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+{"text":"\n\\section{ Introduction}\n\n\n Light curves of supernovae vary significantly. Even within\nthe same supernova type, a spread in the shape of the light curves\n is found. So far, the differences have not been quantified \nin terms of a\nspread in the integrated bolometric light curves, but rather in terms \nof the light curves\nin the various broad--band filters (Hamuy et al. 1996a,b; Riess, Press\n \\& Kirshner 1996).\n   \nFrom the theoretical point of view, several factors can induce changes in the\nevolution in luminosity within a sample of supernovae of the same type:\ndifferent distributions of radioactive material in velocity space resulting\nfrom differences in the burning propagation along the star, a spread in total\nmasses of the ejecta, or a diversity in the configuration and intensity\nof the magnetic field, B, of the stars prior to explosion.\n\nVery few studies have been done on the evolution of the magnetic field of a\nstar which explodes, and how this affects its overall luminosity.\nIn particular, little\nattention has been devoted to the fate of the original magnetic field\nconfiguration, which should experience a drastic change due to the enormous\nexpansion undergone by the supernova ejecta. That can bear observable \nconsequences in the evolution of the luminosity of the supernova.\n\nThe progenitor stars of thermonuclear supernovae are appreciably magnetized\nobjects. Magnetic fields of WDs have been measured and \nrange from 10$^{5}$ to as much as 5 $\\times$ 10$^{8}$ G (Liebert 1995). \nPrior to\nthe explosion, the turbulent motions inside the WD can  alter the original\nintensity and configuration of such field by fast dynamo action. After the\nexplosion, the huge expansion undergone by the ejecta reduces \nthe magnetic field inside\nthe supernova. The evolution with time of the supernova field \nbecomes relevant to the trapping of the\nenergy of the positrons originated in the \nradioactive $\\beta^{+}$--decays of $^{56}$Co.\nSeveral hundreds days after the explosion,  if the magnetic\nfield lines do not contribute to confine the path of the energetic positrons, \nwith kinetic energies in the MeV range, \na fraction of this energy escapes\nthe innermost ejecta and the evolution in luminosity of  \nthe supernova is affected.\n\nColgate, Petschek, \\& Kriese (1980), and more recently Colgate (1991,1997)\nundertook a determination of the ejected mass in supernovae through the\nescape of $\\beta^{+}$ energy in the tail of the light curves. Significant\ndepartures from the $^{56}$Co--decay full--trapping curve are argued\n in those\n works.  Axelrod (1980), on the other hand, suggested in his SN modeling\nthat a chaotic weak magnetic field of B $\\approx$ 10$^{-6}$ G \nafter the explosion would confine the\npositrons up to late phases. Under the later assumption, \nthe late decline in\nluminosity approaches the $^{56}$Co--decay full--trapping line, although\npositron energy is not fully deposited (Chan \\& Lingenfelter 1993).\nWhereas in earlier works a particular configuration of the magnetic \nfield has been assumed, in the\npresent work, we look at the processes undergone by the WD prior to\nexplosion and as it expands, and predict how the magnetic field contents\nmight evolve. The suggested evolution should then be compared with the\nobservations through the predictions of the supernova luminosity. \n\nIn thermonuclear supernovae ( i.e. Type Ia supernovae, or SNe Ia), depending\non whether the diversity among the bolometric tails is moderate (of the order\nof 10---15$\\%$) or larger ($\\ge$ 30--50 $\\%$) in the fraction of energy\ndeposited, one would favor different mechanisms to explain\n this diversity. As will be shown in this work, moderate\ndiversity suggests different distributions of radioactive material\nin velocity space, and a larger diversity implies differences in the magnetic\nfield in the ejecta, and possibly also\ndifferences in the ejected mass. The information on the post--explosion\nmagnetic field derived from the SN late luminosity can be linked to the\nnature of the pre--explosion magnetic field.\n\nIn the case of Type Ibc supernovae, the effect of mixing in the deposition of\nenergy from both $\\gamma$--rays and positrons is examined as an important\nfactor in determining the final shape of their light curves.\n\nIn Sections  2 and 3, $\\gamma$--ray  and $\\beta^{+}$--energy deposition\ncalculations of supernovae are presented. In section 4, it is shown how in\n SNe Ia both mass and nucleosynthetic distribution as well as pre and\npost--explosion magnetic field, can be investigated through the study of the\nbolometric light curves. The present understanding of the evolutionary stages\nprevious to explosion, and the duration of processes such as turbulent\nconvection in accreting WDs, are examined in order to establish the changes\n of the WD magnetic field. In Section 5, the diversity of \nbolometric\ndeclines is presented in terms of the physical processes from which \nit can originate.\nThe influence of different nucleosynthesis and kinematics of SNe Ia \nin the final luminosity is discussed. Mixing in\nsupernovae of core--collapse with small ejected mass such as Type Ibc, is\ndiscussed for its effects on the bolometric luminosity within \nthe  whole core--collapse SN class as well as its incidence on\n the derivation of the ejected mass  for those low-mass ending\nstars. Finally, in Section 6, we infer from the\nstudy of positron escape in supernovae some consequences for \nthe origin of the 511 keV positron annihilation line in our Galaxy. \n\n\n\\section{ Radioactivity from $^{56}$Co: $\\gamma$--rays and positrons}\n\n\nIn supernovae, the radioactive decay $$^{56}Co \\rightarrow  ^{56}Fe$$\nprovides the source of luminosity along the tail of the light curve. Such a\ndecay has a half life of 77 days and 81$\\%$ of the time gives rise to a\n$\\gamma$--ray photon and 19$\\%$ to a e$^{+}$. $\\gamma$--ray photons are\nemitted with a spectrum of energies reaching up to 1.4 MeV, and carry about\n96.5 $\\%$ of the energy of the $^{56}$Co decay. The emitted positrons have an\nenergy spectrum extending up to the endpoint kinetic energy E$_{max}$=1.459\nMeV, and they account for 3.5$\\%$ of the energy of the $^{56}$Co decay.  The\nfate of this 3.5 $\\%$ of energy is crucial at late times.\n\n\nCompton scattering of the emitted $\\gamma$--rays is the main process\ndegrading the energy of the photons as it is transferred to the electrons  of\nthe gas which become nonthermal. The comparative simplicity of the process\ndegrading the energy of the $\\gamma$-rays in expanding ejecta allows us to\ncalculate accurately the energy deposited and the escape of energy as well.\n \nTransport calculations of the $\\gamma$-rays provide the fraction of\nradioactive energy deposited in the supernova ejecta as a function of inner\nmass fraction and time. The deposition function $D_{\\gamma}(t)$ is a \ndecreasing function of time as the supernova expands. The final injection\nof energy in the supernova ejecta takes place at a rate:\n\n\n$$\\xi(t) = (6.76\\times10^{9}\\ D_{\\gamma}(t) + 2.72\\times10^{8}\\ D_{\\beta}\n(t)) \\left(e^{-t\/\\tau_{Co}} - e^{-t\/\\tau_{Ni}}\\right)$$\n$$\\hskip 6.2 true cm +\\ 3.91\\times10^{9}\\ D_{\\gamma}(t) e^{-t\/\\tau_{Ni}}\\\n\\rm {erg\\ g^{-1}\\ s^{-1}}\\eqno(1)$$\n          \n\\noindent\nwhere $\\tau_{Ni}$=8.8 days  and $\\tau_{Co}$=111.26 days are the e--folding\ntimes for radioactive decay of Ni and Co respectively, and D$_{\\beta}$ is the\ndeposition function of  e$^{+}$ energy, whose importance becomes crucial as\nthe ejecta become transparent to $\\gamma$--rays. The term related to the\ndecay $^{56}Ni \\rightarrow  ^{56}Co$ is relevant for the early rise to\nmaximum luminosity.\n\n\nOnce $\\gamma$--ray photons suffer Compton scattering either they do not lose\na significant amount of energy (forward scattering), or they lose\nsignificantly their energy, becoming unable to produce further energetic \nelectrons. This has suggested  the adequacy of treating the Compton\nscattering process as an absorption process, for applications related to the\nenergy deposition of $\\gamma$--rays (Sutherland \\& Wheeler 1984). A similar\napproach to that developed by those authors is used to calculate\n$\\gamma$--ray transport in this work. Two methods of calculation of the\ndeposition of energy were previously compared: the ``absorption'' approach\ngeneralized for an arbitrary $^{56}$Ni distribution was tested against\ndetailed Monte Carlo calculations. As found by previous authors (Swartz,\nSutherland, \\& Harkness 1995) both results gave a very similar deposition\nfunction. Background models such as the W7 model by  Nomoto, Thielemann, \\&\nYokoi (1984)  were used for these tests.\n\nAs one follows the evolution of the bolometric light curve of SNe along the\n$^{56}$Co tail, different phases can be outlined. For SNe Ia, the\npost--maximum decline of the light curve is primarily determined by the\ntemporal evolution of $ D_{\\gamma}(t)$. The luminosity at that phase and its\nrate of decline are related to the degree of escape and deposition from those\nenergetic photons. That degree depends on the distribution of $^{56}$Ni in\nthe velocity--mass  space, and on the total optical depth of the ejecta. This\nsuggests defining a $\\Delta m_{\\gamma}^{100}$ as the number of bolometric\nmagnitudes of decline per day  during the phase when $\\gamma$--rays are the\nmain contributors.\n\nLater on, D$_{\\gamma}$ falls below the contribution of energy  by positrons.\nAt that time a new inflection in the bolometric light curve shape occurs\nlinked to the slower evolution in time of $D_{\\beta}(t)$. The steepness of\nthe decline is then related to the distribution of the radioactive $^{56}$Co,\nthe velocity structure, and to the intensity and configuration of the\nmagnetic field.  As we will see, different behaviors are expected and they\ncan give clues to the mechanism of explosion. What happens in the phase when\npositrons are the dominant luminosity source depends very much on B.\n\nWhen positrons in supernovae start to play a major role in the energy input\n(as soon as $\\tau_{\\gamma}$ becomes very small), the fraction of  escape and\ndeposition of the kinetic energy of those particles establishes the\nluminosity. The most energetic positrons and those emitted in the outer\nlayers may succeed  escaping the ejecta without becoming thermalized. \n A numerical evaluation is required once the supernova\nphysical properties are known.\nThe energy spectrum of the positrons covers a broad range of energies with a\ndistribution of the form:\n\n$$S(\\epsilon)\\propto F(Z, \\epsilon) (\\epsilon_{0} - \\epsilon)^{2} \\epsilon\n\\sqrt{\\epsilon^{2} - 1}\\eqno(2)$$\n\n\\noindent\nwhere $\\epsilon$ is the total positron energy in units of\n $m_{e}c^{2}$; $ \\epsilon_{0}\n= E_{max}\/m_{e}c^{2} + 1$, $E_{max}$ being the maximum kinetic energy; and\n$F(Z,\\epsilon)$ is a correction for the Coulomb interaction with the final\nnucleus of electric charge Z (Segr\\'e 1977):\n\n $$F(Z, \\epsilon) = {2\\pi\\xi\\over 1 - {\\rm exp}(-2\\xi)}\\eqno(3)$$    \n\n\\noindent\nwith\n \n$$\\xi = -{Ze^{2}\\over \\hbar v} = -{Z\\alpha\\over\n\\sqrt{1-\\epsilon^{-2}}}\\eqno(4)$$\n\n\n\\noindent\nwhere $Z = 26$, $v$ is the speed of the positron, and $\\alpha$ is the\nfine--structure constant (Segr\\'e 1977).\n\n\nPositrons with $\\beta$ as large as 0.94 ($\\beta = v\/c$) are produced in the\ndecay.  Given the initial range of kinetic energies -- in the keV and MeV\nrange--, the positrons slow down in the supernova ejecta mainly by ionization\nand excitation losses. At higher energies, bremsstrahlung would be the\ndominant energy loss mechanism, and at lower energies, Coulomb scattering\nwould be dominant (see Segr\\'e 1977, for instance).  \n\nAs the positrons slow down due to their loss of energy in the ejecta, they\ntravel a fraction of the envelope which can be estimated as the {\\it stopping\ndistance, $d_{e}$} due to ionizations and excitations in the SN Ia envelope.\n\nThe full relativistic expression for positron energy loss, per unit length,\nX, due to ionization of atoms is (Heitler 1954; Blumenthal \\& Gould 1970;\nGould 1972):\n\n$${dE\\over dX} = - \\Gamma (E) = - {4\\pi r_{0}^{2}m_{e}c^{2}Z \\rho \\over\n\\beta^{2}Am_{n}} {ln\\left({\\sqrt{\\gamma - 1} \\gamma \\beta \\over\nI\/m_{e}c^{2}}\\right) + {1\\over 2} ln \\ 2 + \\Sigma_{2}(E)}\\eqno(5)$$\n\n\\noindent\nwhere E is the kinetic energy, $r_{0}$ is the classical electron radius,\n$m_{n}$ is the atomic mass unit, $Z$ and $A$ are, respectively, the effective\nnuclear charge and atomic mass of the ejecta material, and $\\Sigma_{2} (E)$\ngives the relativistic factors as a function of the Lorentz factor, $\\gamma$,\nand of $\\beta$  (Berger \\& Seltzer 1954).\n\n\n\n\\noindent\n$I$ is the effective ionization potential for the ambient atoms in the\nejecta. A semiempirical formula for the ionization potential gives (Roy \\&\nReed 1968; Segr\\'e 1977):\n\n$$I = 9.1Z\\left(1 + {1.9\\over Z^{2\/3}}\\right) eV\\eqno(6)$$\n\nDue to the weak dependence of  $dE\/dX$ on $I$, the formula above for I is\naccurate enough for the practical calculation of the energy loss.\n\n The {\\it stopping distance of the positron}\nas result of impact ionization and\nexcitation in a SN Ia envelope is found to be approximately:\n\n$$d_{e} \\equiv {E\\over {-dE\/dX}} \\approx  {3.36\\over \\rho} \\left({E\\over\nm_{e}c^{2}}\\right){A\\over Z}(ln{E\\over I} )^{-1}\\ cm \\eqno(7)$$\n\nIn Table 1 some typical values are given for d$_{e}$,  the \n{\\it stopping distance\nof the positrons} of different energies both for Chandrasekhar WDs\nand WDs of the smallest possible exploding mass.\nSynchrotron losses by the e$^{+}$ in the presence of the magnetic field,\nbremsstrahlung losses, and losses due to Compton scattering off photons\ncontribute to the slowing down of the positrons to a much lesser extent.\n\n\n\nEach magnetic field configuration specifies in a given way  the positron\ntransport in the supernova ejecta. We have specified three  likely\nconfigurations of the field lines, and adopted an efficient way to calculate\nthe deposition function. Three situations which the positrons might encounter\n in exploded ejecta are: a {\\it chaotic magnetic field background} (a likely\nresult of the turbulent motions  prior to explosion), a  {\\it radial field} \n(resulting from expansion of the original dipole  field in fast moving\nejecta), or {\\it the absence of a significant magnetic field}, \nin which case they\nare just subjected to their interactions with ions and electrons along free\ntrajectories.\n\n\nThe deposition calculation consists in determining how efficiently the\nrelativistic positrons transfer their energy to ions and  electrons\nincreasing the kinetic energy of the latter: positrons thermalize if \nthey release most of their kinetic energy. That energy should reappear \nas optical--infrared luminosity through the excitation of a whole range of\ntransitions or through ionization and recombination processes.  In the\npresent work, the confinement of positrons  in a chaotic  magnetic field is \nfirst investigated.  Positrons of different energies are followed through their\ninteractions over time, testing whether they become thermal or whether they\nremain nonthermal within the ever more diluted ejecta. \n The positron\nmean free path is very small as compared with the characteristic radius of\nthe supernova ejecta when the density of the ejecta is still\nhigh enough to produce large losses of the positron energy by ionization and\nexcitation, or  when  the presence of a   turbulent magnetic field inside the\nejecta confines the trajectories of the positrons along the winding field\n lines\nand induces a larger number of interactions. The mean free path of the\npositron becomes large when either the density of the ejecta is too low to slow\ndown the positrons or the energy density of the magnetic field is \nextremely low and therefore the Larmor gyroradius of the particle is a\nsizeable fraction of the radius of the ejecta. In the latter case the escape \nis enhanced. The distance travelled by the positron increases when a strong\nradial  magnetic field confines the positrons to move out in their helical\n motions along the radial field lines. \n\n\nIn the presence of a background chaotic magnetic field, positrons of energy\n$E_{i}$ born at a given radius r$_{i}$ (of mass coordinate $m_{i}$ and\nvelocity $v_{i}$) cannot slow down to thermal energies if they are\nemitted after a critical time $t_{i} > t_{c} (m_{i}, E_{i})$.