diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzdfzx" "b/data_all_eng_slimpj/shuffled/split2/finalzzdfzx"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzdfzx"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction\\label{Intro}}\n\nThere is currently growing interest in theoretical and experimental studies\nof topologial superconductors \\cite{Kallin}-\\cite{MS}. One of the well-known\nexamples is $Sr_2RuO_4$, which is thought to be a chiral triplet p-wave\nsuperconductor \\cite{Maeno}-\\cite{Maeno2016}. There are several features of \ntriplet p-wave superconductivity, among them the key signature is appearence of the odd-frequency states\nin the vicinity of surface, leading to formation undergap bound states \\cite{Zhang}-\\cite{LuBo}. The chiral\nsuperconductivity is highlighted by existence of the spontaneous\nsurface currents being one of most interesting among the predicted features. Unfortunately,\nthese currents have never been experimentally detected yet \\cite{PGB},\n\\cite{Kirtley}.\n\nOne of possible reasons for the absence of the surface currents could be the\npresence of non-superconducting layers on the surface of the samples, which\nmay be formed due to sample degradation. Such layers can be characterized by\nvarious degrees of diffusivity and by range of thickness, $d,$ measured in\nsuperconducting decay length units, $\\xi _{0}$.\nIt was demonstrated recently within the quasiclassical Eilenberger formalism\n\\cite{Eilenberger} that the edge currents in a chiral p -wave\nsuperconductor are robust with respect to surface roughness \\cite{Suzuki} in the regime of small thickness $d$. On the other hand, calculations done\nfor the case of specular surface within the tight-binding model \\cite{Ashby}\n\\cite{Huang} have shown that the spontaneous currents could be suppressed if $d$ is of the same order or greater than the decay length $\\xi _{0}$.\n\nIn order to establish general conditions for the existence of surface currents in chiral p-wave superconductors, in this work we formulate\nthe general approach in the framework of the Eilenberger formalism \\cit\n{Eilenberger} for various experimental realizations of surface layers in\nthese materials. We investigate thin $(d\\ll \\xi _{0})$ and\nthick $(d\\gg \\xi _{0})$ clean and diffusive films in proximity with bulk\np-wave chiral superconductor. We show that, in accordance with the results of\n\\cite{Suzuki}, spontaneous currents are not sensitive to surface roughness\nin the limit of thin surface layer, while in the opposite case $(d\\gg \\xi\n_{0})$ these currents are strongly suppressed. We also discuss the peculiarities of\ncurrent distributions inside the sample for different types of surface layers and the possibility of their experimental observation.\n\n\\section{Model\\label{Sec1}}\n\nThe description of the surface properties of chiral p-wave superconductor\nhas been done within the framework of the quasiclassical Eilenberger equations\n\\cite{Eilenberger}. To solve the problem, we will assume that the conditions\nof the clean limit (scattering time $\\tau \\rightarrow \\infty $ ) are valid\nin the bulk superconductor region $-\\infty<x\\leqslant0$ and the equations\nhave the form\n\\begin{equation}\n2\\omega f(x,\\theta )+v\\cos (\\theta )\\frac{d}{dx}f(x,\\theta )=2\\Delta\ng(x,\\theta ),  \\label{El0}\n\\end{equation\n\\begin{equation}\n2\\omega f^{+}(x,\\theta )-v\\cos (\\theta )\\frac{d}{dx}f^{+}(x,\\theta )=2\\Delta\n^{\\ast }g(x,\\theta ),  \\label{El0a}\n\\end{equation\n\\begin{equation}\n2v\\cos (\\theta )\\frac{d}{dx}g_{\\omega }(x,\\theta )=2\\left( \\Delta ^{\\ast\n}f_{\\omega }-\\Delta f_{\\omega }^{+}\\right) .  \\label{El0b}\n\\end{equation\nHere $g(x,\\theta )$, $f(x,\\theta )$ and $\\ f^{+}(x,\\theta )$ are normal and\nanomalous Eilenberger functions, $\\theta $ is angle between the vector\nnormal to the interface and the direction of the electron Fermi velocity $v;$\n$\\omega =\\pi T(2n+1)$ are Matsubara frequencies and $T$ is temperature, $x$\nis coordinate along the axis normal to the boundary. Pair potential $\\Delta\n(x,\\theta )$ in a chiral p-wave superconductor is separated into two\nterms $\\Delta =\\Delta _{x}\\cos (\\theta )+i~\\Delta _{y}\\sin (\\theta )$.\n\nEquations (\\ref{El0})-(\\ref{El0b}) should be supplemented with\nthe self-consistency equation, which can be written in the form \\cite{Bruder}\n\\begin{eqnarray}\n\\Delta _{x}\\ln \\frac{T}{T_{c}}+2\\pi T\\sum_{\\omega }\\frac{\\Delta _{x}}{\\omega\n}-\\left\\langle 2\\cos (\\theta ^{\\prime }) Re f(x,\\theta ^{\\prime\n})\\right\\rangle &=&0,  \\label{Sc0b} \\\\\n\\Delta _{y}\\ln \\frac{T}{T_{c}}+2\\pi T\\sum_{\\omega }\\frac{\\Delta _{y}}{\\omega\n}-\\left\\langle 2\\sin (\\theta ^{\\prime }) Im f(x,\\theta ^{\\prime\n})\\right\\rangle &=&0.  \\label{Sc0c}\n\\end{eqnarray}\nHere $\\left\\langle ...\\right\\rangle =(1\/2\\pi )\\int_{0}^{2\\pi }(...)d\\theta $\nand $T_{c}$ is the critical temperature.\n\nPreviously, the properties of the superconductor surface have been described \\cit\n{Golubov1}-\\cite{Golubov2} in the frame of Ovchinnikov model \\cit\n{Ovchinnikov}. Within this model it is assumed that the surface is covered by a thin\ndiffusive normal metal layer. In the present study we go beyond this approximation\nand consider the situation when a clean p-wave superconductor is covered\nby a normal metal of finite thickness\nwith respect to the coherence length $\\xi _{0}=v\/2\\pi T_{c} $ and electron mean free\npath $l_{e}$ and having mirror-reflecting surface. Inside this layer, located in the area $0\\leq x\\leq d$,\nEilenberger equations \\cite{Eilenberger} transform to\n\n\\begin{equation}\nv\\cos (\\theta )\\frac{d}{dx}f(x,\\theta )=\\frac{1}{\\tau }\\left( g\\left\\langle\nf\\right\\rangle -f\\left\\langle g\\right\\rangle \\right) -2\\omega f(x,\\theta ),\n\\label{El2a}\n\\end{equation\n\\begin{equation}\nv\\cos (\\theta )\\frac{d}{dx}f^{+}(x,\\theta )=\\frac{1}{\\tau }\\left(\ng\\left\\langle f^{+}\\right\\rangle -f^{+}\\left\\langle g\\right\\rangle \\right)\n+2\\omega f^{+}(x,\\theta ),  \\label{El2b}\n\\end{equation\n\\begin{equation}\n2v\\cos (\\theta )\\frac{d}{dx}g(x,\\theta )=\\frac{1}{\\tau }\\left( f\\left\\langle\nf^{+}\\right\\rangle -f^{+}\\left\\langle f\\right\\rangle \\right).  \\label{El2c}\n\\end{equation\nFollowing the procedure developed in \\cite{Golubov1}-\\cite{Golubov2} we will\nconsider (\\ref{El2a})-(\\ref{El2c}) as the system of linear equations in\nwhich $\\left\\langle f\\right\\rangle $, $\\left\\langle f^{+}\\right\\rangle $ and\n$\\left\\langle g\\right\\rangle $ should be determined self-consistently in the\nprocess of finding a solution to the equations (\\ref{El0})-(\\ref{El2c}). To\nsimplify the problem further we introduce pairing functions of the incident \nf_{+}(\\theta )=f(\\theta )$ and reflected $f_{-}(\\theta )=f(\\pi +\\theta )$\nparticles, which are defined in the segment of angles $-\\pi \/2\\leq \\theta\n\\leq \\pi \/2.$\nThe functions $\\ f_{\\pm }^{+}$ and $\\ g_{\\pm }$ are defined below in a\nsimilar manner.\n\nFor the development of numerical algorithms for solution of the Eilenberger\nequations it is convenient to rewrite them using the Ricatti parametrization\n\\cite{Schopohl,Tanaka6}.\n\\begin{equation}\nf_{\\pm }=\\frac{2a_{\\pm }}{1+a_{\\pm }b_{\\pm }},\\ f_{\\pm }^{+}=\\frac{2b_{\\pm \n}{1+a_{\\pm }b_{\\pm }},\\ g_{\\pm }=\\frac{1-a_{\\pm }b_{\\pm }}{1+a_{\\pm }b_{\\pm \n}.  \\label{Ric1}\n\\end{equation\nFor the Ricatti functions $a_{\\pm }$ and $b_{\\pm }$ from (\\ref{El0})-(\\re\n{El2c}) we have\n\\begin{eqnarray}\nv\\cos (\\theta )\\frac{d}{dx}a_{\\pm } &=&\\Delta -\\Delta ^{\\ast }a_{\\pm\n}^{2}\\mp 2\\omega a_{\\pm },  \\label{El_ric_cl_A} \\\\\nv\\cos (\\theta )\\frac{d}{dx}b_{\\pm } &=&-\\Delta ^{\\ast }+\\Delta b_{\\pm\n}^{2}\\pm 2\\omega b_{\\pm },  \\label{El_ric_cl_B}\n\\end{eqnarray\nin the clean superconducting region and\n\\begin{eqnarray}\n\\frac{da_{\\pm }}{dx} &=&\\mp \\frac{\\left\\langle f\\right\\rangle -a_{\\pm\n}^{2}\\left\\langle f^{+}\\right\\rangle -2a_{\\pm }\\left\\langle g\\right\\rangle }\n2\\tau v\\cos (\\theta )}\\mp \\frac{2\\omega a_{\\pm }}{v\\cos (\\theta )},\n\\label{El_ric_dr_A} \\\\\n\\frac{db_{\\pm }}{dx} &=&\\mp \\frac{\\left\\langle f^{+}\\right\\rangle -b_{\\pm\n}^{2}\\left\\langle f\\right\\rangle -2b_{\\pm }\\left\\langle g\\right\\rangle }\n2\\tau v\\cos (\\theta )}\\pm \\frac{2\\omega b_{\\pm }}{v\\cos (\\theta )},\n\\label{El_ric_dr_B}\n\\end{eqnarray\nin the diffusive layer. The subscript $+$ in (\\ref{Ric1})-(\\ref{El_ric_dr_B\n) indicates that functions are calculated along trajectories in the\ndirection towards the SN boundary, while the subscript $-$ denotes the\nfunctions calculated on trajectories in the direction away from the SN interface\ntowards the bulk superconductor.  Far from the SN interface (at $x\\rightarrow\n-\\infty $) the values of $a_{\\pm }$ and $b_{\\pm }$ equal to their magnitudes\nin a homogeneous superconductor\n\\begin{equation}\na_{\\pm }=\\pm \\frac{\\Delta }{\\omega +\\sqrt{\\omega ^{2}+|\\Delta |^{2}}},\n\\label{As0}\n\\end{equation\n\\begin{equation}\nb_{\\pm }=\\pm \\frac{\\Delta ^{\\ast }}{\\omega +\\sqrt{\\omega ^{2}+|\\Delta |^{2}}\n,  \\label{As1}\n\\end{equation\nwhere $\\Delta $ is the bulk value of the pair potential.\n\nFinally, the problem must be supplemented by the boundary conditions \\cit\n{Bakurskiy}  at the free surface of the diffusion layer $(x=d)\n\\begin{equation}\nb_{+}(d,-\\theta )=b_{-}(d,\\theta ),  \\label{BC1a}\n\\end{equation\n\\begin{equation}\na_{-}(d,-\\theta )=a_{+}(d,\\theta ),  \\label{BC1b}\n\\end{equation}\nand by the requirement of continuity of the $a_{\\pm }$ and $b_{\\pm }$ functions\non the SN interface. \n\nThe boundary problem (\\ref{El_ric_cl_A})-(\\re\n{BC1b}) has been solved numerically.\nTo solve it, we start with some trial functions for order parameter, \n\\left\\langle f\\right\\rangle,$ $\\left\\langle f^{+}\\right\\rangle,$ \n\\left\\langle g\\right\\rangle$ and intergrade numerically equations (\\re\n{El_ric_cl_A}) - (\\ref{El_ric_dr_B}) for $a_{+}(x,\\theta )$ and \nb_{-}(x,-\\theta )$ moving along the trajectory from the bulk values of these\nfunctions (\\ref{As0}) at infinity $(x=-\\infty )$ towards the free N layer\nsurface $(x=d)$ making use of the requirement of continuity of the $a_{+}$\nand $b_{-}$ functions on the SN interface $(x=0)$. Thus obtained solutions \na_{+}(0,\\theta )$ and $b_{-}(0,-\\theta )$ and boundary conditions (\\ref{BC1a\n), (\\ref{BC1b}) determine the magnitudes of $a_{-}(0,\\theta )$ and \nb_{+}(0,\\theta )$. Starting from these solutions, the functions $a_{-}(x,\\theta )$ and \nb_{+}(x,\\theta )$ are obtained by integration along the trajectories going\nacross the diffusive N layer into the bulk superconductor.  The averaged\nfunctions $\\left\\langle f\\right\\rangle,$ $\\left\\langle f^{+}\\right\\rangle,$ \n\\left\\langle g\\right\\rangle$ and the spatial dependence of the order\nparameter $\\Delta (x)$ are determined in an iterative self-consistent way\nusing the definitions (\\ref{Ric1}) and self-consistency equations ( (\\re\n{Sc0b})-(\\ref{Sc0c}).\n\nThe calculated Ricatti functions $a_{\\pm }(x,\\theta )$ and $b_{\\pm }(x,\\theta)\n$ permit one to restore the angular distribution of the Eilenberger functions in\nany cross-sectional plane inside SN bilayer and calculate the spatial\ndistribution of the density of spontaneous edge current $j(x)$\n\\begin{equation}\nj(x)=2evN_{0}T\\sum_{\\omega >0}j_{\\omega}(x), \\\nj_{\\omega}(x)=\\int\\limits_{0}^{2\\pi }j_{\\omega}(x, \\theta)\\sin \\theta\nd\\theta,  \\label{current}\n\\end{equation}\nwhere $N_{0}$ is the density of states and angular resolved supercurrent\ndensity $j_{\\omega}(x, \\theta)=Im(g_{\\omega }(x, \\theta))=Im(\\sqrt\n1-f_{\\omega }(\\theta )f_{\\omega }(\\pi -\\theta )})$. When writing the last\nequality we used\nthe normalization condition for triplet functions\n\\begin{equation}\ng_{\\omega }^{2}(\\theta )-f_{\\omega }(\\theta )f_{\\omega }^{+}(\\theta )=1,\n\\label{norcond}\n\\end{equation}\nand the symmetry relatio\n\\begin{equation}\nf_{\\omega }(\\theta )=-f_{\\omega }^{+}(\\pi -\\theta ).  \\label{symrel}\n\\end{equation\nFor simplicity we will suppose further that the amplitudes $f_{\\omega }(x,\n\\theta )$ are small, and making use of (\\ref{norcond}), (\\ref{symrel})\nwe rewrite (\\ref{current}) in the form\n\\begin{equation}\n\\begin{array}{c}\nj_{\\omega }(x)=\\int\\limits_{-\\pi\/2}^{\\pi\/2 }\\left( Re(f_{\\omega }(x,\\theta\n))Im(f_{\\omega }(x,\\pi -\\theta ))+\\right. \\\\\n\\left. +Re(f_{\\omega }(x,\\pi -\\theta ))Im(f_{\\omega }(x,\\theta ))\\right)\n\\sin \\theta d\\theta \n\\end{array}\n\\label{jsmall}\n\\end{equation}\n\nIt is seen from Eq.(\\ref{jsmall}) that  the appearance of the spontaneous\nsurface current is a result of superposition of anomalous Eilenberger\nfunctions of incident $f_{\\omega }(x,\\theta )$ and reflected $f_{\\omega\n}(x,\\pi -\\theta )$ from the free interface trajectories. In the geometry considered in this work, when the $p_{x}$ principal axis is parallel to the interface normal, there is  general relation $f_{\\omega }(\\theta )=f_{\\omega }^{\\ast }(-\\theta\n)$ providing  the odd symmetry of imaginary part of pair amplitude. Far from the\nSN interface inside the bulk p-wave superconductor there is the odd\nsymmetry of real part of pairing function $Re(f_{\\omega }(\\theta ))=$ \n-Re(f_{\\omega }(\\pi -\\theta ))$ and the terms under integration in (\\re\n{jsmall}) have the same magnitude, but different sign and cancel each other\nduring integration resulting in the absence of a supercurrent. However, in\nthe vicinity of the surface of p-wave superconductor or SN interface the\nsymmetry of $Re(f_{\\omega }(\\theta ))$ on ingoing and outgoing trajectories is broken resulting in generation of a spontaneous supercurrent.\n\n\\section{Spontaneous current \\label{Current}}\n\nIn this Section we present the results of calculations of spatial distributions of the pair potential $\\Delta (x),$, the spontaneous supercurrent $j(x),$ as well as the angular resolved pair\namplitude $f_{1}(x,\\theta)$ and the density of the first spectral component \nj_{1}(x)$ in (\\ref{current})  together with its angular resolved\ndistribution $j_{1}(x,\\theta)$ at $\\omega=\\pi T$ for four different\nsituations.\nThey are specular p-wave surface ($d\\ll \\ell_{e}$, $d\\ll \\xi_{0}$); rough\nsurface ($d\\gg \\ell_{e}$, $d\\ll \\xi_{0}$); clean metallic surface ($d\\geq\n\\xi_{0}$, $\\ell_{e}\\gg \\xi_{0}$) and diffusive metallic surface ($d\\geq\n\\xi_{0}$, $\\ell_{e}\\ll \\xi_{0}$).\n\nAll  calculations have been performed at temperature $T=0.5T_{C}$ and the\nresults are shown in Fig.\\ref{F_Specular}-Fig.\\ref{f_inter}. Numerical analysis shows that at $T=0.5T_{C}$ the calculated value of $j_{1}(x)$ provides the contribution to the full current with accuracy of the order of 10 percent, therefore all the features discussed below reflect the behavior of total spontaneous current.\n\nThe figures\nconsist of several panels. Panel (a) shows the coordinate dependencies of the pair\npotential for a chiral $p_{x}+ip_{y}$ superconductor. Panels (b) and (c)\ndemonstrate the density of the first spectral component $j_{1}(x)$ in (\\re\n{current})  and its angular resolved distribution $j_{1}(x,\\theta)$,\nrespectively. Panels (d) show real (solid lines) and imaginary (dashed lines)\nparts of $f_{1}(x,\\theta)$ calculated at different distances $x\/\\xi_0$ from\nthe SN interface. Color in panels d) specifies the information on the sign\nof $f_{1}(x,\\theta)$ functions: the red color corresponds to the positive\nvalues of $f_{1}(x,\\theta)$, while the blue one represents their negative\nmagnitudes.\n\n\\bigskip \\emph{Specular p-wave surface ($d\\ll \\ell_{e}$, $d\\ll \\xi_{0}$)}.\n\n\\bigskip Figure \\ref{F_Specular} summarize the results obtained for\nspecular p-wave surface under absence of any clean or diffusive N layer on\nthe surface.\n\nIn full agreement with the previously obtained results \\cite{Bakurskiy} in\nthis case bulk pair potential (see Fig\\ref{F_Specular}a) has BCS amplitude\n(at our temperature $\\Delta _{x}=\\Delta _{y}\\approx 1.67T_{C}$) due to\nspherical symmetry. The $\\Delta _{y}$ component increases as we approach the\nsurface and grows up to bulk value for $p_{y}$ symmetry since after\nreflection electrons propagate into the band with the same sign of pair\npotential. The corresponding part of angular resolved distribution of pair\namplitude $f_{1}(x,\\theta)$ (see dashed lines in Fig. \\ref{F_Specular}d)\ndemonstrate a similar behavior. The magnitudes of this imaginary parts of \nf_{1}(x,\\theta)$ only slightly increase with $x$ decrease in the full\naccordance with $\\Delta _{y}$ growth.\n\nContrary to that, in the vicinity of interface $\\Delta _{x}$ is suppressed up\nto zero due to reflection of electrons from the band having positive sign of\npair potential into the band with negative one. The transformations of real\npart of angular resolved distribution of pair amplitude $f_{1}(x,\\theta)$\n(see solid lines in Fig. \\ref{F_Specular}d) is more complex. Starting from\nthe bulk (red curve in Fig. \\ref{F_Specular}d(0)) we first exhibit the\nsuppression of this component with $x$ decrease, as shown in Fig. \\re\n{F_Specular}d(1)) - Fig. \\ref{F_Specular}d(3)). This is a direct consequence\nof the suppression of the $\\Delta _{x}$ component of the order parameter. At\nthe surface the boundary conditions (\\ref{BC1a}), (\\ref{BC1b}) dictate for\nthe outgoing functions $f_{1}(0,\\theta)$ a sign opposite to that existing in\nthe depth of the superconductor. This perturbation relaxes on characteristic\nscale $\\xi=\\xi_{0} \\cos(\\theta)(\\pi T_{C}\/ \\sqrt{\\omega^{2}+|\\Delta |^{2}} )$\nof the equation (\\ref{El_ric_cl_A}), (\\ref{El_ric_cl_B}), which is $\\theta$\ndependent. It is for this reason the sign of $f_{1}(0,\\theta)$ functions\nrecovers the faster the smaller is $\\cos(\\theta)$, that is the larger is\ndeviation of outgoing trajectory from the normal to the superconductor\nsurface direction. This is clearly seen in Fig. \\ref{F_Specular}d(3). It\ndemonstrates that at $x=-0.2\\xi_0$ the sign of functions $f_{1}(-0.2\n\\xi_0,\\theta)$ is different for different $\\theta$. At $x=-0.4\\xi_0$ the\npairing functions are negative in all angle domain. Further decrease of $x$\nresults in increase of magnitude of $f_{1}(x,\\theta)$ and to the full\nrelaxation to the balk values at $x=-2\\xi_0.$\n\nThis lack of symmetry on the trajectories incoming and outgoing from the surface\nleads to generation of spontaneous supercurrent. The angular\nresolved density of its first spectral component $j_{1}(x,\\theta)$ is shown\nin Fig. \\ref{F_Specular}c. In accordance with (\\ref{current}) its\nintegration over $\\theta$ results in nonzero $j_{1}(x),$ which attains a\nmaximum at the surface and monotonically decays into superconductor.\nExactly at the surface plane ($x=0$) the incomig and reflecting electrons make\nequal contributions to $j_{1}(0).$ The component of $j_{1}(x)$ from the\noutgoing electrons decreases monotonically to zero with the decrease of $x$ and\nbecomes negative in the bulk area resulting in full compensation of the\npositive part of $j_{1}(x)$ generated by incoming particles. The maximum of \nj_{1}(x,\\theta)$ dependencies is achieved at $\\theta \\approx \\pi\/4$\nresulting in decay scale of $j_{1}(x)$ of the order of $\\xi_0 \/ \\sqrt{2}.$\n\n\\bigskip \\emph{Rough surface ($d\\gg \\ell_e$ and $d\\ll \\xi_0$.})\n\n\\bigskip  The situation described above changes dramatically when there is\nstrong diffuse scattering of electrons at the free surface of the\nsuperconductor (see Fig. \\ref{f_rough}). Below we model the diffuse\nscattering of electrons by a normal layer placed on the superconductor\nsurface. This layer has thickness $d$ equal to electronic mean free path \n\\ell_e,$ decay length $\\xi_0=3 \\ell_e$ and $\\Delta=0$.\n\nFigure \\ref{f_rough}a shows that in the considered situation in the vicinity\nof SN interface there is suppression of both components of the order\nparameter. The panels Fig.\\ref{f_rough}d(0) - Fig.\\ref{f_rough}d(5)\ndemonstrate that amplitudes of angular resolved distributions of real (solid\nred curves) and imaginary (dashed curves) parts of functions $f_{1}(x,\\theta)\n$ for ingoing trajectories ($-\\pi\/2 \\leq \\theta \\leq \\pi\/2)$ decrease slowly\nthen $x$ goes to SN interface located at $x=0$. In the diffusive layer, the\nimaginary part of $f_{1}(x,\\theta)$ decays rapidly due to averaging of\nits alternating parts.  At $x=d$ (see Fig.\\ref{f_rough}d(8)) the imaginary\npart of $f_{1}(d,\\theta)$ practically disappear and boundary conditions (\\re\n{BC1a}), (\\ref{BC1b}) provide nearly isotropic angular distribution of \nf_{1}(d,\\theta)$. The propogation on outlet trajectories from free surface\nback to SN interface across the diffusive layer results in further\nisotropisation of $f_{1}(x,\\theta)$, as it follows from Fig.\\ref{f_rough\nd(5).\n\nDuring the propagation into the S film (in the angular domain $\\pi\/2 \\leq x\n\\leq 3\\pi\/2$) the isotropic function $f_{1}(0,\\theta)$ distribution rapidly vanished\ndue to strong nucleation of the imaginary component of $f_{1}(x,\\theta)$. As\nit follows from Fig.\\ref{f_rough}d(4) - Fig.\\ref{f_rough}d(2) it recovers\nits bulk distribution practically at $x=-0.5\\xi_0$. Evolution of real part\nof $f_{1}(x,\\theta)$ is nearly the same as discussed in the previous\nparagraph. First it goes to zero with $x$ decrease, then it changes its sign and\ntends monotonically to its bulk behavior.\n\nFigures \\ref{f_rough}b and Fig. \\ref{f_rough}c show the spatial variation of the\nfirst spectral component $j_{1}(x)$ and its angular resolved spectral\ncomponent $j_{1}(x,\\theta)$ in the considered regime. It is seen that\nspontaneous supercurrent is pushed out from the N layer towards the SN\ninterface. It reaches its maximum at the SN boundary and then decreases with\nthe distance deep into the superconductor. Note, that there are two\ncharacteristic scales in this decay. In order to understand these phenomena\nit is enough to look at Fig. \\ref{f_rough}c and use the expression (\\re\n{jsmall}). Expression (\\ref{jsmall}) reads that in the N layer and in a\nvicinity of SN interface the main contribution to $j_{1}(x)$ provided by the\nproduct of isotropic real part of of $f_{1}$ ($p_{x}$ component) on outgoing\ntrajectories and imaginary part of $f_{1}$ ($p_{y}$ component) on ingoing\ntrajectories. The latter is strongly suppressed in the N film (see Fig. \\re\n{f_rough}d(6) - Fig. \\ref{f_rough}d(8)) resulting in the current\nsuppression. At the SN interface, the $p_{y}$ component of $f_{1}$ on ingoing\ntrajectories still exists. It achieves maximum values at $\\theta \\approx\n70^{o}$ thus providing the maximum in the angular resolved spectral\ncomponent $j_{1}(x,\\theta)$ at $\\theta \\approx 75^{o}$. Thus, in the\nvicinity of the SN boundary main contribution to the spontaneous current\ncomes from trajectories nearly parallel to interface. The characteristic\ndecay length of the anomalous functions on these trajectories is \n\\xi_{0}\\cos(75^{o})$, that is much smaller than $\\xi_{0}.$ Further deviation\nfrom the SN boundary is accompanied by deformations in the $f_{1}(x,\\theta)$,\nwhich will eventually lead to recovery of $f_{1}(x,\\theta)$ to the bulk one.\nThese modifications are accompanied by reconstruction of $j_{1}(x,\\theta)$ and by displacement of the\nposition of its maximum to angles close to $\\pi\/4$. As a result, for\nsmaller $x$ the decay length of $j_{1}(x)$  goes to the same value \n\\xi_{0}\\cos(\\pi\/4)$ that has been found for the specular superconductor\ninterface.\n\nIn the case where the thickness of the N layer becomes comparable with the decay length $\\xi_0$ there is\nsignificant attenuation of superconducting correlations in this layer\nresulting in strong suppression of superconductivity at outgoing from SN\ninterface trajectories. This, in turn, leads to further displacement of\nspontaneous supercurrents into the bulk region. We will demonstrate this\neffect below by considering two examples; \"clean metallic surface\"\nand \"diffusive metallic surface\". In both cases we will assume that there is\nno intrinsic superconductivity in the N metal and that its thickness and electronic\nmean free path are $d=\\xi_0$, $l_e=\\infty$ in the clean case and $d= l_e\n$ and $\\xi_0=0.3 l_e$ in the diffusive case.