\\n5 & 702  & [\\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot]    \n \\\\\n5 & 5616 & [\\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 1 \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot]    \n\\\\ \n5 & 5616  &  [\\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 1 \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 2]    \n\\\\\n5 & 5616  & [\\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 2 1 \\cdot \\cdot \\cdot \\cdot 2 \\cdot 2 \\cdot]    \n\\\\\n\\midrule  \n6 & 624 & [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot 1 \\cdot \\cdot]   \n\\\\\n6 & 624 & [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot 1 \\cdot 2]     \n\\\\ \n6 & 5616 & [\\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1]    \n\\\\\n6 & 3744 & [\\cdot \\cdot 1 \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot 2 \\cdot 2 1 1 \\cdot \\cdot \\cdot 2 1 \\cdot 1 1]    \n\\\\\n6 & 3744 & [\\cdot \\cdot 1 \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot 2 \\cdot 2 1 1 \\cdot \\cdot \\cdot 2 1 \\cdot 1 2]    \n \\\\\n \\midrule \n7 & 288  & [\\cdot \\cdot 1 \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot 2 \\cdot 2 1 1 \\cdot \\cdot \\cdot 2 1 \\cdot 1 \\cdot] \n\\\\ \n  \\bottomrule\n \\end{array}\n  \\]\n  \\medskip\n  \\caption{Canonical forms of $3 \\times 3 \\times 3$ symmetric tensors over $\\mathbb{F}_3$}\n  \\label{table333symmetricmod3}\n  \\end{table}\n\n\n\n\n\n\n\\section{Symmetric Tensors of Format $3 \\times 3 \\times 3 \\times 3$}\n\nWe consider $3 \\times 3 \\times 3 \\times 3$ symmetric tensors over $\\mathbb{F}_2$. Since the order of the group is too large for our algorithm, we do not determine the orbits under the group action. Instead, we determine the maximum symmetric rank, the number of symmetric tensors in each rank, and the minimal element.\n\n\n\\subsection{Canonical Forms of $3 \\times 3 \\times 3 \\times 3$ Symmetric Tensors over $\\mathbb{F}_2$}\nThere are $2^{81}$ tensors over $\\mathbb{F}_2$, where 32,768 are symmetric. Only 128 (approximately 0.29\\%) of these symmetric tensors have a symmetric decomposition.\n\\[\n\\begin{array}{lrrrrrrrr}\n\\text{rank} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\\n\\text{number} & 1 & 7 & 21 & 35 & 35 & 21 & 7 & 1 \\\\\n\\text{$\\approx$ $\\%$} & 0.0031\\% & 0.0214\\% & 0.0641\\% & 0.1068\\% & 0.1068\\% & 0.0641\\% & 0.0214\\% & 0.0031 \\%\n\\end{array}\n\\]\n\nThe rank-0 representative is the zero tensor. The rank-1 minimal representative is the symmetric tensor with every entry equal to 0 except for the $(3,3,3,3)$th entry, which equals 1. The rank-2,3,4,5,6,7 minimal representatives are given respectively by \n\\begin{align*}\n&\n\\left[\n\\begin{array}{ccc|ccc|ccc}\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ \\hline\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 \\\\\n0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 \\\\ \\hline\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 \\\\ \n0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 \\\\ \n\\end{array}\n\\right], \n\\quad\n\\left[\n\\begin{array}{ccc|ccc|ccc}\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ \\hline\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 \\\\\n0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 \\\\ \\hline\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 \\\\ \n0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 \\\\ \n\\end{array}\n\\right],\n\\\\\n&\n\\left[\n\\begin{array}{ccc|ccc|ccc}\n0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\\\\n0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\\\ \\hline\n0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\\\\n0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\\\\n1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ \\hline\n0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\\\\n1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ \n0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\\\\n\\end{array}\n\\right],\n\\quad\n\\left[\n\\begin{array}{ccc|ccc|ccc}\n0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\\\\n0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\\\ \\hline\n0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\\\\n0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\\\\n1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ \\hline\n0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\\\\n1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ \n0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 1 \\\\\n\\end{array}\n\\right],\n\\\\\n&\n\\left[\n\\begin{array}{ccc|ccc|ccc}\n0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\\\\n0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\\\ \\hline\n0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\\\\n0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 \\\\ \\hline\n0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\\\\n1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 \\\\ \n0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\\\\n\\end{array}\n\\right],\n\\quad\n\\left[\n\\begin{array}{ccc|ccc|ccc}\n0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\\\\n0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\\\ \\hline\n0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\\\\n0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 \\\\ \\hline\n0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\\\\n1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 \\\\ \n0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n\\end{array}\n\\right].\n\\end{align*}\n\nWe attempted to work over $\\mathbb{F}_3$ but the computations could not be completed. We were able to determine that the maximum symmetric rank is at least 13 (since the program was terminated at this step while many symmetric tensors still had not been found). Thus, larger prime fields and larger tensor formats cannot be dealt with by our computer algorithm in a reasonable amount of time.\n\n\n\\section{Conclusion}\nWe began by summarizing known results, and our own previous work about canonical forms of tensors (symmetric and non-symmetric). For third and fourth order tensors, the simplest cases for the classification of canonical forms is for $2 \\times \\dots \\times 2$ tensor formats. The classification over $\\mathbb{R}$ and $\\mathbb{C}$ has appeared many times throughout the literature. In previous work, we extended the results by determining the canonical forms of $2 \\times 2 \\times 2$ tensors over the prime fields $\\mathbb{F}_p$ for $p = 2, 3, 5$. We also examined the same problem for the larger tensor format $2 \\times 2 \\times 2 \\times 2$ over $\\mathbb{F}_p$ for $p = 2, 3$, where the choice of $p$ is restricted by the memory capabilities of our computer.\n\nNext, we summarized known results about $2 \\times 2 \\times 2$ symmetric and $2 \\times 2 \\times 2 \\times 2$ symmetric tensors. The canonical forms of these tensor formats have been enumerated in the past over $\\mathbb{R}$ and $\\mathbb{C}$ by Weinberg \\cite{Weinberg} and Gurevich \\cite{Gurevich}. In recent work, we extended these results by determining the canonical forms over $\\mathbb{F}_p$. For the third order case, we considered all primes $p \\leq 17$. For the fourth order case, we considered $p \\leq 5$. \n\nIn this paper, we determined the canonical forms of second, third, and fourth order symmetric tensors of format $3 \\times \\dots \\times 3$ over $\\mathbb{F}_p$ for $p \\in \\{2, 3, 5\\}$. We determined that the maximum symmetric rank over finite fields is at least 4 in the second order case, 7 in the third order case, and 13 in the fourth order case. Due to memory limitations we could not consider larger primes or tensor formats. \n\n\n\n\n\\section*{Acknowledgements}\nThe author would like to thank Edoardo Ballico for his insights and suggestions. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}