\n The turbulent magnetic field confines the positrons at their site\nof origin, but as the ejecta expand and decrease in density, the\npossibilities for thermalization decrease. Thus, a fraction of the\n energetic positrons will not\nsuccesfully thermalize even under  confinement\nand survive in the ejecta as a ``fast'', nonthermal\npopulation.\n The critical time for\nthermalization  depends on the gradient of velocity along the ejecta, on the\nenergy of the positrons, and on their rate of energy loss. A useful \nexpression to evaluate such critical time is given by Chan \\& Lingenfelter\n(1993):\n\n\n$$t_{c}(m_{i}, \\gamma_{i}) = \\left[{8\\pi m_{e}cv_{sn}^{2}(m_{i})\\over M}\n\\left({dv_{sn}\\over dm}\\right)_{m_{i}} \\times \\int_{1}^{\\gamma_{i}}\n{\\gamma\\over \\Gamma(\\gamma m_{e}c^{2})\\sqrt{\\gamma^{2} - 1}}\\\nd\\gamma\\right]^{-1\/2}\\eqno(8)$$\n\n\\noindent\nwhere M is the mass of the ejecta, $v_{sn}(m_{i})$ is the velocity of the\nsupernova ejecta at $m_{i}$, $(dv_{sn}\/dm)_{m_{i}}$ is the velocity gradient\nat the location of $m_{i}$, and $\\Gamma(E)$ is the energy loss due to the\ndifferent processes. Chan \\& Lingenfelter (1993) included among those \nprocesses impact ionization and excitation, whereas we found that \none should include as well Coulomb scattering (Bhabha scattering\ninvolving e$^{+}$ e$^{-}$, in this case),\nsince this is also a major process in the deposition of energy of \n positrons. Our algorithm differs from that from those authors in the\ninclusion of this additional process as degrading the positron kinetic \nenergy, and also in the focus of the calculations: the main quantity\nfor light curve calculations is \nthe energy deposited in the supernova, instead of the\nenergy escaping as energetic positrons. \n\n\nIn the second configuration considered here, the confinement of\n positrons in a\nchaotic magnetic field is substituted for a different frame: the particles\ntravel along the lines of a radial magnetic field.  Again, $\\Gamma (E)$, the\nenergy loss function, and the mass of the ejecta will determine the \nfraction of kinetic energy that they deposit. The equation for the\n trajectory has to be\nsolved simultaneously with the energy loss equation.\n\n$$r = v_{sn} (m_{i}) \\  t_{i} + \\int_{t_{i}}^{t} c \\beta (t') cos [\\theta\n(m, t')] dt'\\eqno(9)$$\n  \n\\noindent\ngiven an initial mass coordinate $m_{i}$ and pitch angle $\\theta_{i}$. The\nchanges in pitch angles due to the gradient in B(r,t) outwards, favor a\nforward beaming of the positrons in the radial direction, even if they were\nemitted with $\\theta_{i}$ close to $\\pi\/2$ (Colgate, Petschek, \\& Kriese 1980;\nChan \\& Lingenfelter 1993).\n\nA last option is the\nabsence of any significant magnetic field able to affect the trajectory of\nthe particle. In that case, positrons are not confined to follow \nany trajectories and a treatment similar to $\\gamma$--ray transport \ncan  be used, adopting the appropiate absorption coefficient for the \npositron processes.\n\nThe default values given in the Tables for bolometric magnitude declines\ncorrespond to the chaotic field case, but decline rates in the absence of\nmagnetic field and in the case of a radial field will also be mentioned\nwhen comparing models with observations.\n\n\n\\section{ Time evolution of the bolometric magnitude}\n\n\n\\subsection{ C+O WD ignition and structure of the radioactive source }\n\n\n\nThe degree to which differences observed in \nthe evolution in luminosity of supernovae are linked to the distribution \nof radioactive sources and the kinematic structure of the exploded star has\nhardly been quantified. \nThe central ignition of a C+O WD with a mass close to the Chandrasekhar mass\nproduces a $^{56}$Ni distribution buried from the center up to variable\nmass fractions, depending on the characteristics of the burning front.  \n In the alternative  edge--lit \ndetonations of C+O WDs, \nburning starts at the outermost layers of the star and proceeds towards\nthe center. In those  explosions,  the radioactive material is found at two\ndifferent locations: very near to the surface, where the ignition started, \nand around the center where, after propagation of the burning front,\nthe densities  of the interior favor burning to NSE (Nuclear Statistic\nEquilibrium) products. \n\n  Within both frames for the explosion of a SNIa,\nvariations in the total mass of $^{56}$Ni and of its location in velocity\nspace are found, related to the extent to which the burning front incinerates\nthe material.  Classical Chandrasekhar central ignition models are able to\nincinerate 0.6 M$\\sb\\odot$ of the star to $^{56}$Ni.  The nucleosynthesis\n and density structure of the\nclass is well represented by model W7\n (Nomoto, Thielemann, \\& Yokoi 1984), where 0.63 M$\\sb\\odot$ of\n$^{56}$Ni are buried below the surface of 9000 km s$^{-1}$. This model is\nknown to provide a good spectrum for ``normal SNe Ia''.  In the case of\nChandrasekhar explosions, very $^{56}$Ni--poor explosions can also be found\nwhen the WD undergoes a pulsation that changes  the mode of propagation of\nthe burning front (Khokhlov 1991).  The pulsating delayed detonations can\nproduce a low amount of $^{56}$Ni in the center, which is buried  at low\nvelocities and very low mass fractions. An example of such model  is the here\ndepicted  model WPD1 (Woosley 1997), suggested to account for subluminous SNe\nIa.\n\nAmong sub--Chandrasekhar models, a range of possible explosions corresponds\nto the ignition of WDs of different masses (Woosley \\& Weaver 1994; Livne \\&\nArnett 1995). Results from 1--D and 2--D hydrodynamic calculations give\nsimilar final structures for the ejecta, and a whole range of possible\nstructures corresponding to the ignition of WDs of different masses. An\nexploded WD of mass $\\simeq$ 0.97 M$\\sb\\odot$ synthesizes the same amount of\n$^{56}$Ni as W7, but it contains this radioactive element also  in the\noutermost layers (model 6 by Livne \\& Arnett 1995, for instance).  The\ndetonation of a 0.7 M$\\sb\\odot$ C+O WD synthesizing about 0.15 M$\\sb\\odot$ of\n$^{56}$Ni corresponds to the  lowest end of possible WD masses able to\nexplode by edge-lit detonations (model 2 by Livne \\& Arnett 1995\nrepresents such a structure). It is a candidate to explain very subluminous SNe\nIa. On the highest end in luminosity, the detonation of a 1 M$\\sb\\odot$ C+O\nWD provides the largest amount of $^{56}$Ni ($\\simeq$ 0.97 M$\\sb\\odot$). As\nrepresentative of the top end,  we investigate a model by Nomoto (1994).\nTable 2 summarizes the characteristics of the models investigated here as\npossible structures of exploded WDs.\n\n\n\\subsection{ D$_{\\gamma}$(r) and $\\Delta {m_{\\gamma}^{100}}$ in Type\nIa supernovae }\n\n\nThe bolometric light curve in the phase where the $\\gamma$--rays fuel the\nluminosity is well described by the decrease between one hundred and \ntwo hundred days after\nthe explosion. We can define $M_{bol}^{100}$ as the bolometric magnitude at\n100 days after explosion and  $\\Delta {m_{\\gamma}^{100}}$ {\\it as the number of\nbolometric magnitudes declined per day after 100 days}.  During the period\nwhere $\\Delta {m_{\\gamma}^{100}}$ measures the decline rate\n due to the increasing transparency of\nthe envelope to $\\gamma$--rays, the SNIa models of lower mass do not\nexperience a larger change of magnitude  than the more massive ejecta.\nThis is due to the fact that the $\\gamma$--rays of the outermost $^{56}$Ni\nhave always escaped easily, and, on energy deposition effects they where not\nimportant. The inner structure of the deposition function shows a peak  at\nthe innermost radii as in the Chandrasekhar models, and the  density\nstructure is somewhat flatter in less massive WDs.  The luminosity of\nChandrasekhar models is, however, higher  than that of low--mass models\nbecause the total $^{56}$Ni mass being equal, the effectively buried\n $^{56}$Ni relevant for $\\gamma$--ray trapping is larger\n than in the edge-lit cases, and\nthe optical depth is larger (more mass).  Table 3 gives the values for\n $\\Delta {m_{\\gamma}^{100}}$, and absolute magnitudes for various\nmodels.\n\n\nFigure 1 shows by the example of subluminous SNe Ia resulting from the \nsub--Chandrasekhar explosion of a 0.7 M$\\sb\\odot$ (model 2 by Livne \\& Arnett\n1995) or from a pulsating delayed detonation (model WPD1 by Woosley 1997), the\nlevel of deposition of $\\gamma$--rays of the supernova at late \nphases. The model of pulsating delayed detonation achieves a higher\nluminosity than the sub--Chandrasekhar edge--lit detonation of a 0.7\nM$\\sb\\odot$ WD. The Figure also shows  how most of the emission comes from\nthe inner 20$\\%$  fraction of mass.\n\n\nIf $^{56}$Co would only give $\\gamma$--rays, we would never see the 50--80\n$\\%$ of mass fraction in SNe Ia at late phases. Since the\n$\\gamma$--ray--sphere (if we define it as the sphere which concentrates more\nthan  80$\\%$ of the deposition in energy) has shrunk down to the 20$\\%$\ninner mass fraction or even deeper, we would only see emission at very\nlow velocities. Due to the role of positrons and to their flatter deposition\nfunction, this does not actually happen. Positrons stop the drop in\nluminosity of the supernova ejecta.\n\n \n\n\\subsection{Decline in the positron tail}\n\nLeaving aside the discussion on the magnetic field intensity (it will be\naddressed in the next section), it is clear that some inner structures of\nexploded WDs are more favorable to trapping of positrons than others. The\neffect can start to become evident as early as 200 days after the explosion.\nFor larger velocity gradients, and $^{56}$Ni placed outside the inner\nregions, the escape of positron energy is enhanced. The escape strongly\ndepends on the velocity gradient along the ejecta and on the distribution of\nthe radioactive source. Better trapping structures  are centrally--ignited\nChandrasekhar C+O WDs, as compared with  edge--lit C+O WDs.  If we determine\nthe {\\it number of magnitudes declined per day in the bolometric light curve\nbetween 200 and 400 days},  $\\Delta m_{1 \\beta}$, {\\it and between  400\ndays and 1000 days}, $\\Delta m_{2 \\beta}$, a good clue as to the right\nmodel can be achieved. The expected values for some representative models are\ngiven in Table 3.  Table 3  compares the rate of decline between 200 and 400\ndays when positrons are the main energy source with that later on, between\n400 and 1000 days, when they might fail to fully deposit their energy,\ndepending on the post--explosion  structure of the supernova. This Table\nstresses the intrinsic differences due to the  kinematics and distribution of\nradioactive material in different models. The calculations have been done\nassuming confinement by B within the ejecta. In Section 5 we relax this\nrequirement in view of the existing observations.\n\nThe difference in deposition of positron energy at 300 days and 350 days for\nthe abovementioned SNe Ia models, if the magnetic field succeeds in providing\nenough trapping in the ejecta, is displayed in Table 4. As it can be seen,\nescape in sub--Chandrasekhar WDs can reach up to 20$\\%$ after 200 days. \nIn Figure 2\nthe deposition function of the $\\beta^{+}$ energy shows different behaviors\nfor the different models from 200 to 1000 days. In\na Chandrasekhar model like W7 even with a turbulent configuration of the\nmagnetic field, at 740 days after the explosion the positrons do not deposit\ntheir energy totally. However, the departure is only moderate ($10 \\%$).\n If the configuration of the magnetic field were\nradial, or the magnetic field intensity were very low,  significant\n departures from\nfull trapping could be achieved even earlier.\n\n\n \n\n\\section { Magnetic field of the WD: pre and post-explosion}\n\n\nEither full confinement of positrons in a chaotic magnetic field (Axelrod\n1980), or a enhanced escape through a radially combed out configuration\n(Colgate, Petschek, \\& Kriese 1980; Colgate 1991) have been considered as\nlimiting cases for the magnetic field in the SN ejecta. From \ncurrent knowledge of the pre--explosion structure of the WD, and following\nthe effects of expansion, we can reexamine these issues.\n\nSome WDs are known to host magnetic fields of intensity ranging between\n10$^{5}$ and 10$^{9}$ G (Liebert 1995). Such high magnetic fields are not\ncommon, however. Most WDs probably have fields below the detection limit of\n10$^{4}$--10$^{5}$ G. A number of studies suggest  that the configuration of\nthe magnetic field is generally  more complex than a dipole (non--centered\ndipolar geometry or quadripolar), and that the strength of the magnetic field\nmight be correlated with the mass of the WD in the sense of more massive\nWDs hosting larger magnetic fields. This last point, however, has not been\nestablished on firm statistical basis.\n\nPrior to the huge expansion induced by the explosion, however, the \nmass--accreting WD progenitor of the SNe Ia goes through a stage which \nmight increase the intensity of its initial magnetic field. \nThermonuclear runaway, which initiates the explosion, is preceded by a \nstage of quasistatic C burning that creates a central convective core\n(Woosley et al. 1990; Niemeyer \\& Hillebrandt 1995). As already found\nby Arnett (1969), when C burning accelerates after C ignition, electron\nconduction alone becomes unable to transport the ever larger energy flux\nand the temperature gradient becomes superadiabatic. Therefore, turbulent\nmotions develop, encompassing a sizeable fraction of the WD interior.\nThe interaction of the turbulence with the magnetic field should thus\nbe examined, in order to see whether an initially small $B$ might be \nsignificantly amplified during the steps immediately preceding the \nexplosion.\n\n\n\nA magnetic\nfield wound up by turbulence  increases with time, until the field strength\nbecomes strong enough to resist the turbulent flow. Since we are interested\nin weak fields, we may ignore here such backreaction by the Lorentz\nforces. The precise rate of increase of the field depends on the details of\nthe small--scale flow. For a flow dominated by a single length scale Kraichnan\n(1976) has derived exponential growth on a time scale of the order of the\nturnover time $\\tau$ of the flow. Thus, \nwrapping up of field lines by a small\nscale flow can enhance the intensity of the magnetic field exponentially:\n\n   \n$$ B_{\\rm seed} \\ e^{t\/ \\tau} {\\rm where} \\ \\ \\ \\ \\tau \\approx l\/v_{\\rm\nturb}\\eqno(10)$$\n\n\\noindent\n$\\tau$ being the typical turnover time of the convective cells, which is of\nthe order of the ratio of characteristic length of the turbulent region over\nthe turbulent velocity. If the duration of a turbulent period allows for a\nfew turnovers of the turbulent material  ($t$ larger than $\\tau$),  the lines\nof the magnetic field would be wrapped a few times, and  the intensity would\nincrease. This effect is not the classical dynamo where the field\nis amplified as cyclonic motions twist the lines of the magnetic field in a\nrotating fluid. The action examined here is linked to the turbulence--induced\nwinding up of the lines as the eddies carrying the seed magnetic field undergo\nseveral turnovers.\n\n\n{\\it Therefore, prior to explosion, during the\nquasistatic burning of C}, an enhancement of $B$ can occur. \nThe result depends on\nthe duration of the turbulent quasistatic phase.\n\n \n\n\nIf the turbulence  lasts for more than a few turnovers,\n equipartition of the kinetic energy\ndensity and magnetic field density could be achieved. In equipartition:\n\n$$ {1 \\over 2} \\rho v^{2}= {B^{2} \\over 8 \\pi}\\eqno(11)$$\n\n\\noindent\nTaking characteristic values for the turbulent kinetic energy density,\ni.e 10$^{11}$  erg  g$^{-1}$ and given that $\\rho$ is $\\approx$\n 2--3 \\ 10$^{9}$ g\ncm$^{-3}$, $B$ values for equipartition are $\\approx$ 10$^{10-11}$ G.