\n\n\\bigskip\n\n\\emph{Clean metallic surface ($d\\geq \\xi_{0}$, $\\ell_{e}\\gg \\xi_{0}$).}\n\n\\bigskip\n\nFigure \\ref{f_clean} summarizes the results obtained for the clean metallic\nsurface. The panels Fig.\\ref{f_clean}d(0) - Fig.\\ref{f_clean}d(5)\ndemonstrate that amplitudes of angular resolved distributions of real (solid\nred curves) and imaginary (dashed curves) parts of functions $f_{1}(x,\\theta)\n$ for inlet trajectories ($-\\pi\/2 \\leq \\theta \\leq \\pi\/2)$ behaves nearly in\nthe same manner as in the cases considered previously. They decrease slowly\nthen $x$ goes to SN interface located at $x=0$. Propagation into the N layer\nleads to full suppression of anomalous functions and they are zero in outlet\ntrajectories at SN interface (see Fig.\\ref{f_clean}d(5)). It is for this\nreason the spontaneous supercurrent is zero in the area $0\\leq x \\leq d$.\nFrom Fig.\\ref{f_clean}d(0) - Fig.\\ref{f_clean}d(4.5) it is seen that\nimaginary part of anomalous functions recover faster compare to real one.\nThis imbalance provides nucleation of spontaneous currents. Their density\nincreases (see Fig.\\ref{f_clean}b) and achieves its maximum at $x\\approx-0.5 \\xi_0\n. At this point (see Fig.\\ref{f_clean}d(2)) the imaginary part of anomalous\nfunctions saturates at the bulk value. With further decrease of $x$ the magnitude of the real part of $f_1$ increases and defines the\nsupercurrent component, which compensates the one from inlet electrons. As a\nresult, the density of spontaneous current starts to decrease for $x< -0.5\n\\xi_0$ and goes to zero in the bulk. Figure \\ref{f_clean}c shows that\nmaximum in spectral component $j_{1}(x,\\theta)$ is achieved at $\\theta\\approx\n\\pi\/4$. It means that characteristic scale of $j_{1}(x)$ variation is $\\xi_0\n\\cos(\\pi\/4))$, similar to the case of specular p-wave\nsuperconductor interface.\n\n\\bigskip  \\emph{Diffusive metallic surface ($d\\geq \\xi_{0}$, $\\ell_{e}\\ll\n\\xi_{0}$).}\n\n\\bigskip\n\nThe difference between the diffusive metallic interface from the clean one\nlies in the fact that due to the presence of electron scattering centers\nnear the border, some of electrons can be reflected back at small distances\nfrom the SN interface. This means that in contrast to the clean limit\nconsidered previously, there is a probability for nucleation of small isotropic\ndiffuse component in $f_1 (0, \\theta)$ distribution. Its presence would lead\nto the appearance of a diffuse current peak at $x=0$ (see Fig. \\ref{f_inter\nb). The angular resolved density of spontaneous supercurrent $j_{1}(x,\\theta)\n$ (see Fig. \\ref{f_inter}c) clearly demonstrates the existence of two\ncharacteristic angles. It has one maximum at $\\theta \\approx 75^{o}$\noriginated by diffusive processes and the second one at $\\theta \\approx \\pi\/4$.\nThis leads to the existence of two characteristic scales in $j_1 (x)$\ndependence. They are the \"diffusion scale\" $\\xi_{d}=\\xi_0 \\cos(75^{o})$ and\n\"clean scale\" $\\xi_{c}=\\xi_0 \\cos(\\pi\/4).$ It is seen from Fig. \\ref{f_inter\nb) that associated with this scales maximums in $j_1 (x)$ dependence can be\nresolved due to large difference between $\\xi_{d}$ and $\\xi_{c}$ and the\nshift of order of $0.5 \\xi_{0}$ between the space positions of this\npeculiarities.\n\n\\section{Discussion \\label{Discussion}}\n\nOur studies provide the detailed analysis of the mechanisms leading to the formation of spontaneous currents at the surface of a chiral p-wave superconductor in terms of the Ricatti functions.\n From the structure of the expressions (\\ref{current})-(\\ref{jsmall}), which determine the magnitude and direction of spontaneous currents it follows that these currents  depend on the products of the real and imaginary parts of the anomalous Green's functions on the incoming and outgoing trajectories from the border. These parts strongly depends\n on the requirements of the symmetry of the Green's functions, which, in turn, dictate the form of the boundary conditions on the open surface. For d$_{x^2-y^2}$-wave superconductors, $b_{\\pm }(x,\\theta )=a_{\\mp }(x,\\theta ),$ holds and  the conditions (\\ref{BC1a}), (\\ref{BC1b}) are reduced to the\nrelation $b_{+}(d,-\\theta )=a_{+}(d,\\theta ),$ which under the circumstances that lead to  suppression of the order parameter down to zero at the free interface\n  simultaneously ensures the vanishing of anomalous functions on that surface   \\cite{Golubov2}. \n\nContrary to that, in the considered situation vanishing of $\\Delta_x$ order parameter component is not accompanied by vanishing of anomalous functions at a specular surface, see the plot Fig. \\ref{F_Specular}d(5).  Moreover, due to the conditions (\\ref{BC1a}), (\\ref{BC1b}) the sign of the $f_1 (\\theta)$ functions on outgoing directories is opposite to that in the bulk. It is exactly the mechanism that results in nucleation of spontaneous current at free mirror surface.  Figure 1 gives that normalized amplitude of the first item in the expression for the supercurrent is slightly larger that $0.4.$\n\n  At the rough surface, the redistribution of current density takes place: it is pushed towards the border between the clean and the dirty regions (see Fig. \\ref{f_rough}). It is also seen from Fig. \\ref{f_rough}a that $\\Delta_x$ component at this interface is larger than that at specular surface (see Fig. \\ref{F_Specular}a), while  $\\Delta_y$ is smaller. Due to these competing processes the peak value of the $j_1$ component to the current at $x=0$ is only slightly reduced compared to $j_1 (0)$ calculated in the previous case.\n\n\nThe situation changes drastically when the thickness of the normal layer on the top of p-wave superconductor exceeds $\\xi_0$ irrespective of the degree of purity of the normal metal. As follows from Fig. \\ref{F_Specular}b and Fig. \\ref{f_inter}b, in both considered cases there is nearly one order of magnitude suppression of maximum value of $j_1$ compare to $j_1 (0)$ at specular interface. This result is correlated with that obtained in \\cite{Lederer} in the tight-binding model for the clean N metal. It is also seen from Fig. \\ref{F_Specular}b and Fig. \\ref{f_inter}b that spatial distribution of the density of spontaneous current for the clean and dirty cases are completely different. \n\nThe difference in the current distributions considered above can be accessed experimentally by muon spin rotation technique \\cite{Flokstra}. According to the results of our study, such measurements of surface currents may provide important information on the underlying pairing symmetry in the bulk.\n\nAcknowledgements. The authors acknowledge fruitful discussions with Y. Asano and Y. Tanaka. This work was partially supported by RFBR-JSPS grants 15-52-50054, 17-52-50080, RFBR grant 16-29-09515-ofi-m, by Russian Science Foundation, Project No. 15-12-30030, and by Ministry of Education and Science of the Russian Federation, grants MK-5813.2016.2 and 14Y26.31.0007.\n\n\\bigskip\n\n\\bigskip\n\n\\bigskip\n\n\n\\begin{figure*}[h]\n\\begin{minipage}[h]{0.5\\linewidth}\n\\begin{minipage}[h]{0.99\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{Del_mirror}} \\\\\n\\vspace{-2 mm}\n\\raggedright{a)}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.99\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{j1_mirror}} \\\\\n\\vspace{-2 mm}\n\\raggedright{b)}\n\\end{minipage}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.45\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{ang_mirror}} \\\\\n\\vspace{-8 mm}\n\\raggedright{c)}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{\\linewidth}\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_mirror_0}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(0)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_mirror_1}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(1)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_mirror_2}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(2)}}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_mirror_3}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(3)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_mirror_4}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(4)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_mirror_5}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(5)}}\n\\end{minipage}\n\\end{minipage}\n\\raggedright{d)}\n\\caption{(Color Online) Spatial distribution of set a) - d) of parameters in\np-wave superconductor with specular surface: \\newline\na) the pair potentials $\\protect\\Delta_x$ and $\\protect\\Delta_y$ and b) the\nsurface current density $j_1$, c) contribution of particles with angle \n\\protect\\theta$ in the formation of surface current $j_1$ at the different\npoints of the structure, d) angle dependent pair amplitude $f_1(\\protec\n\\theta)$ at the different points (0)-(5) of the structure. At the panel d)\nsolid lines correspond to the real part of $f$ and dashed lines correspond\nto the imaginary part of $f $; red color means positive value and blue color\nmeans negative one. Set of the points is following: (0) $x = -2 \\protect\\x\n_0 $, (1) $x = -0.8 \\protect\\xi_0 $, (2) $-0.4 \\protect\\xi_0 $, (3) $-0.2\n\\protect\\xi_0 $, (4) $-0.1 \\protect\\xi_0 $, (5) - $x = 0 \\protect\\xi_0 $.\nAll calculations were performed at $T=0.5 T_C$ and $d=0$.}\n\\label{F_Specular}\n\\end{figure*}\n\n\n\\begin{figure*}[h]\n\\begin{minipage}[h]{0.5\\linewidth}\n\\begin{minipage}[h]{0.99\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{Del_rough}} \\\\\n\\vspace{-2 mm}\n\\raggedright{a)}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.99\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{j1_rough}} \\\\\n\\vspace{-2 mm}\n\\raggedright{b)}\n\\end{minipage}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.45\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{ang_rough}} \\\\\n\\vspace{-8 mm}\n\\raggedright{c)}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{\\linewidth}\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_rough_0}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(0)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_rough_1}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(1)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_rough_2}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(2)}}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_rough_3}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(3)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_rough_4}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(4)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_rough_5}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(5)}}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_rough_6}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(6)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_rough_7}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(7)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_rough_8}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(8)}}\n\\end{minipage}\n\\end{minipage}\n\\raggedright{d)}\n\\caption{(Color Online) Spatial distribution of set a) - d) of parameters in\np-wave superconductor with rough surface: \\newline\na) the pair potentials $\\protect\\Delta_x$ and $\\protect\\Delta_y$ and b) the\nsurface current density $j_1$, c) contribution of particles with angle \n\\protect\\theta$ in the formation of surface current $j_1$ at the different\npoints of the structure, d) angle dependent pair amplitude $f_1(\\protec\n\\theta)$ at the different points (0)-(8) of the structure. Set of the points\nis following: (0) $x = -2 \\protect\\xi_0 $, (1) $-0.8 \\protect\\xi_0 $, (2) \n-0.4 \\protect\\xi_0 $, (3) $-0.2 \\protect\\xi_0 $, (4) $-0.1 \\protect\\xi_0 $,\n(5) $0 \\protect\\xi_0 $, (6) $0.1 \\protect\\xi_0 $, (7) $0.2 \\protect\\xi_0 $,\n(8) the surface $x = 0.33 \\protect\\xi_0 $. All calculations were performed\nat $T=0.5 T_C$, $d= 1 l_e$ and $\\protect\\xi_0=3 l_e$ .}\n\\label{f_rough}\n\\end{figure*}\n\n\n\\begin{figure*}[h]\n\\begin{minipage}[h]{0.5\\linewidth}\n\\begin{minipage}[h]{0.99\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{Del_clean}} \\\\\n\\vspace{-2 mm}\n\\raggedright{a)}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.99\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{j1_clean}} \\\\\n\\vspace{-2 mm}\n\\raggedright{b)}\n\\end{minipage}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.45\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{ang_clean}} \\\\\n\\vspace{-8 mm}\n\\raggedright{c)}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{\\linewidth}\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_clean_0}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(0)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_clean_1}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(1)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_clean_2}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(2)}}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_clean_3}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(3)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_clean_4}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(4)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_clean_4p5}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(4.5)}}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_clean_5}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(5)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_clean_6}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(6)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_clean_7}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(7)}}\n\\end{minipage}\n\n\\end{minipage}\n\\raggedright{d)}\n\\caption{(Color Online) Spatial distribution of set a) - d) of parameters in\np-wave superconductor with clean metallic surface: \\newline\na) the pair potentials $\\protect\\Delta_x$ and $\\protect\\Delta_y$ and b) the\nsurface current density $j_1$, c) contribution of particles with angle \n\\protect\\theta$ in the formation of surface current $j_1$ at the different\npoints of the structure, d) angle dependent pair amplitude $f_1(\\protec\n\\theta)$ at the different points (0)-(7) of the structure. Set of the points\nis following: (0) $x = -2 \\protect\\xi_0 $, (1) $-0.8 \\protect\\xi_0 $, (2) \n-0.4 \\protect\\xi_0 $, (3) $-0.2 \\protect\\xi_0 $, (4) $-0.1 \\protect\\xi_0 $,\n(4.5) $-0.03 \\protect\\xi_0 $, (5) $0 \\protect\\xi_0 $, (6) $0.25 \\protect\\x\n_0 $, (7) $0.5 \\protect\\xi_0 $. All calculations were performed at $T=0.5\nT_C $, $d=\\protect\\xi_0$ and $l_e=\\infty$ .}\n\\label{f_clean}\n\\end{figure*}\n\n\n\\begin{figure*}[h]\n\\begin{minipage}[h]{0.5\\linewidth}\n\\begin{minipage}[h]{0.99\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{Del_inter}} \\\\\n\\vspace{-2 mm}\n\\raggedright{a)}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.99\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{j1_inter}} \\\\\n\\vspace{-2 mm}\n\\raggedright{b)}\n\\end{minipage}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.45\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{ang_inter}} \\\\\n\\vspace{-8 mm}\n\\raggedright{c)}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{\\linewidth}\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_inter_0}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(0)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_inter_1}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(1)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_inter_2}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(2)}}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_inter_3}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(3)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_inter_4}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(4)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_inter_5}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(5)}}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_inter_6}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(6)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_inter_7}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(7)}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.3\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{f_inter_8}} \\\\\n\\vspace{-6 mm}\n\\raggedright{\\small{(8)}}\n\\end{minipage}\n\n\\end{minipage}\n\\raggedright{d)}\n\\caption{(Color Online) Spatial distribution of set a) - d) of parameters in\np-wave superconductor with diffusive metallic surface: \\newline\na) the pair potentials $\\protect\\Delta_x$ and $\\protect\\delta_y$ and b) the\nsurface current density $j_1$, c) contribution of particles with angle \n\\protect\\theta$ in the formation of surface current $j_1$ at the different\npoints of the structure, d) angle dependent pair amplitude $f_1(\\protec\n\\theta)$ at the different points (0)-(8) of the structure. Set of the points\nis following: (0) $x = -2 \\protect\\xi_0 $, (1) $-0.8 \\protect\\xi_0 $, (2) \n-0.4 \\protect\\xi_0 $, (3) $-0.2 \\protect\\xi_0 $, (4) $-0.1 \\protect\\xi_0 $,\n(5) $0 \\protect\\xi_0 $, (6) $0.25 \\protect\\xi_0 $, (7) $0.5 \\protect\\xi_0 $,\n(8) $x = 1 \\protect\\xi_0 $. All calculations were performed at $T=0.5 T_C$, \nd= l_e$ and $\\protect\\xi_0=0.3 l_e$.}\n\\label{f_inter}\n\\end{figure*}\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe relativistic multipole moments of asymptotically flat spacetimes have been defined by Geroch \\cite{geroch} for static spacetimes and then generalised to \nthe stationary case by Hansen \\cite{hansen}. Together with Beig's \\cite{beigAPA} generalised definition of centre of mass this gives\na coordinate independent description of all asymptotically flat stationary spacetimes. \n\nHowever, the tensorial recursion which defines the multipole moments \\eqref{orgrec}, is computationally rather complicated as it stands. In the axisymmetric (static or stationary) case, on the other hand, it was shown,\n\\cite{backdahl1}, \\cite{backdahl2} that the recursion can be replaced\nby a scalar recursion on $\\mathbb{R}$, and that all the moments can be collected\ninto one complex valued function $y$ on $\\mathbb{R}$, where the moments are given by the\nthe derivatives if $y$ at $0$.\n\nIn the general case, the multipole of order $2^n$ has $2n+1$ degrees of freedom, as compared to one degree of freedom in the axisymmetric case. Therefore, apart from the technical problems, it is not obvious what form a generalisation to the general case should take. In this paper, we show that also in the general stationary case, the recursion\n\\eqref{orgrec} can be simplified to a scalar recursion, this time on\n$\\mathbb{R}^2$. This is shown using normal coordinates, complex null geodesics, and exploiting the extra conformal freedom of the conformal compactification. \n\nUsing this simplification we can partially confirm an extension of a long standing conjecture by Geroch \\cite{geroch}:\n\\begin{quote}\nGiven any set of multipole moments, subject to the appropriate convergence condition, \nthere exists a static solution of Einstein's equations having precisely those moments.\n\\end{quote}\nThis conjecture has its natural extension to the stationary case.\n\nIn this paper we will state the appropriate convergence condition in the general stationary case,\ni.e., we will prove that this condition is necessary for existence of a stationary solution to Einstein's equations. \n\n\\section{Multipole moments of stationary spacetimes} \\label{tmoments}\nIn this section we quote the definition of multipole moments given by Hansen in \\cite{hansen}, which is an extension \nto stationary spacetimes of the definition by Geroch \\cite{geroch}.\nWe thus consider a stationary spacetime $(M, g_{ab})$ with timelike Killing vector field $\\xi^a$.\nWe let $\\lambda=-\\xi^a\\xi_a$ be the norm, and define the twist $\\omega$ through \n$\\nabla_a\\omega=\\epsilon_{abcd}\\xi^b\\nabla^c\\xi^d$.\nIf $V$ is the 3-manifold of trajectories, the metric $g_{ab}$ (with signature $(-,+,+,+)$) induces the \npositive definite metric\n$$h_{ab}=\\lambda g_{ab}+\\xi_a\\xi_b$$ on $V$.\nIt is required that $V$ is asymptotically flat, i.e.,  there exists a 3-manifold\n$\\hat V$ and a conformal factor $\\Omega$ satisfying\n\\begin{itemize}\n\\item[(i)]{$\\hat V = V \\cup \\Lambda$, \\; where $\\Lambda$ is a single point}\n\\item[(ii)]{$\\hat h_{ab}=\\Omega^2 h_{ab}$ is a smooth metric on $\\hat V$}\n\\item[(iii)]{At $\\Lambda$, $\\Omega=0, \\hat D_a \\Omega =0, \\hat D_a \\hat D_b \\Omega = 2 \\hat h_{ab}$,}\n\\end{itemize}\nwhere $\\hat D_a$ is the derivative operator associated with $\\hat h_{ab}$.\nOn $M$, and\/or $V$ one defines the scalar potential\n$$\\phi=\\phi_M+i\\phi_J, \\quad \\phi_M=\\frac{\\lambda^2+\\omega^2-1}{4\\lambda}, \\,\\phi_J=\\frac{\\omega}{2\\lambda}.$$\nThe multipole moments of $M$ are then defined on $\\hat V$ as certain  \nderivatives of the scalar\npotential $\\hat \\phi=\\phi\/\\sqrt \\Omega$ at $\\Lambda$. More  \nexplicitly, following \\cite{hansen}, let $\\hat R_{ab}$ denote\nthe Ricci tensor of $\\hat V$, and let $P=\\hat \\phi$. Define the  \nsequence $P, P_{a_1}, P_{a_1a_2}, \\ldots$\nof tensors recursively:\n\\begin{equation} \\label{orgrec} P_{a_1 \\ldots a_n}=C[\n\\hat D_{a_1}P_{a_2 \\ldots a_n}-\n\\tfrac{(n-1)(2n-3)}{2}\\hat R_{a_1 a_2}P_{a_3 \\ldots a_n}],\n\\end{equation}\nwhere $C[\\ \\cdot \\ ]$ stands for taking the totally symmetric and  \ntrace-free part. The multipole moments\nof $M$ are then defined as the tensors $P_{a_1 \\ldots a_n}$ at  \n$\\Lambda$. The requirement that all $P_{a_1 \\ldots a_n}$ be totally symmetric and\ntrace-free makes the actual calculations very cumbersome.\n\nIn \\cite{beigPRSL}, \\cite{kundu} it was shown that (when the mass is non-zero) there exist a conformal \nfactor $\\Omega$ and a chart, such that all components of the metric $\\hat h_{ab}$ and the potential $\\hat \\phi$ are analytic in terms of the \ncoordinates, in a neighbourhood of the infinity-point. Expressed in these coordinates, the exponential map becomes analytic. \nTherefore, we can use Riemannian normal coordinates and still have analyticity of the metric components and the potential.\nIf the mass is zero, this analyticity condition will be assumed. Thus, henceforth we assume that $\\Omega$ is chosen such that\nthe (rescaled) metric and potential are analytic in a neighbourhood of $\\Lambda$.\n\n\\section{Multipole moments through a scalar recursion on $\\mathbb{R}^2$}\n\\label{momentsscalar}\nSuppose that $(x^1, x^2, x^3)=(x, y, z)$ are normal coordinates (with respect to \n$\\hat h_{ab}$) centred around $\\Lambda$.\nThis means that for any constants $a=a^1, b=a^2 , c=a^3$ the curve $t \\to (a t,b t,c t)$ is a geodesic,\ni.e. in terms of coordinates that\n$$ \\ddot x^i+\\Gamma^i_{kl}\\dot x^k \\dot x^l=\\Gamma^i_{kl}a^k a^l=0$$\nwhere the Christoffel symbols are evaluated at $(a t,b t,c t)$ for appropriate $t$. Due to analyticity,\nthis relation holds for complex values of $a,b,c$, i.e., we can consider geodesics in the complexification\n$\\hat V_\\mathbb{C}$ of $\\hat V$. Of particular interest is the one-parameter family of curves:\n$$\\gamma_\\varphi: t \\to (t \\cos\\varphi,t \\sin \\varphi, i t),\n \\qquad t \\in [0,t_0), \\varphi\\in [0,2\\pi)$$\n for some suitable $t_0$. The tangent vector $\\eta^a=\\eta^a_\\varphi(t)=\n\\cos\\varphi (\\frac{\\partial}{\\partial x^1})^a+\\sin\\varphi (\\frac{\\partial}{\\partial x^2})^a+i (\\frac{\\partial}{\\partial x^3})^a$ is seen to be a complex null vector along $\\gamma_\\varphi$.\nNamely, from $\\eta^a \\hat D_a \\eta^b=0$, we infer that $\\eta^a \\hat D_a (\\eta^b \\eta_b)=0$.\nThe (constant) value of $\\eta^b \\eta_b$ is then found to be $0$ by evaluation at $t=0$.