\n\n\n\nThe convective phase prior to explosion is found \n in the work by Woosley (1990) \nto last for $\\approx$ 10$^{2-3}$ s. Turbulent velocities are   \nof the order of a 10$^{5-6}$ cm $^{-1}$ s $^{-1}$ and the\nconvective core \nis a fraction of the WD radius  of typical l $\\le$ 10$^{7}$ cm. This\nimplies turnover times of  $\\tau$ $\\approx$ 100 s.\nIn the work by Arnett (1996) and Bravo et al. (1996), the pre--explosion\nevolution of the WD during the accretion process is followed in detail \nseveral thousand years prior to explosion.  Both works find independently\nthat a convective core develops several thousand years prior \nto the explosion. Turnover timescales are of the order of 300s. The \nconvective period is of the order of 10$^{10}$ s, long enough, according \nto (10), to rise the magnetic field strength to equipartition values. \nSuch a convective core is linked to the evolution towards explosion of \ncentrally ignited Chandrasekhar C+O WDs. In edge--lit detonations\nof sub--Chandrasekhar WDs, there is no such development of convection\nin the C+O core in the presupernova evolution according to the \ncalculations by Nomoto (1982) and Hernanz et al. (1997), since \nin  this kind of explosion there is no strongly peaked central\nheating of the WD before the shock wave generated at the surface reaches\n the center and induces the explosion. \nTherefore, the initial field configuration in the core of the WD\n would not have been \nsignificantly distorted. The post--explosion configuration for the two\ntypes of explosion  would then be\ndifferent and its \neffect on the light curves can help to determine the SNIa mechanism. \n\n \n\nThus, depending on the evolution towards explosion,  \nthe dynamo might have had enough time to efficiently increase\nthe intensity of magnetic field by large factors, or fail to do so.  \nThe effects of a failure to increase drastically  the mean\nintensity of the field  will be reflected by the supernova light curve.\n\n\nIf this phase fails to develop an entagled field, the following phases \ndo not favor any major change. \n\n\n\n{\\it When the incineration starts}, the turbulent velocities increase to 10\n$^{7}$ cm s$^{-1}$, and  the characteristic size of the turbulent region\nis of the order of 10$^{7}$ cm (Niemeyer \\& Hillebrandt 1997). However, \nthis phase lasts only $\\approx$ 1\nsec, and it would not be able to provide a sufficient enhancement of the\nmagnetic field in the ejecta.\n\n\nAfter the explosion of the WD, the ejecta undergo a large expansion. The\nhomologous expansion achieved about 1 sec after the explosion suggests the\nfurther conservation of the magnetic flux (there is no compression which\nwould distort the number of  lines crossing a given element of area). In a\nhomologous expansion all components of the field decrease like:\n\n\n\n$${ B_{1} \\over B_{0}} = {(R_{0})^{2} \\over (R_{1})^{2} } = {(R_{0})^{2}\n\\over (v \\times t )^{2}}\\eqno(12)$$\n\n  \n\nThough the overall flow is nearly homologous on a large scale, there is \nlikely to\nbe some form of small scale motion inside the ejecta. For example, this could\nbe a remnant of the convective motions in the pre-SN stage, or the result of\na Rayleigh--Taylor instability at an early stage during the explosion. Such\nmotions wind up field lines, causing again a\n roughly exponential growth of the\nfield strength, in the kinematic (low field strength) limit. The question\nthus arises if such a process could increase the field strength over that\nexpected from a purely homologous expansion. We can now show that this is not\nthe case except in the unlikely event that the overturn time of the small\nscale motions is less than the expansion time scale.\n\nSmall scale motions generated at any time $t_0$ during the expansion expand\nwith the flow, so that their length scale varies as $t\/t_0$. The flow\nvelocity in these motions will remain the same or decrease in the presence of\ndissipation, hence the turnover time scales at least as $t\/t_0$.\n Hence in the absence of overall expansion,\nwe would expect the field to increase as ${\\rm d}\\ln B\/{\\rm d}t\\approx\n1\/\\tau$. Due to the expansion the overturning time varies as $\\tau=\\tau_0\nt\/t_0$, where $\\tau_0$ is the initial overturning time and $t_0=R_0\/v$ the\ninitial expansion time scale. Thus, because the overturning time increases\nwith time, the growth of the field is no longer exponential. The expansion\n(using eq 12) changes the field at a rate ${\\rm d}\\ln B\/{\\rm d}t\\approx\n-2\/t$. Adding these contributions, we get\n\n $$ {{\\rm d}\\ln\nB\\over{\\rm d}t}={1\\over\\tau}-{2\\over t}={1\\over t}\n\\left({t_0\\over\\tau_0}-2\\right)\\eqno(13)$$\n\nThis shows that the decrease of the magnetic field by expansion dominates \nas long as the initial turnover time of the small scale motions is \nsufficiently long compared with the initial expansion time scale. For an \ninitial expansion time scale of less than a second, this condition is easily \nsatisfied by any small scale motions in the pre--SN. We conclude that the \nfirst term on the right in (13) can be ignored, and that the magnetic \nfield decreases according to eq (12), even in the presence of small scale \nmotions in the expansion.\n\n\nWhatever magnetic field was present before the onset, at the quasistatic\nburning phase of C, will be directly left to the effects of the overall\nexpansion of the ejecta, which tends to lower its intensity.\n\n\n{\\it After the explosion}, the flux of the magnetic field would be preserved,\nand should be decreasing  with t$^{-2}$ as the supernova expands. \n{\\it At 100 days the intensity of the magnetic field would have decreased by\na factor of $10^{-15}$}.  At 1000 days it would be decreased by 10$^{-17}$.\nThe ejecta can thus host magnetic fields much lower than the intensity of the\ninterstellar magnetic field, if the magnetic field in the WD has not been \nsignificantly enhanced prior to explosion. \n Taking, for instance, an initial magnetic field of\n10$^{4}$ G, at 100 days it will be as weak as $10^{-11}$ G, and at 1000 days\nit will be $10^{-13}$ G.\nThe magnetic energy density will be among the lowest ones in\nknown astrophysical objects. The supernova becomes a huge de--magnetized\nbubble (with a material density much higher than the surrounding medium,\nthough).\n\nThe low values of B should, however, be compared with the\ndimensions which the ejecta have achieved.\n\n\nFor a charged particle of charge $q$, moving in a magnetic field, B, with\nvelocity $v$ whose component perpendicular to the direction of the magnetic\nfield is $v_{trans}$, the gyroradius (or Larmor radius) is:\n\n$$r_{\\rm gy} = {mc\\gamma v_{\\rm trans}\\over qB}\\eqno(14)$$\n\n\\noindent\nwhere $m$ is the mass of the particle and $\\gamma = [1 -\n(v^{2}\/c^{2})]^{-1\/2}$.\n\n\nFor a positron of energy 1 MeV,  the gyroradius would be:\n     \n$$r_{gy} = { 4.7 \\times 10^{3} {\\rm} \\over B}\\eqno(15)$$\n\n\nThus, the gyroradii of the e$^{+}$, as compared with the size of the\nenvelope would be:\n    \n$$ {x_{\\xi}} = {r_{gy} \\over R} =  { 4.7 \\times 10^{3}{\\rm cm} \\ B_{0}^{-1}\nR_{0}^{-2} \\ ( v \\ t)}\\eqno(16)$$\n\n\nThis is of the order of:\n\n$${x_{\\xi}} = 10^{-5} \\ \\ t_{7} \\ \\ B_{08}^{-1}\\eqno(17)$$\n\n\n\n\\noindent \nwhere $B_{08}$ is the magnetic field before expansion in units of 10$^{8}$ G\nand $t_{7}$, the elapsed time since the explosion in 10$^{7}$ s. \n Depending on the \n$B_{0}$ value, the positrons can be more or less trapped inside the ejecta.\nLess energetic particles have smaller gyroradius.  When the magnetic field \n has\ndiluted down to very low  values, the gyroradius  encompasses a high fraction\nof the expanded ejecta.\n\n   \nA pre--expansion  magnetic field of 10$^{11}$ G would prevent large\ndepartures from full--trapping of $^{56}$Co at phases even later than 1000\ndays after explosion. The confinement requirements are \nseen from equation (17). The available observations on the late bolometric\ndecline of the light curve allow us to estimate if  a departure from the\nfull-trapping decay line occurs in SN. {\\it This leads to an estimate of the\nintensity of the magnetic field achieved before the homologous\nexpansion.}\n\n\\bigskip\n\n\\noindent{\\it 4.1. Possible magnetic shield around the supernova ejecta}\n\n\\bigskip\n\n    \n The field strength in the ejecta \n may become \nsufficiently low after 100--1000d to allow the positrons to travel \nthrough the ejecta without interacting with the field.\n If the magnetic field of the WD is a dipole\n magnetic field, and the ISM around the ejecta is dense enough to \n present significant opposition \n to the magnetic field expansion, the conditions on final escape\n of positrons might change. \n\n\\smallskip   \n\n Before the particles can be regarded \nas having \nsuccessfully escaped from the ejecta, it must be shown that \nthey are not \nreflected back into the ejecta by an external medium of sufficient density. \n This external medium \nconsists \nof two regions. Immediately outside the ejecta is a magnetic `shell', a region \ndominated by the external magnetic field of the original pre-SN core, \nnow expanded \nbut still containing the original amount of magnetic flux. Outside this \nshell is the \nISM, modified by the SN shock that has passed through it. The magnetic \npressure in \nthe shell has to balance the ram pressure of the ISM relative \nto the ejecta. This \nyields its field strength:\n\n$$ B_{\\rm s}=v(4\\pi\\rho_{\\rm I})^{1\/2}=0.04 n_{\\rm I}^{1\/2}v_9 {\\rm G}\\eqno\n{(18)}$$\nwhere $v_9$ is defined such that \n $v=10^9v_9$ is the velocity of the ejecta and $n_{\\rm I}$ the\n ISM particle \ndensity. The flux of field lines crossing the magnetic equator \nis conserved during \nthe expansion, and is of the order $2\\pi B_0 R_0^2$, where  $R_0$ \nand $B_0$ are the \nradius and surface field strength of the WD.\n At time $t$, the \nthickness $d$ of the \nshell is therefore: \n$$ \nd=R_0^2B_0\/(vtB_{\\rm s})=2\\,10^7 {B_{04} R_9^2\\over\n n_{\\rm I}^{1\/2}v_9^2t_7}\\quad{\\rm \ncm} \\eqno{(19)}$$\nwhere $B_{04}$ is the\nmagnetic field in units of 10$^{4}$ Gauss, $R_{\\rm 9}$ is the radius of\nthe WD in 10$^{9}$ cm.\nThe gyroradius of a 1MeV positron in this field is:\n$$ {r_{\\rm L}\\over d}=5\\,10^{-3} B_{04}^{-1}v_9t_7\/R^2_9 \\eqno{(20)}$$\nSince the field is parallel to the interface with \nthe ejecta, the shell presents an \neffective `shield' which reflects the positrons. \n\n\\bigskip\n\n\nThus we may reasonable assume the shell around the ejecta to be a near \nperfect reflector. If the positrons inside it are unconfined\ndue to the absence of a tangled field, they will\nspread uniformly through the ejecta.  The mass of the ejecta is \nconcentred towards the innermost radii. If the half--mass radius is at a \nfraction $f\\sim 0.1-0.2$ of the radius of the envelope, the central density\nof the uniformly spread positrons is of the order $\\sim 2f^3$ times \nwhat it would be if the positrons stayed trapped near their source in the\nhigh--density regions. Since the luminosity is proportional to the \ndensity of positrons in the region containing most of the mass, the\nreduction of the luminosity is likely to be a large factor, even if none of\nthe positrons actually escape from the ejecta. \n\n\n\\bigskip  \n\n\n\n\\section{Bolometric declines and their interpretation}\n\n\n\n\\subsection{Physical conditions in the SN envelope and their effects\non the departures in the bolometric light curve}                        \n\n\n \n  Both a weak magnetic field and the progressive thinning out of the ejecta\n  produce a departure in the bolometric light curve of supernovae from \n  the full--trapping of the $^{56}$Co--decay energy. \n  There is no possibility of having 100 $\\%$ trapping of positron energy\n  as time goes by. The case where this departure occurs at the earliest, is\n  when there is no confinement of positrons at their site of origin, as\n  discussed before.  \n  If the positrons are freely streaming or escaping \n  through a radial magnetic field, the time \n  required by the relativistic  e$^{+}$\n  to cross the ejecta is shorter than any relevant timescale for modifications \n  in the physical conditions of the envelope, such as the\n  the radioactive timescale  for $^{56}Co$--decay (111.26 days) \n  or the expansion timescale ($n \/ {\\dot n} = t\/3$  days), both being \n  of the order of 100 days. \n  In the free--streaming condition, \n  the moment when the departure occurs is early enough to ensure that any \n  deposited energy is radiated in a short timescale through collisional\n  excitation and emission in a large number of forbidden transitions of \n  iron ions. Freeze--out conditions for re--radiation of that energy occur\n  much later.  \n\n\n\\smallskip\n\n In the confinement phase, the\n  positrons do not move from their site of origin. There is a time when the    \n  probability of interaction with the ions through impact ionization and\n  excitation becomes very low. Although the follow--up of the\n  deposition of energy by the energetic particles involves to keep \n  track of the old positrons during the whole expansion history of the envelope \n  while injecting the new ones at each site, the fact of neglecting them \n  when they start to become very \n  inefficient, i.e  after $ t > t_{c}$ (see section 2),  \n  is a fair approximation. The\n  probability for interactions decreases with t$^{-3}$ and the contribution of\n  all those positrons in the diluted medium is much smaller than before, at\n  $ t > t_{c}$. At the latest times, \n  the  quantitative prediction can underestimate somehow the luminosity. \n  The estimate of the time at which the departure\n  occurs according to the physical SN model and magnetic field configuration,\n  is, however, very precise. \n \n\\bigskip \n \n\\noindent{\\it Timescales  and expected departures}\n\n\n\\noindent\n The frequency of impact ionizations or excitations by the \n $e^{+}$\n becomes lower as $n_{e}^{+}$, the density of \n the energetic positrons, and that of the target ions decrease.\n\n  \n\\noindent\n The timescale for impact ionizations by positrons, $\\tau_{e^{+} coll}$,\n can be expressed as: \n\n$$\\tau_{e^{+} coll} = \\left[\\int_{E_{min}}^{E_{max}} n_{e^{+}} (\\tilde E)\n\\int_{0}^{\\tilde E} \\sum_{ij} \\sigma_{ij} (E)\\ v_{\\!E}\\ f_{ij}\\\ndE\\right]^{-1}\\eqno(21)$$\n\n\\noindent\nwhere  $n_{e^{+}} (\\tilde E) $ is the number density of positrons \nof a given energy $\\tilde E$ originated in the $^{56}$Co decay, \n $\\sigma_{ij}$ are the impact ionization cross sections with each \n target ion $\\it i$ of species $\\it j$ and $f_{ij}$ are the relative \n abundances of those ions. $\\rm E_{min}$ and $\\rm E_{max}$ are the \n minimum and maximum kinetic energy of the positrons, and $\\rm v_{\\!E}$\n the velocity. $n_{e^{+}} (\\tilde E) $ is much lower than $n_{e}$, \n the electron density.  Such timescale increases due to the \n thinning out of the ejecta, and at a given point it becomes \n larger than the radioactive \n timescale  and the expansion timescale.\n\n Equivalently, the timescale for the positrons\n to lose half of their energy, $\\tau_{e^{+} loss}$, becomes\n also large (this quantity is related to the stopping distance \n of the positron, which grows with time, see Table 1): \n\n   $${\\tau_{e^{+} loss}} \\sim  { E  \\over   \\dot E}  \\eqno(22) $$\n\n\n\\noindent\n  In {\\it the confinement regime}, the rate at which interactions occur \n is favored by the presence of a uniform density of target ions\n along the positron path. The departure occurs much later than in\n {\\it the free--streaming regime} for all models. \n  This departure signals a point of ``breakout\n of nonthermal ionization balance'' or  ``non--steady state for the \n nonthermal processes''. However, the collisional processes and radiative\n transitions between levels still occur at a fast rate.\n Reemission is occuring through the large number of forbidden transitions\n of iron ions. The deposited energy is reemitted, to a very \n high degree, in steady state. \n\n\\bigskip\n \n\\noindent\n In the {\\it positrons free streaming regime}, \n $\\tau_{e^{+} loss}$ for the most energetic\n positrons is longer than \n the crossing time of the envelope. \n Those positrons, which are not confined, \n will escape with high kinetic energies. The free streaming favors much\n earlier departures from full--trapping of the $^{56}Co$, at epochs when\n the recombination and collisional processes still occur at high rates. \n \n\n\\bigskip\n\n\n\\noindent\n{\\it The infrared catastrophe: freeze--out of the supernova ejecta}\n\n\\noindent\n The observational requirements to \n extract information from the bolometric light curve of a supernova \n for the prospects given in this work are\n to rigurously reconstruct this light curve placing estimates\n on the infrared emission, and to  complement this task with a spectrum\n at the time where a change in the decline rate is observed. \n Here, \n we present \n our predictions to be compared with future observational data. \n By obtaining those data, it should be possible to \n distinguish observationally between the various effects entering \n in the bolometric light curve decline.\n\n\\bigskip\n\n\\noindent\n Along this work, when addresing the departure in the luminosity, we \n assume that the emission at ultraviolet, optical and infrared\n broad bands is recovered, and \n an estimate of the temperature or evaluation of  \n how much luminosity has gone into far infrared wavelengths has been done. \n \n\n\\bigskip\n\n\\noindent\n  Axelrod (1980) first pointed out that the supernova ejecta in their\n  late--time evolution would reach a point when temperatures would \n  fall below a critical temperature,\n  T$_{c}$ $\\simeq$ 2000 K, in their innermost layers. When this occurs,\n  most of the emission of the supernova, would come out in the\n  fine structure forbidden transitions at infrared wavelengths. \n  This is named as the Infrared Catastrophe (IRC) since it will imply an\n  inflation of the emission at very long wavelengths\n  while a depletion in the optical and ultraviolet emission occurs. \n  It is possible to calculate for each model when $T_{\\it core}$ (in the \n  innermost dense ejecta)   \n  falls below T$_{c}$, and determine it as well observationally.\n   \n\\bigskip\n\n\\noindent\n  The departure, when an IRC occurs, affects the B, V, R monochromatic light \n  curves, but it is a temperature effect and does not imply a proper \n  departure of the overall emissivity. \n\\bigskip\n\n\n Limitations in the accuracy of the results of the following sections\n arise from the uncertainties \n in the observations and\n reconstructions of those bolometric light curves (if a limited number of\n photometric data are available) and from the growing time--dependence\n of the reemission processes well after two years.    \n The different predictions related to the two evolutionary \n histories of centrally ignited Chandrasekhar WDs and edge--lit \nlow mass C+O WDs seem worth testing.   \n  Combining the analysis of the\n $\\gamma$--ray tail and the positron tail helps to complement information \n on the physical models. This information will be addressed in the next \n sections. \n \n\n\\subsection {Physical models and the rate of the late decline of SNe Ia}\n\n As a general\ntrend,  C+O centrally ignited Chandrasekhar WDs tend to trap\nsignificantly the positrons and give a bolometric light curve decline close to\nthe full trapping line drawn by the exponential decay of $^{56}$Co. The\nbolometric light curves of sub--Chandrasekhar models tend to fall below the\nfull--trapping line after 400 days even if $B$ confines the e$^{+}$, or even\nearlier in massive edge--lit detonations (model NIDD by Nomoto 1995). A\nfollow--up of those bolometric light curves is a good tool to clarify the\nnature of the explosion. The bolometric light curve in the earlier $\\gamma$--ray\ndominated tail (before 200--300 days) is different for those models and\nallows a first discrimination. The positron tail informs further about the\nejected mass and the magnetic field configuration. \nIn Figure 4 we present M$_{bol}$ decline rates \nfor different models during the first 400 days under the\nconfinement hypothesis. This sort of figure can be useful for \ncomparison with observations. \n  Departures from the full--trapping of\n$^{56}$Co--decay  of the order of 10--15$\\%$ at about 400 days can be explained\nby the distribution of radioactive material. Larger departures, of \n30--40 $\\%$ or\nlarger,  have to be interpreted in terms of lack of magnetic field\nconfinement of the positrons, or even enhancement of the escape.\n\n\n\\subsection{Trapping and departure: the magnetic field and the \n mass of the ejecta}\n\n  \nA way to evaluate from its physical effects the real effectivity of positron\ntrapping is to compare the calculations with observed bolometric light curves\nof SNe Ia. Very few bolometric light curves of supernovae are, however, \navailable. For SN 1972E in  NGC 5253, about two years of  bolometric\nfollow--up after the explosion is provided by Kirshner \\& Oke (1975).  \nSince this exceptional long--lasting coverage, the general trend has\n been to concentrate \nthe observations of supernovae  to the first year after\nthe explosion. Suntzeff (1996) obtained the \nbolometric light curve of SN 1992A \n up to 300 days after explosion. More recently, the \nfast declining bolometric light curve of SN\n1991bg covering the first two hundred days after explosion  was \npresented by Turatto et al. (1995).\n\n\n    \nWhat can we learn from the observations? SN 1972E  might represent a SN Ia\nclose to ``normality'' in its luminosity and spectral characteristics (i.e.\nsimilar to SN 1981B, SN 1990N). The spectral scans obtained by Kirshner \\&\nOke (1975) and integrated along wavelength by Axelrod (1980) provide a\nbolometric (or quasi--bolometric) light curve which can usefully be compared \nwith model calculations. \nThe distance to the supernova is known from the Cepheids period--luminosity\nrelationship (Sandage et al. 1994), and it is known that the supernova\nwas not substantially reddened ($ E(B-V) \\approx 0.05$). We scale the\nabsolute luminosity values according to the known distance \nto NGC 5253 and a low E(B--V) (Axelrod had assumed E(B--V) =0.22), \nand compare them with the model predictions. Figure 3 shows that the \nSN 1972E bolometric light curve follows\nwell  the behavior of centrally ignited C+O Chandrasekhar WD with a\nturbulent magnetic field configuration. In particular, model W7 seems \nto be giving a very good\naccount of the bolometric light curve.  The turbulent configuration of the\nmagnetic field is thus favored by the level of deposition of energy  suggested\nby  the bolometric curve of SN 1972E prior to 500 days. A lack of  magnetic\nfield as well as a radial strong magnetic field would produce larger\ndepartures from the full--trapping $^{56}$Co--decay line.\n\nThe model light curve follows well the observed one until at least 500 days.\nThen, a departure at 720 days after maximum is observed, of the order of  70\n$\\%$ enhancement of escape as compared with the confinement prediction (only 30\n$\\%$ of $^{56}$Co positron kinetic \nenergy is deposited). The confinement prediction gives\nonly 10 $\\%$ of departure at 740 days for model W7. Taken at face value, \nthis reported departure, and the \nelapsed time in the\nconfinement regime (in the case of SN 1972E up to 500 days according to our\nanalysis), tell us  \nabout the magnetic field intensity prior to explosion, as\ndiscussed above.  According to equation (17), the confinement of positrons \nstarts\nto fail at about 700 days for WD magnetic fields of $B \\approx 10^{5} G$, a\nplausible value. If the magnetic field intensity of the initial WD\nwould have been lower, the departure would have occured earlier. A very \nmagnetized WD prior\nto explosion (B $\\ge$ 10$^{10-11}$ G) would not give any\nsignificant departure until much\nlater on. Unfortunately, it can not be discarded that the very \nlast point in the light curve is more unaccurate\nthan the rest of the data (Kirshner 1997), \nand that the bolometric light curve keeps \nfalling not far from full--trapping. If that were the case, and the bolometric\nlight curve would follow within a 10$\\%$ of departure the \nfull--trapping curve, that would\nconfirm the theoretical expectations of a chaotic magnetic field enlarged\nup to equipartition during the long convective accreting period expected \nin centrally ignited C+O WDs. The safest conclusion given here is that\n{\\it SN 1972E represents the case of a centrally ignited C+O WD with a likely\ntangled magnetic field of at least 10$^{5}$G prior to the expansion}. The \nlight curve and luminosity is well reproduced by model W7. \n\n\nFuture observations, describing the bolometric light curves of SNe Ia, should \nprovide information about the post--explosion magnetic field. Strong\ndust obscuration can cause deviation in the bolometric declines, but it is\naccompanied by shifts in the centroids of the emission lines at late phases.\nThus, there is a way to point out when dust obscuration occurs. The light\ncurve can be corrected by observing the far infrared and including the energy\nemitted at those wavelengths. The bolometric light curve of SN 1992A seems\nto follow the same decline rate as SN 1972E, although data are only\navailable up to 300 days. Given the uncertainties in the distance to this\nsupernova, we have shifted arbitrarily in the figures\n the absolute scale of the luminosity\ngiven by Suntzeff (1996).\n   \n\n\\subsection{ A fast decaying bolometric light curve}\n\nA much faster decline than in SN 1972E is seen in SN 1991bg. Observations\nwere presented in terms of the {\\it uvoir} bolometric light curve \nby Turatto et al.(1995). They followed the same procedure to \nintegrate the luminosity in the\ndifferent bands as Suntzeff (1996) for SN 1992A. Light from the\nfar--infrared is not included in their luminosity count. However, {\\it JHK}\nobservations by Porter et al. (1992) showed a fast decline of 3 mag in the\nfirst month and no secondary maximum. This suggested that the supernova was\nnot emitting strongly in the infrared.\n\n\nCalculated bolometric light curves are displayed in Figure 5 and compared\nwith SN 1991bg. The decline after 200 days (and even earlier) is faster than\npredicted for confinement by a chaotic magnetic field in a wide variety of\nmodels. The early decline shows that the two opposite models proposed to\nexplain this supernova: an edge--lit detonation (Livne \\& Arnett 1995) and a\npulsating delayed detonation model in a Chandrasekhar--mass WD \n(Woosley 1997) fail\nto give the right luminosity and evolution in time of the bolometric light\ncurve in the $\\gamma$--ray dominated phase already. Shifting the absolute \nmagnitude\nof both models to agree with the observed one would imply as well uncomfortable\ndistances to the core of the Virgo Cluster (well beyond the current\ndiscussions on it). The bolometric light curve of SN 1991bg requires a \nsmall mass of\n$^{56}$Ni, of the order of 0.07 M$\\sb\\odot$, as found by spectral modeling\n(Ruiz--Lapuente et al. 1993). In addition to requiring a small mass of\n$^{56}$Ni, further considerations are needed to explain the unusual late\nbehavior.  At day 200 there is a departure from the confinement prediction: \nthe deposition is only 50$\\%$ of what would be expected in the confinement\ncase. In this case the observational basis for such departure is firmly\nestablished. Different options have to be considered: 1) a low magnetic\nfield of the original WD precludes confinement. To evaluate this option,\npositron escape in the absence of a magnetic field is calculated for both a\nChandrasekhar and a sub--Chandrasekhar explosion model (model W7 and model\nWD065 of the detonated WD of 0.65 M$\\sb\\odot$). 2) A radially combed--out\nmagnetic field enhances escape in a low--mass WD explosion and in a\nChandrasekhar WD explosion (same models and model 2 of the edge--lit\ndetonation of a 0.7 M$\\sb\\odot$ WD). None of the hypotheses combining a\nChandrasekhar mass model and enhanced escape can account for a 50$\\%$ of\ndeposition of the $^{56}$Co--decay energy at 200 days. As shown in Fig. 5,\nthe option of lack of a significant magnetic field  and a small ejecta mass,\nas in model WD065, gives a reasonable agreement  with the observations. In\nthe absence of a magnetic field, positrons lose their energy according to the\ninteractions undergone along their free trajectories. The sort of\ncalculations done here rescale the $\\gamma$--ray results to an opacity\nappropiate for the processes undergone by the e$^{+}$: $\\kappa_{e} \\sim 10\\\ng^{-1}\\ cm^{2}$ (Axelrod 1980; Colgate, Petschek, \\& Kriese 1980). The\ncalculation then reproduces the observed decline rates.\n\nThe picture described here of positron transport in the ejecta\ncorresponds to  zero confinement or negligible action of the magnetic field.\nSuch condition occurs when the magnetic field intensity of the WD prior to\nexplosion is lower than 10$^{3-4}$G. Thus, this looks like a plausible\nexplanation for SN 1991bg: {\\it low--mass and weak magnetic field prior to\nexpansion. Lack of confinement but a Chandrasekhar--mass explosion gives a\nlight curve \nmuch closer to full-trapping and it departs much later than\nobserved}.\n\nThe option of the enhanced escape through radial magnetic fields had to be\nevaluated by integrating the trajectories of the positrons. Escape occurs in\nChandrasekhar--mass explosions at a level lower than observed. Low--mass and a\nradially combed but strong magnetic field is thus another possible explanation,\nalthough less likely from the implications of evolution in time of the\nmagnetic field. \n\nTo summarize, the bolometric light curve of SN 1991bg can be well accounted\nfor if positrons are not confined by the post--explosion magnetic field,\n due to a low initial magnetic field or\na radially combed--out magnetic field in low--density ejecta (small mass). \n The observed\nlight curve seems  hard to fit with Chandrasekhar--mass WD explosions, \nsince even for the most favorable configuration of the magnetic field\nto enhance escape, a Chandrasekhar mass of ejecta would be enough to produce \nsignificant deposition\nof  $\\beta^{+}$ energy.\n\n    \n\n\\subsection{ Bolometric light curves and the mass in Type Ibc}\n\n \nThe mass of the star at the time of the explosion in Type Ibc is a matter of\ndiscussion, and it is linked to the identification of their progenitors. The\nwell observed bolometric light curve of SN 1994I (Richmond et al. 1995)\nallows us to discuss models for Type Ibc SNe as compared with the\nobservations.   The precursors of Type Ibc could be Wolf--Rayet stars, with\nmain sequence masses in the range of 30--40 M$\\sb\\odot$. Those stars undergo\nstrong winds and also mass transfer, if they are in binary systems. Mass\ntransfer and winds might produce the loss of the  H envelope in the star\nwithout removing completely the He envelope.   After undergoing gravitational\ncollapse at the end of their evolution,  a total ejected mass close to  2\nM$\\sb\\odot$ is expected from this massive progenitor case (Woosley,  Langer,\n\\& Weaver 1996).  In other scenarios, the initial mass of the progenitor is\nsmaller--i.e. in the range of 10--20 M$\\sb\\odot$--, and the star ends up its\nevolution, after having lost both the H envelope and the He mantle, as a bare\nC+O core. The ejecta mass could be below 1 M$\\sb\\odot$.  Both $\\gamma$--ray\nand energy deposition by positrons should be different in the two cases.\nSpectral calculations show the need of enhanced mixing in SN 1994I, and in\nother SNe Ic (Eastman \\& Woosley 1997; Ruiz--Lapuente 1997). Large--scale\nmixing, required for  SNe Ic, will affect very much the deposition of energy\nby positrons, and thus the bolometric  luminosity.  Figure 6 shows the\ndifference that mixing induces in the deposition of $\\gamma$-rays in Type Ibc\nmodels. Mixing enhances  as well the escape of energy from  positrons. In\nejecta with masses lower than 1 M$\\sb\\odot$ it leads to a departure from the\nfull--trapping curve of $^{56}$Co decay.  