\n \nNext, consider the mapping $F: \\mathbb{R}^2 \\to V_\\mathbb{C}: \n(\\xi,\\zeta) \\to (\\xi,\\zeta,i \\sqrt{\\xi^2+\\zeta^2})$. We let $S$ denote the 2-surface $F(U) \\subset \\hat V_\\mathbb{C}$, where $U \\subset \\mathbb{R}^2$ is a suitable neighbourhood of $(\\xi,\\zeta)=(0,0)$. \n$S$ is then a smooth surface, except at $\\Lambda$ where it has a vertex point, closely resembling\na null cone in a three dimensional Lorentzian space.\nThe curves $\\gamma_\\varphi$ are given by $\\gamma_\\varphi(t)=F(t\\cos\\varphi,t\\sin\\varphi)$, and\nin particular $\\eta^a$ lies along $S$.\nThis suggests that we use the polar coordinates $\\rho,\\varphi$ around $\\Lambda$ on $S$ defined via \n$\\xi=\\rho\\cos\\varphi$, $\\zeta=\\rho\\sin\\varphi$.\nWe now follow the approach from \\cite{backdahl2}, where a useful vector field $\\eta^a$ on $\\hat V$ was introduced. \nIn \\cite{backdahl2}, where the spacetime was axisymmetric, $\\eta^a$ was explicitly expressed in \nterms of the metric cast in the Weyl-Papapetrou form \\cite{wald}, and was defined on the whole of $\\hat V$ except on the symmetry axis. \nIn this paper the axisymmetry condition is dropped, which makes the construction of a corresponding $\\eta^a$ more difficult. \nIn addition, a general spacetime has $2n+1$ degrees of freedom for the multipole moment of order $2^n$, \ncompared to one degree of freedom in the axisymmetric case \\cite{herb}.\nNevertheless, it will turn out to be sufficient to know the potential $\\hat \\phi$ on $S$ to determine all the moments. \nOn $S$, $\\eta^a$ has the following properties.\n\n\\begin{lemma}\\label{etalemma}\nSuppose $\\hat V$ and $S$ are defined as above. Then there exists a\nregularly direction dependent (at $\\Lambda$) vector field $\\eta^a$ on $S$ with the following properties:\\\\\na) $\\eta^a  \\hat D_a \\eta^b$ is parallel to $\\eta^b$.\\\\\nb) For all tensors $T_{a_1 \\ldots a_n}$, $\\eta^{a_1} \\ldots\n\\eta^{a_n}T_{a_1 \\ldots a_n}=\\eta^{a_1} \\ldots \\eta^{a_n}C[T_{a_1\n  \\ldots a_n}]$,\\\\\nc) At $\\Lambda$, $P_{a_1 \\ldots a_n}$ (in $\\hat V$) is determined by \n$\\eta^{a_1} \\ldots \\eta^{a_n}P_{a_1 \\ldots a_n}$ (on $S$)\n\\end{lemma}\n\n\\begin{proof}\na) was demonstrated above, b) follows as in \\cite{backdahl2}, while c) requires a different argument.\nA totally symmetric and trace-free tensor $P_{a_1\\dots a_n}$ has $2n+1$ degrees of freedom, and\nin Cartesian coordinates $(x,y,z)$ it can be expressed via the components \n$P\\underbrace{{}_{x  \\ldots\\ldots x}}_{j}\\underbrace{{}_{y ..\\ldots\\ldots y}}_{n-j-1}{}_z$\nand $P\\underbrace{{}_{x\\ldots\\ldots x}}_{j}\\underbrace{{}_{y\\ldots\\ldots y}}_{n-j}$.\n\\footnote{In brief, any index occurrence of several z's can be removed via $P_{zz\\dots}+P_{xx\\dots}+P_{yy\\dots}=0$. }\nTherefore, at $\\Lambda$, we can write\n\\begin{equation}\n\\label{reconstrtensor}\n\\begin{split}\nP_{a_1\\dots a_n}=\n\\sum_{j=0}^n{a_{j}C[(dx)_{a_1}\\dots(dx)_{a_j}(dy)_{a_{j+1}}\\dots(dy)_{a_{n}}]}\\\\\n+\\sum_{j=0}^{n-1}{b_{j}C[(dx)_{a_1}\\dots(dx)_{a_j}(dy)_{a_{j+1}}\\dots(dy)_{a_{n-1}}(dz)_{a_{n}}]}\n\\end{split}\n\\end{equation}\nContracting with $\\eta^{a_1}\\dots\\eta^{a_n}$, and using lemma \\ref{etalemma}b we find that\n\\begin{equation}\\label{trigpol}\n\\eta^{a_1}\\dots\\eta^{a_n}P_{a_1\\dots a_n}=\\sum_{j=0}^n{a_{j}\\cos^j\\varphi\\sin^{n-j}\\varphi}\n+i\\sum_{j=0}^{n-1}{b_{j}\\cos^j\\varphi\\sin^{n-1-j}\\varphi}\n\\end{equation}\nIf the left hand side is zero, the trigonometric polynomial to the right must be identically zero.\nThis means that all the coefficients $a_n$ and $b_n$ are zero, and by \\eqref{reconstrtensor}\nthat $P_{a_1\\dots a_n}$ is zero. In particular the $2n+1$ components in the RHS of\n\\eqref{reconstrtensor} are linearly independent. This proves c).\n\\end{proof}\nNote that although the moments are encoded in the coefficients $a_n$ and $b_n$, this encoding\nis dependent on the choice of normal coordinates, i.e., the orientation of the coordinate axes in $T_\\Lambda \\hat V_\\mathbb C$.\n\nWe can now replace the recursion \\eqref{orgrec} on  $\\hat V$ with a scalar recursion on $S$.\nAgain, we follow \\cite{backdahl2}, and define\n\\begin{equation} \\label{fn}\nf_n=\\eta^{a_1}\\eta^{a_2}\\dots\\eta^{a_n}P_{a_1a_2\\dots a_n}, \\ n=0,1,2,\\ \\ldots\n\\end{equation}\non $S$. In particular, $f_0=P=\\hat\\phi=\\phi\/\\sqrt{\\Omega}$. The moments \n$P_{a_1a_2\\dots a_n}(\\Lambda)$ will now be encoded in the trigonometric polynomial\ngiven by the direction dependent limit\n$\\lim_{\\rho \\to 0}f_n(\\rho,\\phi)$, which takes the form in \\eqref{trigpol}. See also lemma\n\\ref{Slemma}.\nNote that although $P_{a_1a_2\\dots a_n}$ is analytic on $\\hat V$, $f_n$ will not\nbe analytic in terms of $\\xi,\\eta$ since $\\eta^a$ is direction dependent at $\\Lambda \\in S$.\nIn general we have the following lemmas:\n\\begin{lemma}\\label{Slemma1}\nSuppose that $f$ is an analytic function,\non a ball of radius $r_0$ around $\\Lambda$ on $\\hat V$. \nThen the restriction of $f$ to $S$, $f_L$,  can be decomposed as $f_L(\\xi,\\zeta)=f_1(\\xi,\\zeta)+i\\rho\nf_2(\\xi,\\zeta)$, where $f_1$ and $f_2$ are analytic in terms of $\\xi$ and $\\zeta$ \non the disk $\\xi^2+\\zeta^2<\\frac{r_0^2}{2}$, and where $\\rho=\\sqrt{\\xi^2+\\zeta^2}$.\nFurthermore, if $f$ is real-valued then $f_1$ and $f_2$ are real-valued.\n\\end{lemma}\n\\begin{proof}\nWe start by splitting $f=f(x,y,z)$ into its even, $f_e$, and odd, $f_o$, part with respect to $z$.\nWe can now rewrite $f_e(x,y,z)=\\tilde f_e(x,y,z^2)$ and $f_o(x,y,z)=z \\tilde f_o(x,y,z^2)$,\nwhere both $\\tilde f_e$ and $\\tilde f_o$ are analytic in their arguments (at least near \n$(0,0,0)$). The restriction of $\\tilde f_e$ to $S$ gives $f_1$: $f_1(\\xi,\\zeta)=\\tilde f_e(\\xi,\\zeta,-(\\xi^2+\\zeta^2))$,\nwhile the restriction of $\\tilde f_o$ gives $f_2$: $f_2(\\xi,\\zeta)=\\tilde f_o(\\xi,\\zeta,-(\\xi^2+\\zeta^2))$. Adding these,\nand also noting that $z \\to i \\rho$ gives the required decomposition.\nOn $S$ we have $|z|^2=\\xi^2+\\zeta^2$, hence $\\xi^2+\\zeta^2< \\frac{r_0^2}{2}$ implies $x^2+y^2+z^2 < r_0^2$.\nThis gives the domain of analyticity. The reality follows from the construction.\n\\end{proof}\n\nRemark: In $f_L$, the subscript $L$ stands for the 'leading term'.\n\nAlthough tensor fields on $\\hat V$ can be pulled back to $S \\setminus \\{\\Lambda\\}$, we will only need their\ncontractions with the appropriate number of $\\eta^a$ vectors. This contraction will introduce a direction dependence\nwhich shows up in the following lemma.\n\n\\begin{lemma}\\label{Slemma}\nSuppose $T_{a\\ldots b}$ is an analytic tensor field on a ball of radius $r_0$ around $\\Lambda$ on $\\hat V$. \nThen the scalar field\n$f_L=\\eta^a \\ldots \\eta^b T_{a\\ldots b}$ on $S$ can be written as $f_L(\\xi,\\zeta)=\n\\frac{1}{\\rho^n}(f_1(\\xi,\\zeta)+i\\rho f_2(\\xi,\\zeta))$, where $f_1$ and $f_2$ are analytic \n(on the disk with radius $\\frac{r_0}{\\sqrt{2}}$ around the origin) in terms of $\\xi$ and $\\zeta$,\nand where $\\rho=\\sqrt{\\xi^2+\\zeta^2}$.\nFurthermore, if $T_{a\\ldots b}$ is real-valued then $f_1$ and $f_2$ are real-valued.\n\\end{lemma}\n\\begin{proof}\nConsider the scalar field $g=x^a \\ldots x^b T_{a\\ldots b}$ on $\\hat V$, where \n$$\nx^a=x\\left ( \\frac{\\partial}{\\partial x}\\right )^a+y\\left ( \\frac{\\partial}{\\partial y}\\right )^a+z\\left ( \\frac{\\partial}{\\partial z}\\right )^a .\n$$\nThis scalar field is analytic on the same ball as $T_{a\\ldots b}$. Hence we can use lemma \\ref{Slemma1} \nto obtain analytic functions $f_1$ and $f_2$ such that $g_L=f_1+i\\rho f_2$. The reality also follows from lemma \\ref{Slemma1}.\nFurthermore, $x^a=\\rho\\eta^a$ on $S$. Thus $g_L=\\rho^n f_L$. But $f_L$ is bounded near the origin, \nthus we can divide $g_L$ by $\\rho^n$ and get the lemma.\n\\end{proof}\nWe remark that the boundedness of  $f_L(\\xi,\\zeta)=\n\\frac{1}{\\rho^n}(f_1(\\xi,\\zeta)+i\\rho f_2(\\xi,\\zeta))$ when $\\rho \\to 0$ implies that\nboth $f_1$ and $f_2$ have zeros of sufficient order at $(\\xi,\\zeta)=(0,0)$. It also implies\nthat $f_L$ will be direction dependent there.\n\nWe can now contract \\eqref{orgrec}\nwith $\\eta^a$ and get the following theorem.\n\\begin{theorem}\\label{scalrarrec2d}\nLet $\\hat V$ and $S$ be defined as in sections (\\ref{tmoments}) and (\\ref{momentsscalar}).\nLet $\\eta^a$ have the properties given by lemma \\ref{etalemma}, and let $f_n$\nbe defined by \\eqref{fn}.Then the recursion \\eqref{orgrec} on $\\hat V$ takes the form\n\\begin{equation}\\label{scalarrec}\nf_n=\\eta^a\\hat D_af_{n-1}\n-\\tfrac{(n-1)(2n-3)}{2}\\eta^a\\eta^b\\widehat R_{ab}f_{n-2}\n\\end{equation}\non $S$. The moments of order $2^n$ are captured in the direction dependent limit \n$\\lim_{\\rho \\to 0} f_n(\\rho,\\varphi)$.\n\\end{theorem}\n\\begin{proof}\nThat \\eqref{orgrec} takes the form \\eqref{scalarrec} \n follows exactly as in \\cite{backdahl2} using that $\\eta^a\\hat D_a \\eta^b=0$, although the recursion is defined only on $S$ rather than on $\\hat V$. The last statement is the content of\n lemma \\ref{etalemma}c.\n\\end{proof}\n\n\\subsection{Simplified calculation of the moments}\nIn this section, we will show that it is possible to obtain the recursion \\eqref{scalarrec}\nwithout the term involving the Ricci tensor. This will be accomplished by using the conformal\nfreedom at hand, i.e. change $\\Omega$. The conformal freedom is $\\Omega \\to\n\\tilde\\Omega=\\tfrac{\\Omega}{\\alpha}$ where $\\alpha$ is analytic near $\\Lambda$ with $\\alpha(\\Lambda)=1$\\footnote{A more natural condition is the equivalent statement\n$\\Omega \\to \\tilde\\Omega=\\Omega \\alpha$. However, this formulation gives slightly neater calculations.}.\n$\\hat D_a (\\tfrac{1}{\\alpha})$ at $\\Lambda$ gives a shift of the moments which corresponds to a \n'translation' of the physical space, \\cite{geroch}. Hence we can assume that $\\hat D_a (\\tfrac{1}{\\gamma})=0$\nat $\\Lambda$.\nIt is to be noted that a change of $\\Omega$ changes the (rescaled) potential\n$\\phi\/\\sqrt{\\Omega}$. It also changes the normal coordinates on $\\hat V$, and hence all conclusions\nmust be made with some care. In order to derive the simplified recursion \\eqref{scalarreducedrec}, we will specify a $\\alpha$ through $\\alpha_L$, the restriction of $\\alpha$ to $S$. However, in order to deduce that there exists a \n{\\em real-valued} function $\\alpha$ which the prescribed values of $\\alpha_L$, we need to say more\non the representation of $\\alpha_L$. This result and a useful estimate is the content of lemma \\ref{serieslemma}.\n\n\\begin{lemma}\\label{serieslemma}\nLet $f_L=f_1(\\xi,\\zeta)+i\\rho f_2(\\xi,\\zeta)$ where $f_1$ and $f_2$ are analytic \non the ball $U=\\{ |\\xi|^2+|\\zeta|^2<r_0^2 \\}$, and where $\\rho=\\sqrt{\\xi^2+\\zeta^2}$.\nWe can then write\n\\begin{equation}\\label{fcomplex}\nf_L(\\rho\\cos\\varphi,\\rho\\sin\\varphi)=\n\\sum_{l=0}^\\infty\\sum_{m=-l}^l c_{l,m} e^{i m \\varphi}\\rho^l ,\n\\end{equation}\nwhere \n\\begin{equation}\\label{complexconv}\n\\sum_{l=0}^\\infty\\sum_{m=-l}^l |c_{l,m}| \\rho^l < \\infty \\; , \\; \\rho < r_0 .\n\\end{equation}\nFurthermore, the converse is true; if $f_L$ is a function satisfying \\eqref{fcomplex} and \\eqref{complexconv}\nthen there are functions $f_1$ and $f_2$ analytic in $U$ such that $f_L=f_1+i\\rho f_2$. \nThe functions $f_1$ and $f_2$ are real-valued if and only if the coefficients\n$c_{l,m}$ satisfy $\\bar c_{l,m}=(-1)^{l-m}c_{l,-m}$.\n\\end{lemma}\n\\begin{proof}\nForm the functions $f_1(\\frac{x+y}2,\\frac{x-y}{2i})$ and $f_2(\\frac{x+y}2,\\frac{x-y}{2i})$.\\footnote{Although we will always take $\\xi$ and $\\zeta$ to be real, they are temporarily\ncomplexified in this proof.}  These functions \nare analytic in terms of $x$ and $y$. Therefore there exist coefficients $c_{l,m}$ such that\n\\begin{equation}\\label{f1f2}\nf_j(\\frac{x+y}2,\\frac{x-y}{2i})=\\sum_{l=0}^\\infty\\sum_{k=0}^l i^{1-j} c_{l+j-1,2k-l}x^ky^{l-k} ,\n\\end{equation}\nwhere $j=1$ or $2$, and\n$$\n\\sum_{l=0}^\\infty\\sum_{k=0}^l |c_{l+j-1,2k-l}||x|^k|y|^l < \\infty \\; , \\; |\\tfrac{x+y}2|^2+|\\tfrac{x-y}{2i}|^2\n=\\tfrac{1}{2}(|x|^2+|y|^2) < r_0^2 .\n$$\nNow let $x=\\rho e^{-i\\varphi}$ and $y=\\rho e^{i\\varphi}$. For all $\\rho^2=\\tfrac{1}{2}(|x|^2+|y|^2) < r_0^2$ this shows that\n\\begin{equation}\\label{flc}\nf_L(\\rho\\cos\\varphi,\\rho\\sin\\varphi)=\n\\sum_{l=0}^\\infty\\sum_{k=0}^l c_{l,2k-l} e^{i (2k-l) \\varphi}\\rho^l +\n\\sum_{l=0}^\\infty\\sum_{k=0}^l c_{l+1,2k-l} e^{i (2k-l) \\varphi}\\rho^{l+1} .\n\\end{equation}\nThe reality condition follows easily from \\eqref{f1f2} and \\eqref{flc}.\nA reorganisation gives the first part of the lemma. \nThe converse is given by the substitution $x=\\xi+i\\zeta$ and $y=\\xi-i\\zeta$ in \\eqref{f1f2}. The series \nconverges absolutely for $|\\xi|^2+|\\zeta|^2=\\rho^2 < r_0^2$.\n\\end{proof}\nWe now show that we can choose $\\alpha$ such that the Ricci term in \\eqref{scalarrec} vanishes.\n\\begin{lemma} \\label{nnRnoll}\nThere exists a real-valued real analytic function $\\alpha$ on $\\hat V$ such that\nthe Ricci tensor $\\tilde R_{ab}$ of the new metric $\\tilde h_{ab}=\\alpha^{-2}\\hat h_{ab}$  \nsatisfies $\\tilde \\eta^a \\tilde \\eta^b \\tilde R_{ab}=0$ on $\\tilde S$, where both $\\tilde \\eta^a$, $\\tilde S$\nand the mapping $F$ are defined\nin terms of coordinates which are normal with respect to the new metric $\\tilde h_{ab}$.\n\\end{lemma}\n\\begin{proof} We first demonstrate that we can make $\\eta^a \\eta^b \\tilde R_{ab}=0$ on $S$.\n{}From \\cite{wald}\n$$\n\\tilde R_{ab}=\\hat R_{ab}+\\tfrac{1}{\\alpha}\\hat D_a \\hat D_b \\alpha+\n\\alpha^{-1}\\hat h_{ab}\\hat h^{cd}\\hat D_c \\hat D_d\\alpha-\n2\\alpha^{-2}\\hat h_{ab}\\hat h^{cd}(\\hat D_c \\alpha) \\hat D_d \\alpha .\n$$\nOn $S$, (using $\\eta^a$ and $S$ belonging to $\\hat h_{ab}$), this becomes\n\\begin{equation}\\label{rkappa}\n\\eta^a \\eta^b \\tilde R_{ab}=\n\\eta^a\\eta^b\\hat R_{ab}+\\tfrac{1}{\\alpha_L}\\eta^a\\eta^b\\hat D_a \\hat D_b \\alpha_L=\n\\eta^a\\eta^b\\hat R_{ab}+\\tfrac{1}{\\alpha_L}\\tfrac{\\partial^2 \\alpha_L}{\\partial\\rho^2}\n\\end{equation}\nwhere we have used $\\eta^a \\hat D_a \\eta^b=0$. In order to make $\\eta^a \\eta^b \\tilde R_{ab}=0$,\nwe require\n\\begin{equation} \\label{omegaeq}\n\\alpha_{L \\rho\\rho}+\\eta^a\\eta^b\\hat R_{ab}\\alpha_L=0 .\n\\end{equation}\nNote that, on $\\hat V$, $\\hat R_{ab}$ satisfies the conditions\nof lemma \\ref{Slemma}, therefore from lemma~\\ref{Slemma} and lemma~\\ref{serieslemma}\nit follows that\n\\begin{equation}  \\label{rserie}\n\\eta^a\\eta^b\\hat R_{ab}=\\sum_{n=0}^\\infty b_{n+2}(e^{i\\varphi},e^{-i\\varphi})\\rho^n ,\n\\end{equation}\nwhere each $b_n$ is a polynomial of degree at most $n$. \nIt is easy to see that the reality condition from lemma \\ref{serieslemma} is equivalent to\n$\\overline{b_{n}(e^{i\\varphi},e^{-i\\varphi})}=(-1)^n b_n(-e^{i\\varphi},-e^{-i\\varphi})$.\nHence the reality of $\\hat R_{ab}$ implies this condition. Moreover, since\n$\\hat R_{ab}$ is analytic in a neighbourhood of $\\Lambda$, there exists a $\\rho_0$ such\nthat for any fixed $\\varphi$, the series \\eqref{rserie} converges for all $\\rho < \\rho_0$.\nWell known ODE theory gives that for each fixed $\\varphi$ there exists\na solution $\\alpha_L$, analytic in $\\rho$ for $0 \\leq \\rho<\\rho_0$, i.e., for all $\\varphi$ \nwe have \n$$\n\\alpha_L=\\sum_{n=0}^\\infty a_n(\\varphi)\\rho^n , \\quad \\rho < \\rho_0 .\n$$\nWe note that $a_0=1$ while $a_1$\ncan be chosen to be 0 (translation). \n$\\alpha_L$ will have the right regularity if we can show that each\n$a_n(\\varphi)$ is a polynomial in $e^{i\\varphi}$ and $e^{-i\\varphi}$ of\ndegree at most $n$. Equation \\eqref{omegaeq} becomes\n$$\n-\\sum_{n=0}^\\infty n(n-1) a_n\\rho^{n-2}=\n\\sum_{n=0}^\\infty \\sum_{j=0}^\\infty b_{n+2} a_j\\rho^{n+j}.\n$$\nEquating powers of $\\rho$ we get\n\\begin{equation}\\label{reckappa}\n-(n+2)(n+1)a_{n+2}=\\sum_{m=0}^n b_{m+2} a_{n-m},\\quad n \\geq 0 .\n\\end{equation}\nThe polynomials $a_0$ and $a_1$ has maximal degree $0$ respective $1$. The polynomials $b_n$ has maximal degree $n$.\nStraightforward induction shows that $a_n$ has maximal degree $n$. \nInduction and \\eqref{reckappa} also implies $\\overline{a_n(e^{i\\varphi},e^{-i\\varphi})}=(-1)^n a_n(-e^{i\\varphi},-e^{-i\\varphi})$.\nThus, due to lemma \\ref{serieslemma}, there are real-valued analytic functions $\\alpha_1$ and $\\alpha_2$ such that \n$\\alpha_L=\\alpha_1+i\\rho \\alpha_2$. The function $\\alpha=\\alpha_1(x,y)+z\\alpha_2(x,y)$ is then a real-valued analytic \nextension of $\\alpha_L$ to a ball in $\\hat V$. This shows that there exist an $\\alpha$\nsuch that $\\eta^a \\eta^b \\tilde R_{ab}=0$ on $S$. \n\nHowever, it then also follows that $\\tilde \\eta^a \\tilde \\eta^b \\tilde R_{ab}=0$ on $\\tilde S$.\n{}From $\\alpha$ we get $\\tilde \\Omega=\\Omega\/\\alpha$, and the corresponding new metric $\\tilde h_{ab}=\\alpha^{-2}\\hat h_{ab}$. \nNote that in \\eqref{scalarrec}, where now $\\eta^a\\eta^b\\widehat R_{ab}=0$,\nthe recursion is stated in terms of $\\hat D_a$, i.e., it is expressed in terms of $\\hat h_{ab}$\ninstead of $\\tilde h_{ab}$. From $\\tilde h_{ab}$\nwe get new normal coordinates, i.e., $(x,y,z)$ in $T_\\Lambda \\hat V_\\mathbb C$,\nare mapped into $\\hat V_\\mathbb C$ using the exponential map belonging to $\\tilde h_{ab}$. We then construct $\\tilde \\eta^a$ and the mapping\n$F$ with respect to these coordinates. Now, null geodesics, of which $S$ consists, are conformally invariant,\nalthough they become non-affinely parametrised. This means that\n$\\tilde \\eta^a \\propto \\eta^a$, and that points in $S$ are mapped into points in $\\tilde S$.\nThus $\\tilde \\eta^a \\tilde \\eta^b\\tilde R_{ab} \\propto  \\eta^a  \\eta^b\\tilde R_{ab}=0$ on $\\tilde S$ (or $S$). \n\\end{proof}\n\nHenceforth we denote all entities defined via $\\tilde h_{ab}$ instead of $\\hat h_{ab}$ with\n a tilde. In particular, $\\tilde D_a$ will denote the derivative operator associated with $\\tilde h_{ab}$. Applying lemma \\ref{nnRnoll} to theorem \\ref{scalrarrec2d} we immediately get the following theorem.\n\\begin{theorem}\\label{scalrarreducedrec2d}\nLet $\\hat V$ and $\\tilde S$ be defined as in sections \\eqref{tmoments} and \\eqref{momentsscalar}, where\n$\\tilde S$ is defined in terms of normal coordinates connected to $\\tilde h_{ab}$.\nLet $\\tilde \\eta^a$ have the properties given by lemma \\ref{etalemma} with respect to\n$\\tilde h_{ab}$, and let $\\tilde f_n$\nbe defined by \\eqref{fn} with $\\tilde \\eta^a$ replacing $\\eta^a$.\nThen the recursion \\eqref{orgrec} on $\\tilde V$ takes the form\n\\begin{equation}\\label{scalarreducedrec}\n\\tilde f_n=\\tilde \\eta^a\\tilde D_a\\tilde f_{n-1}=(\\tilde \\eta^a \\tilde D_a)^n \\tilde f_0\n=\\frac{\\partial^n}{\\partial \\tilde \\rho^n}\\tilde f_0\n\\end{equation}\non $\\tilde S$. The moments of order $2^n$ are captured in the direction dependent limit \n$\\lim_{\\tilde \\rho \\to 0} \\tilde f_n(\\tilde \\rho,\\varphi)$.\n\\end{theorem}\n\\begin{proof}\nSince $\\tilde \\eta^a \\tilde D_a$ is $\\frac{\\partial}{\\partial \\tilde \\rho}$, each $\\tilde f_n$ is easily \nderived from $\\tilde f_0$. \nAlso, from lemma \\ref{etalemma}c), we (again) know that the $2n+1$ degrees of freedom\nof $P_{a_1 \\ldots a_n}$ at $\\Lambda$ are encoded in $\\tilde f_n$.\n\\end{proof}\n\n\\section{Bounds on the moments} \nAll multipole moments are encoded in $\\tilde f_0$, and we note that the recursion\n\\eqref{scalarreducedrec} is identical to the recursion emanating from a scalar function\nin $\\mathbb{R}^3$ (after inversion). It is clear that many different functions on \n$\\mathbb{R}^3$ will produce the same moments, but if we also require that the function, $g$ say,\nis harmonic, $g$ is uniquely determined by the moments.\n\nThus, provided that we can connect a function which is harmonic in \na neighbourhood of ${\\mathbf 0} \\in \\mathbb{R}^3$\nto each $\\tilde f_0$, we have the following theorem: \n\n\\begin{theorem}  \\label{bounds}\nSuppose that $(M,g_{ab})$ is a stationary asymptotically flat spacetime, admitting an analytic (rescaled) potential and an analytic\nchart\\footnote{As discussed before, the analyticity has been proved for the case with non-zero mass.}  on the conformally compactified manifold of timelike Killing \ntrajectories, around the infinity point $\\Lambda$.\nThen there exist a (flat-)harmonic function $g$ in a neighbourhood of \n${\\mathbf 0}\\in T_{\\Lambda}\\hat V \\cong \\mathbb{R}^3$, such that all \nmultipole moments of $M$ are given by\n\\begin{equation}\\label{harmrec}\nP_{a_1\\dots a_n}(\\Lambda)=(\\nabla_{a_1} \\dots \\nabla _{a_n} g)(0) ,\n\\end{equation}\nwhere $\\nabla_a$ in the LHS of \\eqref{harmrec} is the flat derivative operator in $\\mathbb{R}^3$.\nThis puts a bound on the multipole moments, since the Taylor expansion \n$\\sum_{|\\alpha|\\geq 0}\\frac{\\mathbf{r}^\\alpha}{\\alpha!}(\\partial_\\alpha g)(\\mathbf 0)$\nof  $g$ converges in a neighbourhood of  the origin in $\\mathbb{R}^3$.\n\\end{theorem}\n\n\\begin{proof}\nBy lemma \\ref{serieslemma} we have\n$$\n\\tilde f_L(\\rho\\cos\\varphi,\\rho\\sin\\varphi)=\n\\sum_{l=0}^\\infty\\sum_{m=-l}^l c_{l,m} e^{i m \\varphi}\\rho^l, \\quad \\rho < r_0\n$$\nfor some $r_0>0$, and we know that $\\tilde f_L$ fully determines the moments of $M$.\nWe will now define the function $g$, which will be shown to be harmonic in a neighbourhood\nof the origin. Finally we will argue that $g$ has the correct derivatives at $\\mathbf 0$, i.e.,\nthat the equality \\eqref{harmrec} is valid (for all $n \\geq 0$).\nFirst we define the coefficients \n$$\na_{l,m}=c_{l,m}i^{-m-l} 2^{1-l}\\pi \\frac{\\sqrt{(l+m)!(l-m)!}}{\\Gamma(l+\\tfrac{1}2)\\sqrt{2l+1}} .\n$$\nand the function\n$$\ng(r,\\theta,\\varphi)=\\sum_{l=0}^\\infty\\sum_{m=-l}^l a_{l,m} Y^m_l(\\theta,\\varphi) r^l .\n$$\nDue to the construction, $g$ is harmonic at those (interior) points for which the\nsum converges. We now establish convergence.\n{}From the identity\n$$\n\\sum_{m=-l}^l |Y_l^m(\\theta,\\varphi) |^2=\\frac{2l+1}{4\\pi}\n$$\nwe get\n\\begin{gather*}\n\\left | \\sum_{m=-l}^l a_{l,m}Y^m_l \\right | \\leq \n\\sum_{m=-l}^l \\left | a_{l,m}Y^m_l \\right | \\leq\n\\left ( \\sum_{m=-l}^l | a_{l,m}|^2 \\right )^{\\frac{1}{2}}\n\\left ( \\sum_{m=-l}^l |Y_l^m |^2 \\right )^{\\frac{1}{2}} \\\\\n=\\left ( \\sum_{m=-l}^l | \\sqrt{\\tfrac{2l+1}{4\\pi}} a_{l,m}|^2 \\right )^{\\frac{1}{2}}\n=\\frac{\\sqrt{\\pi}}{2^l\\Gamma(l+\\tfrac{1}2)} \\left ( \n\\sum_{m=-l}^l \\left | c_{l,m} \\sqrt{(l+m)!(l-m)!}\\right |^2 \\right )^{\\frac{1}{2}}\n\\end{gather*}\nFurthermore,\nthe inequality $(l+m)!(l-m!)\\leq (2l)!$, when $-l\\leq m\\leq l$ follows from the convexity of $\\ln(\\Gamma(x))$.\nTherefore\n$$\n\\left | \\sum_{m=-l}^l a_{l,m}Y^m_l \\right | \n\\leq \\frac{\\sqrt{\\pi}\\sqrt{(2l)!}}{2^l\\Gamma(l+\\tfrac{1}2)} \n\\left ( \\sum_{m=-l}^l | c_{l,m} |^2 \\right )^{\\frac{1}{2}}\n$$\nNext, the inequality $\\frac{\\pi (2l)!}{4^{l}\\Gamma\\left(l+\\tfrac{1}2\\right)^2}\n=\\frac{(2l)!!}{(2l-1)!!}=(2l+1)\\frac{(2l)!!}{(2l+1)!!}\\leq 2l+1$ gives\n$$\n\\left | \\sum_{m=-l}^l a_{l,m}Y^m_l \\right | \n\\leq \\sqrt{2l+1}\\left ( \\sum_{m=-l}^l | c_{l,m} |^2 \\right )^{\\frac{1}{2}}\n\\leq \\sqrt{2l+1}\\sum_{m=-l}^l | c_{l,m} |\n$$\nBut for all $\\epsilon>0$ we have \n$\\sqrt{2l+1}(1+\\epsilon)^{-l} \\rightarrow 0$ as  $l\\rightarrow \\infty$.\nSo the factor $\\sqrt{2l+1}$ will not affect the radius of convergence.\n\nHence \n$$\n \\sum_{l=0}^\\infty\\sum_{m=-l}^l a_{l,m}Y^m_l r^l \\; \\text{ converges if } r< r_0 .\n$$\nThis shows that $g$ is well defined in a neighbourhood of $\\mathbf 0$ in $T_\\Lambda \\hat V$,\nand we must now show that we have equality in \\eqref{harmrec}. We will do this by forming $g_L$\nand then compare with $\\tilde f_L$. Note, however, that $\\tilde f_L$ is defined on $\\tilde S \\subset V_\\mathbb{C}$,\nwhile $g_L$ will be defined on the corresponding surface $\\bar S=S_{\\tilde h_{ab}(\\Lambda)}$ in $T_\\Lambda \\hat V_\\mathbb{C}$. (By $\\bar S$ we denote the surface\ndefined by $F$ as previously, but where $F$ now maps $(\\xi,\\zeta)$ into $T_\\Lambda \\hat V_\\mathbb{C}$ via the flat metric $\\tilde h_{ab}(\\Lambda)$.)\nThis means that $f_L$ and $g_L$\nreally can be compared only at $\\Lambda$ (i.e. $\\mathbf 0 \\in T_\\Lambda \\hat V$), where also the equality \\eqref{harmrec} is evaluated.\nOn the other hand, the radial derivatives of both entities are well defined and comparable at $\\Lambda$.