Figure 7 shows the deposition of\nenergy from e$^{+}$ and its evolution in time for different models of SNe Ibc\nand the chaotic configuration of the magnetic field. Model 7A corresponds to\nthe more massive progenitor option mentioned above, and model 7A mixed has\nthe  same ejected mass, but with large--scale  mixing  (Woosley, Langer, \\&\nWeaver 1996; Eastman \\& Woosley 1996), as required from the spectra. Model\nCO21 (Nomoto et al. 1996) corresponds to the less massive progenitor, and a\nmixed version has also been calculated.\n\nIt can be seen, that the positron energy is not deposited in the ejecta of\nexploded C+O stars already at 200 days after explosion. A more massive\nejecta than 1 M$\\sb\\odot$ seems to be \nrequired to produce effective trapping of the energy\nfrom $\\gamma$--rays and positrons and preclude a fast decline of the\nluminosity. As shown in Figure 8, such requirement gives a good account \nof the observed bolometric light\ncurve of SN 1994 I.\n\n\\section{ Positron escape and the Galactic 511 keV line}\n\nThe positron annihilation radiation towards the Galactic center (Haymes et\nal. 1975) is believed to originate from two contributions: a time--variable\ncompact source located in the Galactic center and a diffuse component along\nthe Galactic plane. Supernovae have been identified as a likely origin for\nthe diffuse component: positrons escaping from supernovae and annihilating in\nthe surrounding regions would give rise to that emission (Lingenfelter \\&\nRamaty 1989). The width of the diffuse 511 keV radiation places strong\nconstraints on the temperature and density of the region where the\nannihiliation takes place (Ramaty \\& M\\'ez\\'aros 1981; Guessoum, Ramaty \\&\nLingenfelter 1991; Wallyn et al. 1993). It is found that electron--positron\npairs need to lose their energy in a dense medium before annihilating, or the\n511 keV would be broadened and blueshifted (Ramaty \\& M\\'ez\\'aros 1981). Our\nfindings about positron confinement suggest that the SN ejecta are a first\nsite where positron can be confined, as the ejecta evolve into the remnant\nphase. The way in which the nonthermal positrons are retained in the\nincreasingly diluted ejecta until they escape to the neighbouring ISM depends\non the intensity and configuration of the WD magnetic field prior to\nexplosion, which is determined by the WD evolutionary path.  Type Ia\nsupernovae exploding as centrally ignited Chandrasekhar WDs would favor\nconfinement through a chaotic magnetic field, whereas  sub--Chandrasekhar\nedge--lit WDs present an environment more favorable to escape of positrons\nfrom their site of origin, although those particles find in the region of\ninteraction between the supernova and the interstellar medium a shield\nprecluding further escape.\n\n\n\n\\section{Conclusions}\n\nThis work  has shown that the bolometric light curves of \nSNe Ia trace a\npoorly investigated property of the supernova progenitors: their magnetic\nfield. Through a well tracked departure from the full--trapping curve of\n$^{56}$Co decay, insights on the pre--expansion magnetic field of the star\ncan be obtained. It can be investigated whether the convective turbulence\nprevious to the explosion in accreting WDs succeeds in amplifying the mean\nintensity of the original magnetic field of the WD by winding up the magnetic\nfield lines. Or whether, on the contrary, the original WD magnetic field\nprevails without growing significantly before the explosion and its intensity\nsimply decreases as the supernova expands. The consequences of these two\nextreme hypothesis have appreciable different impacts on the supernova \nluminosity. It is possible that Chandrasekhar WDs develop a highly tangled\nmagnetic field which would favor a bolometric light curve close to \nfull--trapping of $^{56}$Co decay positrons energy. The important phase\nwhen this can occur is the period when accretion and gain in mass of the\nC+O WDs lead to a compression and quasistatic C burning in the center. A\ncentral convective core is developed several thousands years before \nthe explosion. This should be a distintive signature of C+O accreting WDs\nprecursors of centrally ignited Chandrasekhar explosions. It is \nnot expected to occur in sub--Chandrasekhar WDs ignited through He detonations.\nIn those explosions, the initial magnetic field of the WD would have \npreserved its initial configuration enhancing the escape of the energy\nof the positrons.  \n\nOn the other hand, differences in the distribution of radioactive material in\nvelocity space and in the mass of the exploding WDs give rise as well to\ndifferent declines in the late--time bolometric luminosity of SNe Ia. A\ntabulation of typical decline rates resulting from different explosion\nmechanisms is given to facilitate comparisons with observations.  Three\ndifferent epochs can be considered in the deposition of energy from\n$^{56}$Co: a first phase where $\\gamma$--ray deposition is sustaining the\nearly $^{56}$Co tail. A second phase where the $\\gamma$--ray contribution is\nstarting to become negligible and $\\beta^{+}$--rays start to provide the\nluminosity. In this second phase the density of the ejecta  (provided that\nthe ejected masses are close to 1.4 M$\\sb\\odot$) is still high enough to\nensure the effectivity of positron energy deposition. And a third phase in\nwhich the density of the ejecta is not high enough to trap significantly the\npositron energy and the configuration and intensity of the magnetic field\ndetermines the fate of the released energy. A very wide difference in the\ntails of the bolometric luminosity at this phase could easily be linked to\ndifferent intensities of the magnetic field previous to the enormous\nexpansion resulting from the explosion. If a wide diversity (larger than $30\n\\%$) is found even earlier, in the second phase --i.e, as soon as\n$\\gamma$-rays become a negligible energy contribution--, since at that time\nthe density of the ejecta should still be high enough to slow down the\npositrons, the departure points towards an enhanced escape favored by the\npreserved but expanded dipole structure of the original $B$ of the WD, and,\nalso to possible low mass of the ejecta in case of a extreme escape occuring\nas early as in SN 1991bg.  A lesser degree of diversity (lower than $10-15\n\\%$ in deposition) in the bolometric decline at this intermediate phase could\nbe interpreted as differences in the kinematics and $^{56}$Co--distribution\nin the supernovae.\n\nOur analysis of SN 1972E suggests that full--trapping lasted at least\n400--500 days. The Chandrasekhar model W7 accounts well for the overall shape\nof the bolometric light curve. The  decline of the supernova so close to\nfull--trapping and the late departure taking place at about 740 days after\nexplosion indicates that a tangled magnetic field of at least 10$^{5}$G had\ndeveloped in this supernova. In the case of SN 1991bg, the early  and large\nescape of energy suggests a lower $B_{0}$ previous to explosion (lower than\n10$^{3-4}$ G),  or a dipole magnetic field expanded towards a radial\nstructure.   Whereas in SN 1972E there is no evidence pointing towards a mass\nlower than the Chandrasekhar mass, in SN 1991bg even within magnetic field\nconfigurations maximally favoring escape, a Chandrasekhar--mass WD would\nstill show larger trapping of radioactive energy than observed. A very good\nagreement with the observations of SN 1991bg  is found if confinement is\nnegligible and the total mass of the ejecta is {\\it a half of a\nChandrasekhar mass}.\n\nOn the other hand, in determining the evolution of the bolometric luminosity\nof core--collapse supernovae (coming from massive stars) and their ejected\nmass, mixing plays a fundamental role. A study of the mixing in each type of\ncore--collapse SN through spectral modeling is needed before deriving any\nconclusions on the ejected mass and on the amount of $^{56}$Ni synthesized in\nthose explosions. The mass of the star at the time of the explosion in Type\nIbc is a subject of discussion, even in such well--observed cases as SN 1994\nI. Examining the bolometric light curve of this supernova, and the mixing\nconstraints from the spectra, a better agreement with the more massive\nprogenitor is suggested. The diversity in the bolometric luminosity of Type\nIbc can be linked both to mixing and ejected mass differences among the\nexploded stars giving rise to this supernova class. A longer follow--up of\ntheir bolometric light curves will help to determine the trapping of\n$^{56}$Co energy and to clarify the actual mass range of the stars which\nexplode as supernovae of different types.\n\n\\bigskip\n\nP.R.L thanks the very estimulating discussions with Wolfgang Hillebrandt in\nrelation to this work, and useful information provided by him, Dave Arnett\nand Jens Niemeyer on convection in WDs. Thanks go as well to Peter Milne for\ninteresting exchanges on positron transport in supernovae. Financial support\nfor this work has been provided by the Spanish DGICYT.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $(X^n,Y^n)$ be a random data sequence where $X$ and $Y$ represent the public and private sections of the data respectively, and are drawn from an i.i.d. distribution $P_{X,Y}$. Each entry $(X_i,Y_i)$ represents a row of the dataset. We wish to find a privacy mechanism, i.e. a random mapping, that reveals a sequence $\\hat{X}^n$ such that (i) statistical information about $X^n$ can be learned from $\\hat{X}^n$, and (ii) as little information as possible about private data $Y^n$ should be revealed by $\\hat{X}^n$. These two goals are in conflict, since typically $X$ and $Y$ are correlated (especially when $X=Y$). Thus, we wish to characterize the privacy utility tradeoff (PUT) while being careful to choose meaningful utility and privacy metrics.\n\nOur focus is on inferential adversaries that can learn the hidden features $Y^n$ from the released dataset $\\hX^n$. To this end, we motivate the use of mutual information between the private features $Y^n$ and the revealed version of the dataset $\\hX^n$ as a metric for privacy leakage. Mutual information as a measure captures an adversary that refines its posterior belief of the private data from the released data, i.e., the adversary's loss function is the log-loss function \\cite{CalmonFawaz2013}. Indeed other measures such as maximal leakage \\cite{Issa}, maximal correlation \\cite{Calmon2013}, and the recently introduced alpha leakage \\cite{Liao} can capture the adversary's ability to learn any function of the dataset. However, we restrict our focus here to a belief-refining adversary.\n\n\nFor the choice of utility metric, the average distortion constraint in the form of $\\mathbb{E}[d(X^n,\\hX^n)] \\le D$ has been used in many works, where $D$ is a distortion threshold and $d(\\cdot,\\cdot)$ is a given distortion function between public data and released data. However, this utility metric does not capture all aspects of distortion distribution.\nOne possible step in order to capture more aspects of the distortion distribution, is via the \\textit{tail probability constraint} (or equivalently called \\textit{excess distortion constraint}). This has been of much interest in source coding (see for example \\cite{Yeung,Kostina,Kostina2012c,YannisVerdu,Shkel}), channel coding (see for example \\cite{YucelISIT2015,YucelAllerton2015,YucelISIT2016}) and studied in the context of privacy in \\cite{KKSAllerton2016}. For a more detailed survey on finite blocklength approaches see \\cite{tan2014asymptotic}.\n\nHowever, even the tail probability constraint does not capture the full spectrum of possibilities on applying bounds on distortion distribution. In this paper, we generalize the tail probability constraint in two ways:\n\\begin{itemize}\n\t\\item A bound $t$ on the average distortion cost, where the distortion cost is a non-decreasing function $f$ applied on a separable distortion measure $d$ between $X^n$ and $\\hat{X}^n$. The resulting PUT is given by\n\t\\begin{equation}\\label{eq:optimization} \n\t\\begin{array}{ll}\n\t{\\substack{\\text{\\normalsize{minimize}}\\\\ P_{\\hX^n|X^n,Y^n} }} & \\displaystyle  \\frac{1}{n}I(Y^n;\\hat{X}^n)\\\\\n\t\\text{subject to} & \\mathbb{E}[f(d(X^n,\\hat{X}^n))]\\le t,\n\t\\end{array}\n\t\\end{equation}\n\t\n\t\\item A non-increasing function $g$ to bound the complementary CDF of the distortion measure $d$ between $X^n$ and $\\hat{X}^n$. The resulting PUT is given by\n\t\\begin{equation}\\label{eq:optimization 2} \n\t\\begin{array}{ll}\n\t{\\substack{\\text{\\normalsize{minimize}}\\\\ P_{\\hX^n|X^n,Y^n} }} & \\displaystyle  \\frac{1}{n}I(Y^n;\\hat{X}^n)\\\\\n\t\\text{subject to} &  \\mathbb{P}[d(X^n,\\hX^n) > D] \\le g(D), \\forall D.\n\t\\end{array}\n\t\\end{equation}\n\\end{itemize}\nThe cost constraint in \\eqref{eq:optimization} imposes increasing penalties on higher levels of distortion in general, and reduces to a tail probability constraint when $f(D)=\\boldsymbol{1}(D> D_0)$, for some constant $D_0$. The distortion distribution bound in \\eqref{eq:optimization 2} allows arbitrarily fine-tuned bounds on the complementary CDF of the distortion, and reduces to a tail probability constraint when $g(D)= 1 - (1-\\epsilon) \\boldsymbol{1}(D\\ge D_0)$, for some constant $D_0$. Note that these two types of constraint are not equivalent in general and can capture different requirements on the distortion distribution.\n\\subsection{Contributions}\nA privacy mechanism could be applied to a dataset as a whole, or to each individual entry of the dataset independently. We label the mechanisms for the two approaches as \\textit{general} and \\textit{memoryless} mechanisms. In this paper:\n\\begin{itemize}\n\t\\item We derive precise expressions for the asymptotic leakage distortion-cost function tradeoff in \\eqref{eq:optimization}. For memoryless mechanisms, it is equal to the single letter leakage function evaluated at the inverse of the cost function applied to the cost threshold $t$, and for general mechanisms, it is the lower convex envelope of the leakage tradeoff curve under memoryless mechanisms.\n\t\\item We also give the exact formulation of the asymptotic leakage in \\eqref{eq:optimization 2} for memoryless and general mechanisms. For memoryless mechanisms, it is equal to the single letter leakage function evaluated at the largest distortion value that $g(.)$ is equal to $1$. For general mechanisms, it is the integral of single letter leakage function with respect to the Lebesgue---Stieltjes measure defined based on the constraint function $g$.\n\t\\item In both cases, the optimal general mechanisms are mixtures of memoryless mechanisms.\n\\end{itemize} \nThe formulations in \\eqref{eq:optimization} and \\eqref{eq:optimization 2} include the dependence on both the public and private aspects of the dataset. In cases where the private data is not directly available, but the statistics are known, the private ($Y^n$), public ($X^n$), and revealed data ($\\hX^n$) form a Markov chain $Y^n \\rightarrow X^n \\rightarrow \\hX^n$. In this paper, we focus on the general case with both public and private data being available to the mechanism, but the results here generalize in a straightforward manner to the case when private data is not available.\n\n\\subsection{Related Work}\nAn alternative approach to more general distortion constraints is considered in \\cite{Shkel} and referred to as \\textit{$\\tilde{f}$-separable distortion measures} \\setcounter{footnote}{0} \\footnote{We have changed their notation from $f$-separable to $\\tilde{f}$-separable, in order to avoid confusion with our notation.}.\nIn \\cite{Shkel}, a multi-letter distortion measure $\\tilde{d} (\\cdot,\\cdot)$ is defined as $\\tilde{f}$-separable if\n\\begin{equation}\n\\tilde{d}(x^n, \\hx^n) = \\tilde{f}^{-1}\\left(\\frac{1}{n} \\sum_{i=1}^{n} \\tilde{f}(\\tilde{d}(x_i,\\hx_i))\\right),\n\\end{equation}\nfor an increasing function $\\tilde{f}$. The distortion cost constraints that we consider are more general in the sense that our notion of cost function $f$ applied to the distortion measure $d(\\cdot,\\cdot)$ covers a broader class of distortion constraints than an average bound on $\\tilde{f}$-separable distortion measures studied in \\cite{Shkel}. Specifically, the average constraint on an $\\tilde{f}$-separable distortion measure has the form\n\\begin{equation}\n\\mathbb{E}\\left[\\tilde{f}^{-1} \\left( \\frac{1}{n} \\sum_{i=1}^{n} \\tilde{f}(\\tilde{d}(x_i,\\hx_i)) \\right)\\right] \\le D,\n\\end{equation}\nwhich clearly is a specific case for our formulation in \\eqref{eq:optimization} that results from choosing $f= \\tilde{f}^{-1}$ and $d(x,\\hx) = \\tilde{f}(\\tilde{d}(x,\\hx))$, such that $d(x^n,\\hx^n) = \\frac{1}{n} \\sum_{i=1}^{n} d(x_i,\\hx_i)$. Moreover, we allow for non-decreasing functions $f$, which means that $\\tilde{f}$ does not have to be strictly increasing. We also note that our focus is on privacy rather than source coding.\n\nIn the context of privacy, the privacy utility tradeoff with distinct $X$ and $Y$ is studied in \\cite{Yamamoto} and more extensively in \\cite{Sankar_TIFS_2013}, but the utility metric is only restricted to identity cost functions, i.e. $f(D)=D$. Generalizing this to the excess distortion constraint was considered by \\cite{KKSAllerton2016}. In \\cite{KKSAllerton2016}, we also differentiated between explicit availability or unavailability of the private data $Y$ to the privacy mechanism. Information theoretic approaches to privacy that are agnostic to the length of the dataset are considered in \\cite{CalmonFawaz2013,CalmonMM15,Asoodeh2015}.\n\nIn \\cite{KKSAllerton2016}, we also allow the mechanisms to be either memoryless (also referred to as \\textit{local privacy}) or general. This approach has also been considered in the context of differential privacy (DP) (see for example \\cite{KoushaISIT2016,Warner,KLNRS11,Kairouz, Duchi}). In the information theoretic context, it is useful to understand how memoryless mechanisms behave for more general distortion constraints as considered here. Furthermore, even less is known about how general mechanisms behave and that is what this paper aims to do.\n\n\n\nIn this paper, we first setup the problem formulation in Section \\ref{sec: prelim}. Then, in Section \\ref{sec: main results} we present our main results for the asymptotic leakage for general and memoryless mechanisms, under the average distortion cost and complementary CDF bounds on distortion. Finally, we provide all the proofs in Sections \\ref{sec: proofs}.\n\n\\subsection{Notation}\nThroughout this paper we use $D$ as the distortion value, and $d(\\cdot,\\cdot)$ to indicate the distortion function used for measuring utility. We also use $D_{\\text{KL}}(\\cdot||\\cdot)$ for the KL-divergence between two distributions. The mutual information between two variables $X$ and $Y$ is denoted by $I(X;Y)$ and the base for all the logarithm and exponential functions are the same, but can be any numerical value. We denote binary entropy by $H_b(\\cdot)$, and use $\\mathbb{E}_{P}[\\cdot]$ for expectation with respect to distribution $P$, where the subscript $P$ is dropped when it is clear from context. We denote random variables with capital letters, and their corresponding alphabet set by calligraphic letters. The lower convex envelope of a function $r(\\cdot)$ for any point $t$ in its domain is given by\n\n\t\\begin{equation}\n\tr^{**}(t) \\triangleq  \\sup\\left\\{  s(t) { \\bigg | } \n\t\\begin{array}{ll}\n\t&{ s \\text{ is convex}},\\\\\n\t&\\normalsize{ s(x) \\le r(x), \\forall x \\in \\text{Dom } r} \n\t\\end{array}\n\t\\right\\}.\n\t\\end{equation}\n\n\n\\section{Problem Definition and Preliminaries \\label{sec: prelim}}\nLet the source data $(X^n,Y^n)$ be a dataset of $n$ independently and identically distributed (i.i.d.) random variables, where $(X_i,Y_i) \\sim P_{X,Y}$, for all $i=1,\\ldots,n$. The revealed data is an $n$-length sequence $\\hX^n$ drawn from the alphabet $\\hat{\\mathcal{X}}^n$, and all the alphabet sets $\\mc{X}, \\mc{Y},\\mc{\\hX}$ are assumed to be finite sets. A random mechanism is used to generate the revealed data $\\hX^n$ given the source data $(X^n,Y^n)$.\n\nIn order to quantify the utility of the revealed data, consider the single letter distortion measure as a function $d: \\mathcal{X} \\times \\hat{\\mathcal{X}} \\rightarrow [D_{\\text{min}},D_{\\text{max}}]$. Then, the distortion between $n$-length sequences is given by $d(x^n,\\hx^n) = \\frac{1}{n}\\sum_{i=1}^{n} d(x_i,\\hx_i)$. \nThe following definitions represent our main quantities of interest, given by the minimum leakage for a dataset subject to a distortion cost constraint and a complementary CDF bound on distortion.\nWe differentiate between the memoryless and general mechanisms by the superscripts $M$ and $G$, respectively.\n\n\n\t\\begin{definition}[Information Leakage under a Cost Function]\n\t\t\tGiven a left-continuous and non-decreasing cost function $f: [D_{\\text{min}},D_{\\text{max}}] \\rightarrow [0, \\infty)$ and $t > f(D_\\text{min})$, the minimal leakage under an expected distortion cost constraint is defined as follows:\n\t\\begin{equation}\n\tL^{(\\cdot)}(n,t,f) \\triangleq \\min_{\\substack{P_{\\hX^n|X^n,Y^n} :\\\\ \\mathbb{E}[f(d(X^n,\\hat{X}^n))]\\le t}} \\frac{1}{n}I(Y^n;\\hat{X}^n), \\label{eq: LG definition}\n\t\\end{equation}\nand\n\t\\begin{equation}\n\tL^{(\\cdot)}(t,f) \\triangleq \\lim_{n \\rightarrow \\infty} L^{(\\cdot)}(n,t,f),\n\t\\end{equation}\n\twhen the limits exist. The superscript $(\\cdot)$ takes values $M$ or $G$, where for $L^{(M)}$ the $n$-letter mechanism $P_{\\hX^n|X^n,Y^n}$ is restricted to be stationary and memoryless, i.e. given by $P_{\\hX^n|X^n,Y^n} = (P_{\\hX|X,Y})^n$, but for $L^{(G)}$ it can be any mechanism satisfying the distortion constraint.\n\\end{definition}\n\n\n\\begin{definition}[Information Leakage with Distortion CDF Bound]\n\tGiven a right-continuous and non-increasing function $g: [D_{\\text{min}}, D_{\\text{max}}] \\rightarrow (0,1] $, the minimal leakage with a cumulative distortion distribution bounded by $g$ is defined as follows:\n\t\t\\begin{equation}\n\t\tL^{(\\cdot)}(n,g) \\triangleq \\min_{\\substack{P_{\\hX^n|X^n,Y^n} : \\\\ \\mathbb{P}[d(X^n,\\hX^n) > D] \\le g(D), \\forall D }} \\frac{1}{n}I(Y^n;\\hat{X}^n),\n\t\t\\label{eq: g bounded nonasymptotic leakage}\n\t\t\\end{equation}\nand\n\t\t\\begin{equation}\n\t\tL^{(\\cdot)}(g) \\triangleq \\lim_{n \\rightarrow \\infty} L^{(\\cdot)}(n,g),\n\t\t\t\t\\label{eq: g bounded asymptotic leakage}\n\t\t\\end{equation}\n\t\twhen the limits exist. The superscript $(\\cdot)$ takes values $M$ or $G$, where for $L^{(M)}$ the $n$-letter mechanism $P_{\\hX^n|X^n,Y^n}$ is restricted to be stationary and memoryless, i.e. given by $P_{\\hX^n|X^n,Y^n} = (P_{\\hX|X,Y})^n$, while for $L^{(G)}$ it can be any mechanism satisfying the distortion constraint.\n\t\\label{def: g bounded asymptotic leakage}\n\\end{definition}\n\nWe now define the optimal single letter information leakage under a constraint on the expected value of the distortion. This is analogous to the single-letter rate-distortion function, and has appeared in earlier works on privacy \\cite{Sankar_TIFS_2013}. As we will show later, this quantity appears as a key element in first-order leakage.\n\\begin{definition}[Single Letter Information Leakage]\n\t\\begin{align}\n\tL(D) &\\triangleq \\min_{P_{\\hat{X}|X,Y} : \\mathbb{E}\\left[ d(X,\\hat{X})\\right] \\le D} I(Y;\\hat{X})\\label{eq: L}.\n\t\\end{align}\n\\end{definition}\nNote that $L(\\cdot)$ is convex, non-increasing, and thus continuous on $(D_{\\text{min}},D_{\\text{max}}]$.\n\\begin{remark}\nFor $f(D)=D$, and any $n$, the optimization in \\eqref{eq: LG definition} reduces to \\eqref{eq: L} for both memoryless and general mechanisms. \n\\end{remark}\n\nWe now define functions that will be critical in expressing asymptotic leakage with the expected distortion cost bound under stationary memoryless and general mechanisms.\n\n\\begin{definition}\n\tFor any cost function $f$, and a distortion cost threshold $t > f(D_{\\text{min}})$, let\n\t\\begin{align}\n\tf^{-1}_l(t) \\triangleq \\sup \\{D \\in [D_{\\text{min}}, D_{\\text{max}}]: f(D) < t\\},\\label{eq: f inv l}\\\\\n\tf^{-1}_u(t) \\triangleq \\sup \\{D \\in [D_{\\text{min}}, D_{\\text{max}}]: f(D) \\le t\\}, \\label{eq: f inv h}\n\t\\end{align}\n\tand define\n\t\\begin{equation}\n\t\\mc{T}_{f} \\triangleq \\{t: f^{-1}_l(t) \\neq f^{-1}_u(t)  \\}.\n\t\\label{eq: T_f definition}\n\t\\end{equation}\n\tConsequently, for any $t \\notin \\mc{T}_{f}$, we have $f^{-1}_l(t) = f^{-1}_u(t)$, and thus, the inverse function for $f$ can be uniquely determined as\n\t\\begin{equation}\n\tf^{-1}(t) \\triangleq f^{-1}_l(t) = f^{-1}_u(t).\n\t\\end{equation}\n\\end{definition}\n\n\n\n\n\n\n\n\n\n\n\n\\section{Main Results \\label{sec: main results}}\n\\subsection{Distortion Cost Constraint}\n\\begin{theorem}\nLet $t > f(D_{\\text{min}})$. If $t \\notin \\mc{T}_f$, then the asymptotic minimum leakage under stationary memoryless mechanisms is given by\n\\begin{equation}\nL^{(M)}(t,f) = (L \\circ f^{-1})(t),\n\\label{eq: first order leakage memoryless 1}\n\\end{equation}\nand for any $t \\in \\mc{T}_f$, we have\n\n\n\t\\begin{equation}\n\t\\begin{array}{rll}\n\t\tL^{(M)}(t,f) = &\\min_{P_{\\hX|X,Y}} & \\hspace{-3mm}I(Y;\\hX),\\vspace{3mm}\\\\\n\t\t& \\text{subject to} & \\hspace{-3mm} \\substack{ R^{l} (P_{X,\\hX},f^{-1}_l(t)) \\\\ \\le R^{u} (P_{X,\\hX},f^{-1}_u(t))},\n\t\\end{array}\n\t\t\\label{eq: first order leakage memoryless 2}\n\t\\end{equation}\n\twhere for any $P_{X,\\hX}$ and constant $c$, \n\t\\begin{align}\n\t\\displaystyle\n\tR^{u} (P_{X,\\hX},c) &\\triangleq \\min_{Q_{X,\\hX} : \\mathbb{E}_{Q} [d(X,\\hX)] \\ge c} D_{\\text{KL}}(Q_{X,\\hX}||P_{X,\\hX}),\t\\label{eq: definition of P_c h}\\\\\n\tR^{l} (P_{X,\\hX},c) &\\triangleq \\min_{Q_{X,\\hX} : \\mathbb{E}_{Q} [d(X,\\hX)] \\le c} D_{\\text{KL}}(Q_{X,\\hX}||P_{X,\\hX}).\n\n\t\\label{eq: definition of P_c l}\n\t\\end{align}\n\tFurthermore, the inequality constraint in \\eqref{eq: first order leakage memoryless 2} reduces to equality if $L(f^{-1}_u(t)) >0$.\n\t\\label{theorem: f approximation memoryless}\n\\end{theorem}\n\n\\textit{Proof sketch:}\nFrom the law of large numbers, applying a memoryless mechanism concentrates the distortion around a particular $D$, typically to the expected value, as $n \\rightarrow \\infty$. Therefore, the distortion cost constraint roughly translates to choosing an expected distortion $D$ such that $f(D) \\le t$, or equivalently $f^{-1}(t) \\ge D$. If $f^{-1}(t)$ is uniquely determined, then we have the asymptotic leakage in the form of $L(f^{-1}(t))$. Otherwise, our desired $D$ lies somewhere between $f^{-1}_l(t)$ and $f^{-1}_u(t)$. For a more detailed proof, see Section \\ref{proof: f approximation memoryless}.\n\n\\begin{remark}\n\tNote that $R^{u} (P_{X,\\hX},c)$ and $R^{l} (P_{X,\\hX},c)$ are continuous in $c$ because the feasible set in \\eqref{eq: definition of P_c h} and \\eqref{eq: definition of P_c l} are convex and $D(Q||P)$ is convex in $Q$. They are also continuous in $P_{X,\\hX}$ due to continuity of $D_{\\text{KL}} (\\cdot||\\cdot)$ in both arguments.\n\t\\label{remark: Ru and Rl are continuous in c}\n\\end{remark}\n\n\n\\begin{remark}\n\tIf $f(\\cdot)$ is strictly increasing, then $\\mc{T}_f = \\emptyset$, and $L^{(M)}(t,f)$ is given by \\eqref{eq: first order leakage memoryless 1} for any $t$. \n\\end{remark}\n\n\\begin{remark}\n\tFor any $t > f(D_\\text{min})$, since the closure of the convex hull of epigraphs of $L \\circ f^{-1}_l$ and $L \\circ f^{-1}_u$ are equal, their lower convex envelopes are equal too. Therefore, $(L \\circ f^{-1}_l)^{**} (t) = (L \\circ f^{-1}_u)^{**} (t)$, and we refer to this value as $(L \\circ f^{-1})^{**} (t)$.\n\t\\label{remark: upper and lower g** are equal}\n\\end{remark}\n\n\\begin{theorem}\n\tFor $t > f(D_{\\text{min}})$, the asymptotic minimum leakage under general mechanisms is given by\n\t\\begin{equation}\n\tL^{(G)}(t,f) = (L \\circ f^{-1})^{**} (t).\n\t\\label{eq: first order leakage}\n\t\\end{equation}\n\t\\label{theorem: f approximation}\n\\end{theorem}\n\\textit{Proof sketch:} Since $L^{(G)}(t,f)$ is convex in $t$, a convex combination of any two feasible mechanisms is also feasible. Hence, we can always design convex combinations of memoryless mechanisms to achieve the lower convex envelope of $(L \\circ f^{-1}) (t)$, and therefore $L^{(G)}(t,f) \\le (L \\circ f^{-1})^{**} (t)$. Conversely, we show that it is not possible to achieve a smaller leakage. For proof details, we refer the reader to Section \\ref{proof: f approximation}.\n\\begin{remark}\n\tNote that for $t\\ge f(D_{\\text{max}})$, we have $L^{(M)}(t,f)=L^{(G)} (t,f)=0$, where the minimum is achieved by any mechanism with output independent from the input.\n\t\\label{remark: for large t}\n\\end{remark}\n\\begin{remark}\n\tIf $f$ is convex, then for $t>f(D_{\\text{min}})$ we have $(L \\circ f^{-1})^{**} (t) = L(f^{-1}(t))$. Therefore, from Theorem \\ref{theorem: f approximation memoryless} we have\n\t\\begin{equation}\nL^{(G)}(t,f) = L^{(M)}(t,f) = L(f^{-1}(t)).\n\t\\end{equation}\n\\end{remark}\n\n\\begin{remark}\nNote that if $L(f^{-1}(t))$ is not equal to its lower convex envelope for some $t$, then the optimal mechanism is formed by a convex combination of the optimal memoryless mechanisms for distortion costs $t_1$ and $t_2$, where $t_1$ is the largest threshold smaller than $t$ and $t_2$ is the smallest threshold larger than $t$, such that $L(f^{-1}(\\cdot))$ is equal to its lower convex envelope at $t_1$ and $t_2$.\n\\end{remark}\n\n\\subsection{Complementary CDF Bound}\nWe now proceed to the result on information leakage with distortion CDF bound. In the following, we give closed form results for the asymptotic information leakage with the distortion CDF bounded by a function $g$.\n\n\\begin{theorem}\n\tFor a non-increasing right-continuous function $g: [D_{\\text{min}}, D_{\\text{max}}] \\rightarrow (0,1] $, the asymptotic information leakage for memoryless mechanisms under distortion CDF bounded by $g(\\cdot)$ is given by\n\t\\begin{equation}\n\tL^{(M)}(g) = L(D_g),\n\t\\end{equation}\n\twhere $D_g \\triangleq \\inf \\{D \\in [D_{\\text{min}},D_{\\text{max}}]: g(D)<1\\}$.\n\\end{theorem}\n\\begin{IEEEproof}\n\tSuppose $D_g > D_{\\text{min}}$. Then, for any fixed $\\delta >0$ and $n$, choose $P_{\\hX^n|X^n,Y^n} = \\left( P^{*^{(n)}}_{\\hX|X,Y} \\right ) ^{n}$, where $P^{*^{(n)}}_{\\hX|X,Y}$ is the optimal single letter mechanism achieving $L(D_g - \\delta)$. Note that by definition $g$ is bounded away from zero, because it is right continuous and considered to be positive over $[D_{\\text{min}},D_{\\text{max}}]$. Therefore, $\\mathbb{P}[d(X^n, \\hX^n) > D_g]$ goes to zero as $n$ goes to infinity and the distortion constraint $\\mathbb{P}[d(X^n, \\hX^n) > D] \\le g(D)$ is satisfied for all $D$ for sufficiently large $n$. Then, as $\\delta \\rightarrow 0$, continuity of $L(\\cdot)$ implies $L(D_g)$ is achievable.\n\t\n\tConversely, according to the law of large numbers, the distortion $d(X^n, \\hX^n)$ concentrates around its expected value as $n$ goes to infinity. In other words, we have $\\mathbb{P}[d(X^n,\\hX^n) > D] \\rightarrow 1$, if $D < \\mathbb{E}[d(X^n, \\hX^n)]$. This, in turn, implies that for any $D$ such that $g(D)<1$, we must have $\\mathbb{E} [d(X^n, \\hX^n)] \\le D$. Therefore, a feasible memoryless mechanism has to satisfy $\\mathbb{E}[d(X^n, \\hX^n)] \\le D_g$.\n\t\n\tFinally, for $D_g=D_{\\text{min}}$, we have to satisfy $\\mathbb{P}[d(X^n, \\hX^n) = D_{\\text{min}}]=1$. Note that in this case, the constraint for $L(D_g)$, i.e. $\\mathbb{E}[d(X^n, \\hX^n)] \\le D_g$, is also equivalent to $\\mathbb{P}[d(X^n, \\hX^n) = D_{\\text{min}}]=1$. Therefore, the set of feasible memoryless mechanisms for $L^{(M)}(g)$ is equal to those for $L(D_g)$, and thus, $L^{(M)}(g)=L(D_g)$.