\nIn other words, if both $\\tilde f_L$ and $g_L$ are equal when expressed in terms of the\ncoordinates $\\xi, \\zeta$, they will produce the same derivatives\/moments. To summarise; in the case of $g$, \nthe function $F$ and the vector $\\eta^a$ are simply interpreted in $T_{\\Lambda} \\hat V_\\mathbb{C}$\nrather than in $V_\\mathbb{C}$.\n\nTo proceed, we recall that \n$$\nY^m_l(\\theta,\\varphi)=i^{m-|m|}\\sqrt{\\frac{(2l+1)(l-|m|)!}{4\\pi(l+|m|)!}}P^{|m|}_l(\\cos\\theta)e^{im\\varphi}\n$$\nfor $-l\\leq m\\leq l$, and that\n$$\nP^m_l(\\cos\\theta)=(-1)^m2^{-l} \\sin^m\\theta \\sum_{k=0}^{\\bigl \\lfloor \\tfrac{l-m}{2} \\bigr \\rfloor} \n\\frac{(-1)^k (2l-2k)!}{k!(l-k)!(l-2k-m)!}\\cos^{l-m-2k}\\theta \n$$\nwhere $0\\leq \\theta \\leq \\pi$ and $m\\geq 0$.\nTherefore,\n$$\ng(r,\\theta,\\varphi)=\\sum_{l=0}^\\infty\\sum_{m=-l}^l \\sum_{k=0}^{\\bigl \\lfloor \\tfrac{l-|m|}{2} \\bigr \\rfloor}\n\\frac{ c_{l,m}\\sqrt{\\pi}(l-|m|)!(2l-2k)! e^{i m \\varphi}\\rho^{|m|} z^{l-|m|-2k} r^{2k}}\n{i^{l-|m|}(-1)^k 4^lk!(l-k)!(l-|m|-2k)!\\Gamma(l+\\tfrac{1}2)} , \n$$\nwhere $z=r\\cos\\theta$, $\\rho=r\\sin\\theta$.\nWhen we take the restriction of $g$ to $\\bar S$, i.e., form $g_L$, only the terms with $k=0$ survives since $r_L=0$. Thus\n$$\ng_L(\\rho\\cos\\theta,\\rho\\sin\\theta)\n=\\sum_{l=0}^\\infty\\sum_{m=-l}^l\n\\frac{ c_{l,m}\\sqrt{\\pi}(2l)! e^{i m \\varphi}\\rho^l}\n{4^l l!\\Gamma(l+\\tfrac{1}2)}\n=\\sum_{l=0}^\\infty\\sum_{m=-l}^l c_{l,m}e^{i m \\varphi}\\rho^l=\\tilde f_L,\n$$\nwhich means that we have equality in \\eqref{harmrec}.\n\\end{proof}\n\n\\section{Discussion}\nIn this paper we have studied the multipole moments of stationary asymptotically flat spacetimes. By using normal coordinates, and by exploiting the conformal freedom, we could show that the tensorial recursion \\eqref{orgrec} could be replaced by the scalar\nrecursion \\eqref{scalarreducedrec}. This recursion is a direction dependent\nrecursion on $\\mathbb{R}^2$, where the moments are encoded in the direction dependent\nlimits at $\\Lambda$.\n\nUsing this setup, we could also show that the multipole moments cannot grow too fast.\nIn essence, the rescaled potential behaves (locally) \nin the manner of a harmonic function on $\\mathbb{R}^3$. The bounds on the moments given in theorem\n\\ref{bounds} gives the necessary part in a conjecture due to Geroch \\cite{geroch},\nand it is of course tempting to conjecture that this condition on the moments\nalso will be sufficient (as long as the monopole is real-valued).\n\nWhether this can be proved using the techniques presented here is still an open question.\n\nWe also remark that similar questions concerning the convergence of asymptotic expansions in the static case are currently being studied by Friedrich, using a different technique, \\cite{friedrich}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nResource Description Framework (RDF) is a general method for conceptual description or modeling of information that is implemented \nin web resources. RDF Schema (RDFS) extends RDF to classes providing basic elements for the description of vocabularies. OWL adds \nmore vocabulary for describing properties and classes i.e. relations between classes, cardinality, and richer typing of properties.\nUnfortunately, OWL has high worst-case complexity results for key inference problems. To overcome this problem we propose a\nlightweight OWL profile called OWL-P.\n\nA rule is perhaps one of the most understandable notion in computer science. It consists of the condition and the conclusion. \nIf some condition that is checkable in some dataset holds, then the conclusion is processed. In the same way RDF(S) and OWL\nentailments work.\n\nThe paper is constructed according to sections. Section~\\ref{sec:preliminaries} presents RDF and Notation 3 Logic concepts. \nIn Section~\\ref{sec:rules} we present inference rules for RDF. RDFS and OWL in N3Logic. Section~\\ref{sec:relatedwork} is devoted \nto related work. The paper ends with conclusions.\n\n\\section{Preliminaries}\n\\label{sec:preliminaries}\nThe RDF data model rests on the concept of creating web-resource statements in the form of subject-predicate-object expressions, \nwhich in the RDF terminology, are referred to as \\emph{triples} (or \\emph{statements}).\n\nAn RDF triple comprises a subject, a predicate, and an object. In \\cite{Wood2014}, the meaning of subject,\npredicate and object is explained. The \\emph{subject} denotes a resource, the \\emph{object} fills the value of the relation, \nthe \\emph{predicate} refers to the resource's characteristics or aspects and expresses a subject -- object relationship. \nThe predicate denotes a binary relation, also known as a property.\n\nFollowing \\cite{Wood2014}, we provide definitions of RDF triples below.\n\n\\begin{mydef}[RDF triple]\n\\label{def:rdftriple}\nAssume that $\\mathcal{I}$ is the set of all Internationalized Resource Identifier (IRI) \nreferences, $\\mathcal{B}$ (an infinite) set of blank nodes, $\\mathcal{L}$ the set of literals.\nAn \\emph{RDF triple} $t$ is defined as a triple $t = \\langle s, p, o\\rangle$ where $s \\in \\mathcal{I} \\cup \\mathcal{B}$ is \ncalled the \\emph{subject}, $p \\in \\mathcal{I}$ is called the \\emph{predicate} and $o \\in \\mathcal{I} \\cup \\mathcal{B} \\cup \\mathcal{L}$ \nis called the \\emph{object}.\n\\end{mydef}\n\nThe elemental constituents of the RDF data model are RDF terms that can be used in reference to resources: \nanything with identity. The set of RDF terms is divided into three disjoint subsets: IRIs, literals, and blank nodes.\n\n\\begin{mydef}[IRIs]\n\\label{def:rdfterms}\n\\emph{IRIs} serve as global identifiers that can be used to identify any resource.\n\\end{mydef}\n\n\\begin{mydef}[Literals]\n\\label{def:rdfliterals}\n\\emph{Literals} are a set of lexical values. It can be a set of plain strings, such as \\verb|\"Apple\"|, optionally \nwith an associated language tag, such as \\verb|\"Apple\"@en|.\n\\end{mydef}\n\n\\begin{myrem}\nIn RDF 1.1 literals comprise a lexical string and a datatype, such as \\verb|\"1\"^^http:\/\/www.w3.org\/2001\/XMLSchema#int|. \n\\end{myrem}\n\n\\begin{myrem}\nIn literals datatypes are identified by IRIs, where RDF borrows many of the datatypes defined in XML Schema 1.1 \\cite{Sperberg-McQueen2012}.\n\\end{myrem}\n\n\\begin{mydef}[Blank nodes]\n\\label{def:bnodes}\n\\emph{Blank nodes} are defined as existential variables used to denote the existence of some resource for which an IRI or \nliteral is not given.\n\\end{mydef}\n\n\\begin{myrem}\nBlank nodes are inconstant or stable identifiers and are in all cases locally scoped to the RDF store or the RDF file.\n\\end{myrem}\n\nA collection of RDF triples intrinsically represents a labeled directed multigraph. The nodes are the subjects and objects of \ntheir triples. RDF is often referred to as being \\emph{graph structured data} where each $\\langle s, p, o\\rangle$ triple can be \ninterpreted as an edge $s \\xrightarrow{p} o$.\n\n\\begin{mydef}[RDF graph]\n\\label{def:rdfgraph}\nLet $\\mathcal{O} = \\mathcal{I} \\cup \\mathcal{B} \\cup \\mathcal{L}$ and\n$\\mathcal{S} = \\mathcal{I} \\cup \\mathcal{B}$, then $G \\subset \\mathcal{S} \\times \\mathcal{I} \\times \\mathcal{O}$ \nis a finite subset of \\emph{RDF triples}, which is called \\emph{RDF graph}.\n\\end{mydef}\n\nOn the other hand, in the Semantic Web environment there is a Notation3 format, which offers a new human-readable serialization of \nRDF model but it also extended RDF by logical symbols and created a new Semantic Web logic called Notation3 Logic (N3Logic). \nFollowing \\cite{Arndt2015}, we provide definitions of N3Logic below.\n\n\\begin{mydef}[N3Logic alphabet]\nA \\emph{N3Logic alphabet} $A_{N3}$ consists of the following disjoint classes of symbols:\n\\begin{enumerate}\n \\item a set $\\mathcal{I}$ of IRI symbols beginning with \\verb|<| and ending with \\verb|>|,\n \\item a set $\\mathcal{L}$ of literals beginning and ending with \\verb|\"|,\n \\item a set $\\mathcal{V}$ of variables, $\\mathcal{V} = \\mathcal{B} \\cup V_U$, where $B$ is a set of existential variables \n (blank nodes in RDF-sense) start with \\verb|_:| and $V_U$ is a set of universal variables start with \\verb|?|,\n \\item brackets \\verb|{|, \\verb|}|,\n \\item a logical implication \\verb|=>|,\n \\item a period \\verb|.|,\n \\item a period \\verb|@false|.\n\\end{enumerate}\n\\end{mydef}\n\n\\begin{myrem}\nNotation3 allows to abbreviate IRIs by using prefixes. Instead of writing \\verb|<http:\/\/example.com>|, we can write \\verb|ex:|.\n\\end{myrem}\n\n\\begin{myrem}\nEach IRI, variable and literal is an expression.\n\\end{myrem}\n\n\\begin{myrem}\n\\texttt{\\{$f$\\}} is an expression called formula.\n\\end{myrem}\n\n\\begin{myrem}\n\\texttt{$e_1$ => $e_2$} is a formula called implication.\n\\end{myrem}\n\nIn Notation3 literals, IRIs, variables or even formula expressions can be subjects, objects or predicates.\n\n\\section{Inference rules}\n\\label{sec:rules}\nIn this section, we introduce inference rules for RDF, RDFS and OWL. Inference rules connected with RDF(S) and OWL are basis of the \ndeductive RDF graph stores.\n\n\\begin{mydef}[Deductive RDF graph store]\nA \\emph{deductive RDF graph store} is an entity which remembers RDF triples and can generate new ones under certain conditions \nthrough deduction or inference. It can answer queries about the combined given and inferred triples.\n\\end{mydef}\n\n\\subsection{RDF and RDFS}\nIn Table~\\ref{tab:rulesrdf} we present patterns which hold by RDF and RDFS entailments. All rules are tested in reasoning engines such as \nFuXi\\footnote{\\url{https:\/\/github.com\/RDFLib\/FuXi}} and cwm\\footnote{\\url{http:\/\/www.w3.org\/2000\/10\/swap\/doc\/cwm.html}}.\n\n\\begin{table*}[]\n\\caption{Inference rules for RDF and RDFS}\n\\label{tab:rulesrdf}\n\\centering\n\\begin{tabular}{|lll|}\n\\hline\nConditions &  & Conclusions \\\\\n\\hline\n\\verb|{?S ?P ?O}| & \\verb|=>| & \\verb|{?P rdf:type rdf:Property}.| \\\\\n\\verb|{?P rdfs:domain ?C. ?S ?P ?O}| & \\verb|=>| & \\verb|{?S rdf:type ?C}.| \\\\ \n\\verb|{?P rdfs:range ?C. ?S ?P ?O}| & \\verb|=>| & \\verb|{?O rdf:type ?C}.| \\\\ \n\\verb|{?S ?P ?O}| & \\verb|=>| & \\verb|{?S rdf:type rdfs:Resource}.| \\\\ \n\\verb|{?S ?P ?O}| & \\verb|=>| & \\verb|{?O rdf:type rdfs:Resource}.| \\\\ \n\\verb|{?Q rdfs:subPropertyOf ?R. ?P rdfs:subPropertyOf ?Q}| & \\verb|=>| & \\verb|{?P rdfs:subPropertyOf ?R}.| \\\\\n\\verb|{?Q rdf:type rdf:Property}| & \\verb|=>| & \\verb|{?Q rdfs:subPropertyOf ?Q}.| \\\\ \n\\verb|{?P rdfs:subPropertyOf ?R. ?S ?P ?O}| & \\verb|=>| & \\verb|{?S ?R ?O}.| \\\\ \n\\verb|{?C rdf:type rdfs:Class}| & \\verb|=>| & \\verb|{?C rdfs:subClassOf rdfs:Resource}.| \\\\ \n\\verb|{?A rdfs:subClassOf ?B. ?S rdf:type ?A}| & \\verb|=>| & \\verb|{?S rdf:type ?B}.| \\\\ \n\\verb|{?Q rdf:type rdfs:Class}| & \\verb|=>| & \\verb|{?Q rdfs:subClassOf ?Q}.| \\\\ \n\\verb|{?B rdfs:subClassOf ?C. ?A rdfs:subClassOf ?B}| & \\verb|=>| & \\verb|{?A rdfs:subClassOf ?C}.| \\\\ \n\\verb|{?X rdf:type rdfs:ContainerMembershipProperty}| & \\verb|=>| & \\verb|{?X rdfs:subPropertyOf rdfs:member}.| \\\\ \n\\verb|{?X rdf:type rdfs:Datatype}| & \\verb|=>| & \\verb|{?X rdfs:subClassOf rdfs:Literal}.| \\\\ \n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{OWL}\n\nIn Table~\\ref{tab:surowl} we analyze existing proposals for different OWL2 profiles: RDFS++ \\cite{Rdfspp2016}, \nL2 \\cite{Fischer2010}, RDF 3.0\/OWLPrime \\cite{Hendler2010}, OWLSIF\/pD* \\cite{Terhorst2005}, OWL-LD \\cite{Glimm2011} and \nOWL-RL \\cite{Motik2012}. We check which terms are most commonly used and propose a new version of OWL 2 called OWL-P. \nWe also considered time complexity for detecting a required rule application and frequently used vocabulary terms in \nour corpus. The snapshot (Table \\ref{tab:fowl}) is built by \\cite{Isele2010} and use seeds from \\cite{Schmachtenberg2014}. \n\nThis profile of OWL2 is simpler that OWL-RL. It drops support for restriction and cardinality classes, class \nrelationships and list-based axioms. In the Table~\\ref{tab:rulesowl1}, Table~\\ref{tab:rulesowl2} and we present \ninference rules of OWL-P.\n\n\\begin{table*}[]\n\\caption{Comparison of OWL profiles}\n\\label{tab:surowl}\n\\centering\n\\begin{tabular}{|l|c|c|c|c|c|c|c|}\n\\hline\n                          & OWL-P & RDFS++ & L2  & RDFS 3.0              & OWLSIF         & OWL-LD & OWL-RL \\\\\n                          &       &        &     & OWLPrime              & pD*            &        &        \\\\\n\\hline\n\\verb|owl:AllDifferent|              & \\XBox         & \\XBox          & \\XBox       & \\CheckedBox                    & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:AllDisjointClasses|        & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:AllDisjointProperties|     & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:allValuesFrom|             & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\CheckedBox             & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:assertionProperty|         & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:AsymmetricProperty|        & \\CheckedBox   & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:cardinality|               & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:complementOf|              & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:DatatypeProperty|          & \\CheckedBox   & \\XBox          & \\XBox       & \\CheckedBox                    & \\XBox                   & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:differentFrom|             & \\CheckedBox   & \\XBox          & \\XBox       & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:disjointUnionof|           & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:disjointWith|              & \\CheckedBox   & \\XBox          & \\XBox       & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:equivalentClass|           & \\CheckedBox   & \\XBox          & \\CheckedBox & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:equivalentProperty|        & \\CheckedBox   & \\XBox          & \\CheckedBox & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:FunctionalProperty|        & \\CheckedBox   & \\XBox          & \\XBox       & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:hasKey|                    & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:hasSelf|                   & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:hasValue|                  & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\CheckedBox             & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:intersectionof|            & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:InverseFunctionalProperty| & \\CheckedBox   & \\XBox          & \\XBox       & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:inverseOf|                 & \\CheckedBox   & \\CheckedBox    & \\CheckedBox & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:IrreflexiveProperty|       & \\CheckedBox   & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:maxCardinality|            & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:minCardinality|            & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:ObjectProperty|            & \\CheckedBox   & \\XBox          & \\XBox       & \\CheckedBox                    & \\XBox                   & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:oneOf|                     & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:propertyChainAxiom|        & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:propertyDisjointWith|      & \\CheckedBox   & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:qualifiedCardinality|      & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:qualifiedMaxCardinality|   & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:qualifiedMinCardinality|   & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:sameAs|                    & \\CheckedBox   & \\CheckedBox    & \\CheckedBox & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:someValuesFrom|            & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\CheckedBox             & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:sourceIndiviual|           & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:SymmetricProperty|         & \\CheckedBox   & \\XBox          & \\CheckedBox & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:targetIndividual|          & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:targetValue|               & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|owl:TransitiveProperty|        & \\CheckedBox   & \\CheckedBox    & \\CheckedBox & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|owl:unionof|                   & \\XBox         & \\XBox          & \\XBox       & \\XBox                          & \\XBox                   & \\XBox          & \\CheckedBox    \\\\\n\\verb|rdfs:domain|                   & \\CheckedBox   & \\CheckedBox    & \\CheckedBox & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|rdfs:range|                    & \\CheckedBox   & \\CheckedBox    & \\CheckedBox & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|rdfs:subClassOf|               & \\CheckedBox   & \\CheckedBox    & \\CheckedBox & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\verb|rdfs:subPropertyOf|            & \\CheckedBox   & \\CheckedBox    & \\CheckedBox & \\CheckedBox                    & \\CheckedBox             & \\CheckedBox    & \\CheckedBox    \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{table}[]\n\\caption{Vocabulary terms used in LOD snapshot 2015}\n\\label{tab:fowl}\n\\centering\n\\begin{tabular}{|l|c|}\n\\hline\n                                     &  voc terms \\\\\n\\hline\n\\verb|owl:AllDifferent|              & 111       \\\\\n\\verb|owl:AllDisjointClasses|        & 21        \\\\\n\\verb|owl:AllDisjointProperties|     & 13        \\\\\n\\verb|owl:allValuesFrom|             & 126330    \\\\\n\\verb|owl:assertionProperty|         & 0         \\\\\n\\verb|owl:AsymmetricProperty|        & 0         \\\\\n\\verb|owl:cardinality|               & 23910     \\\\\n\\verb|owl:complementOf|              & 873       \\\\\n\\verb|owl:DatatypeProperty|          & 27471     \\\\\n\\verb|owl:differentFrom|             & 784       \\\\\n\\verb|owl:disjointUnionOf|           & 0         \\\\\n\\verb|owl:disjointWith|              & 3743      \\\\\n\\verb|owl:equivalentClass|           & 29708     \\\\\n\\verb|owl:equivalentProperty|        & 201       \\\\\n\\verb|owl:FunctionalProperty|        & 3730      \\\\\n\\verb|owl:hasKey|                    & 5         \\\\\n\\verb|owl:hasSelf|                   & 3         \\\\\n\\verb|owl:hasValue|                  & 1877      \\\\\n\\verb|owl:intersectionOf|            & 2681      \\\\\n\\verb|owl:InverseFunctionalProperty| & 94        \\\\\n\\verb|owl:inverseOf|                 & 1341      \\\\\n\\verb|owl:IrreflexiveProperty|       & 0         \\\\\n\\verb|owl:maxCardinality|            & 257371    \\\\\n\\verb|owl:minCardinality|            & 455203    \\\\\n\\verb|owl:ObjectProperty|            & 40330     \\\\\n\\verb|owl:oneOf|                     & 853       \\\\\n\\verb|owl:propertyChainAxiom|        & 68        \\\\\n\\verb|owl:propertyDisjointWith|      & 4         \\\\\n\\verb|owl:qualifiedCardinality|      & 109       \\\\\n\\verb|owl:qualifiedMaxCardinality|   & 2         \\\\\n\\verb|owl:qualifiedMinCardinality|   & 20        \\\\\n\\verb|owl:sameAs|                    & 3967150   \\\\\n\\verb|owl:someValuesFrom|            & 4446      \\\\\n\\verb|owl:sourceIndiviual|           & 0         \\\\\n\\verb|owl:SymmetricProperty|         & 194       \\\\\n\\verb|owl:targetIndividual|          & 0         \\\\\n\\verb|owl:targetValue|               & 11        \\\\\n\\verb|owl:TransitiveProperty|        & 267       \\\\\n\\verb|owl:unionOf|                   & 53735     \\\\\n\\verb|rdfs:domain|                   & 111865    \\\\\n\\verb|rdfs:range|                    & 59252     \\\\\n\\verb|rdfs:subClassOf|               & 1339391   \\\\\n\\verb|rdfs:subPropertyOf|            & 13416     \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table*}[]\n\\caption{Inference rules for OWL-P properties}\n\\label{tab:rulesowl1}\n\\centering\n\\begin{tabular}{|lll|}\n\\hline\nConditions &  & Conclusions \\\\\n\\hline\n\\verb|{?S ?P ?O}| & \\verb|=>| & \\verb|{?S owl:sameAs ?S. ?P owl:sameAs ?P.| \\\\\n                  &           &  \\verb| ?O owl:sameAs ?O}.| \\\\\n\\verb|{?S owl:sameAs ?O}| & \\verb|=>| & \\verb|{?O owl:sameAs ?S}.| \\\\\n\\verb|{?Q owl:sameAs ?R. ?R owl:sameAs ?P}| & \\verb|=>| & \\verb|{?Q owl:sameAs ?P}.| \\\\\n\\verb|{?S owl:sameAs ?S2. ?S ?P ?O}| & \\verb|=>| & \\verb|{?S2 ?P ?O}.| \\\\\n\\verb|{?P owl:sameAs ?P2. ?S ?P ?O}| & \\verb|=>| & \\verb|{?S ?P2 ?O}.| \\\\\n\\verb|{?O owl:sameAs ?O2. ?S ?P ?O}| & \\verb|=>| & \\verb|{?S ?P ?O2}.| \\\\\n\\verb|{?Q owl:sameAs ?R. ?Q owl:differentFrom ?R}| & \\verb|=>| & \\verb|{@false}.| \\\\\n\\verb|{?P rdf:type owl:FunctionalProperty.| & & \\\\\n\\verb| ?Q ?P ?R1. ?Q ?P ?R2}| & \\verb|=>| & \\verb|{?R1 owl:sameAs ?R2}.| \\\\\n\\verb|{?P rdf:type owl:InverseFunctionalProperty.| & & \\\\\n\\verb| ?Q1 ?P ?R. ?Q2 ?P ?R}| & \\verb|=>| & \\verb|{?Q1 owl:sameAs ?Q2}.| \\\\\n\\verb|{?P rdf:type owl:IrreflexiveProperty. ?Q ?P ?Q}| & \\verb|=>| & \\verb|{@false}.| \\\\\n\\verb|{?P rdf:type owl:SymmetricProperty. ?Q ?P ?R}| & \\verb|=>| & \\verb|{?R ?P ?Q}.| \\\\\n\\verb|{?P rdf:type owl:AsymmetricProperty. ?Q ?P ?R. ?R ?P ?Q }| & \\verb|=>| & \\verb|{@false}.| \\\\\n\\verb|{?P rdf:type owl:TransitiveProperty. ?Q ?P ?R. ?R ?P ?P}| & \\verb|=>| & \\verb|{?Q ?P ?P}.| \\\\\n\\verb|{?P1 owl:equivalentProperty ?P2. ?Q ?P1 ?R}| & \\verb|=>| & \\verb|{?Q ?P2 ?R}.| \\\\\n\\verb|{?P1 owl:equivalentProperty ?P2. ?Q ?P2 ?R}| & \\verb|=>| & \\verb|{?Q ?P1 ?R}.| \\\\\n\\verb|{?P1 owl:propertyDisjointWith ?P2. ?Q ?P1 ?R. ?Q ?P2 ?R}| & \\verb|=>| & \\verb|{@false}.| \\\\\n\\verb|{?P1 owl:inverseOf ?P2. ?Q ?P1 ?R}| & \\verb|=>| & \\verb|{?R ?P2 ?Q}.| \\\\\n\\verb|{?P1 owl:inverseOf ?P2 . ?Q ?P2 ?R}| & \\verb|=>| & \\verb|{?R ?P1 ?Q}.| \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}[]\n\\caption{Inference rules for OWL-P classes}\n\\label{tab:rulesowl2}\n\\centering\n\\begin{tabular}{|lll|}\n\\hline\nConditions &  & Conclusions \\\\\n\\hline\n\\verb|{?A owl:equivalentClass ?B . ?x rdf:type ?A}| & \\verb|=>| & \\verb|{?x a ?B}.| \\\\\n\\verb|{?A owl:equivalentClass ?B . ?x rdf:type ?B}| & \\verb|=>| & \\verb|{?x a ?A}.| \\\\\n\\verb|{?A owl:disjointWith ?B.| & & \\\\\n\\verb|{ ?x rdf:type ?A. ?x rdf:type ?B}| & \\verb|=>| & \\verb|{@false}| \\\\\n\\verb|{?C rdf:type owl:Class}| & \\verb|=>| & \\verb|{?C rdfs:subClassOf ?C. ?C owl:Thing.| \\\\\n                               &           & \\verb| ?C owl:equivalentClass ?C.| \\\\\n                               &           & \\verb| owl:Nothing rdfs:subClassOf ?C}.| \\\\\n\\verb|{?A owl:equivalentClass ?B}| & \\verb|=>| & \\verb|{?A rdfs:subClassOf ?B. ?B rdfs:subClassOf ?A}.| \\\\\n\\verb|{?A rdfs:subClassOf ?B.| & & \\\\\n\\verb| ?B rdfs:subClassOf ?A}| & \\verb|=>| & \\verb|{?A owl:equivalentClass ?B}.| \\\\\n\\verb|{?P rdf:type owl:ObjectProperty}| & \\verb|=>| & \\verb|{?P rdfs:subPropertyOf ?P.| \\\\\n                                        &           & \\verb| ?P owl:equivalentProperty ?P}.| \\\\\n\\verb|{?P rdf:type owl:DatatypeProperty}| & \\verb|=>| & \\verb|{?P rdfs:subPropertyOf ?P.| \\\\\n                                          &           & \\verb| ?P owl:equivalentProperty ?P}.| \\\\\n\\verb|{?P owl:equivalentProperty ?R}| & \\verb|=>| & \\verb|{?P rdfs:subPropertyOf ?R.| \\\\\n                                        &           & \\verb| ?R rdfs:subPropertyOf ?P}.| \\\\\n\\verb|{?P rdfs:subPropertyOf ?R.| & & \\\\\n\\verb| ?R rdfs:subPropertyOf ?P}| & \\verb|=>| & \\verb|{?P owl:equivalentProperty ?R}.