\n\n\\end{IEEEproof}\n\n\n\\begin{theorem}\nLet $g: [D_{\\text{min}}, D_{\\text{max}}] \\rightarrow (0,1] $ be a non-increasing right-continuous function. If the single letter leakage function $L(\\cdot)$ is bounded on $[D_{\\text{min}}, D_{\\text{max}}]$, then the asymptotic information leakage for general mechanisms under distortion CDF bounded by $g(\\cdot)$ is given by\n\\begin{equation}\nL^{(G)}(g)= \\int_{D_{\\text{min}}}^{D_{\\text{max}}} L(D) d(g(D)),\n\\label{eq: g bounded asymptotic leakage result}\n\\end{equation}\nwhere the integral is a Lebesgue\u2013--Stieltjes integral of the single letter leakage function $L(\\cdot)$ with respect to the Lebesgue\u2013--Stieltjes measure associated with the constraint function $g$.\n\\label{theorem: g bounded asymptotic leakage}\n\\end{theorem}\n\n\\textit{Proof sketch:}\nWe first prove this result for simple constraint functions $g$, which are in the form of a finite sum of step functions. Then, we show that any non-increasing right-continuous constraint function $g$ can be upper and lower bounded by such simple functions, and therefore, the corresponding leakage can be upper and lower bounded by that of the simple functions. For a more detailed proof, see Section \\ref{proof: g bounded asymptotic leakage}.\n\n\\begin{remark}\nAn alternative way of describing the result in Theorem \\ref{theorem: g bounded asymptotic leakage} is that the asymptotically optimal mechanism behaves as if it first chooses a random $D$ drawn from a distribution with a complementary CDF exactly equal to $g(\\cdot)$, and then applies the single letter optimal mechanism achieving the single letter optimal leakage $L(D)$ in a stationary and memoryless fashion.\nThus, averaging over the random choice of $D$, the resulting leakage is given as the integral in \\eqref{eq: g bounded asymptotic leakage result}.\n\\end{remark}\n\n\\subsection{Auxiliary Result}\nWe now present a result characterizing the asymptotic optimal privacy leakage subject to multiple excess probability constraints. This can be seen as a special case of complementary CDF bound in which the $g$ function is a simple function, i.e. it takes finitely many values. The following results will also be used in the proof of Theorem \\ref{theorem: f approximation}.\n\nFor vectors $\\boldsymbol{D} = (D_1, D_2, \\ldots, D_k)$ and $\\boldsymbol{\\epsilon} = (\\epsilon_1, \\epsilon_2, \\ldots, \\epsilon_k)$, where $D_{\\text{min}} \\le D_1 < \\cdots < D_k \\le D_{\\text{max}}$ and $1 \\ge \\epsilon_1 > \\cdots > \\epsilon_k > 0$, a simple function $g_{\\boldsymbol{\\epsilon},\\boldsymbol{D}}$ is illustrated in Fig. \\ref{fig: simple g function} and formally defined as\n\\begin{equation}\ng_{\\boldsymbol{\\epsilon},\\boldsymbol{D}}(D) \\triangleq \\begin{cases}\n1, & D_{\\text{min}} \\le D < D_1,\\\\\n\\epsilon_i, & D_{i} \\le D < D_{i+1}, i=1,\\ldots,k-1,\\\\\n\\epsilon_k, & D_k \\le D \\le D_{\\text{max}}.\n\\end{cases}\n\\label{eq: simple function}\n\\end{equation}\n\\begin{figure}[htb!]\n\t\\centering\n\t\\includegraphics[width= 0.4 \\columnwidth]{gdeps.pdf}\n\t\\caption{A simple $g_{\\boldsymbol{\\epsilon},\\boldsymbol{D}}(D)$.}\n\t\\label{fig: simple g function}\n\t\\vspace{-10pt}\n\\end{figure}\nOne can verify that for a constraint function of this form, the minimization in \\eqref{eq: g bounded asymptotic leakage} is equivalent to the \\textit{information leakage with multiple excess distortion constraints}, defined as follows.\n\\begin{definition}[Information Leakage with Multiple Excess Probability Constraints]\n\tGiven a distortion vector $\\boldsymbol{D} = (D_1, D_2, \\ldots, D_k)$ and a tail probability vector $\\boldsymbol{\\epsilon} = (\\epsilon_1, \\epsilon_2, \\ldots, \\epsilon_k)$, where $D_{\\text{min}} \\le D_1 < \\cdots < D_k \\le D_{\\text{max}}$ and $1 \\ge \\epsilon_1 > \\cdots > \\epsilon_k > 0$, the minimal leakage with multiple excess distortion constraints is defined as\n\t\\begin{align}\n\tL^{(G)}(n,\\boldsymbol{D},\\boldsymbol{\\epsilon}) &\\triangleq \\min_{\\substack{P_{\\hX^n|X^n,Y^n} :\\\\ \\mathbb{P}[d(X^n,\\hX^n) > D_i] \\le \\epsilon_i, \\\\ \\forall 1 \\le i  \\le k}} \\frac{1}{n}I(Y^n;\\hat{X}^n),\n\t\\label{eq: multi constraint}\n\n\n\t\\end{align}\n\twhere the $n$-letter mechanisms in \\eqref{eq: LG definition} are not constrained to be memoryless or stationary, and\n\t\\begin{equation}\n\tL^{(G)}(\\boldsymbol{D},\\boldsymbol{\\epsilon}) \\triangleq \\lim_{n \\rightarrow \\infty} L^{(G)}(n,\\boldsymbol{D},\\boldsymbol{\\epsilon}),\n\t\\end{equation}\n\t\\label{def: multi constraint}\n\\end{definition}\nwhen the limit exists. In the following lemma, we provide the asymptotic optimal leakage under general mechanisms for the class of distortion CDF bound functions defined in Definition \\ref{def: multi constraint}.\n\\begin{lemma}\n\t\\begin{align}\n\tL^{(G)}(\\boldsymbol{D},\\boldsymbol{\\epsilon})\t&= \\sum_{i=1}^{k} (\\epsilon_{i-1}-\\epsilon_i) L(D_i) \\nonumber\\\\\n\t&= \\int_{D_{\\text{min}}}^{D_{\\text{max}}} L(D) d(g_{\\boldsymbol{\\epsilon},\\boldsymbol{D}}(D)),\n\t\\label{eq: multi constraint result}\n\t\\end{align}\n\twhere $\\epsilon_0 = 1$. In particular, we have\n\t\\begin{equation}\n\tL^{(G)}(n,\\boldsymbol{D},\\boldsymbol{\\epsilon}) = \\sum_{i=1}^{k} (\\epsilon_{i-1}-\\epsilon_i) L(D_i) + \\theta (k,n),\n\t\\end{equation}\n\twhere\n\t\\begin{equation}\n\t-\\frac{\\log(k+1)}{n} \\le  \\theta (k,n) \\le O\\left(\\sqrt {\\frac{\\log n}{n}}\\right).\n\t\\label{eq: multi constraint LB and UB}\n\t\\end{equation}\n\t\\label{lemma: g bounded asymptotic leakage - simple}\n\\end{lemma}\n\n\\textit{Proof sketch:}\nThe proof hinges on choosing a combination of memoryless mechanisms, each of them being the single letter optimal mechanism for a separate $D_i$ applied in a stationary and memoryless fashion. The weights of this combination will be chosen such that all the excess distortion probabilities are met. For a detailed proof see section \\ref{proof: g bounded asymptotic leakage - simple}. \n\n\\section{Illustration of Results}\nIn this section, we first examine the generic cases of single and double step $f$ and $g$ functions. Then, we consider a doubly symmetric binary source and derive its corresponding single letter leakage function. Finally, we use the single letter leakage function to find the asymptotically optimal leakage under specific examples of the average distortion cost constraint and complementary CDF bound.\n\\subsection{Distortion Cost Function}\n\\begin{example}\n\t$f(D)=\\boldsymbol{1} (D>D_0)$ as shown in Fig. \\ref{fig: f one step}. In this case, $\\mathcal{T}_f = \\{1\\}$, and we have\n\t\\begin{figure}[htb!]\n\t\t\\centering\n\t\t\\includegraphics[width= 0.4 \\columnwidth]{example2fnoline.pdf}\n\t\t\\caption{The single step cost function $f(D)=\\boldsymbol{1} (D>D_0)$.}\n\t\t\\label{fig: f one step}\n\t\\end{figure}\n\t\\begin{align}\n\tf^{-1}_u(t) &= \\begin{cases}\n\tD_0, & t<1,\\\\\n\tD_{\\text{max}},& t \\ge 1,\n\t\\end{cases}\\\\\n\tf^{-1}_l(t) &= \\begin{cases}\n\tD_0, & t \\le 1,\\\\\n\tD_{\\text{max}},& t > 1.\n\t\\end{cases}\n\t\\end{align}\nTherefore, according to Theorem \\ref{theorem: f approximation memoryless} for stationary memoryless mechanisms we have\n\t\\begin{equation}\n\tL^{(M)}(t,f) = \\begin{cases}\n\tL({D_0}), &  0 < t < 1,\\\\\n\t0,& t\\ge 1,\n\t\\end{cases}\n\t\\end{equation}\nand for general mechanisms, according to Theorem \\ref{theorem: f approximation} we have\n\t\\begin{equation}\n\tL^{(G)}(t,f) = \\begin{cases}\n\t(1-t) L({D_0}), & 0 \\le t < 1\\\\\n\t0, & t \\ge 1.\n\t\\end{cases}\n\t\\end{equation}\n\tThis exactly matches our earlier results in \\cite{KKSAllerton2016} and for the special case of $X=Y$ simplifies to the result in \\cite{Kostina}. The leakages $L^{(G)}$ and $L^{(M)}$ are depicted in Fig. \\ref{fig: example2}.\n\tNote that for $t=1$, we have $L^{(G)}(t,f) = L^{(M)}(t,f) = 0$ due to Remark \\ref{remark: for large t}.\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[width= 0.4 \\columnwidth]{example2.pdf}\n\t\t\\caption{The leakage functions $L^{(M)}(t,f)$ and $L^{(G)}(t,f)$ for $f(D)=\\boldsymbol{1} (D > D_0)$.}\n\t\t\\label{fig: example2}\n\t\t\\vspace{-10pt}\n\t\\end{figure}\n\\label{example: 1}\n\\end{example}\n\\begin{example}\n\t$f(D)=a_1 \\boldsymbol{1} (D>D_1) + a_2 \\boldsymbol{1} (D > D_2)$, $D_1 < D_2$ as shown in Fig. \\ref{fig: f two step}. In this case, $\\mathcal{T}_f = \\{a_1,a_1+a_2\\}$, and we have\n\t\\begin{align}\n\tf^{-1}_u(t) &= \\begin{cases}\n\tD_1, & t<a_1,\\\\\n\tD_2, & a_1 \\le t < a_1+ a_2,\\\\\n\tD_{\\text{max}},& t \\ge a_1 + a_2,\n\t\\end{cases}\\\\\n\tf^{-1}_l(t) &= \\begin{cases}\n\tD_1, & t\\le a_1,\\\\\n\tD_2, & a_1 < t \\le a_1+ a_2,\\\\\n\tD_{\\text{max}},& t > a_1 + a_2.\n\t\\end{cases}\n\t\\end{align}\nHence, according to Theorem \\ref{theorem: f approximation memoryless} for stationary memoryless mechanisms we have\n\\begin{equation}\n\tL^{(M)}(t,f) = \\begin{cases}\n\tL(D_1), & t < a_1,\\\\\n\tL(D_2), & a_1 < t< a_1 + a_2,\\\\\n\t0, & a_1 + a_2 \\le t.\n\t\\end{cases}\n\\end{equation}\nNote that for $t=a_1$, the exact value for $L^{(M)}(t,f)$ is derived by \\eqref{eq: first order leakage memoryless 2}, and for $t=a_1+a_2$, we have $L^{(M)}(t,f) =0$ due to Remark \\ref{remark: for large t}.\nFrom Theorem \\ref{theorem: f approximation}, we know that $L^{(G)}(t,f)$ is the lower convex envelope of $L^{(M)}(t,f)$. If $a_2 L(D_1) \\ge  (a_1 + a_2) L(D_2)$, then it is given by\n\\begin{equation}\n\tL^{(G)}(t,f) = \\begin{cases}\n\tL(D_2)+(1 - \\frac{t}{a_1}) L(D_1) & t \\le a_1,\\\\\n\t(1-\\frac{t-a_1}{a_2}) L(D_2) & a_1 \\le t \\le a_1+a_2,\\\\\n\t0 & a_1 + a_2 \\le t,\n\t\\end{cases}\n\t\\end{equation}\n\tand otherwise,\n\t\\begin{equation}\n\tL^{(G)}(t,f) = \\begin{cases}\n\t(1-\\frac{t}{a_1+a_2}) L(D_1)& t \\le a_1 + a_2 , \\\\\n\t0 & a_1 + a_2 \\le t.\n\t\\end{cases}\n\t\\end{equation}\n\tThese two cases together with their corresponding $L^{(M)}(t,f)$ are shown in Figs. \\ref{fig: example3a} and \\ref{fig: example3b}, respectively.\n\t\\begin{figure}[htb!]\n\t\t\\centering\n\t\t\\includegraphics[width= 0.4 \\columnwidth]{example3afnoline.pdf}\n\t\t\\caption{The double step cost function $f(D)=a_1 \\boldsymbol{1} (D>D_1) + a_2 \\boldsymbol{1} (D > D_2)$, $D_1 < D_2$.}\n\t\t\\label{fig: f two step}\n\t\\end{figure}\n\t\\begin{figure}[htb!]\n\t\t\\centering\n\t\t\\includegraphics[width= 0.4 \\columnwidth]{example3anoline.pdf}\n\t\t\\caption{$L^{(M)}(t,f)$ and $L^{(G)}(t,f)$ for $f(D)=a_1 \\boldsymbol{1} (D>D_1) + a_2 \\boldsymbol{1} (D>D_2)$, if $a_2 L(D_1) \\ge  (a_1 + a_2) L(D_2)$.}\n\t\t\\label{fig: example3a}\n\t\\end{figure}\n\t\\begin{figure}[htb!]\n\t\t\\centering\n\t\t\\includegraphics[width= 0.4 \\columnwidth]{example3bnoline.pdf}\n\t\t\\caption{$L^{(M)}(t,f)$ and $L^{(G)}(t,f)$ for $f(D)=a_1 \\boldsymbol{1} (D>D_1) + a_2 \\boldsymbol{1} (D>D_2)$, if $a_2 L(D_1) <  (a_1 + a_2) L(D_2)$.}\n\t\t\\label{fig: example3b}\n\t\\end{figure}\n\\end{example}\n\n\\subsection{Distortion CDF Constraints}\nWe now proceed to complementary CDF bounds on distortion. First, we consider a single step function $g$ (hard tail probability constraint), and then generalize to a sum of two step functions.\n\n\\begin{example}\n\t$g(D) = 1- (1-\\epsilon) \\boldsymbol{1}(D \\ge D_0)$ as shown in Fig. \\ref{fig: example4}, where $D_{\\text{min}}<D_0<D_{\\text{max}}$. For stationary memoryless mechanisms we have\n\t\\begin{equation}\n\tL^{(M)}(g) = L(D_0),\n\t\\label{eq: LM for tail prob const}\n\t\\end{equation}\n\twhile for the general mechanisms, we have\n\t\\begin{equation}\n\tL^{(G)}(g) = \\int_{D_{\\text{min}}}^{D_{\\text{max}}} L(D) d(g(D)) = (1-\\epsilon)L(D_0).\n\t\\label{eq: LG for tail prob const}\n\t\\end{equation}\t\n\t\\begin{figure}[htb!]\n\t\t\\centering\n\t\t\\includegraphics[width= 0.45 \\columnwidth]{example4noline.pdf}\n\t\t\\caption{The single step complementary CDF bound function $g(D)=1- (1-\\epsilon) \\boldsymbol{1}(D \\ge D_0)$.}\n\t\t\\label{fig: example4}\n\t\\end{figure}\n\tNote that this is equivalent to Example \\ref{example: 1}. Therefore, \\eqref{eq: LM for tail prob const} and \\eqref{eq: LG for tail prob const} verify the results in \\cite{Kostina} and \\cite{KKSAllerton2016}, wherein the tail probability constraint is used as a utility metric.\n\\end{example}\n\n\\begin{example}\n\t$g(D) = \\boldsymbol{1}(D < D_1) + \\epsilon_1 \\boldsymbol{1}( D_1 \\le D < D_2) + \\epsilon_2 \\boldsymbol{1}( D_2 \\le D) $ as shown in Fig. \\ref{fig: example5}. For stationary memoryless mechanisms we have\n\t\\begin{equation}\n\tL^{(M)}(g) = L(D_1),\n\t\\label{eq: LM for example 5}\n\t\\end{equation}\n\twhile for the general mechanisms, we have\n\t\\begin{equation}\n\tL^{(G)}(g) = (1-\\epsilon_1)L(D_1) + (\\epsilon_1 - \\epsilon_2) L(D_2).\n\t\\label{eq: LG for example 5}\n\t\\end{equation}\n\t\\begin{figure}[htb!]\n\t\t\\centering\n\t\t\\includegraphics[width= 0.45 \\columnwidth]{example5noline.pdf}\n\t\t\\caption{The double step complementary CDF bound function $g(D) = \\boldsymbol{1}(D < D_1) + \\epsilon_1 \\boldsymbol{1}( D_1 \\le D < D_2) + \\epsilon_2 \\boldsymbol{1}( D_2 \\le D)$.}\n\t\t\\label{fig: example5}\n\t\\end{figure}\n\\end{example}\n\n\\subsection{Doubly Symmetric Binary Source (DSBS)}\nWe now consider a doubly symmetric source with parameter $q$ as depicted in Fig. \\ref{fig: binary source} with Hamming distortion, i.e. $d(x,\\hx) = \\boldsymbol{1}(x \\neq \\hx)$, as the utility metric. \nIn the following lemma, proved in Section \\ref{proof: single letter leakage for doubly symmetric source}, we derive the single letter leakage function for this source.\n\t\\begin{figure}[htb!]\n\t\t\\centering\n\t\t\\includegraphics[width= 0.25 \\columnwidth]{binarysource.pdf}\n\t\t\\caption{A doubly symmetric source with parameter $q$.}\n\t\t\\label{fig: binary source}\n\t\\end{figure}\n\\begin{lemma}\nFor a doubly symmetric source with $q \\le 0.5$, the single letter leakage function is given by\n\\begin{equation}\nL(D) = \\begin{cases}\n1 - H_b(q + D), & D < 0.5 - q,\\\\\n0, &D\\ge 0.5 - q.\n\\end{cases} \n\\label{eq: single letter leakage for doubly symmetric source}\n\\end{equation}\n\\label{lemma: single letter leakage for doubly symmetric source}\n\\end{lemma}\n\\begin{remark}\nDue to the inherent symmetry of the problem, for all $q > 0.5$, Lemma \\ref{lemma: single letter leakage for doubly symmetric source} holds with $q$ replaced by $1-q$.\n\\end{remark}\n\nGiven the single letter leakage function for a doubly symmetric source, we provide numerical examples for the asymptotically optimal leakages under both distortion cost constraints and complementary CDF bounds.\n\n\\begin{example}\nFor a doubly symmetric source with parameter $q=0.1$ and Hamming distortion, consider the cost function\n\\begin{equation}\nf(D) = \\begin{cases}\n\\frac{4(8x - i -0.5)^5+16x+1-2i}{32},& x \\in [\\frac{i}{8}, \\frac{i+1}{8}), i \\in \\{0,\\ldots,7\\}, \\\\\n1,& x=1,\n\\end{cases}\n\\end{equation}\nas shown in Fig. \\ref{fig: f}. Then, the corresponding leakage functions $L^{(M)}(t,f)$ and $L^{(G)}(t,f)$ are shown in Fig. \\ref{fig: example f}.\n\n\\begin{figure}[htb!]\n\t\\centering\n\t\\includegraphics[width= 0.45 \\columnwidth]{f.pdf}\n\t\\caption{The cost function $f(D)$ for Example \\ref{example: 3}.}\n\t\\label{fig: f}\n\\end{figure}\n\n\t\\begin{figure}[htb!]\n\t\t\\centering\n\t\t\\includegraphics[width= 0.45 \\columnwidth]{LMLG.pdf}\n\t\t\\caption{Memoryless and general leakage functions $L^{(M)}(t,f)$ and $L^{(G)}(t,f)$ for Example \\ref{example: 3}.}\n\t\t\\label{fig: example f}\n\t\\end{figure}\n\t\\label{example: 3}\n\\end{example}\n\nWe now proceed to an examples that resemble a \\textit{soft} single step complementary CDF bound. We choose functions that are parametrized with a parameter $\\lambda$ such that they converge to a hard single step CDF bound as $\\lambda \\rightarrow \\infty$. \n\n\\begin{example}\n\tConsider a doubly symmetric source with parameter $q=0.1$. Then, for any $\\lambda \\ge 0 $ define\n\t\\begin{align}\n\tg_\\lambda(D) =& \\epsilon + (1-\\epsilon)  \\boldsymbol{1} (D \\le D_0) \\nonumber \\\\\n\t& + (1-\\epsilon)\\left(\\frac{1}{2} - \\boldsymbol{1} (D \\le D_0)\\right) e^{-\\lambda |D - D_0|}.\n\t\\end{align}\n\tIn Fig. \\ref{fig: g soft}, this function is plotted for $D_0 = 0.2$, $\\epsilon = 0.1$, and four different values of $\\lambda$. Note that in Fig. \\ref{fig: LGg}, the value of $L^{(G)}(g_\\lambda)$ converges to the asymptotic value of $(1-\\epsilon) L(D_0)$ as $\\lambda \\rightarrow \\infty$, and $L^{(G)}(g_\\lambda)$ is non-monotonic in $\\lambda$.