| \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\section{Related Work}\n\\label{sec:relatedwork}\nOne of the most important general purpose logic programming language is Prolog \\cite{Clocksin2003}. It is declarative, which means\nthat the program logic is declared in terms of relations, represented as facts and rules. Yet anoder declarative language\nis Datalog \\cite{Eiter1997}, which is syntactically a subset of Prolog. Apart from the Notation3, there are other rule-based \ninference engines formats for the Semantic Web, such as: FOL-RuleML \\cite{Boley2004}, SWRL \\cite{Horrocks2004}, RIF \\cite{Kifer2008}, \nR-DEVICE \\cite{Bassiliades2004}, TRIPLE \\cite{Sintek2001}, Jena rule\\footnote{\\url{http:\/\/jena.apache.org\/documentation\/inference}} and\nSPIN \\cite{Ryman2014}.\n\nFOL-RuleML (First-order Logic Rule Markup Language) \\cite{Boley2004} is a rule language for expressing first-order \nlogic for the web. It is a sublanguage of RuleML \\cite{Boley2012}. In FOL-RuleML each of rules consists of a set of statements \ncalled an atom. The atom is a form which consists of objects which are individuals or variables, and a relation between them.\n\nSWRL (Semantic Web Rule Language) \\cite{Horrocks2004} is based on OWL \\cite{Parsia2012} and Unary\/Binary Datalog RuleML, which sublanguage of \nthe RuleML. It extends the set of OWL axioms to include Horn-like rules. Logical operators and quantifications supports \nof SWRL are the same as in RuleML. Moreover, RuleML contents can be parts of SWRL content. Axioms may consist of OWL, RDF\nand rule axioms. A relation can be an IRI, a data range, an OWL property or a built-in relation. An object can be a variable, \nan individual, a literal value or a blank node.\n\nRIF (Rule Interchange Format) \\cite{Kifer2008} is a standard for exchanging rules among disparate systems. It focused on \nexchange rather than developing a single one-fits-all rule language. It can be separated into a number of parts, \nRIF-core \\cite{Reynolds2013} which is the common core of all RIF dialects, RIF-BLD (Basic Logic Dialect) \\cite{Kifer2013} comprising \nbasic dialects (i.e. Horn rules) for writing rules, RIF-PRD \\cite{Marie2013} (Production Rule Dialect) for representing production rules and \nRIF-DTB (Datatypes and Built-in Functions) \\cite{Polleres2013} comprising a set of datatypes and built-in functions.\n\nR-DEVICE \\cite{Bassiliades2004} is a deductive rule language for reasoning about RDF data. In R-DEVICE resources are represented \nas objects and RDF properties are realized as multi-slots. It supports a second-order syntax, where variables can range over \nclasses and properties. It provides a RuleML-like syntax.\n\nTRIPLE \\cite{Sintek2001} is an RDF rule (query, inference, and transformation) language, with a layered and modular nature. \nIt is based on Horn Logic \\cite{Horn1951} and F-Logic \\cite{Kifer1989}. Rules in TRIPLE are used for transient querying and \ncannot be used for defining and maintaining views.\n\nSPIN (SPARQL Inferencing Notation) \\cite{Ryman2014} is a constraint and SPARQL-based rule language for RDF. It can link class with \nqueriesto capture constraints and rules which describe the behavior of those classes. SPIN is also a method to represent queries as \ntemplates. It can represent SPARQL statement as RDF triples. That proposal allows to declare new SPARQL functions.\n\nJena rule is a rule format used only by inference engine in the Jena framework \\cite{Mcbride2002}. The rule language \nsyntax is based on RDF. It uses the triple representation, which is similar to Notation3 except that a rule name can be \nspecified in a rule. There are not any formula notation, and built-in functions are written in function terms. \n\n\\section{Conclusions}\nThis paper define how knowledge and logic might be handled on the Semantic Web environment. We present inference rules \nRDF, RDF Schema and OWL. All rules are tested in reasoning engines. Our formalization is based on Notation 3 Logic, which \nextended RDF by logical symbols and created a new Semantic Web logic. Moreover, we propose a lightweight OWL profile called OWL-P.\nOur proposed rule will be useful for deductive \nRDF graph stores.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nA fractional derivative is an operator that generalizes the ordinary derivative, in the sense that if\n\n\n\\begin{eqnarray*}\n\\dfrac{d^\\alpha}{d x^\\alpha},\n\\end{eqnarray*}\n\n\n\n\ndenotes the differential of order $ \\alpha $, then $\\alpha$ may be considered a parameter, with $\\alpha\\in \\nset{R}$, such that the first derivative corresponds to the particular case $\\alpha=1$. On the other hand, a fractional differential equation is an equation that involves at least one differential operator of order $ \\alpha $, with $(n-1)< \\alpha \\leq n$ for some positive integer $n$, and it is said to be a differential equation of order $\\alpha$ if this operator is the highest order in the equation. The fractional operators have many representations, but one of their fundamental properties is that they allow retrieving the results of conventional calculus when $\\alpha \\to n$. So, considering a scalar function $h: \\nset{R}^m \\to \\nset{R}$ and the canonical basis of $\\nset{R}^m$ denoted by $\\set{\\hat{e}_k}_{k\\geq 1}$, it is possible to define the following fractional operators of order $\\alpha$ using Einstein notation\n\n\n\n\\begin{eqnarray}\no^\\alpha h(x):=\\hat{e}_k o_k^\\alpha h(x).\n\\end{eqnarray}\n\n\n\n\n\nTherefore, denoting by  $\\partial_k^n$ the partial derivative of order $n$ applied with respect to the $k$-th component of the vector $x$, using the previous operator it is possible to define the following set of fractional operators\n\n\n\n\\begin{eqnarray}\\label{eq:0}\n\\Op_{x,\\alpha}^n(h):=\\set{o^\\alpha \\ : \\ \\exists \\partial_k^n h(x)  \\  \\mbox{ and } \\ \\lim_{\\alpha \\to n}o_k^\\alpha h(x)=\\partial_k^n h(x) \\ \\forall k\\geq 1 },\n\\end{eqnarray}\n\n\n\nwhich may be considered as a generating set of sets of fractional tensor operators. For example, considering $\\alpha,n\\in \\nset{R}^m$ with $\\alpha=\\hat{e}_k\\alpha_k$ and $n=\\hat{e}_k n_k$, it is possible to define the following set of fractional tensor operators\n\n\n\n\\begin{eqnarray}\n\\Op_{x,\\alpha}^{n}(h):=\\set{o^\\alpha \\ : \\ o^\\alpha \\in \\Op_{x,\\alpha_1}^{n_1}(h) \\times \\Op_{x,\\alpha_2}^{n_2}(h)\\times \\cdots \\times \\Op_{x,\\alpha_m}^{n_m}(h) }.\n\\end{eqnarray}\n\n\nOne of the most famous fixed point methods is the well-known Newton-Raphson method. However, it sometimes goes unnoticed that this method has the following problem related to finding roots of polynomials in the complex space: If it is necessary to find a complex root $\\xi$ of a polynomial using the Newton-Raphson method, with $\\xi\\in \\nset{C}\\setminus\\nset{R}$, a complex initial condition $x_0$ must be provided, and if a suitable initial condition is selected, this will lead to a complex solution, but there is also the possibility that this may lead to a real solution. If the root obtained is real, it is necessary to change the initial condition and expect that this will lead to a complex solution, otherwise, it is necessary to change the value of the initial condition again, this process is repeated until it finally converges to a complex solution. The process described above is very similar to what happens when different values $\\alpha$ are used in fractional operators until we find a solution that meets some established criterion. Considering the Newton-Raphson method from the perspective of fractional calculus, it is possible to consider that an order $\\alpha$ remains fixed, in this case $\\alpha=1$, and the initial conditions $x_0$ are varied until found a solution $\\xi$ that fulfills an established criterion. It is necessary to mention that considering a relationship between fractional calculus and the Newton-Raphson method may seem somewhat forced at first, but the latter is characterized by the fact that when it generates divergent sequences of complex numbers, it can sometimes lead to the creation of fractals \\cite{tatham}, and this feature is complemented quite well with the fact that the orders of the fractional derivatives seem to be closely related to the fractal dimension  \\cite{brambila2017fractal}. Based on the above, it is possible to consider inverting the behavior of the order $\\alpha=1$ of the derivative  and the initial condition $x_0$, that is, leaving the initial condition $x_0$ fixed and varying the order $\\alpha$ of the derivative, thus obtaining the \\textbf{fractional Newton-Raphson method}  \\cite{torres2021fracnewrap}, which is nothing other than the Newton-Raphson method using any definition of the fractional derivative that fits the function whose zeros want to be determined. \n\n\n\n\\begin{figure}[!ht]\n        \\begin{subfigure}[c]{0.35\\textwidth}\n        \\centering\n \\includegraphics[width=\\textwidth, height=0.6\\textwidth]{rN077}      \n    \\caption*{a) $\\alpha=-0.77$}\n    \\end{subfigure}\n        \\begin{subfigure}[c]{0.35\\textwidth}\n        \\centering\n \\includegraphics[width=\\textwidth, height=0.6\\textwidth]{rN032}     \n    \\caption*{b) $\\alpha=-0.32$}\n    \\end{subfigure} \n    \\centering\n    \\begin{subfigure}[c]{0.35\\textwidth}\n    \\centering\n \\includegraphics[width=\\textwidth, height=0.6\\textwidth]{r019}      \n    \\caption*{c) $\\alpha=0.19$}\n    \\end{subfigure}\n    \\begin{subfigure}[c]{0.35\\textwidth}\n    \\centering\n \\includegraphics[width=\\textwidth, height=0.6\\textwidth]{r187}      \n    \\caption*{d) $\\alpha=1.87$}\n    \\end{subfigure}        \n        \\caption{Illustrations of some trajectories generated by the fractional Newton-Raphson method for the same initial condition $x_0$ but with different orders $\\alpha$ of the fractional operator used \\cite{torres2021fracnewrap}.}\\label{fig:01}\n\\end{figure}\n\n\nBefore continuing, it is necessary to mention that due to the large number of fractional operators that can exist, some sets need to be defined to fully characterize the fractional Newton-Raphson method. So, considering a function $h: \\Omega \\subset \\nset{R}^m \\to \\nset{R}^m$, it is possible to define the following set of fractional operators\n\n\n\\begin{eqnarray}\n{}_m\\Op_{x,\\alpha}^n(h):=\\set{ o^\\alpha \\ : \\ o^\\alpha\\in \\Op_{x,\\alpha}^n\\left([h]_j \\right) \\ \\forall j \\mbox{ with } 1\\leq j\\leq m},\n\\end{eqnarray}\n\n\n\nwhere $[h]_k: \\Omega \\subset \\nset{R}^m \\to \\nset{R}$ denotes the $k$-th component of the function $h$. So, it is possible to define the following set of matrices\n\n\n\\begin{eqnarray}\n{}_{m}\\Ma_{x,\\alpha}^{n}(h):=\\set{A=A(o^\\alpha) \\ : \\   o^\\alpha \\in {}_m\\Op_{x,\\alpha}^n(h) \\ \\mbox{ and } \\ A(x)=\\left([A]_{jk}(x) \\right):= \\left( o_k^\\alpha [h]_j(x)\\right)  },\n\\end{eqnarray}\n\n\nand therefore, the fractional Newton-Raphson method can be defined and classified through the following set of  matrices\n\n\n\\begin{eqnarray}\n{}_{m}\\IMa_{x,\\alpha}^{1}(h):=\\set{A \\ : \\   \\exists A^{-1} \\in  {}_mM_{x,\\alpha}^{1}(h) },\n\\end{eqnarray}\n\nand as a consequence, if $\\Phi_{FNR}$ denotes the iteration function of the fractional Newton-Raphson method, it is possible to obtain the following result:\n\n\n\n\n\\begin{eqnarray}\n\\mbox{Let }\\alpha_0\\in \\nset{R}\\setminus \\nset{Z} \\Rightarrow \\forall A \\in {}_{m}\\IMa_{x,\\alpha_0}^{1}(h) \\hspace{0.1cm}  \\exists\\Phi_{FNR}=\\Phi_{FNR}(\\alpha_0, A) \\  \\therefore  \\ \\forall A \\hspace{0.1cm}  \\exists  \\set{\\Phi_{FNR}(\\alpha, A) \\ : \\ \\alpha \\in \\nset{R}\\setminus \\nset{Z}}.  \n\\end{eqnarray}\n\n\n\n\nThe change from leaving the initial condition $x_0$ fixed and varying the order $\\alpha$ of the derivative, although seemingly simple, gives the fractional Newton-Raphson method the ability to partially solve the intrinsic problem associated with classical fixed point methods, which is that in general, to find $N$ zeros of a function, $N$ initial conditions must be provided. This is because by varying the order $\\alpha$ of the fractional derivative, the fractional Newton-Raphson method can find $N$ zeros of a function using a single initial condition as shown in the Figure \\ref{fig:01}. It is necessary to consider that mentioned above is also valid for any fixed point method that implements fractional operators in some way, which may be named as \\textbf{fractional fixed point methods} or \\textbf{fractional iterative methods}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nTo finish this section, it is necessary to mention that the applications of fractional operators have spread to different fields of science such as finance \\cite{safdari2015radial,sabatelli2002waiting}, economics \\cite{traore2020model,torres2020nonlinear}, number theory through the Riemann zeta function \\cite{guariglia2021fractional,torres2021zeta} and in engineering with the study for the manufacture of hybrid solar receivers \\cite{de2021fractional,torres2020reduction,torres2020fracpseunew,torres2021codefracpseudo}. It should be mentioned that there is also a growing interest in fractional operators and their properties for the solution of nonlinear systems \\cite{wang2021derivative,gdawiec2019visual,akgul2019fractional,torres2021fracnewrap,torres2020fracsome,torres2021fracnewrapaitken}, which is a classic problem in mathematics, physics and engineering, which consists of finding the set of zeros of a function $f:\\Omega \\subset \\nset{R}^n \\to \\nset{R}^n$, that is,\n\n\n\\begin{eqnarray}\\label{eq:1-001}\n\\set{\\xi \\in \\Omega \\ : \\ \\norm{f(\\xi)}=0},\n\\end{eqnarray}\n\n\nwhere $\\norm{ \\ \\cdot \\ }: \\nset{R}^n \\to \\nset{R}$ denotes any vector norm, or equivalently\n\n\n\\begin{eqnarray}\n\\set{\\xi \\in \\Omega \\ : \\ [f]_k(\\xi)=0 \\ \\forall k\\geq 1}.\n\\end{eqnarray}\n\n\n\n\n\n\n\n\n\nAlthough finding the zeros of a function may seem like a simple problem, it is generally necessary to use numerical methods of the iterative type to solve it. So, considering that fractional iterative methods can find $N$ solutions of a system using a single initial condition, this article shows an alternative way to the Aitken's method to accelerate the order of convergence of a family of fractional fixed point methods, which consists of implementing a function in the order of the fractional operators involved, with which it is possible to obtain an order of convergence at least quadratic.\n\n\n\n\n\n\n\n\n\\section{Fixed Point Method}\n\n\n\nLet  $\\Phi:\\nset{R}^n \\to \\nset{R}^n$ be a function. It is possible to build a sequence $\\set{x_i}_{\\geq 1}$  by defining the following iterative method\n\n\\begin{eqnarray}\\label{eq:2-001}\nx_{i+1}:=\\Phi(x_i), & i=0,1,2,\\cdots,\n\\end{eqnarray}\n\n\n\n\nif it is fulfilled that $x_i\\to \\xi\\in \\nset{R}^n$ and if the function $\\Phi$ is continuous around $\\xi$, we obtain that\n\n\n\\begin{eqnarray}\\label{eq:2-002}\n\\ds \\xi=\\lim_{i\\to \\infty}x_{i+1}=\\lim_{i\\to \\infty}\\Phi(x_i)=\\Phi\\left(\\lim_{i\\to \\infty}x_i \\right)=\\Phi(\\xi),\n\\end{eqnarray}\n\n\n\n\nthe above result is the reason by which the method \\eqref{eq:2-001} is known as the \\textbf{fixed point method}. Furthermore, the function $\\Phi$ is called an \\textbf{iteration function}. The following corollary allows characterizing the order of convergence of an iteration function $ \\Phi $ through its \\textbf{Jacobian matrix} $\\Phi^{(1)}$ \\cite{torres2021fracnewrapaitken}:\n\n\\begin{corollary}\\label{cor:2-001}\nLet $\\Phi:\\nset{R}^n \\to \\nset{R}^n$ be an iteration function. If $\\Phi$ defines a sequence $\\set{x_i}_{\\geq 1}$ such that $x_i\\to \\xi\\in \\nset{R}^n$. So, $\\Phi$ has an \\textbf{order of convergence}  of order (at least) $p$  in $B(\\xi;\\delta)$, where \n\n\n\n\n\n\\begin{eqnarray}\\label{eq:2-003}\np:=\\left\\{\n\\begin{array}{cc}\n1 , &\\ds \\mbox{If } \\lim_{x\\to \\xi}\\norm{\\Phi^{(1)}(x)}\\neq 0  \\vspace{0.2cm}\\\\\n2, &\\ds \\mbox{If } \\lim_{x\\to \\xi}\\norm{\\Phi^{(1)}(x)}= 0  \\\\\n\\end{array}\\right. .\n\\end{eqnarray}\n\n\n\\end{corollary}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Riemann-Liouville Fractional Operators}\n\n\nOne of the fundamental operators of fractional calculus is the operator \\textbf{Riemann-Liouville fractional integral},  which is defined as follows \\cite{hilfer00,oldham74}\n\n\n\n\\begin{eqnarray}\n\\ifr{}{a}{I}{x}{\\alpha}f(x):=\\dfrac{1}{\\gam{\\alpha}}\\int_a^x (x-t)^{\\alpha-1}f(t)dt,\n\\end{eqnarray}\n\n\n\n\n\nwhich is a fundamental piece to build the operator \\textbf{Riemann-Liouville fractional derivative},  which is defined as follows \\cite{hilfer00,kilbas2006theory}\n\n\\begin{eqnarray}\\label{eq:3-001}\n\\normalsize\n\\begin{array}{c}\n\\ifr{}{a}{D}{x}{\\alpha}f(x) := \\left\\{\n\\begin{array}{cc}\n\\ds \\ifr{}{a}{I}{x}{-\\alpha}f(x), &\\mbox{if }\\alpha<0 \\vspace{0.1cm}\\\\  \n\\ds \\dfrac{d^n}{dx^n}\\left( \\ifr{}{a}{I}{x}{n-\\alpha}f(x)\\right), & \\mbox{if }\\alpha\\geq 0\n\\end{array}\n\\right.\n\\end{array}, \n\\end{eqnarray}\n\n\n\n\nwhere  $ n = \\lceil \\alpha \\rceil$ and $\\ifr{}{a}{I}{x}{0}f(x):=f(x)$. Applying the operator \\eqref{eq:3-001} with $a=0$ to the  function $ x^{\\mu} $, with $\\mu> -1$, we obtain the following result \\cite{torres2021fracnewrapaitken}:\n\n\n\\begin{eqnarray}\\label{eq:3-002}\n\\ifr{}{0}{D}{x}{\\alpha}x^\\mu = \n \\dfrac{\\gam{\\mu+1}}{\\gam{\\mu-\\alpha+1}}x^{\\mu-\\alpha}, & \\alpha\\in \\nset{R}\\setminus \\nset{Z}.\n\\end{eqnarray}\n\n\n\n\n\n\\section{Fractional Fixed Point Method}\n\nLet $f:\\Omega \\subset \\nset{R}^n \\to \\nset{R}^n$ be a function with a point $\\xi\\in \\Omega$ such that $\\norm{f(\\xi)}=0$ . So, considering an iteration function $\\Phi:(\\nset{R}\\setminus \\nset{Z})\\times \\nset{R}^n\\to \\nset{R}^n$, the iteration function of a fractional iterative method may be written in general form as follows\n\n\n\\begin{eqnarray}\\label{eq:4-001}\n\\Phi(\\alpha,x):=x-A_{g,\\alpha}(x)f(x),& \\alpha\\in \\nset{R}\\setminus \\nset{Z},\n\\end{eqnarray}\n\n\n\nwhere $A_{g,\\alpha}$ is a matrix that depends, in at least one of its entries, on fractional operators of order $\\alpha$ applied to some function $g:\\nset{R}^n \\to \\nset{R}^n$, whose particular case occurs when $g=f$. So, it is possible to define in a general way a fractional fixed point method as follows\n\n\n\\begin{eqnarray}\\label{eq:4-002}\nx_{i+1}:=\\Phi(\\alpha,x_i), & i=0,1,2,\\cdots.\n\\end{eqnarray}\n\nIf it is fulfilled that $x_i\\to \\xi\\in \\Omega$, it is possible to define the following set\n\n\\begin{eqnarray}\n\\Conv(\\xi):=\\set{\\Phi \\ : \\ \\lim_{x\\to \\xi}\\Phi(\\alpha,x)=\\xi},\n\\end{eqnarray}\n\n\nwhich may be interpreted as the set of fractional fixed point methods that define a convergent sequence $\\set{x_i}_{i\\geq 1}$ to the value $\\xi \\in \\Omega$. Considering the \\mref{Corollary}{\\ref{cor:2-001}}, as well as the \\textbf{Proposition 1} from the reference \\cite{torres2021fracnewrapaitken}, it is possible to define the following sets to classify the order of convergence of some fractional iterative methods\n\n\n\n\n\n\\begin{eqnarray}\n\\Ord^1(\\xi):=\\set{\\Phi \\in \\Conv(\\xi) \\ : \\ \\lim_{x \\to \\xi}\\norm{\\Phi^{(1)}(\\alpha,x)}\\neq 0 },\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\n\\Ord^2(\\xi):=\\set{\\Phi \\in \\Conv(\\xi) \\ : \\ \\lim_{x \\to \\xi}\\norm{\\Phi^{(1)}(\\alpha,x)}= 0 },\n\\end{eqnarray}\n\n\n\n\\begin{eqnarray}\n\\ord^1(\\xi):=\\set{\\Phi \\in \\Conv(\\xi) \\ : \\ \\lim_{x \\to \\xi}A_{g,\\alpha}(x)\\neq \\left(f^{(1)}(\\xi)\\right)^{-1} \\ \\mbox{ or } \\ \\lim_{\\alpha \\to 1} A_{g,\\alpha}(\\xi)\\neq \\left(f^{(1)}(\\xi)\\right)^{-1} },\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\n\\ord^2(\\xi):=\\set{\\Phi \\in \\Conv(\\xi) \\ : \\ \\lim_{x \\to \\xi}A_{g,\\alpha}(x)= \\left(f^{(1)}(\\xi)\\right)^{-1} \\ \\mbox{ or } \\ \\lim_{\\alpha \\to 1} A_{g,\\alpha}(\\xi)= \\left(f^{(1)}(\\xi)\\right)^{-1}}.\n\\end{eqnarray}\n\n\n\nOn the other hand, considering that depending on the nature of the function $f$ there exist cases in which the Newton-Raphson method can present an order of convergence (at least) linear \\cite{torres2021fracnewrapaitken}, it is possible to obtain the following relations between the previous sets\n\n\n\\begin{eqnarray}\n\\ord^1(\\xi) \\subset \\Ord^1(\\xi)& \\mbox{ and } &\\ord^2(\\xi)\\subset \\Ord^1(\\xi)\\cup \\Ord^2(\\xi),\n\\end{eqnarray}\n\nwith which it is possible to define the following sets\n\n\n\n\\begin{eqnarray}\n\\Ord_2^1(\\xi) := \\ord^2(\\xi)   \\cap  \\Ord^1(\\xi) & \\mbox{ and } & \\Ord_2^2(\\xi):= \\ord^2(\\xi) \\cap \\Ord^2(\\xi). \n\\end{eqnarray}\n\n\n\n\n\n\\subsection{Acceleration in the Order of Convergence of the Set $\\Ord_2^1(\\xi)$}\n\n\nLet $f:\\Omega \\subset \\nset{R}^n \\to \\nset{R}^n$ be a function with a point $\\xi\\in \\Omega$ such that $\\norm{f(\\xi)}=0$, and denoting by $\\Phi_{NR}$ to the iteration function of the Newton-Raphson method, it is possible to define the following set of functions\n\n\n\n\\begin{eqnarray}\n\\Ord_{NR}^2(\\xi):=\\set{ f \\ : \\  \\lim_{x\\to \\xi}\\norm{\\Phi_{NR}^{(1)}(x)}=0 }.\n\\end{eqnarray}\n\n\nSo, it is possible to define the following corollary:\n\n\n\\begin{corollary}\\label{cor:4-001}\nLet $f:\\Omega \\subset \\nset{R}^n \\to \\nset{R}^n$ be a function with a point $\\xi\\in \\Omega$ such that $f\\in \\Ord_{NR}^2(\\xi)$, and let $\\Phi$ be a iteration function given by the equations \\eqref{eq:4-001} such that $\\Phi\\in \\Ord_2^1(\\xi)$. So, it is possible to replace the order $\\alpha$ of the fractional operators of the matrix $A_{g,\\alpha}$ by the following function\n\n\n\n\n\\begin{eqnarray}\\label{eq:4-005}\n\\alpha_f([x]_k,x):=\\left\\{\n\\begin{array}{cc}\n\\alpha ,& \\mbox{if \\hspace{0.1cm}}\\abs{[x]_k}\\neq 0 \\mbox{ \\hspace{0.1cm}and\\hspace{0.1cm} } \\norm{f(x)}> \\delta \\vspace{0.1cm}\\\\\n1,& \\mbox{if \\hspace{0.1cm}}\\abs{[x]_k} = 0 \\mbox{ \\hspace{0.1cm}or\\hspace{0.1cm} } \\norm{f(x)}\\leq \\delta\n\\end{array}\n\\right. ,\n\\end{eqnarray}\n\n\n\n\nobtaining a new matrix that may be denoted as follows\n\n\n\n\\begin{eqnarray}\\label{eq:4-006}\nA_{g,\\alpha_f}(x)=\\left([A_{g,\\alpha_f}]_{jk}(x) \\right), & \\alpha\\in \\nset{R}\\setminus \\nset{Z},\n\\end{eqnarray}\n\nand that guarantees that there exists a set $\\Omega_\\xi \\subset B(\\xi;\\delta)$ such that $\\Phi\\in \\Ord_2^2(\\xi)$ in $\\Omega_\\xi$.\n\n\n\\end{corollary}\n\n\n\nIt is necessary to mention that the origin of the function \\eqref{eq:4-005} arises from the need to accelerate the order of convergence of the fractional Newton-Raphson method, which generated the method known as the \\textbf{fractional Newton method}, whose matrix $A_{g,\\alpha_f}$ corresponds to a particular case in which $g=f$ \\cite{torres2020fracsome,torres2021fracnewrap,torres2021fracnewrapaitken}. Finally, for practical purposes, it may be defined that if a fractional iterative method $\\Phi \\in \\Ord_2^1(\\xi)$ uses the function \\eqref{eq:4-005}, it may be called a \\textbf{fractional iterative method accelerated}.\n\n\n\n\n\\section{Equations of a Hybrid Solar Receiver}\n\n\nConsidering the notation\n\n\n\\begin{eqnarray*}\ns=(T_{cell},T_{hot},T_{cold},\\eta_{cell},\\eta_{TEG})^T:=\\left([x]_1,[x]_2,[x]_3,[x]_4,[x]_5 \\right)^T,\n\\end{eqnarray*}\n\n\n\n\nthe following expressions\n\n\n\n\\begin{eqnarray*}\n\\begin{array}{c}\n\\begin{array}{lll}\na_0=\\dfrac{2*r_{intercon}}{\\cdot \\sqrt{f^*\\cdot A_{TEG}}\\left(b\\cdot \\sqrt{f^*}+\\sqrt{A_{TEG}} \\right) }, & a_1=\\eta_{opt}\\cdot C_g \\cdot DNI, & a_2=r_{cell}+r_{sol}+A_{cell}\\left(\\dfrac{r_{cop}+r_{cer}}{A_{TEG}}+a_0 \\right)\n\\end{array}\\vspace{0.2cm}\\\\\n\\begin{array}{llll}\na_3=\\dfrac{A_{cell}\\cdot l}{f^*\\cdot A_{TEG}\\cdot k_{TEG}}, & a_4=T_{air}, &a_5=A_{cell}\\left( \\dfrac{r_{cer}}{A_{TEG}}+R_{heat\\_exch}+a_0 \\right),&a_6=-\\eta_{cell,ref}\\cdot \\gamma_{cell}\n\\end{array}\\vspace{0.2cm}\\\\\n\\begin{array}{lll}\n a_7=\\eta_{cell,ref}\\left(1+25 \\cdot \\gamma_{cell} \\right), &a_8=\\sqrt{1+ZT}, &a_9=273.15\n\\end{array}\n\\end{array},\n\\end{eqnarray*}\n\n\nand the following particular values \\cite{rodrigo2019performance}\n\n\\begin{eqnarray*}\n\\left\\{\n\\begin{array}{lll}\n\\eta_{opt}=0.85, & r_{intercon}=2.331\\times 10^{-7} ,&C_g=800\\\\\n    A_{cell}=9\\times 10^{-6}     , &  R_{heat\\_exch}=0.5   , & A_{TEG}=5.04 \\times 10^{-5}     \\\\\n \\eta_{cell,ref}=0.43,& r_{cell}=3\\times 10^{-6}   , &    f^*=0.7  \\\\\n \\gamma_{cell}=4.6\\times 10^{-4}, &  r_{sol}=1.603\\times 10^{-6}  , &  b=5\\times 10^{-4}    \\\\\nr_{cop}=7.5\\times 10^{-7}, & r_{cer}=8 \\times 10^{-6},&    l=5\\times 10^{-4}   \\\\\nk_{TEG}=1.5 ,&   ZT=1   & \n\\end{array}\n\\right..\n\\end{eqnarray*}\n\n\n\n\nit is possible to define the following system of equations that corresponds to the combination of a solar photovoltaic system with a thermoelectric generator system \\cite{bjork2015performance,bjork2018maximum}, which is named as a \\textbf{hybrid solar receiver}\n\n\n\\begin{eqnarray}\\label{eq:5-001}\n\\left\\{\n\\begin{array}{l}\n\\left[x\\right]_1=[x]_2+a_1\\cdot a_2\\left( 1-[x]_4 \\right)\\\\\n\\left[x\\right]_2=[x]_3+a_1\\cdot a_3 \\left( 1-[x]_4\\right)\\left(1-[x]_5\\right)\\\\\n\\left[x\\right]_3=a_4+a_1\\cdot a_5 \\left( 1-[x]_4\\right)\\left(1-[x]_5\\right)\\\\\n\\left[x\\right]_4 =a_6[x]_1+a_7\\\\\n\\left[x\\right]_5=(a_8-1)\\left(1-\\dfrac{[x]_3+a_9}{[x]_2+a_9} \\right)\\left(a_8+ \\dfrac{[x]_3+a_9}{[x]_2+a_9}\\right)^{-1}\n\\end{array}\\right.,\n\\end{eqnarray}\n\n\nwhose deduction, as well as details about its interpretation, may be found in the reference \\cite{rodrigo2019performance}. Using the system of equations \\eqref{eq:5-001}, it is possible to define a  function $f_1:\\Omega \\subset \\nset{R}^5\\to \\nset{R}^5$, that is,\n\n\n\\begin{eqnarray}\\label{eq:5-002}\nf_1(s):=\\begin{pmatrix}\n\\left[x\\right]_1-[x]_2-a_1\\cdot a_2\\left( 1-[x]_4 \\right)\\\\\n\\left[x\\right]_2-[x]_3-a_1\\cdot a_3 \\left( 1-[x]_4\\right)\\left(1-[x]_5\\right)\\\\\n\\left[x\\right]_3-a_4-a_1\\cdot a_5 \\left( 1-[x]_4\\right)\\left(1-[x]_5\\right)\\\\\n\\left[x\\right]_4 -a_6[x]_1-a_7\\\\\n\\left[x\\right]_5-(a_8-1)\\left(1-\\dfrac{[x]_3+a_9}{[x]_2+a_9} \\right)\\left(a_8+ \\dfrac{[x]_3+a_9}{[x]_2+a_9}\\right)^{-1}\n\\end{pmatrix}, \n\\end{eqnarray}\n\n\nwhich depends on two parameters, the direct normal irradiance ($DNI$) and the ambient temperature ($T_{air}$). These parameters are measured in real-time at certain times of the day \\cite{rodrigo2019performance}, and it is necessary to calculate a new solution of the system \\eqref{eq:5-001} for each new pair of parameters, that is,\n\n\n\\begin{eqnarray*}\n(DNI,T_{air})\\overset{f_1}{\\longrightarrow} s \\in \\nset{R}^5.\n\\end{eqnarray*}\n\n\n\nHowever, to simplify the task of finding the solutions of the function \\eqref{eq:5-002}, it is possible through the consecutive substitution of the variables $[x]_1, \\ [x]_4, \\ [x]_5$ and some algebraic simplifications, to obtain the following transcendental system \\cite{torres2020reduction}\n\n\n\\begin{eqnarray}\\label{eq:5-003}\n\\left\\{\n\\begin{array}{l}\n\\left[x\\right]_2=[x]_3-a_1\\cdot a_3 \\dfrac{\\left( a_6 [x]_2 + a_7 - 1 \\right) \\left( a_8 \\left([x]_3+ a_9 \\right) + \\left([x]_2+ a_9\\right)  \\right)  }{(1+a_1 a_2 a_6 ) \\left( a_8 \\left( [x]_2 + a_9  \\right) + \\left([x]_3+ a_9\\right) \\right) } \\vspace{0.1cm} \\\\\n\\left[x \\right]_3=a_4-a_1\\cdot a_5 \\dfrac{\\left( a_6 [x]_2 + a_7 - 1 \\right) \\left( a_8 \\left([x]_3+ a_9 \\right) + \\left([x]_2+ a_9\\right)  \\right)  }{(1+a_1 a_2 a_6 ) \\left( a_8 \\left( [x]_2 + a_9  \\right) + \\left([x]_3+ a_9\\right) \\right) } \n\\end{array}\\right.,\n\\end{eqnarray}\n\nwhose solution allows to know the values of the variables $[x]_1,[x]_4$ and $[x]_5$ through the following equations\n\n\\begin{eqnarray}\\label{eq:5-004}\n\\left\\{\n\\begin{array}{l}\n\\left[x\\right]_1=\\dfrac{[x]_2 - a_1 a_2 (a_7 - 1)}{1+a_1 a_2 a_6} \\vspace{0.1cm}\\\\\n\\left[ x \\right]_4=\\dfrac{a_6 \\left( a_1 a_2 + [x]_2\\right) + a_7}{1+ a_1 a_2 a_6 } \\vspace{0.1cm}\\\\\n\\left[ x \\right]_5=\\dfrac{(a_8 - 1)\\left( [x]_2 - [x]_3 \\right)}{a_8 \\left([x]_2+a_9\\right) + \\left( [x]_3+a_9 \\right)}\n\\end{array}\\right..\n\\end{eqnarray}\n\n\n\n\n\n\n\n\nUsing the system of equations \\eqref{eq:5-003}, it is possible to define a  function $f_2:\\Omega \\subset \\nset{R}^2\\to \\nset{R}^2$, that is,\n\n\n\\begin{eqnarray}\\label{eq:5-005}\nf_2(x):=\\begin{pmatrix}\n\\left[x\\right]_2-[x]_3+a_1\\cdot a_3 \\dfrac{\\left( a_6 [x]_2 + a_7 - 1 \\right) \\left( a_8 \\left([x]_3+ a_9 \\right) + \\left([x]_2+ a_9\\right)  \\right)  }{(1+a_1 a_2 a_6 ) \\left( a_8 \\left( [x]_2 + a_9  \\right) + \\left([x]_3+ a_9\\right) \\right) } \\vspace{0.1cm} \\\\\n\\left[x \\right]_3-a_4+a_1\\cdot a_5 \\dfrac{\\left( a_6 [x]_2 + a_7 - 1 \\right) \\left( a_8 \\left([x]_3+ a_9 \\right) + \\left([x]_2+ a_9\\right)  \\right)  }{(1+a_1 a_2 a_6 ) \\left( a_8 \\left( [x]_2 + a_9  \\right) + \\left([x]_3+ a_9\\right) \\right) } \n\\end{pmatrix}, \n\\end{eqnarray}\n\n\nand then finding the solutions of the function \\eqref{eq:5-005}, through the equations \\eqref{eq:5-004}, it is possible to construct the solutions of the function \\eqref{eq:5-002}.\n\n\n\n\\subsection{Solutions of the Equations of a Hybrid Solar Receiver}\n\nTo solve the equation \\eqref{eq:5-005} and at the same time solve the equation \\eqref{eq:5-002}, a fractional fixed point method will be used as well as its accelerated version through the function \\eqref{eq:4-005}. Before continuing, it is necessary to mention that for some definitions of fractional operators it is fulfilled that the derivative of order $\\alpha$ of a constant is different from zero (for example: Riesz, Gr\u00fcnwald\u2013Letnikov, Riemann-Liouville, etc.\\cite{miller93,hilfer00,oldham74,kilbas2006theory,brambila2017fractal}), that is, \n\n\n\\begin{eqnarray}\\label{eq:5-006}\n\\partial_k^\\alpha c :=\\der{\\partial}{[x]_k}{\\alpha}c \\neq 0 , & c=constant.\n\\end{eqnarray}\n\n\nSo, considering a function $f:\\Omega \\subset \\nset{R}^n \\to \\nset{R}^n$ with a point $\\xi \\in \\Omega$ such that $\\norm{f(\\xi)}=0$, the Riemann-Liouville fractional derivative given by the equation \\eqref{eq:3-002}, and an iteration function $\\Phi:(\\nset{R}\\setminus \\nset{Z})\\times \\nset{R}^n \\to \\nset{R}^n$, it is possible to define the following fractional fixed point method\n\n\\begin{eqnarray}\\label{eq:5-007}\nx_{i+1}:=\\Phi(\\alpha,x_i)=x_i-A_{g_f,\\beta}(x_i)f(x_i), & i=0,1,2,\\cdots,\n\\end{eqnarray}\n\nwhere $ A_{g_f,\\beta}(x_i) $ is given by the following expression\n\n\\begin{eqnarray}\\label{eq:5-008}\nA_{g_f,\\beta}(x_i)=\\left([A_{g_f,\\beta}]_{jk}(x_i) \\right) :=\\left( \\partial_k^{\\beta(\\alpha,[x]_k)}[g_f]_{j}(x)  \\right)_{x_i}^{-1}, & \\alpha\\in \\nset{R}\\setminus \\nset{Z},\n\\end{eqnarray}\n\nwith $ g_{f}(x) $  and $ \\beta (\\alpha, [x]_k) $ functions defined as follows\n\n\n\\begin{eqnarray}\ng_{f}(x):=f(x_i)+f^{(1)}(x_i)x& \\mbox{ and }\n&\n\\beta(\\alpha,[x]_k):=\\left\\{\n\\begin{array}{cc}\n\\alpha, &\\mbox{if \\hspace{0.1cm} } \\abs{  [x]_k }\\neq 0 \\vspace{0.1cm}\\\\\n1,& \\mbox{if \\hspace{0.1cm}  }  \\abs{  [x]_k  }=0\n\\end{array}\\right..\n\\end{eqnarray}\n\n\n\nThe fractional iterative method given by the equation \\eqref{eq:5-007} is named the \\textbf{fractional quasi-Newton method} \\cite{torres2020fracsome}, if it is assumed that $\\Phi \\in \\Conv(\\xi)$ then $\\Phi \\in \\ord^1(\\xi)$. Furthermore, the method fulfills the following condition\n\n\n\\begin{eqnarray}\n\\lim_{\\alpha \\to 1}\\partial_k^{\\beta(\\alpha,[x_i]_k)} [g]_j(x_i)=\\partial_k[f]_j(x_i), & 1\\leq j,k\\leq n,\n\\end{eqnarray}\n\n\nand as a consequence $\\Phi \\in \\Ord_{2}^1(\\xi)$. So, if it is assumed that $f\\in \\Ord_{NR}^2(\\xi)$, by the \\mref{Corollary}{\\ref{cor:4-001}} it is possible to construct the \\textbf{fractional quasi-Newton method accelerated} using the following matrix\n\n\n\\begin{eqnarray}\\label{eq:5-009}\nA_{g_f,\\alpha_f}(x_i)=\\left([A_{g_f,\\alpha_f}]_{jk}(x_i) \\right) :=\\left( \\partial_k^{\\alpha_f([x]_k,x)}[g_f]_{j}(x)  \\right)_{x_i}^{-1}, & \\alpha\\in \\nset{R}\\setminus \\nset{Z}.\n\\end{eqnarray}\n\n\n\nBefore continuing, it is necessary to mention that a description of the algorithm that must be implemented when working with a fractional iterative method given by the equation \\eqref{eq:4-002} may be found in the reference \\cite{torres2020fracsome}. On the other hand, simplified examples of how the methods given by the matrices \\eqref{eq:5-008} and \\eqref{eq:5-009} should be programmed may be found in the references \\cite{torres2021codefracquasi,torres2021codeaccelfracquasi}. Using the fractional fixed point methods defined by the matrices \\eqref{eq:5-008} and \\eqref{eq:5-009}, we proceed to find three solutions of the function \\eqref{eq:5-005}  leaving the following fixed values\n\n\n\\begin{eqnarray*}\n\\delta=13& \\mbox{ and } & x_0=(3000,3000)^T.\n\\end{eqnarray*}\n\n\n\n\n\\begin{example}\n\nConsidering by hypothesis that $f_2\\in \\Ord_{NR}^2(\\xi)$, and using the following values\n\n\n\\begin{eqnarray*}\nDNI=900, &\nTair=20, & \\alpha=0.89825,\n\\end{eqnarray*}\n\n\nthe following iterations are obtained by using the fractional iterative methods given by the matrices  \\eqref{eq:5-008} and \\eqref{eq:5-009}.\n\n\n\\begin{itemize}\n\\item[i)] $A_{g_f,\\beta} \\Rightarrow \\Phi \\in \\Ord_2^1(\\xi)$.\n\n\n\\begin{footnotesize}\n\\centering\n\\begin{longtable}{c|cccc|cccccccc}\n\\toprule\n$i$& $[ x_i ]_2$ &$[ x_i ]_3$ &$\\norm{x_i-x_{i-1}}_2$&$\\norm{f_2(x_i)}_2$&$[ x_i ]_1$&$[ x_i ]_4$&$[ x_i ]_5$& $\\norm{f_1(s_i)}_2$ \\\\\n\\midrule\n    1     & 2048.526273 & 2036.688326 & 1.35E+03 & 2.01E+03 & 2052.245932 & 0.02901075 & 0.00087668 & 2.01E+03 \\\\\n    2     & 1378.380727 & 1357.837031 & 9.54E+02 & 1.33E+03 & 1381.592211 & 0.16166606 & 0.00214528 & 1.33E+03 \\\\\n    3     & 914.5756647 & 887.7554749 & 6.60E+02 & 8.65E+02 & 917.4354426 & 0.25347627 & 0.00391089 & 8.65E+02 \\\\\n    4     & 599.7868499 & 568.5654338 & 4.48E+02 & 5.46E+02 & 602.4079218 & 0.31578871 & 0.00622874 & 5.46E+02 \\\\\n    5     & 390.7721777 & 356.5990844 & 2.98E+02 & 3.34E+02 & 393.2347526 & 0.35716317 & 0.0090235 & 3.34E+02 \\\\\n    6     & 255.3927888 & 219.4044444 & 1.93E+02 & 1.97E+02 & 257.7527048 & 0.38396151 & 0.0120214 & 1.97E+02 \\\\\n    7     & 170.1536777 & 133.2761535 & 1.21E+02 & 1.11E+02 & 172.4489564 & 0.4008346 & 0.01478215 & 1.11E+02 \\\\\n    8     & 118.188164 & 81.23045449 & 7.35E+01 & 5.95E+01 & 120.4440369 & 0.41112117 & 0.01686287 & 5.95E+01 \\\\\n    9     & 87.62585188 & 51.33933793 & 4.27E+01 & 2.99E+01 & 89.85854925 & 0.41717098 & 0.01800683 & 2.99E+01 \\\\\n    10    & 70.31181026 & 35.35034092 & 2.36E+01 & 1.43E+01 & 72.5313783 & 0.42059829 & 0.01823343 & 1.43E+01 \\\\\n    11    & 60.85889363 & 27.58761689 & 1.22E+01 & 6.66E+00 & 63.07129347 & 0.4224695 & 0.01782623 & 6.66E+00 \\\\\n    12    & 55.92035933 & 24.22121073 & 5.98E+00 & 3.07E+00 & 58.12901425 & 0.42344708 & 0.01721438 & 3.07E+00 \\\\\n    13    & 53.49709311 & 22.89305436 & 2.76E+00 & 1.38E+00 & 55.70391046 & 0.42392677 & 0.01672394 & 1.38E+00 \\\\\n    14    & 52.38726485 & 22.39252245 & 1.22E+00 & 6.02E-01 & 54.59324061 & 0.42414646 & 0.01643587 & 6.02E-01 \\\\\n    15    & 51.90534374 & 22.20463447 & 5.17E-01 & 2.55E-01 & 54.11095406 & 0.42424185 & 0.01629349 & 2.55E-01 \\\\\n    16    & 51.70286627 & 22.13313077 & 2.15E-01 & 1.06E-01 & 53.90832305 & 0.42428193 & 0.01622933 & 1.06E-01 \\\\\n    17    & 51.61937072 & 22.10548244 & 8.80E-02 & 4.32E-02 & 53.82476418 & 0.42429846 & 0.01620181 & 4.32E-02 \\\\\n    18    & 51.58529753 & 22.09465371 & 3.58E-02 & 1.76E-02 & 53.79066515 & 0.42430521 & 0.01619031 & 1.76E-02 \\\\\n    19    & 51.5714752 & 22.09037372 & 1.45E-02 & 7.12E-03 & 53.77683234 & 0.42430794 & 0.01618559 & 7.12E-03 \\\\\n    20    & 51.56588734 & 22.0886717 & 5.84E-03 & 2.87E-03 & 53.77124024 & 0.42430905 & 0.01618366 & 2.87E-03 \\\\\n    21    & 51.56363304 & 22.08799215 & 2.35E-03 & 1.16E-03 & 53.76898424 & 0.42430949 & 0.01618288 & 1.16E-03 \\\\\n    22    & 51.56272473 & 22.08772013 & 9.48E-04 & 4.67E-04 & 53.76807524 & 0.42430967 & 0.01618256 & 4.67E-04 \\\\\n    23    & 51.56235904 & 22.08761106 & 3.82E-04 & 1.88E-04 & 53.76770927 & 0.42430975 & 0.01618243 & 1.88E-04 \\\\\n    24    & 51.56221188 & 22.08756728 & 1.54E-04 & 7.56E-05 & 53.767562 & 0.42430978 & 0.01618238 & 7.58E-05 \\\\\n    25    & 51.56215268 & 22.0875497 & 6.18E-05 & 3.04E-05 & 53.76750275 & 0.42430979 & 0.01618236 & 3.05E-05 \\\\\n    26    & 51.56212886 & 22.08754263 & 2.48E-05 & 1.22E-05 & 53.76747891 & 0.42430979 & 0.01618235 & 1.20E-05 \\\\\n    27    & 51.56211928 & 22.08753979 & 9.99E-06 & 4.92E-06 & 53.76746933 & 0.42430979 & 0.01618235 & 4.72E-06 \\\\\n \\bottomrule\n\\caption{Iterations generated by the fractional quasi-Newton method.}\n\\end{longtable}\n\\end{footnotesize}\n\n\\item[ii)] $A_{g_f,\\alpha_f} \\Rightarrow \\Phi \\in \\Ord_2^2(\\xi)$ in $\\Omega_\\xi \\subset B(\\xi;\\delta)$.\n\n\n\n\\begin{footnotesize}\n\\centering\n\\begin{longtable}{c|cccc|cccccccc}\n\\toprule\n$i$& $[ x_i ]_2$ &$[ x_i ]_3$ &$\\norm{x_i-x_{i-1}}_2$&$\\norm{f_2(x_i)}_2$&$[ x_i ]_1$&$[ x_i ]_4$&$[ x_i ]_5$& $\\norm{f_1(s_i)}_2$ \\\\\n\\midrule\n    1     & 2048.526273 & 2036.688326 & 1.35E+03 & 2.01E+03 & 2052.245932 & 0.02901075 & 0.00087668 & 2.01E+03 \\\\\n    2     & 1378.380727 & 1357.837031 & 9.54E+02 & 1.34E+03 & 1381.592211 & 0.16166606 & 0.00214528 & 1.34E+03 \\\\\n    3     & 914.5756647 & 887.7554749 & 6.60E+02 & 8.65E+02 & 917.4354426 & 0.25347627 & 0.00391089 & 8.65E+02 \\\\\n    4     & 599.7868499 & 568.5654338 & 4.48E+02 & 5.46E+02 & 602.4079218 & 0.31578871 & 0.00622874 & 5.46E+02 \\\\\n    5     & 390.7721777 & 356.5990844 & 2.98E+02 & 3.34E+02 & 393.2347526 & 0.35716317 & 0.0090235 & 3.34E+02 \\\\\n    6     & 255.3927888 & 219.4044444 & 1.93E+02 & 1.97E+02 & 257.7527048 & 0.38396151 & 0.0120214 & 1.97E+02 \\\\\n    7     & 170.1536777 & 133.2761535 & 1.21E+02 & 1.11E+02 & 172.4489564 & 0.4008346 & 0.01478215 & 1.11E+02 \\\\\n    8     & 118.188164 & 81.23045449 & 7.36E+01 & 5.95E+01 & 120.4440369 & 0.41112117 & 0.01686287 & 5.95E+01 \\\\\n    9     & 87.62585188 & 51.33933793 & 4.28E+01 & 2.99E+01 & 89.85854925 & 0.41717098 & 0.01800683 & 2.99E+01 \\\\\n    10    & 70.31181026 & 35.35034092 & 2.36E+01 & 1.43E+01 & 72.5313783 & 0.42059829 & 0.01823343 & 1.43E+01 \\\\\n    11    & 60.85889363 & 27.58761689 & 1.22E+01 & 6.66E+00 & 63.07129347 & 0.4224695 & 0.01782623 & 6.66E+00 \\\\\n    12    & 51.56100988 & 22.08746493 & 1.08E+01 & 1.04E-03 & 53.76635909 & 0.42431001 & 0.01618182 & 1.04E-03 \\\\\n    13    & 51.56211284 & 22.08753788 & 1.11E-03 & 4.13E-09 & 53.76746288 & 0.4243098 & 0.01618235 & 3.03E-07 \\\\\n    \\bottomrule\n\\caption{Iterations generated by the fractional quasi-Newton method accelerated.}\n\\end{longtable}\n\\end{footnotesize}\n\n\\end{itemize}\n\n\\end{example}\n\n\n\n\n\n\\begin{example}\n\nConsidering by hypothesis that $f_2\\in \\Ord_{NR}^2(\\xi)$, and using the following values\n\n\\begin{eqnarray*}\nDNI=574.319,&\nTair=16.832,& \\alpha=0.8996,\n\\end{eqnarray*}\n\nthe following iterations are obtained by using the fractional iterative methods given by the matrices  \\eqref{eq:5-008} and \\eqref{eq:5-009}.\n\n\\begin{itemize}\n\\item[i)] $A_{g_f,\\beta} \\Rightarrow \\Phi \\in \\Ord_2^1(\\xi)$.\n\n\n\n\\begin{footnotesize}\n\\centering\n\\begin{longtable}{c|cccc|cccccccc}\n\\toprule\n$i$& $[ x_i ]_2$ &$[ x_i ]_3$ &$\\norm{x_i-x_{i-1}}_2$&$\\norm{f_2(x_i)}_2$&$[ x_i ]_1$&$[ x_i ]_4$&$[ x_i ]_5$& $\\norm{f_1(s_i)}_2$ \\\\\n\\midrule\n    1     & 2029.854772 & 2022.247443 & 1.38E+03 & 2.00E+03 & 2032.218723 & 0.03297214 & 0.00056752 & 2.00E+03 \\\\\n    2     & 1351.035349 & 1337.861649 & 9.64E+02 & 1.32E+03 & 1353.07091 & 0.16730757 & 0.00139631 & 1.32E+03 \\\\\n    3     & 884.5286725 & 867.3584839 & 6.63E+02 & 8.49E+02 & 886.3385526 & 0.25962723 & 0.00256042 & 8.49E+02 \\\\\n    4     & 570.3098992 & 550.3428213 & 4.46E+02 & 5.32E+02 & 571.9677708 & 0.32180977 & 0.00410184 & 5.32E+02 \\\\\n    5     & 363.4003476 & 341.5519319 & 2.94E+02 & 3.23E+02 & 364.9581233 & 0.36275628 & 0.00597385 & 3.23E+02 \\\\\n    6     & 230.608119 & 207.585416 & 1.89E+02 & 1.89E+02 & 232.1016543 & 0.38903529 & 0.00799251 & 1.89E+02 \\\\\n    7     & 147.8561494 & 124.2274595 & 1.18E+02 & 1.06E+02 & 149.3096521 & 0.40541155 & 0.0098586 & 1.06E+02 \\\\\n    8     & 98.01126796 & 74.27302588 & 7.06E+01 & 5.63E+01 & 99.44065736 & 0.41527564 & 0.01127184 & 5.63E+01 \\\\\n    9     & 69.13937735 & 45.768905 & 4.06E+01 & 2.79E+01 & 70.5547995 & 0.42098926 & 0.01205541 & 2.79E+01 \\\\\n    10    & 53.13057813 & 30.57994962 & 2.21E+01 & 1.29E+01 & 54.53825576 & 0.42415733 & 0.01220761 & 1.29E+01 \\\\\n    11    & 44.6597286 & 23.22992376 & 1.12E+01 & 5.69E+00 & 46.06330831 & 0.42583368 & 0.01190151 & 5.69E+00 \\\\\n    12    & 40.41192388 & 20.07409231 & 5.29E+00 & 2.44E+00 & 41.81344865 & 0.4266743 & 0.01143556 & 2.44E+00 \\\\\n    13    & 38.42463196 & 18.85806286 & 2.33E+00 & 1.02E+00 & 39.82519534 & 0.42706758 & 0.01106236 & 1.02E+00 \\\\\n    14    & 37.56159752 & 18.41580876 & 9.70E-01 & 4.16E-01 & 38.9617434 & 0.42723837 & 0.01084908 & 4.16E-01 \\\\\n    15    & 37.20752766 & 18.25657936 & 3.88E-01 & 1.65E-01 & 38.60750225 & 0.42730844 & 0.01074838 & 1.65E-01 \\\\\n    16    & 37.06714943 & 18.19864095 & 1.52E-01 & 6.44E-02 & 38.46705611 & 0.42733622 & 0.01070538 & 6.44E-02 \\\\\n    17    & 37.01251861 & 18.17727243 & 5.87E-02 & 2.49E-02 & 38.41239886 & 0.42734703 & 0.01068795 & 2.49E-02 \\\\\n    18    & 36.99146859 & 18.16930635 & 2.25E-02 & 9.54E-03 & 38.39133866 & 0.42735119 & 0.01068108 & 9.54E-03 \\\\\n    19    & 36.98340172 & 18.16631458 & 8.60E-03 & 3.65E-03 & 38.38326789 & 0.42735279 & 0.01067841 & 3.65E-03 \\\\\n    20    & 36.98031974 & 18.16518558 & 3.28E-03 & 1.39E-03 & 38.38018441 & 0.4273534 & 0.01067738 & 1.39E-03 \\\\\n    21    & 36.97914434 & 18.16475823 & 1.25E-03 & 5.31E-04 & 38.37900845 & 0.42735363 & 0.01067699 & 5.30E-04 \\\\\n    22    & 36.97869653 & 18.16459616 & 4.76E-04 & 2.02E-04 & 38.37856042 & 0.42735372 & 0.01067684 & 2.02E-04 \\\\\n    23    & 36.97852602 & 18.16453462 & 1.81E-04 & 7.69E-05 & 38.37838983 & 0.42735375 & 0.01067678 & 7.67E-05 \\\\\n    24    & 36.97846112 & 18.16451124 & 6.90E-05 & 2.93E-05 & 38.3783249 & 0.42735377 & 0.01067676 & 2.93E-05 \\\\\n    25    & 36.97843643 & 18.16450235 & 2.62E-05 & 1.11E-05 & 38.37830019 & 0.42735377 & 0.01067675 & 1.10E-05 \\\\\n    26    & 36.97842703 & 18.16449898 & 9.99E-06 & 4.23E-06 & 38.37829079 & 0.42735377 & 0.01067675 & 4.12E-06 \\\\ \\bottomrule\n\\caption{Iterations generated by the fractional quasi-Newton method.}\n\\end{longtable}\n\\end{footnotesize}\n\n\\item[ii)] $A_{g_f,\\alpha_f} \\Rightarrow \\Phi \\in \\Ord_2^2(\\xi)$ in $\\Omega_\\xi \\subset B(\\xi;\\delta)$.\n\n\n\\begin{footnotesize}\n\\centering\n\\begin{longtable}{c|cccc|cccccccc}\n\\toprule\n$i$& $[ x_i ]_2$ &$[ x_i ]_3$ &$\\norm{x_i-x_{i-1}}_2$&$\\norm{f_2(x_i)}_2$&$[ x_i ]_1$&$[ x_i ]_4$&$[ x_i ]_5$& $\\norm{f_1(s_i)}_2$ \\\\\n\\midrule\n    1     & 2029.854772 & 2022.247443 & 1.38E+03 & 2.00E+03 & 2032.218723 & 0.03297214 & 0.00056752 & 2.00E+03 \\\\\n    2     & 1351.035349 & 1337.861649 & 9.64E+02 & 1.32E+03 & 1353.07091 & 0.16730757 & 0.00139631 & 1.32E+03 \\\\\n    3     & 884.5286725 & 867.3584839 & 6.63E+02 & 8.49E+02 & 886.3385526 & 0.25962723 & 0.00256042 & 8.49E+02 \\\\\n    4     & 570.3098992 & 550.3428213 & 4.46E+02 & 5.32E+02 & 571.9677708 & 0.32180977 & 0.00410184 & 5.32E+02 \\\\\n    5     & 363.4003476 & 341.5519319 & 2.94E+02 & 3.23E+02 & 364.9581233 & 0.36275628 & 0.00597385 & 3.23E+02 \\\\\n    6     & 230.608119 & 207.585416 & 1.89E+02 & 1.89E+02 & 232.1016543 & 0.38903529 & 0.00799251 & 1.89E+02 \\\\\n    7     & 147.8561494 & 124.2274595 & 1.18E+02 & 1.06E+02 & 149.3096521 & 0.40541155 & 0.0098586 & 1.06E+02 \\\\\n    8     & 98.01126796 & 74.27302588 & 7.06E+01 & 5.63E+01 & 99.44065736 & 0.41527564 & 0.01127184 & 5.63E+01 \\\\\n    9     & 69.13937735 & 45.768905 & 4.06E+01 & 2.79E+01 & 70.5547995 & 0.42098926 & 0.01205541 & 2.79E+01 \\\\\n    10    & 53.13057813 & 30.57994962 & 2.21E+01 & 1.29E+01 & 54.53825576 & 0.42415733 & 0.01220761 & 1.29E+01 \\\\\n    11    & 36.97715447 & 18.16441312 & 2.04E+01 & 1.19E-03 & 38.37701761 & 0.42735403 & 0.0106761 & 1.19E-03 \\\\\n    12    & 36.97842127 & 18.1644969 & 1.27E-03 & 7.75E-09 & 38.37828503 & 0.42735378 & 0.01067675 & 2.15E-07 \\\\\n     \\bottomrule\n\\caption{Iterations generated by the fractional quasi-Newton method accelerated.}\n\\end{longtable}\n\\end{footnotesize}\n\n\\end{itemize}\n\\end{example}\n\n\n\\begin{example}\nConsidering by hypothesis that $f_2\\in \\Ord_{NR}^2(\\xi)$, and using the following values\n\n\\begin{eqnarray*}\nDNI=94.3555,&\nTair=28.373,& \\alpha=0.89964,\n\\end{eqnarray*}\n\n\nthe following iterations are obtained by using the fractional iterative methods given by the matrices  \\eqref{eq:5-008} and \\eqref{eq:5-009}.\n\n\\begin{itemize}\n\\item[i)] $A_{g_f,\\beta} \\Rightarrow \\Phi \\in \\Ord_2^1(\\xi)$.\n\n\n\\begin{footnotesize}\n\\centering\n\\begin{longtable}{c|cccc|cccccccc}\n\\toprule\n$i$& $[ x_i ]_2$ &$[ x_i ]_3$ &$\\norm{x_i-x_{i-1}}_2$&$\\norm{f_2(x_i)}_2$&$[ x_i ]_1$&$[ x_i ]_4$&$[ x_i ]_5$& $\\norm{f_1(s_i)}_2$ \\\\\n\\midrule\n    1     & 2026.948258 & 2025.698601 & 1.38E+03 & 2.00E+03 & 2027.336246 & 0.03393789 & 0.00009324 & 2.00E+03 \\\\\n    2     & 1346.157858 & 1343.993285 & 9.63E+02 & 1.32E+03 & 1346.49176 & 0.16860893 & 0.00022947 & 1.32E+03 \\\\\n    3     & 878.348027 & 875.5257474 & 6.62E+02 & 8.47E+02 & 878.644763 & 0.26114907 & 0.00042095 & 8.47E+02 \\\\\n    4     & 563.2981664 & 560.0145307 & 4.46E+02 & 5.31E+02 & 563.5698729 & 0.32347088 & 0.00067464 & 5.31E+02 \\\\\n    5     & 355.8897756 & 352.2947261 & 2.94E+02 & 3.24E+02 & 356.1450042 & 0.36449952 & 0.00098289 & 3.24E+02 \\\\\n    6     & 222.8349909 & 219.0449879 & 1.88E+02 & 1.90E+02 & 223.0796488 & 0.39081985 & 0.00131521 & 1.90E+02 \\\\\n    7     & 139.9975258 & 136.1076542 & 1.17E+02 & 1.08E+02 & 140.2356025 & 0.4072064 & 0.00162172 & 1.08E+02 \\\\\n    8     & 90.22025428 & 86.31555624 & 7.04E+01 & 5.77E+01 & 90.45437636 & 0.41705312 & 0.00185193 & 5.77E+01 \\\\\n    9     & 61.57917913 & 57.74236258 & 4.05E+01 & 2.92E+01 & 61.81102578 & 0.42271878 & 0.00197603 & 2.92E+01 \\\\\n    10    & 45.9860483 & 42.29258563 & 2.20E+01 & 1.37E+01 & 46.21665614 & 0.42580335 & 0.00199523 & 1.37E+01 \\\\\n    11    & 38.07727079 & 34.56957756 & 1.11E+01 & 5.99E+00 & 38.3072503 & 0.42736783 & 0.00194279 & 5.99E+00 \\\\\n    12    & 34.38320509 & 31.04574209 & 5.11E+00 & 2.46E+00 & 34.61289112 & 0.42809857 & 0.00187038 & 2.46E+00 \\\\\n    13    & 32.785517 & 29.5629105 & 2.18E+00 & 9.76E-01 & 33.0150761 & 0.42841462 & 0.0018152 & 9.76E-01 \\\\\n    14    & 32.13122049 & 28.97016965 & 8.83E-01 & 3.80E-01 & 32.36072761 & 0.42854405 & 0.00178421 & 3.80E-01 \\\\\n    15    & 31.87136629 & 28.7388988 & 3.48E-01 & 1.47E-01 & 32.10085277 & 0.42859545 & 0.00176952 & 1.47E-01 \\\\\n    16    & 31.7697037 & 28.64948568 & 1.35E-01 & 5.67E-02 & 31.9991821 & 0.42861556 & 0.00176316 & 5.67E-02 \\\\\n    17    & 31.73020749 & 28.61501378 & 5.24E-02 & 2.19E-02 & 31.95968275 & 0.42862337 & 0.00176054 & 2.19E-02 \\\\\n    18    & 31.71491342 & 28.60173066 & 2.03E-02 & 8.43E-03 & 31.94438747 & 0.4286264 & 0.00175949 & 8.43E-03 \\\\\n    19    & 31.70900063 & 28.59661144 & 7.82E-03 & 3.25E-03 & 31.93847421 & 0.42862757 & 0.00175907 & 3.25E-03 \\\\\n    20    & 31.70671661 & 28.59463794 & 3.02E-03 & 1.25E-03 & 31.93619001 & 0.42862802 & 0.00175891 & 1.25E-03 \\\\\n    21    & 31.70583474 & 28.59387694 & 1.17E-03 & 4.84E-04 & 31.93530807 & 0.4286282 & 0.00175885 & 4.84E-04 \\\\\n    22    & 31.70549433 & 28.59358344 & 4.50E-04 & 1.87E-04 & 31.93496763 & 0.42862826 & 0.00175882 & 1.87E-04 \\\\\n    23    & 31.70536295 & 28.59347022 & 1.73E-04 & 7.20E-05 & 31.93483624 & 0.42862829 & 0.00175881 & 7.20E-05 \\\\\n    24    & 31.70531225 & 28.59342654 & 6.69E-05 & 2.78E-05 & 31.93478553 & 0.4286283 & 0.00175881 & 2.78E-05 \\\\\n    25    & 31.70529268 & 28.59340969 & 2.58E-05 & 1.07E-05 & 31.93476596 & 0.4286283 & 0.00175881 & 1.07E-05 \\\\\n    26    & 31.70528513 & 28.59340319 & 9.96E-06 & 4.13E-06 & 31.93475841 & 0.4286283 & 0.00175881 & 4.13E-06 \\\\\n    \\bottomrule\n\\caption{Iterations generated by the fractional quasi-Newton method.}\n\\end{longtable}\n\\end{footnotesize}\n\n\\item[ii)] $A_{g_f,\\alpha_f} \\Rightarrow \\Phi \\in \\Ord_2^2(\\xi)$ in $\\Omega_\\xi \\subset B(\\xi;\\delta)$.\n\n\\begin{footnotesize}\n\\centering\n\\begin{longtable}{c|cccc|cccccccc}\n\\toprule\n$i$& $[ x_i ]_2$ &$[ x_i ]_3$ &$\\norm{x_i-x_{i-1}}_2$&$\\norm{f_2(x_i)}_2$&$[ x_i ]_1$&$[ x_i ]_4$&$[ x_i ]_5$& $\\norm{f_1(s_i)}_2$ \\\\\n\\midrule\n    1     & 2026.94826 & 2025.6986 & 1.38E+03 & 2.00E+03 & 2027.33625 & 0.03393789 & 0.0000932 & 2.00E+03 \\\\\n    2     & 1346.15786 & 1343.99329 & 9.63E+02 & 1.32E+03 & 1346.49176 & 0.16860893 & 0.00022947 & 1.32E+03 \\\\\n    3     & 878.348027 & 875.525747 & 6.62E+02 & 8.47E+02 & 878.644763 & 0.26114907 & 0.00042095 & 8.47E+02 \\\\\n    4     & 563.298166 & 560.014531 & 4.46E+02 & 5.31E+02 & 563.569873 & 0.32347088 & 0.00067464 & 5.31E+02 \\\\\n    5     & 355.889776 & 352.294726 & 2.94E+02 & 3.24E+02 & 356.145004 & 0.36449952 & 0.00098289 & 3.24E+02 \\\\\n    6     & 222.834991 & 219.044988 & 1.88E+02 & 1.90E+02 & 223.079649 & 0.39081985 & 0.00131521 & 1.90E+02 \\\\\n    7     & 139.997526 & 136.107654 & 1.17E+02 & 1.08E+02 & 140.235603 & 0.4072064 & 0.00162172 & 1.08E+02 \\\\\n    8     & 90.2202543 & 86.3155562 & 7.04E+01 & 5.77E+01 & 90.4543764 & 0.41705312 & 0.00185193 & 5.77E+01 \\\\\n    9     & 61.5791791 & 57.7423626 & 4.05E+01 & 2.92E+01 & 61.8110258 & 0.42271878 & 0.00197603 & 2.92E+01 \\\\\n    10    & 45.9860483 & 42.2925856 & 2.20E+01 & 1.37E+01 & 46.2166561 & 0.42580335 & 0.00199523 & 1.37E+01 \\\\\n    11    & 38.0772708 & 34.5695776 & 1.11E+01 & 5.99E+00 & 38.3072503 & 0.42736783 & 0.00194279 & 5.99E+00 \\\\\n    12    & 31.7052694 & 28.5933984 & 8.74E+00 & 1.03E-05 & 31.9347427 & 0.42862831 & 0.0017588 & 1.03E-05 \\\\\n    13    & 31.7052804 & 28.5933991 & 1.10E-05 & 2.80E-09 & 31.9347537 & 0.42862831 & 0.00175881 & 3.17E-08 \\\\\n     \\bottomrule\n\\caption{Iterations generated by the fractional quasi-Newton method accelerated.}\n\\end{longtable}\n\\end{footnotesize}\n\n\\end{itemize}\n\\end{example}\n\n\n\n\n\nFrom the previous results it is observed that there is a considerable improvement in the order of convergence between the matrices \\eqref{eq:5-008} and \\eqref{eq:5-009}. Therefore, it may be established that it is more efficient to solve the function \\eqref{eq:5-002} by implementing the fractional quasi-Newton method accelerated in the function \\eqref{eq:5-005}. So, by providing multiple values of the parameters $DNI$ and $T_{air}$, it is possible to obtain a histogram of the efficiencies of a hybrid solar receiver analogous to the one shown in the Figure \\ref{fig:02}. Finally, it is necessary to mention that the \\mref{Corollary}{\\ref{cor:4-001}} can also be implemented in the \\textbf{generalized fractional quasi-Newton method}, which is obtained by using the matrix \\eqref{eq:5-008} with the following function\n\n\\begin{eqnarray}\ng_{a,b,f}(x):=af(x_i)+f^{(1)}(x_i)(x-bx_i), & a,b\\in \\nset{R},\n\\end{eqnarray}\n\n\nas a consequence, denoting by $c$ an arbitrary constant, it is possible to define the following set of matrices\n\n\n\n\\begin{eqnarray}\n\\set{A_{g,\\alpha} \\ : \\    A_{g,\\alpha} \\in  {}_n\\IMa_{x,\\alpha}^{1}(g) }\\cap\\set{A_{g,\\alpha} \\ : \\  \\left([A_{g,\\alpha}]_{jk}(x) \\right) :=\\left( o_k^{\\alpha}[g]_{j}(x)  \\right)^{-1} }\\cap \\set{o^\\alpha \\ : \\ o_k^{\\alpha} c  \\neq 0 \\ \\forall k\\geq 1 } ,\n\\end{eqnarray}\n\n\n\n\nand therefore, it is possible to define the following sets of fractional iterative methods\n\n\n\n\n\\begin{eqnarray}\n\\set{\\Phi \\ : \\ A_{g,\\beta}  \\mbox{ uses }  g=g_{a,0,f} \\mbox{ with }  0<a\\leq 1}, \\vspace{0.1cm}\\\\\n\\set{\\Phi \\ : \\ A_{g,\\beta} \\mbox{ uses }  g=g_{1,b,f} \\mbox{ with } 0< b \\leq 1},\n\\end{eqnarray}\n\nwhich correspond to two uncountable families of fractional fixed point methods in which the  \\mref{Corollary}{\\ref{cor:4-001}} can be implemented.\n\n\n\n\n\n\\section{Conclusions}\n\n\nIn all the examples shown, a decrease in the number of iterations necessary to converge to the solutions is observed when implementing the function \\eqref{eq:4-005} in the fractional quasi-Newton method, which means that the generated sequences show an acceleration in their speed of convergence, which was to be expected given the \\mref{Corollary}{\\ref{cor:4-001}}. The fractional iterative methods, such as the fractional Newton-Raphson method, can find multiple zeros of a function using a single initial condition, this partially solves the intrinsic problem of classical iterative methods, which is that in general, to find $N$ zeros of a function, $N$ initial conditions must be provided. Due to the fractional operators implemented, these methods can be considered \\textbf{non-local parametric iterative methods}, so they have two important characteristics: \n\n\n\n\n\\begin{itemize}\n\\item[i)] The initial condition does not necessarily have to be close to the sought values due to the non-local nature of fractional operators \\cite{torres2020nonlinear}. \n\n\n\\item[ii)] When working in a space of $N$ dimensions, in the case that it is necessary to change the initial condition, unlike the classic iterative methods where in the worst case it is necessary to vary the $N$ components of the initial condition until obtaining a suitable value, in the fractional fixed point methods it is enough to vary the parameter $\\alpha$ of the fractional operators until found an adequate value that allows generating a sequence that converges to a sought value \\cite{torres2020fracsome}.\n\\end{itemize}\n\n\n\nThe above features make fractional iterative methods an ideal numerical tool for working with systems of nonlinear algebraic equations that vary with time-dependent parameters, as is the case of the functions \\eqref{eq:5-002} and \\eqref{eq:5-005}, which allows studying the behavior of temperatures and efficiencies of a hybrid solar receiver \\cite{de2021fractional,torres2020reduction,torres2020fracpseunew}. \nDue to many nonlinear algebraic systems related to engineering and physics are often related to time-dependent parameters. Having a way of classifying and accelerating the order of convergence of fractional fixed point methods through the \\mref{Corollary}{\\ref{cor:4-001}} may become a fundamental piece to continue expanding the applications of the fractional operators.\n\n\n\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.6\\textwidth, height=0.37\\textwidth]{hist}\n\\caption{Histogram and density curve of the efficiency of a hybrid solar receiver obtained from a simulation corresponding to a period of thirty days, which is equivalent to $2410$ pairs of parameters ($DNI,T_{air}$) randomly generated on the domain $[12,958]\\times [11,45]$. The selected domain is based on data measured in real-time at the Center for Advanced Studies in Energy and Environment (CEAEMA) \\cite{rodrigo2019performance,de2021fractional}. The values generated for the simulation presented the mean values $mean(DNI)=662.35$ and $mean(T_{air})=31.28$ with sample standard deviations $std(DNI)=257.83$ and $std(T_{air})=6.11\n$, while the values of the efficiencies were obtained through the solutions of the function \\eqref{eq:5-005} using the fractional quasi-Newton method accelerated.}\\label{fig:02}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{introduction} \nOn August 17, 2017, LIGO-Virgo Collaboration detected a GW signal from a binary neutron star (NS) merger event (GW170817). The total gravitational mass of the system at infinite binary separation is $2.74^{+0.04}_{-0.01}M_{\\odot}$, with a mass ratio in the range of $(0.7-1)$ \\citep{abbott17a}. Limited by the sensitivity of current GW detectors at high frequencies, the GW signals from the oscillations of the post-merger product are undetectable, leaving the merger product unidentified \\citep{abbott17b,abbott19}. \n\nTheoretically, an NS-NS merger can in principle give a variety of post-merger products depending on the remnant (gravitational) mass ($M_{\\rm rem}$, which depends on its spin)\\footnote{Throughout the paper, without specification, masses refer to gravitational masses. The baryonic masses are denoted as $M_b$. For the same $M_b$, the gravitational mass can adopt a range of values, which depend on the spin state of the NS.} and the neutron star equation of state (EoS), which defines the maximum gravitational mass, $M_{\\rm max}$, of an NS without collapsing into a black hole (BH) \\citep{rezzolla10,bartos13,lasky14,ravi14,gao16, margalit17}. \nGiven an EoS, the maximum mass of a non-rotational NS (denoted as $M_{\\rm TOV}$) can be derived by solving the TOV equations \\citep{oppenheimer39}. With rotation, the mass can be enhanced by a factor \n\\begin{eqnarray}\n\\chi={(M-M_{\\rm TOV}) \\over M_{\\rm TOV}}.\n\\label{eq:chi}\n\\end{eqnarray} \nIn the following, $(1+\\chi)$ is defined by the enhancement factor due to uniform rotation. Differential rotation can also enhance the mass, and we denote the enhancement factor as $(1+\\chi_d)$. Typically one has $\\chi < \\chi_d$. \nThe degree of enhancement has been extensively studied in the literature \\citep{cook94,lasota96,breu16,studzinska16,gondek17,bauswein17,bozzola18,weih18}. \nOne can define \n\\begin{eqnarray}\n    M_{\\rm max} & \\equiv & (1+\\chi_{\\rm max}) M_{\\rm TOV}, \\label{eq:Mmax} \\\\\n    M_{\\rm max,d} & \\equiv & (1+\\chi_{\\rm d,max}) M_{\\rm TOV},\n\\end{eqnarray}\nwhere $\\chi_{\\rm max} \\sim 0.2$ \\citep{cook94,lasota96,breu16} and $\\chi_{\\rm d,max} \\sim (0.3-0.6)$\n\\footnote{The quoted $\\chi_{\\rm d,max}$ range is roughly derived from the numerical result that $M_{\\rm tot}>kM_{\\rm TOV}$ (where $M_{\\rm tot}=M_1+M_2$ and $k=1.3-1.6$) is required to directly form a BH as the merger remnant \\citep{shibata05,shibata06,hotokezaka11,bauswein13,margalit17}. Since $M_{\\rm TOV}<M_{\\rm tot}$, the true $\\chi_{\\rm d,max}$ should be somewhat smaller than 0.3-0.6.}\nare the maximum enhancement factor for uniform and differential rotations, respectively. Let us define $M_{\\rm rem}^0$, $M_{\\rm rem}^k$, and $M_{\\rm rem}^\\infty$ as the gravitational masses of the remnant right after the merger, at the beginning of uniform rotation (which is assumed to carry a Keplerian rotation), and with no rotation ($P=\\infty$), respectively. The fate of the merger product can be then determined as follows: If $M_{\\rm rem}^0 > M_{\\rm max,d}$, the merger remnant would directly collapse to a BH. Otherwise the remnant would go through a hypermassive NS (HMNS) phase,  during which the merger remnant loses angular momentum as well as mass. If $M_{\\rm max} < M_{\\rm rem}^k < M_{\\rm rem}^0 \\leq M_{\\rm max,d}$, the merger remnant would collapse into a BH after the HMNS phase. If, however, $M_{\\rm rem}^k \\leq M_{\\rm max}$, a uniformly rotating NS would be formed after the differential rotation is damped. Whether it is an SMNS or an SNS depends on the comparison between $M_{\\rm TOV}$ and $M_{\\rm rem}^{\\infty}$. If $M_{\\rm rem}^\\infty > M_{\\rm TOV}$, the merger remnant is a supramassive NS (SMNS), which would eventually collapse into a BH. If $M_{\\rm rem}^\\infty \\leq M_{\\rm TOV}$, the remnant would never collapse, which is a stable NS (SNS). Either an SMNS or an SNS can be called as a massive NS (MNS). \n\nFor GW170817, although GW data cannot determine the nature of the merger product, it has been suggested that the EM counterpart observations may provide some clues. Unfortunately, owing to the messy physics involved in producing EM counterparts, all the claims on the constraints on $M_{\\rm TOV}$ rely on some assumptions, so that no consensus can be reached. All investigators agree on that the remnant cannot be a promptly formed black hole, since the observed kilonova is too bright to be explained by the dynamical ejecta only. The disagreement comes to the lifetime of the NS produced during the merger. Many authors assumed that in order to produce the short gamma-ray burst (GRB 170817A) \\citep{goldstein17,zhangbb18} following GW170817, a BH engine is needed \\citep[e.g.][]{margalit17,rezzolla18,ruiz18}. Within this picture, the remnant is an HMNS, which must have collapsed before the GRB trigger time, which is 1.7 s after the merger \\citep{abbott17c,goldstein17}. In particular, a good fraction of the observed 1.7 s delay has to be attributed to the HMNS phase, during which significant mass ejection is warranted to account for the observed bright kilonova emission  \\citep{siegel17,gill19}.\nIf this is the case, the multi-messenger observations of GW170817 can be used to provide an upper bound on $M_{\\rm TOV}$ \\citep{margalit17,rezzolla18,ruiz18,shibata19}, e.g. $M_{\\rm TOV}\\lesssim2.16~M_{\\odot}$ by \\cite{margalit17}. \n\nOn the other hand, in order to explain the extended engine activities (flares, extended emission, and internal X-ray plateaus) of short GRBs, it has been long proposed that at least some NS-NS merger systems can produce both a short GRB and an MNS \\citep{dai06,gao06,metzger08,dessart09,lee09,zhang10,fernandez13}. Indeed, when interpreting the rapid decay at the end of internal X-ray plateau observed in a good fraction of short GRBs \\citep{rowlinson10,rowlinson13,lu14,lu15}, $M_{\\rm TOV}$ has to be (much) greater than $2.16~M_{\\odot}$\\citep{gao16,li16}. So there is a direct conflict between the upper limit on $M_{\\rm TOV}$ derived from GW170817 (assuming that a BH is formed before 1.7 s after the merger) and the short GRB X-ray plateau data. Indeed, since GRB 170817A did not trigger Swift, there was no early X-ray afterglow data to check whether there was an early X-ray plateau phase similar to other short GRBs. The fact that the delay time (1.7 s) is comparable to the GRB duration itself ($\\sim 2$ s) also suggests that the jet launching waiting time $\\Delta t_{\\rm jet}$ as well as the shock breakout time $\\Delta t_{\\rm bo}$ may be small \\citep{zhang19}. If so, the launch of a jet may not demand a BH, and the bright kilonova emission may benefit from energy injection of a long-lived remnant \\citep{yu18,li18}. Indeed, the ``blue'' component of the kilonova peaks at $\\sim 1$ d  with a peak luminosity  $\\sim 10^{42}{\\rm erg~s^{-1}}$, which requires a $>0.02M_{\\odot}$ mass of lanthanide-free ejecta ($Y_e \\gtrsim 0.25$) with $v_{\\rm ej,blue}\\approx 0.2-0.3c$ \\citep{kasen17,cowperthwaite17,chornock17,kilpatrick17,villar17,tanvir17,gao17,shappee17}. In order to fit both the peak time and peak luminosity, a small opacity $\\kappa \\sim (0.3-0.5) \\ {\\rm cm^2 \\ g^{-1}}$ is required, which is in conflict with the value $\\kappa \\sim 1 \\ {\\rm cm^2 \\ g^{-1}}$ derived from the most detailed calculations for Fe group elements \\citep{tanaka19}. Energy injection from a long-lived MNS that survives for at least 1 day can help to ease the conflict and interpret the blue component \\citep{li18}. Finally, \\cite{piro19} claimed a low-significance temporal feature at 155 days in the X-ray afterglow of GW170817 which carries properties of GRB X-ray flares. If such a feature is not due to a statistical fluctuation, one would demand an active central engine at such a late epoch. The putative MNS should at least survive for 155 days after the merger.\n\nIn this work, instead of claiming the identity of  the merger remnant of GW170817, we leave it as an open question and generally discuss what constraints on $M_{\\rm TOV}$ one can place for different possible merger remnants. We estimate the remnant mass of GW170817 based on the gravitational wave data and NS EoSs (either for individual EoSs or using some EoS-independent universal relations). We identify the separation lines among three types of products (HMNS\/BH, SMNS, and SNS) in Section 2. In Section, 3, we discuss the cases of SMNS and SNS and place constraints on the NS parameters (surface magnetic field at the pole, $B_p$, and ellipticity, $\\epsilon$) assuming that the remnant can survive for 300 s (typical ending time of internal X-ray plateaus), 1 d (peak time of the blue kilonova component), and 155 d (time of the putative X-ray flare), respectively.\n\n\n\\section{Constraints on $M_{\\rm TOV}$ for different types of merger product}\n\n\\subsection{General approach}\\label{sec:general-approach}\n\nOur goal is to address the following question: given the information provided by the gravitational wave data from GW170817, i.e. the total gravitational mass at infinite binary separation $M_{\\rm tot}= M_1 + M_2 = 2.74^{+0.04}_{-0.01}M_{\\odot}$ and mass ratio $q = m_1\/m_2 = (0.7-1)$ under the low dimensionless NS spin prior \\citep{abbott17a}, what can one say about the maximum mass of the non-spinning NS $M_{\\rm TOV}$? Our general approach is as follows:\n\\begin{itemize}\n    \\item Assume the values of $M_1$ and $M_2$. In our case, we assume that $M_1 = M_2 = M_{\\rm tot} \/ 2$, noticing that the total baryonic mass is highly insensitive to the binary mass ratio;\n    \\item Convert the gravitational masses to baryonic masses $M_{1,b}$ and $M_{2,b}$, either based on the public code \\texttt{RNS} code (for individual EoSs) \\citep{stergioulas95} or some EoS-independent universal relations \\citep[e.g.][]{gao19};\n    \\item {Conserve the total baryonic mass of the binary system throughout the merger. In the post-merger phase, the baryonic mass in the remnant could be derived by subtracting the baryonic mass of various ejecta components from the total baryonic mass}, i.e. $M_{\\rm rem,b}=(M_{1,b} + M_{2,b}) - M_{\\rm ejc}$. Based on the EM counterpart observations, the total ejected mass is estimated as $M_{\\rm ejc} \\sim (0.06\\pm0.01) M_\\odot$ \\cite[][and reference therein]{metzger17};\n    \\item Convert $M_{\\rm rem,b}$ to the gravitational mass of the central object $M_{\\rm rem}$, which depends on its spin state \\citep{gao19}. In particular, we care mostly about $M_{\\rm rem}^k$ and $M_{\\rm rem}^\\infty$.\n    \\item Compare $M_{\\rm rem}^k$ against $M_{\\rm max}$\n    or \n    \\begin{equation}\n      M_{\\rm TOV}^k = (1+\\chi_{\\rm TOV}^k) M_{\\rm TOV}  \n      \\label{eq:MTOVk}\n    \\end{equation}\n    to determine whether the final merger product is an HMNS\/BH, SMNS, or SNS. Here $M_{\\rm TOV}^k$ is the gravitational mass at Keplerian rotation for an NS whose non-spin gravitational mass is $M_{\\rm TOV}$.\n\\end{itemize}\n\nIn the following, we discuss the results for individual EoSs (\\S2.2) and for general cases using universal relations (\\S2.3).\n\n\\subsection{Individual EoSs}\nWe adopt 10 realistic (tabulated) EoSs (as listed in Table 1): SLy \\citep{douchin01}, WFF1 \\citep{wiringa88}, WFF2 \\citep{wiringa88}, AP3 \\citep{akmal97}, AP4 \\citep{akmal97}, BSK21 \\citep{goriely10}, DD2 \\citep{typel10}, MPA1 \\citep{muther87}, MS1 \\citep{muller96}, MS1b \\citep{muller96} with $M_{\\rm TOV}$ ranging from $2.05M_{\\odot}$ to $2.78M_{\\odot}$. For each EoS, we use \\texttt{RNS} to calculate $M_{\\rm TOV}$. We also calculate the allowed minimum Keplerian period (marked as $P_{\\rm k,min}$), which is related to the Keplerian period of the NS at $M_{\\rm max}$. The results are collected in Table 1.\n\nFor each EoS, we derive the $M_{\\rm rem,b}$ following the approach described in Section \\ref{sec:general-approach}. Since the $M_b-M$ relation is somewhat different for different EoSs, the derived $M_{\\rm rem,b}$ is different even for the same event GW170817. With the derived $M_{\\rm rem,b}$ for each EoS, we apply the \\texttt{RNS} code to see whether there is a uniformly rotating NS solution. If not, the merger product would be a BH (likely preceded by a brief HMNS phase). If a solution is available, we further test whether there is a solution in the non-spinning case. The remnant would be an SMNS or SNS if the answer is ``no'' or ``yes'', respectively. As shown in Table 1, among the 10 EoSs studied, one (SLy) with $M_{\\rm TOV} = 2.05 M_{\\odot}$ forms an HMNS\/BH, three (MPA1, Ms1, Ms1b) with minimum $M_{\\rm TOV} = 2.48 M_{\\odot}$ (MPA1) form an SNS, and the other six (with $M_{\\rm TOV}$ between $2.14M_{\\odot}$ (WFF1) and $2.42M_{\\odot}$ (DD2)) form an SMNS. According to this small sample investigation, the $M_{\\rm TOV}$ separation line for HMNS\/BH and SMNS may be between $2.05 M_{\\odot}$ and $2.14M_{\\odot}$, and that between SMNS and SNS may be between $2.42M_{\\odot}$ and $2.48 M_{\\odot}$.\n\n\n\\begin{table*}                                                                      \n\\begin{center}{\\scriptsize                                                                                       \n\\caption{The 10 EoSs investigated in this paper.}                                                                             \n\\begin{tabular}{cccccccccc}                                                                                          \n\\hline                                                                                                                   \n\\hline                                                                                                                                                                                                               \n    & $M_{\\rm TOV}$            &$P_{\\rm k,min}$   &$M_{\\rm b,tot}$               &$M_{\\rm b,rem}$              &$M_{\\rm rem}^k$          &$P_k$      &$1+\\chi_{\\rm TOV}^{k}$      &$1+\\chi_{\\rm max}$     &Product type      \\\\\n    &$\\left(M_{\\odot}\\right)$  &${\\rm ms}$        &$\\left(M_{\\odot}\\right)$      &$\\left(M_{\\odot}\\right)$     &$\\left(M_{\\odot}\\right)$ &${\\rm ms}$ &                        &                 &                  \\\\\n\\hline                                                                                                                                                                                                               \nSLy &2.05                      &$0.55$            &$3.01^{+0.05}_{-0.01}$                          &$2.95^{+0.06}_{-0.02}$                           &$--$                     &$--$       &1.039                   &1.184            &BH                 \\\\\n\\hline                                                                                                                                                                                                               \nWFF1&2.14                      &$0.47$          &$3.07^{+0.05}_{-0.01}$                               &$3.01^{+0.06}_{-0.02}$                        &$2.51^{+0.03}_{-0.01}$                     &$0.52$     &1.051                   &1.201            &SMNS               \\\\\n\\hline                                                                                                                                                                                                               \nWFF2&2.20                      &$0.50$          &$3.04^{+0.05}_{-0.01}$                               &$2.98^{+0.06}_{-0.02}$                         &$2.51^{+0.04}_{-0.01}$                    &$0.58$     &1.048                   &1.192            &SMNS               \\\\\n\\hline                                                                                                                                                                                                               \nAp4 &2.22                      &$0.51$            &$3.03^{+0.05}_{-0.01}$                            &$2.97^{+0.06}_{-0.02}$                        &$2.52^{+0.03}_{-0.01}$                     &$0.60$     &1.047                   &1.194            &SMNS               \\\\\n\\hline                                                                                                                                                                                                               \nBSk21&2.28                     &$0.60$          &$2.99^{+0.05}_{-0.01}$                               &$2.93^{+0.06}_{-0.02}$                         &$2.54^{+0.03}_{-0.02}$                     &$0.74$     &1.044                   &1.205            &SMNS               \\\\\n\\hline                                                                                                                                                                                                               \nAP3 &2.39                      &$0.55$            &$3.01^{+0.05}_{-0.01}$                             &$2.95^{+0.06}_{-0.02}$                         &$2.54^{+0.03}_{-0.02}$                     &$0.70$     &1.049                   &1.202            &SMNS               \\\\\n\\hline                                                                                                                                                                                                               \nDD2 &2.42                      &$0.65$            &$2.99^{+0.05}_{-0.01}$                             &$2.93^{+0.06}_{-0.02}$                        &$2.55^{+0.04}_{-0.01}$                     &$0.82$     &1.042                   &1.208            &SMNS               \\\\\n\\hline                                                                                                                                                                                                               \nMPA1&2.48                      &$0.59$           &$3.00^{+0.05}_{-0.01}$                              &$2.94^{+0.06}_{-0.02}$                        &$2.54^{+0.04}_{-0.01}$                     &$0.76$     &1.048                   &1.208            &SNS               \\\\\n\\hline                                                                                                                                                                                                               \nMs1 &2.77                      &$0.72$            &$2.95^{+0.05}_{-0.01}$                             &$2.89^{+0.06}_{-0.02}$                         &$2.56^{+0.04}_{-0.01}$                    &$1.00$     &1.043                   &1.207            &SNS                \\\\\n\\hline                                                                                                                                                                                                               \nMs1b&2.78                      &$0.71$           &$2.96^{+0.05}_{-0.01}$                             &$2.90^{+0.06}_{-0.02}$                         &$2.56^{+0.04}_{-0.01}$                     &$0.99$     &1.042                   &1.212            &SNS                \\\\\n\\hline                                                                                                                                                                                                               \n\\hline                                                                          \n \\end{tabular}                                                                      \n }                                                                                                                                                              \n\\end{center}                                                                        \n\\end{table*} \n\n\\subsection{Universal Approach}\n\nSince there are many more EoSs discussed in the literature \\citep[e.g.][]{lattimer12}, it is impossible to make a self-consistent check for all the proposed EoSs. Some general constraints on $M_{\\rm TOV}$ (with large uncertainties) may be obtained by applying some EoS-independent empirical relations for GW170817. \n\nGenerally, the type of the merger product is best determined by comparing $M_{\\rm rem}^k$ with $M_{\\rm max}$ (Eq.(\\ref{eq:Mmax})) and $M_{\\rm TOV}^k$ (Eq.(\\ref{eq:MTOVk})), both are highly dependent on the EoS. Fortunately, when the gravitational mass of an NS is normalized to $M_{\\rm TOV}$ and when the rotation period $P$ is normalized to $P_{\\rm k,min}$, the evolution of the separation boundaries among HMNS\/BH, SMNS, and SNS in the ${\\cal M} - {\\cal P}$ plane (where ${\\cal M} \\equiv M \/ M_{\\rm TOV} \\equiv (1+\\chi)$ and ${\\cal P} \\equiv P \/ P_{\\rm k,min}$) is highly EoS-insensitive. This is shown in Figure \\ref{fig:type}. The orange bunch of lines denote the Keplerian lines, which denote the normalized Keplerian period ${\\cal P}_k$ as a function of the gravitational mass of the NS at that period. For the EoSs we investigate, one can get the best-fit line as \n\\begin{eqnarray}\n{\\cal P}_k&=&(-2.697\\pm0.355)\\times({M \\over M_{\\rm TOV}})^2 \\nonumber\\\\\n&+&(4.355\\pm0.764)\\times({M \\over M_{\\rm TOV}})\\nonumber\\\\\n&-&(0.303\\pm0.409),~~P>P_{\\rm k,min}.\n\\end{eqnarray}\n One can see that $P_k$ becomes progressively longer when $M < M_{\\rm max}$. This line is the starting point for the evolution of any NS after the differentiation rotation is damped. \n\nLet us now consider a spinning NS. Its baryonic mass never changes with the spin period while the gravitational mass would decrease as it spins down. These constant $M_b$ curves are examplified as the two black lines (for two particular $M_b$ values) and the red bunch of lines, which show constant $M_{\\rm TOV,b}$ lines for different EoSs. The best-fit line for the 10 EoSs leads to\n\\begin{eqnarray}\n\\log_{10}\\chi_{\\rm TOV}&=&(1.804\\pm0.268)\\times({\\rm log_{10}}{\\cal P})^2 \\nonumber\\\\\n&+&(-3.661\\pm0.190)\\times{\\rm log_{10}}{\\cal P} \\nonumber\\\\\n&+&{\\rm log_{10}(0.101\\pm0.007)}, ~P>P_{\\rm k,TOV}.\n\\end{eqnarray}\nThe intersection of this line and the best-fit orange line gives $P_{\\rm k,TOV}$, which represents the Kepler period when $M_b=M_{\\rm b,TOV}$, and the enhancement factor is $(1+\\chi_{\\rm TOV}^k)$.   \n\n\n\n\nA uniformly rotating NS with $M_b>M_{\\rm TOV,b}$ would eventually collapse into a BH when $M >(1+\\chi_{\\rm col})M_{\\rm TOV}$, where $\\chi_{\\rm col}$ is defined as the maximally allowed enhancement gain at a particular $P$. The values of $(1+\\chi_{\\rm col})$ with different $P$ values can serve as the separation line between SMNS and HMNS\/BH regimes. This corresponds to the green bunch of lines in Figure \\ref{fig:type}, which corresponds to the best fit as \n\\begin{eqnarray}\n\\log_{10}\\chi_{\\rm col}&=&(-2.740\\pm0.045)\\times{\\rm log_{10}} {\\cal P} \\nonumber\\\\\n&+&{\\rm log_{10}(0.201 \\pm 0.005)}\n\\label{eq:chi_ucol}\n\\end{eqnarray}\nWhen $P \\rightarrow P_{\\rm k,min}$, one has $\\chi_{\\rm col}^k \\rightarrow \\chi_{\\rm max}$.\n\nThe region below the orange bunch of lines in Figure \\ref{fig:type} (the white region) is not well defined, since $P$ cannot be defined for an differentially rotating object. The \\texttt{RNS} code we employ can only be used in uniformly rotating case. We therefore indicate the evolution trajectories in the white region using dashed lines.\n\nWe plot several evolutionary trajectories of the GW170817 remnant within the framework or several EoSs: BSk21 (diamond), AP3 (star), MPA1 (upward triangle), and SLy (downward triangle). The symbols are solid or open in the uniformly or differentially rotating regimes, respectively. \n\nThe mass at the starting point of rigid rotation, i.e. $M_{\\rm rem}^k$, is crucial to determine the remnant type through its comparison with $M_{\\rm max}$ (Eq.(\\ref{eq:Mmax})) and $M_{\\rm TOV}^k$ (Eq.(\\ref{eq:MTOVk})). \nThe values of $\\chi_{\\rm TOV}^k$ and $\\chi_{\\rm max}$ of each EoS can be calculated utilizing the \\texttt{RNS} code, and may be also generally estimated as $\\chi_{\\rm TOV}=0.046 \\pm 0.008 (\\pm 0.004)$ and\n$\\chi_{\\rm max}=0.201\\pm 0.017 (\\pm 0.008)$ with 2$\\sigma$ (1$\\sigma$) errors, respectively.\n\n \n\n\\begin{figure}[ht!]\n\\resizebox{90mm}{!}{\\includegraphics[]{type.eps}} \n\\caption{The allowed parameter space of different types of merger products. The orange bunch of lines denote the mass-dependent normalized Keplerian period ${\\cal P}_k$; the red bunch of lines denote of constant $M_{\\rm TOV,b}$ lines; the green bunch of lines denote the boundary line for the SMNS to collapse into a BH. Each bunch includes 10 lines corresponding to 10 different EoSs. The black lines stand for the evolving trajectory of the merger product of GW170817. The markers show the points where the evolution phase changes. The dashed lines and hollow markers are schematic since the period of differential rotating NS is undefined. The vertical dark regions denote the separation lines for three regions at $P=P_k$.} \n\\label{fig:type}\n\\end{figure}\n\nAs shown in Table 1, the value of $M_{\\rm rem}^k$ only weakly depends on EoSs and is slightly correlated with $M_{\\rm TOV}$. We show the relationship between $M_{\\rm rem}^k$ and $M_{\\rm TOV}$ in Figure \\ref{fig:M}, which reads \\begin{eqnarray}\nM_{\\rm rem}^{k}=(2.354 \\pm 0.074)+(0.076\\pm 0.032)M_{\\rm TOV},\n\\label{eq:Mg}\n\\end{eqnarray}\nwith 2$\\sigma$ error. Combining Equations \\ref{eq:chi} and \\ref{eq:Mg}, one can derive the critical values to separate the HMNS\/BH vs. SMNS and SMNS vs. SNS, as also shown in Figure \\ref{fig:M}. \n\nThe following results can be obtained for GW170817: If the merger remnant is a HMNS\/BH, $M_{\\rm TOV}$ should be smaller than $2.09^{+0.11}_{-0.09}(^{+0.06}_{-0.04})M_{\\odot}$ at the 2$\\sigma$ (1$\\sigma$) level\\footnote{After mass shedding from the initial torus due to viscous and neutrino cooling processes, a quasi-stationary torus is supposed to exist surrounding the central core \\citep{siegel17,hanauske17,fujibayashi18}. The mass remaining in the torus is in Keplerian orbits, thus would not add to the gravitational mass in the core. Considering this effect, the values of the critical $M_{\\rm TOV}$ to separate the HMNS\/BH and SMNS cases would be smaller. However, since a significant fraction of mass in the torus has fallen back or ejected within $\\sim 0.3s$ \\citep{siegel17,fujibayashi18} (comparable to the lifetime of HMNS \\citep{metzger18}), the influence of the quasi-stationary torus to the critical $M_{\\rm TOV}$ is small.\n}; if the merger remnant is an SMNS, $M_{\\rm TOV}$ should be in the range from \n$2.09^{+0.11}_{-0.09}(^{+0.06}_{-0.04})M_{\\odot}$ to \n$2.43^{+0.10}_{-0.08}(^{+0.06}_{-0.04})M_{\\odot}$ at the 2$\\sigma$ (1$\\sigma$) level; if the merger remnant is a SNS, $M_{\\rm TOV}$ should be greater than $2.43^{+0.10}_{-0.08}(^{+0.06}_{-0.04}M_{\\odot})$ at the 2$\\sigma$ (1$\\sigma$) level. These results are generally consistent with previous results assuming an HMNS\/BH remnant in GW170817 \\citep[e.g.][]{margalit17,ruiz18,rezzolla18,shibata19}, even though some details differ. For example, \\cite{ruiz18} and \\cite{rezzolla18} both adopted the measured $M_{\\rm tot} \\sim 2.74 M_\\odot$ deducting the mass loss to estimate $M_{\\rm rem}$. This over-estimated $M_{\\rm rem}$ by $\\sim 0.2 M_\\odot$, which would over-estimate the upper limit of $M_{\\rm TOV}$. \\cite{shibata19} performed the most detailed analysis numerically and derived a more conservative upper limt $\\sim 2.3 M_\\odot$ for the HMNS\/BH case, but for the majority of the EOSs studied, the range of $M_{\\rm TOV}$ that form an HMNS\/BH product is still consistent with our estimate. Our derived separation line between SMNS and SNS products is also consistent with theirs.\n\n\n\n\n\\begin{figure}[ht!]\n\\resizebox{90mm}{!}{\\includegraphics[]{M.eps}} \n\\caption{Constraints on the range of $M_{\\rm TOV}$ for three different merger products in the case of GW170817. The colored data points represent the values of $M_{\\rm TOV}$ and the estimated $M_{\\rm rem}$ at the Keplerian period for different EoSs. The solid line is the best fitting relation of $M_{\\rm TOV}$ and $M_{\\rm rem}$, whereas the blue and grey dashed lines showing the $1\\sigma$ and $2\\sigma$ error range, respectively.  The slanted deep grey shadows are the allow regions when $\\chi=\\chi_{\\rm TOV}^k$ and $\\chi=\\chi_{\\rm max}$, respectively. The dot-dashed vertical lines are the central $M_{\\rm TOV}$ values of the separation lines of different merger products, which are surrounded by the $1\\sigma$ (light blue shadow) and $2\\sigma$ (light grey shadow) regions. The hollow circle is the predictive value for the EoS SLy, which forms a BH rather than a MNS.} \n\\label{fig:M}\n\\end{figure}\n\n\n\\section{Constraints on the NS Properties in the MNS cases}\n\nIn the case of SMNS, it would be interesting to investigate under what conditions the SMNS can survive for a particular duration of time, e.g. $\\sim$ 300 s for the typical duration of the X-ray internal plateau, $\\sim$ 1 d to power the blue component of the kilonova, and $\\sim$ 155 d to power the putative X-ray flare. This depends on the NS EoS and the spindown history of the putative MNS remnant.\n\nAt a certain spin period $P$, $\\chi_{\\rm TOV}<\\chi<\\chi_{\\rm col}$, the remnant would be an SMNS. For a particular object, $\\chi$  decreases as the NS loses its rotation energy through magnetic dipole radiation and GW radiation \\citep{shapiro83,zhang01}, i.e.\n\\begin{eqnarray}\n\\dot{\\Omega}=-\\frac{32GI\\epsilon^2\\Omega^5}{5c^5}-\\frac{B_p^2R^6\\Omega^3}{6Ic^3},\n\\label{dotE}\n\\end{eqnarray} \nwhere $\\Omega=2\\pi \/P$ is the angular frequency and $\\dot{\\Omega}$ is its derivative, $B_p$ stands for the surface dipole magnetic field strength at the pole, and $\\epsilon$ presents the ellipticity of the NS. As the NS spins down, it would collapse into a BH when $\\chi>\\chi_{\\rm col}$.\n  \n\nFor a rigidly rotating NS,  the maximum enhancement factor $\\chi_{\\rm col}$ at any given $P$ can be estimated with the EoS-independent relation Equation (\\ref{eq:chi_ucol}), which could be up to $\\chi_{\\rm max} \\sim 20\\%$ when the NS spin period equals to the allowed minimum Keplerian period. Given the initial spin period (Keplerian) and a particular desired lifetime of the MNS, it is possible to follow the spindown evolution and constrain $B_p$ and $\\epsilon$. \n\nTo be conservative, we assume the initial period $P_i=P_{\\rm k,min}$ for GW170817 and use the moment of inertia and radius of a non-rotating NS to constrain $B_p$ and $\\epsilon$ parameters\\footnote{The initial spin period of the merger product, $P_k$ should not be smaller than $P_{\\rm k,min}$, which means that smaller $\\epsilon$ and $B_p$ values than constrained are required for the SMNS to spin down to a certain period. Our derived upper limits on $B_p$ and $\\epsilon$ are therefore safe upper limits. Similarly, if one take larger $I$ and $R$ values for spinning NSs, one also requires smaller $B_p$ and $\\epsilon$ values than derived to reach the same spindown effect.}. Given a value of $\\chi_{\\rm col}$, we plot the boundary lines in the $B_p-\\epsilon$ plane for the region which allows an SMNS to survive for a certain lifetime, e.g. 300 s, 1 day and 155 days (see Figure \\ref{fig:universal}). During a particular time span, part of the spin-down power of the SMNS would be released in the EM channel, so that the EM counterpart observations of GW170817 could be used to make constraints on the spin-down power of the remnant SMNS, and hence,  on $B_p$ and $\\epsilon$. \n\nIf the SMNS can survive for 300s, we consider two constraints from the EM counterpart observations: the EM channel spin-down power integrated within 300s should be less than the kinetic energy of the merger ejecta [$\\beta<0.3$, inferred from the spectrum observation of the optical counterpart \\citep{kasen17,cowperthwaite17,chornock17,kilpatrick17,shappee17}] and less than the kinetic energy of the GRB jet [$E_k<10^{51}{\\rm erg~s^{-1}}$, inferred from the radio afterglow emission \\citep{nakar18,dobie18}]. If the SMNS can survive for 1 d, in addition to the constraints from the kinetic energy, since the merger ejecta already became optically thin at that time, the luminosity of the magnetic dipole spin-down should be smaller than the peak luminosity of the optical counterpart. If the SMNS can survive for 155d, besides the constraints from the kinetic energy and optical peak luminosity, the late time X-ray observations could also serve as the upper limit of the luminosity of the magnetic dipole spin-down. These constraints on $B_p$ and $\\epsilon$ from EM counterpart observations of GW170817 are  shown in Figure \\ref{fig:universal}.  We can see that the EM observations tend to constrain $B_p$ and $\\epsilon$ to small values. Nonetheless, if an SMNS is formed, there always exits a suitable $(B_p, \\epsilon)$ parameter space to allow the SMNS to survive for 300 s, 1 d, 155 d, or even longer,\nwithout violating the observational constraints.\nThese constraints are clearly displayed in Figure \\ref{fig:universal}.\n\n\n\nSome of our selected EoSs, e.g. WFF1, WFF2, AP4, BSK21, AP3 and DD2, are supposed to support an SMNS. In principle, with a certain EoS adopted, the properties of the SMNS could be constrained more precisely (see Figure \\ref{fig:eos}). If the lifetime of SMNS is 300 s or longer,  without violating the constraints from EM observations, we should have  $B_p<2.1 \\times 10^{14} G$ and $\\epsilon<2.0\\times 10^{-4}$ for WFF1, $B_p<2.2 \\times 10^{14}G$ and $\\epsilon<2.8\\times 10^{-4}$ for WFF2, $B_p<1.1\\times 10^{14}G$ and $\\epsilon<1.4 \\times 10^{-4}$ for AP4, $B_p<7.9 \\times 10^{14} G$ and $\\epsilon<1.2\\times 10^{-3}$ for BSK21, $B_p<3.4 \\times 10^{15}G$ and $\\epsilon<1.4\\times 10^{-2}$ for AP3, $B_p<4.2\\times 10^{15}G$ and $\\epsilon<3.3 \\times 10^{-2}$ for DD2. If the lifetime of SMNS is 1 d or longer,  without violating the EM observational constraints, we should have  $B_p<8.7 \\times 10^{10} G$ and $\\epsilon<1.3\\times 10^{-5}$ for WFF1, $B_p<8.7 \\times 10^{10}G$ and $\\epsilon<1.7\\times 10^{-5}$ for WFF2, $B_p<8.7\\times 10^{10}G$ and $\\epsilon<8.9 \\times 10^{-6}$ for AP4, $B_p<1.5 \\times 10^{11} G$ and $\\epsilon<7.4\\times 10^{-5}$ for BSK21, $B_p<1.6 \\times 10^{12}G$ and $\\epsilon<1.1\\times 10^{-3}$ for AP3 and $B_p<3.2\\times 10^{12}G$ and $\\epsilon<2.1 \\times 10^{-3}$ for DD2. If the lifetime of SMNS is 155 d or longer,  without violating the EM observational constraints, we should have  $B_p<8.5 \\times 10^{10} G$ and $\\epsilon<1.0\\times 10^{-6}$ for WFF1, $B_p<8.5 \\times 10^{10}G$ and $\\epsilon<1.4\\times 10^{-6}$ for WFF2, $B_p<8.5\\times 10^{10}G$ and $\\epsilon<7.1 \\times 10^{-7}$ for AP4, $B_p<8.7 \\times 10^{10} G$ and $\\epsilon<5.9\\times 10^{-6}$ for BSK21, $B_p<9.3 \\times 10^{10}G$ and $\\epsilon<2.2\\times 10^{-5}$ for AP3 and $B_p<1.0\\times 10^{11}G$ and $\\epsilon<3.4 \\times 10^{-5}$ for DD2. The constraints on $B_p$ mainly come from the EM observations, which are roughly consistent with (slightly looser than) the constraints derived in our previous work \\citep{ai18}. In this paper, we only used the peak luminosity (or total kinetic energy) rather than the full lightcurve (used in \\cite{ai18}) to constrain the parameters. The ellipticity $\\epsilon$, which was a free parameter in \\cite{ai18}, is now constrained by the lifetime of the SMNS.\n\n\\begin{figure*}[ht!]\n\\begin{center}\n\\begin{tabular}{l}\n\\resizebox{115mm}{!}{\\includegraphics[]{300s.eps}} \\\\ \n\\resizebox{115mm}{!}{\\includegraphics[]{1d.eps}} \\\\\n\\resizebox{115mm}{!}{\\includegraphics[]{155d.eps}} \n\\end{tabular}\n\\caption{Constraints on the allowed parameter space in the $B_p$-$\\epsilon$ plane for three survival times: 300 s (upper), 1 d (middle), and 155 d (lower).\nThe dashed lines are the boundaries for different putative EoSs (which correspond to different $\\chi$) within which the SMNS can survive for a certain timescale. The red and green lines show the constraints from the kinetic energy of ejecta, which are deduced from observations of afterglow and mergernova, respectively. The blue and purple lines stand for the constraints from the luminosity of the mergernova and the late-time X-ray signal. }\n\\label{fig:universal}\n\\end{center}\n\\end{figure*}\n\n\n\\begin{figure*}[ht!]\n\\begin{center}\n\\begin{tabular}{l}\n\\resizebox{115mm}{!}{\\includegraphics[]{eos_300s.eps}} \\\\\n\\resizebox{115mm}{!}{\\includegraphics[]{eos_1d.eps}} \\\\\n\\resizebox{115mm}{!}{\\includegraphics[]{eos_155d.eps}}\n\\end{tabular}\n\\caption{Same as Figure 3, but for concrete EoSs studied in this paper.} \n\\label{fig:eos}\n\\end{center}\n\\end{figure*}\n\n\n\\section{Conclusions and Discussion}\n\nA tight constraint on the NS maximum mass $M_{\\rm TOV}$ is helpful to constrain the NS EOS. Before GW170817, the constraints on $M_{\\rm TOV}$ mainly comes from the observations of Galactic pulsars. Some massive pulsars have been observed, e.g. PSR J1614-2230 with $1.97 \\pm 0.04M_{\\odot}$ \\citep{demorest10} and PSR J0348+0432 with $2.01 \\pm 0.04M_{\\odot}$ \\citep{antoniadis13}), which set a lower limit to $M_{\\rm TOV}$ around $\\sim 2M_{\\odot}$. Recently, the most massive NS PSR J0740+6620 was measured to have a mass $2.14^{+0.10}_{-0.09}~M_{\\odot}$ at $68.3\\%$ confidence level \\citep{cromartie19}. This sets an even more stringent lower limit to $M_{\\rm TOV}$.\n\nNS-NS merger events could in principle give tighter constraints on $M_{\\rm TOV}$ if the merger product can be unambiguously identified. Unfortunately, without post-merger GW signal (which may not be obtained in the near future), the EM signals are not clean enough to draw definite conclusions. For the case of GW170817, even though the existence of GRB 170817A $\\sim 1.7$ s later was regarded by some authors as evidence of the formation of a BH before the onset of the GRB, some other authors argued for a long-lived NS remnant that may exist for an extended period of time. As a result, we cannot place a constraint on $M_{\\rm TOV}$. Rather, in this paper, we discuss the range of $M_{\\rm TOV}$ for different assumed merger products. We applied two approaches: case studies for 10 individual EoSs and an EoS-independent approach adopting some universal relations. We reached the following self-consistent results: If the merger product was a short-lived HMNS, one has $M_{\\rm TOV}<2.09^{+0.11}_{-0.09}(^{+0.06}_{-0.04})M_{\\odot}$; If the merger product was a long-lived SMNS, the constraint should be $2.09^{+0.11}_{-0.09}(^{+0.06}_{-0.04})M_{\\odot} \\leq M_{\\rm TOV}<2.43^{+0.10}_{-0.08}(^{+0.06}_{-0.04})M_{\\odot}$; If the merger product was a stable NS, the constraints should be $M_{\\rm TOV} \\geq 2.43^{+0.10}_{-0.08}(^{+0.06}_{-0.04})M_{\\odot}$. The quoted uncertainties are at the 2$\\sigma$ (1$\\sigma$) level. \n\n\nIf the merger remnant is a long-lived MNS, the next question is whether the MNS can survive for a desired period of time, e.g. 300 s, 1 d or 155 d, to interpret various observations. This depends on the spindown history of the remnant, which critically depends on two NS parameters, $B_p$ that defines the dipole spindown and $\\epsilon$ that defines the secular GW spindown. For an SNS remnant, this is never a problem. For an SMNS remnant, the survival time actually constrain $B_p$ and $\\epsilon$ to be smaller than certain values. We have derived these constraints for the case of GW170817 for different EoSs (which have different $M_{\\rm TOV}$ and hence require different $\\chi$ values). These constraints, together with those posed from the EM observations, define the parameter space in the $(B_p, \\epsilon)$ plane that satisfies the desired lifetime. In general, we find that for any EoS that forms an SMNS in the case of GW170817, without violating the EM observational constraints, there always exist a set of ($B_p, \\epsilon$) parameters that makes the SMNS survive for 300 s, 1 d, 155 d or even longer. {In particular, for EoSs in Table 1 with $M_{\\rm TOV}$ from $2.14M_{\\odot}$ to $2.42M_{\\odot}$, we have $\\epsilon\\lesssim 1.4\\times10^{-4}-3.3\\times10^{-2}$ and $B_p\\lesssim (1.1\\times10^{14}-4.2\\times10^{15})$G if the SMNS survives for 300 s; $\\epsilon\\lesssim 8.9\\times10^{-6}-2.1\\times10^{-3}$ and $B_p\\lesssim(8.7\\times10^{10}-3.2\\times10^{12})$G  if the SMNS survives for 1 d; and $\\epsilon\\lesssim 7.1\\times10^{-7}-3\\times1.0\\times10^{-6}$ and $B_p\\lesssim(8.5\\times10^{10}-9.3\\times10^{12})$G if the SMNS survives for 155 d.}  \n\n\nFuture joint GW\/EM observational campaigns of NS-NS merger events may identify more definite observational criteria to identify the nature of the merger remnants. The similar approach proposed in this paper can be applied in those events, which will lead to tighter constraints on $M_{\\rm TOV}$ and NS EoS.\n\n\\acknowledgments\nWe thank Luciano Rezzolla and Masaru Shibata for helpful discussion and the anonymous referee for useful comments.\nHG acknowleges support by National Natural Science Foundation of China under Grant No. 11722324, 11603003, 11633001 and 11690024, the Strategic Priority Research Program of the Chinese Academy of Sciences, Grant No. XDB23040100 and the Fundamental Research Funds for the Central Universities.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}