\t\n\t\\begin{figure}[htb!]\n\t\t\\centering\n\t\t\\includegraphics[width= 0.45 \\columnwidth]{g.pdf}\n\t\t\\caption{$g(d)$ as described in Example \\ref{example: 6}, for $D_0=0.2$ and $\\epsilon=0.1$, parametrized by $\\lambda$.}\n\t\t\\label{fig: g soft}\n\t\\end{figure}\n\t\\begin{figure}[htb!]\n\t\t\\centering\n\t\t\\includegraphics[width= 0.45 \\columnwidth]{LGg.pdf}\n\t\t\\caption{$L^{(G)}(g)$ for the $g$ function given in Example \\ref{example: 6}.}\n\t\t\\label{fig: LGg}\n\t\\end{figure}\n\t\\label{example: 6}\n\\end{example}\n\n\\section{Proofs \\label{sec: proofs}}\nBefore proving our main results, we first review Hoeffding's inequality, a version of Chernoff bound used for bounded random variables.\n\\begin{lemma} [Hoeffding's inequality {\\cite[Theorem~2]{Hoeffding}}]\n\tLet $X_1, \\ldots , X_n$ bounded independent random variables, i.e. $a_i \\le X_i \\le b_i$ for each $1\\le i\\le n$. We define the empirical mean of these variables by\n\t${\\displaystyle {\\bar{X}}={\\frac {1}{n}}(X_{1}+ \\ldots +X_{n})}$. Then\n\t\\begin{align}\n\t{\\displaystyle {\\begin{aligned}\\mathbb {P} \\left({\\bar{X}}-\\mathbb {E} \\left[{\\bar{X}}\\right]\\geq t\\right)&\\leq \\exp \\left(-{\\frac {2n^{2}t^{2}}{\\sum _{i=1}^{n}(b_{i}-a_{i})^{2}}}\\right),\\end{aligned}}}\n\t\\end{align}\n\twhere $t$ is positive, and $E[X]$ is the expected value of $X$.\n\t\\label{lemma: Hoeffding}\n\\end{lemma}\n\n\n\n\n\n\\subsection{Proof of Theorem \\ref{theorem: f approximation memoryless}} \\label{proof: f approximation memoryless}\nAssuming a stationary memoryless mechanism, we provide upper and lower bounds on $\\mathbb{E}(f(d(X^n,\\hX^n)))$ in terms of $f(\\mathbb{E}[d(X^n,\\hX^n)]$. This in turn allows us to bound $L^{(M)}(\\cdot)$ in terms of $L(f^{-1}(\\cdot))$.\nLet $\\delta_n=(D_\\text{max} -D_\\text{min})\\sqrt{\\log n \/ n}$. Then, for large enough $n$ we have\n\\begin{subequations}\n\n\t\\label{eq: upper bound on E(d)}\n\t\\allowdisplaybreaks\n\\begin{align}\n& \\mathbb{E} \\left[ f(d(X^n,\\hX^n)) \\right]\\\\\n& \\le \\mathbb{P}\\left(d(X^n,\\hX^n) \\le \\mathbb{E}[d(X^n,\\hX^n)]  + \\delta_n \\right) \\nonumber\\\\\n& \\quad \\cdot f\\left(\\mathbb{E}[d(X^n,\\hX^n)] + \\delta_n\\right)  \\nonumber\\\\\n& \\quad +  \\mathbb{P}\\left(d(X^n,\\hX^n) > \\mathbb{E}[d(X^n,\\hX^n) ] + \\delta_n \\right) f(D_{\\text{max}})\\\\\n& \\le f(\\mathbb{E}[d(X^n,\\hX^n)] + \\delta_n) + f(D_{\\text{max}}) e^{-n \\frac{\\delta^2_n}{(D_{\\text{max}} - D_{\\text{min}})^2 }} \\label{eq:feasibility of memoryless mech 1}\\\\\n& \\le f\\left(\\mathbb{E}[d(X^n,\\hX^n)] + \\delta_n\\right) + \\frac{f(D_{\\text{max}})}{n}, \\label{eq:feasibility of memoryless mech 2}\n\\end{align}\n\\end{subequations}\nwhere \\eqref{eq:feasibility of memoryless mech 1} is due to Lemma \\ref{lemma: Hoeffding} and \\eqref{eq:feasibility of memoryless mech 2} follows from the definition of $\\delta_n$. If $\\mathbb{E} \\left[ d(X^n,\\hX^n) \\right] \\le f^{-1}_l\\left(t-\\frac{f(D_{\\text{max}})}{n}\\right) - \\delta_n$, then $\\mathbb{E} \\left[ f \\left (d(X^n,\\hX^n)\\right ) \\right] \\le t$, and we have\n\t\\begin{equation}\n\tL^{(M)} (n,t,f) \\le L\\left(f^{-1}_l\\left(t-\\frac{f(D_{\\text{max}})}{n}\\right) - \\delta_n\\right) \\label{eq: LM upper}.\n\t\\end{equation}\nSince $f^{-1}_l(\\cdot)$ is left-continuous, and $L(\\cdot)$ is continuous, taking the limit as $n \\rightarrow \\infty$ gives\n\\begin{equation}\nL^{(M)} (t,f) \\le L\\left(f^{-1}_l\\left(t\\right) \\right) \\label{eq: LM upper limit}.\n\\end{equation}\nWith a similar argument and using the negative of the distortion function in Lemma \\ref{lemma: Hoeffding}, we have\n\\begin{subequations}\n\n\t\\label{eq: lower bound on E(d)}\n\t\\allowdisplaybreaks\n\t\\begin{align}\n\t& \\mathbb{E} \\left[ f(d(X^n,\\hX^n)) \\right]\\\\\n\t& \\ge \\mathbb{P}\\left(d(X^n,\\hX^n) \\ge \\mathbb{E}[d(X^n,\\hX^n)]  - \\delta_n \\right) \\nonumber\\\\\n\t& \\quad \\cdot f\\left(\\mathbb{E}[d(X^n,\\hX^n)] - \\delta_n\\right)  \\nonumber\\\\\n\t& \\quad +  \\mathbb{P}\\left(d(X^n,\\hX^n) < \\mathbb{E}[d(X^n,\\hX^n) ] - \\delta_n \\right) f(D_{\\text{min}})\\\\\n\t& \\ge \\left(1 - \\mathbb{P}\\left(d(X^n,\\hX^n) < \\mathbb{E}[d(X^n,\\hX^n) ] - \\delta_n \\right)\\right)\\nonumber\\\\\n\t&\\quad \\cdot f(\\mathbb{E}[d(X^n,\\hX^n)] - \\delta_n) \\\\\n\t& \\ge \\left(1- \\frac{1}{n}\\right)  f\\left(\\mathbb{E}[d(X^n,\\hX^n)] - \\delta_n\\right) \\label{eq: converse of memoryless mech 1},\n\t\\end{align}\n\n\\end{subequations}\nwhere \\eqref{eq: converse of memoryless mech 1} is due to Lemma \\ref{lemma: Hoeffding}.\nTherefore, if \n\\begin{equation}\n\\mathbb{E} \\left[ f \\left (d(X^n,\\hX^n)\\right ) \\right] \\le t,\n\\end{equation}\nthen\n\\begin{equation}\n\\mathbb{E} \\left[ d(X^n,\\hX^n) \\right] \\le f^{-1}_u\\left(t\\left(1+\\frac{1}{n-1}\\right)\\right)+\\delta_n,\n\\end{equation}\nand we have\n\\begin{equation}\nL\\left(f^{-1}_u\\left(t\\left(1+\\frac{1}{n-1}\\right)\\right)+\\delta_n\\right) \\le L^{(M)} (n,t,f). \\label{eq: LM lower} \n\\end{equation}\nSince $f^{-1}_u(\\cdot)$ is right-continuous, and $L(\\cdot)$ is continuous, taking the limit as $n \\rightarrow \\infty$ gives\n\\begin{equation}\nL\\left(f^{-1}_u(t)\\right) \\le L^{(M)} (t,f).  \\label{eq: LM lower limit} \n\\end{equation}\n\nRecall the definition of $\\mc{T}_f$ in \\eqref{eq: T_f definition}. If $t \\notin \\mc{T}_f$, then $f^{-1}_l(t)=f^{-1}_u(t)$. Hence, \\eqref{eq: LM upper limit} and \\eqref{eq: LM lower limit} imply \\eqref{eq: first order leakage memoryless 1}. Otherwise, if $t \\in \\mc{T}_f$, fix some $\\delta >0$. Then, we bound the expected distortion cost for the function $f$ under any mechanism $P_{\\hX|X,Y}$. Specifically, as an upper bound we have\n\\begin{subequations}\n\n\t\\label{eq: upper bound on E(d) 2}\n\t\\allowdisplaybreaks\n\t\\begin{align}\n\t&\\mathbb{E} \\left[ f(d(X^n,\\hX^n)) \\right] \\nonumber\\\\\n\t&\\le  \\mathbb{P}\\left [ d(X^n,\\hX^n) \\le f^{-1}_l(t)-\\delta \\right] f(f^{-1}_l(t) - \\delta) \\nonumber\\\\\n\t& \\quad +\\mathbb{P}\\left[ f^{-1}_l(t) - \\delta < d(X^n,\\hX^n) < f^{-1}_u(t) \\right] t \\nonumber\\\\\n\t& \\quad +\\mathbb{P}\\left[ f^{-1}_u(t) \\le d(X^n,\\hX^n) \\right] f(D_{\\text{max}})\\\\\n\t&= t + \\left(f(D_{\\text{max}}) - t\\right) \\mathbb{P}\\left[ f^{-1}_u(t) \\le d(X^n,\\hX^n) \\right] \\nonumber\\\\\n\t& \\quad - \\left(t-f(f^{-1}_l(t)-\\delta)\\right) \\mathbb{P}\\left [ d(X^n,\\hX^n) \\le f^{-1}_l(t)-\\delta \\right]\\\\\n\t&\\le t + \\left(f(D_{\\text{max}}) - t\\right) e^{-n \\left( R^u \\left(P_{X,\\hX},f^{-1}_u(t) \\right)  - \\gamma_n \\right)} \\nonumber\\\\\n\t& \\quad - \\left(t-f(f^{-1}_l(t)-\\delta)\\right) e^{-n \\left(R^l\\left(P_{X,\\hX},f^{-1}_l(t)-\\delta\\right) + \\gamma_n\\right) }, \\label{eq: Sanov}\n\t\\end{align}\n\n\\end{subequations}\nwhere \\eqref{eq: Sanov} is due to Sanov's theorem \\cite[Theorem~11.4.1]{Cover}, $P_{X,\\hX}^{h}(\\cdot)$ and $P_{X,\\hX}^{l}(\\cdot)$ are defined in \\eqref{eq: definition of P_c h} and \\eqref{eq: definition of P_c l} respectively, and $\\gamma_n = \\frac{|\\mc{X}||\\mc{\\hX}|\\log(n+1)}{n}$. Therefore, if\n\\begin{align}\n&\\left(f(D_{\\text{max}}) - t\\right) e^{-n \\left( R^u \\left(P_{X,\\hX},f^{-1}_u(t) \\right) -\\gamma_n \\right)} \\nonumber\\\\\n& \\le \\left(t-f(f^{-1}_l(t)-\\delta)\\right) e^{-n \\left( R^l\\left(P_{X,\\hX},f^{-1}_l(t)-\\delta\\right) + \\gamma_n \\right)},\\label{eq: Sanov ineq upper bound}\n\\end{align}\nthen  $\\mathbb{E} [ f(d(X^n,\\hX^n)) ] \\le t$. Note that \\eqref{eq: Sanov ineq upper bound} holds for sufficiently large $n$ if\n\\begin{equation}\nR^u \\left(P_{X,\\hX},f^{-1}_u(t) \\right) >  R^l\\left(P_{X,\\hX},f^{-1}_l(t)-\\delta\\right).\n\\end{equation}\nTherefore, for any $\\delta>0$, $L^{(M)}(t,f)$ is upper bounded by\n\\begin{equation}\n\\begin{array}{rl}\n&L^{(M)}_{\\text{UB}}(t,f,\\delta) \\triangleq \\\\&\n\\end{array}\n\\hspace{-2mm}\n\\begin{array}{rll}\n&\\inf_{P_{\\hX|X,Y}} &  I(Y;\\hX)\\\\\n&\\text{subject to:} & {\\substack{R^l\\left(P_{X,\\hX},f^{-1}_l(t)-\\delta\\right) < R^u \\left(P_{X,\\hX},f^{-1}_u(t) \\right)}}.\n\\end{array}\n\\label{eq: UB for L_M t f and special case t}\n\\end{equation}\n\nNote that $L^{(M)}_{\\text{UB}}(t,f,\\delta)$ is increasing in $\\delta$ because $R^l(\\cdot,\\cdot)$ and $R^u(\\cdot,\\cdot)$ are decreasing and increasing in their second arguments, respectively. Therefore, taking the infimum over $\\delta$ gives\n\\begin{subequations}\n\t\\begingroup\n\t\\allowdisplaybreaks\n\t\\label{eq: UB for L_M t f and special case t 2}\n\\begin{align}\n&L^{(M)}(t,f) \\nonumber\\\\\n& \\le \\inf_{\\delta>0} L^{(M)}_{\\text{UB}}(t,f,\\delta) \\\\\n& = \\inf_{\\substack{\n\\delta>0, P_{\\hX|X,Y} :\\\\\nR^l\\left(P_{X,\\hX},f^{-1}_l(t)-\\delta\\right) \\\\< R^u \\left(P_{X,\\hX},f^{-1}_u(t) \\right)\n}} \\hspace{4mm} I(Y;\\hX)\\\\\n&\\le\n\\inf_{\\substack{P_{\\hX|X,Y}: \\;  \\forall \\delta > 0\\\\ R^l\\left(P_{X,\\hX},f^{-1}_l(t)-\\delta\\right) \\\\< R^u \\left(P_{X,\\hX},f^{-1}_u(t) \\right)}} \\hspace{4mm} I(Y;\\hX)\n\\label{eq: UB for L_M t f and special case t 2a} \\\\\n&\\le \\inf_{\\substack{P_{\\hX|X,Y}: \\\\R^l\\left(P_{X,\\hX},f^{-1}_l(t)\\right) \\\\< R^u \\left(P_{X,\\hX},f^{-1}_u(t) \\right)}}  I(Y;\\hX)\n\\label{eq: UB for L_M t f and special case t 2b}\\\\\n&= \\quad L^{(M)}_{\\text{UB}}(t,f,0)\\nonumber,\n\\end{align}\n\\endgroup\n\\end{subequations}\nwhere \\eqref{eq: UB for L_M t f and special case t 2a} is derived by restricting the feasible set to those distributions that satisfy the constraint for all $\\delta >0$, and \\eqref{eq: UB for L_M t f and special case t 2b} is due to the fact that $R^l(\\cdot)$ is decreasing and continuous in its second argument by Remark \\ref{remark: Ru and Rl are continuous in c}.\n\n\nConversely, we can lower bound the expected distortion cost using a similar argument used in \\eqref{eq: upper bound on E(d) 2}. Thus, we have:\n\\begin{subequations}\n\n\t\\label{eq: lower bound on E(d) 2}\n\t\\allowdisplaybreaks\n\t\\begin{align}\n\t&\\mathbb{E} \\left[ f(d(X^n,\\hX^n)) \\right] \\nonumber\\\\\n\t&\\ge  \\mathbb{P}\\left [ d(X^n,\\hX^n) \\le f^{-1}_l(t) \\right] f(D_{\\text{min}})  \\nonumber\\\\\n\t& \\quad +\\mathbb{P}\\left[ f^{-1}_l(t)  < d(X^n,\\hX^n) < f^{-1}_u(t) + \\delta \\right] t \\nonumber\\\\\n\t& \\quad +\\mathbb{P}\\left[ f^{-1}_u(t) + \\delta \\le d(X^n,\\hX^n) \\right] f(f^{-1}_u(t) + \\delta)\\\\\n\t&= t + \\left(f(f^{-1}_u(t) + \\delta) - t\\right) \\mathbb{P}\\left[ f^{-1}_u(t) + \\delta \\le d(X^n,\\hX^n) \\right] \\nonumber\\\\\n\t& \\quad - \\left(t-f(D_{\\text{min}}))\\right) \\mathbb{P}\\left [ d(X^n,\\hX^n) \\le f^{-1}_l(t) \\right]\\\\\n\t&\\ge t + \\left(f(f^{-1}_u(t) + \\delta) - t\\right) e^{-n \\left( R^u \\left(P_{X,\\hX},f^{-1}_u(t) + \\delta \\right) + \\gamma_n \\right)} \\nonumber\\\\\n\t& \\quad - \\left(t-f(D_{\\text{min}})\\right) e^{-n \\left(R^l \\left(P_{X,\\hX},f^{-1}_l(t) \\right) - \\gamma_n\\right) }, \\label{eq: Sanov 2}\n\t\\end{align}\n\n\\end{subequations}\nIf a mechanism $P_{\\hX|X,Y}$ satisfies $\\mathbb{E} \\left[ f(d(X^n,\\hX^n)) \\right] \\le t$, then\n\\begin{align}\n&\\left(f(f^{-1}_u(t)+\\delta) - t\\right) e^{-n \\left(  R^u \\left(P_{X,\\hX},f^{-1}_u(t) + \\delta \\right) + \\gamma_n \\right)} \\nonumber\\\\\n&- \\left(t- f(D_{\\text{min}})\\right) e^{-n \\left((R^l \\left(P_{X,\\hX},f^{-1}_l(t) \\right) - \\gamma_n\\right) } \\le 0.\\label{eq: Sanov ineq lower bound}\n\\end{align}\nIf \\eqref{eq: Sanov ineq lower bound} holds for sufficiently large $n$, then\n\\begin{equation}\nR^u \\left(P_{X,\\hX},f^{-1}_u(t) + \\delta \\right) \\ge  R^l \\left(P_{X,\\hX},f^{-1}_l(t) \\right).\n\\end{equation}\nHence, for any $\\delta>0$, $L^{(M)}(t,f)$ is lower bounded by\n\\begin{equation}\n\\begin{array}{rl}\n&L^{(M)}_{\\text{LB}}(t,f,\\delta) \\triangleq \\\\&\n\\end{array}\n\\hspace{-2mm}\n\\begin{array}{rll}\n&\\inf_{P_{\\hX|X,Y}} & I(Y;\\hX)\\\\\n&\\text{subject to:} & { \\substack{R^l \\left(P_{X,\\hX},f^{-1}_l(t) \\right)  \\le R^u \\left(P_{X,\\hX},f^{-1}_u(t) + \\delta \\right) }}.\n\\end{array}\n\\label{eq: LB for L_M t f and special case t}\n\\end{equation}\n\n\nNote that $L^{(M)}_{\\text{LB}}(t,f,\\delta)$ is decreasing in $\\delta$ because $R^l(\\cdot,\\cdot)$ and $R^u(\\cdot,\\cdot)$ are decreasing and increasing in their second arguments, respectively. Note that the feasible set in \\eqref{eq: LB for L_M t f and special case t} is closed, due to continuity of $R^l(\\cdot,\\cdot)$ and $R^u(\\cdot,\\cdot)$ in their first arguments and having a non-strict inequality. Therefore, the infimum in \\eqref{eq: LB for L_M t f and special case t} can be replaced with minimum, and minimizing mechanisms exist. For any $\\delta >0$, let $P^{(\\delta)}_{\\hX|X,Y}$(or simply $P^{(\\delta)}$) be a mechanism that achieves $L^{(M)}_{\\text{LB}}(t,f,\\delta)$. Let $\\{\\delta_n\\}_{n=1}^{\\infty}$ be convergent to zero from above. Then, $\\{P^{(\\delta_n)}\\}_{n=1}^{\\infty}$ is a sequence in a compact set, and therefore has a converging subsequence $\\{P^{(\\delta_{n_m})}\\}_{m=1}^{\\infty}$, which converges to some $P^*$. Due to continuity of $R^l(\\cdot, \\cdot)$ and $R^u(\\cdot, \\cdot)$ in their first argument (see Remark \\ref{remark: Ru and Rl are continuous in c}), $P^*$ satisfies $R^l \\left(P^*_{X,\\hX},f^{-1}_l(t) \\right)  \\le R^u \\left(P^*_{X,\\hX},f^{-1}_u(t) \\right)$, and therefore, it is in the feasible set of $L^{(M)}_{\\text{LB}}(t,f,0)$. Hence, by continuity of mutual information we get\n\\begin{equation}\nL^{(M)}_{\\text{LB}}(t,f,0) \\le \\lim_{\\delta \\rightarrow 0^+} L^{(M)}_{\\text{LB}}(t,f,\\delta).\n\\label{eq: limit of LB side 1}\n\\end{equation}\nOn the other hand, note that $R^l(\\cdot,\\cdot)$ is non-increasing and $R^u(\\cdot,\\cdot)$ is non-decreasing  in their second arguments, based on their definition in \\eqref{eq: definition of P_c h} and \\eqref{eq: definition of P_c l}, respectively. Therefore, $L^{(M)}_{\\text{LB}}(t,f,\\delta)$ is non-increasing in $\\delta$. Thus, we have\n\\begin{equation}\nL^{(M)}_{\\text{LB}}(t,f,0) \\ge \\lim_{\\delta \\rightarrow 0^+} L^{(M)}_{\\text{LB}}(t,f,\\delta).\n\\label{eq: limit of LB side 2}\n\\end{equation}\nTherefore, according to \\eqref{eq: limit of LB side 1} and \\eqref{eq: limit of LB side 2} $L^{(M)}_{\\text{LB}}(t,f,\\delta)$ is continuous at $\\delta=0$, and\n\\begin{subequations}\n\t\\begingroup\n\t\\label{eq: LB for L_M t f and special case t 2}\n\t\\allowdisplaybreaks\n\\begin{align}\nL^{(M)}(t,f) &\\ge \\sup_{\\delta >0} L^{(M)}_{\\text{LB}}(t,f,\\delta) \\\\\n&=\\lim_{\\delta \\rightarrow 0^+} L^{(M)}_{\\text{LB}}(t,f,\\delta)\\\\\n&=L^{(M)}_{\\text{LB}}(t,f,0).\n\\end{align}\n\\endgroup\n\\end{subequations}\n\n\nWe now show that the strict inequality in \\eqref{eq: UB for L_M t f and special case t} can be replaced with a non-strict inequality when $\\delta=0$, and consequently, \n\\begin{equation}\nL^{(M)}_{\\text{UB}}(t,f,0) = L^{(M)}_{\\text{LB}}(t,f,0).\n\\label{eq: equality of LB and UB}\n\\end{equation}\nWe do so by showing that for any $P_{\\hX|X,Y}$ that satisfies $R^l \\left(P_{X,\\hX},f^{-1}_l(t) \\right) = R^u \\left(P_{X,\\hX},f^{-1}_u(t) \\right)$ there exists an arbitrarily close $\\bar{P}_{\\hX|X,Y}$ that satisfies $R^l \\left(\\bar{P}_{X,\\hX},f^{-1}_l(t) \\right) < R^u \\left(\\bar{P}_{X,\\hX},f^{-1}_u(t) \\right)$. We first show that for some $(x_0,\\hx_0) \\in \\mathcal{X} \\times \\hat{\\mathcal{X}}$ we have\n\\begin{equation}\n\\frac{\\partial}{\\partial P_{X,\\hX}(x_0,\\hx_0)} \\left( R^l \\left(P_{X,\\hX},f^{-1}_l(t) \\right) - R^u \\left(P_{X,\\hX},f^{-1}_u(t) \\right)\\right) \\neq 0.\n\\label{eq: derrivative of difference is nonzero}\n\\end{equation}\nApplying the results in \\cite[Chapter 14]{PolyanskiNotes} yields that for any given $P_{X,\\hX}$ with $f^{-1}_l(t)\\le E_{P}[d(X,\\hX)] \\le f^{-1}_u(t)$, the minimizing distributions for $R^l \\left(P_{X,\\hX},f^{-1}_l(t) \\right)$ and $R^u \\left(P_{X,\\hX},f^{-1}_u(t) \\right)$ are in the form of $P^{(\\lambda^*_l)}_{X,\\hX}$ and $P^{(\\lambda^*_u)}_{X,\\hX}$ for some $\\lambda^*_l , \\lambda^*_u \\in \\mathbb{R}$, satisfying $\\lambda^*_l \\le 0 \\le \\lambda^*_u$, where\n\\begin{equation}\nP^{(\\lambda)}_{X,\\hX} (x,\\hx) \\triangleq \\frac{e^{\\lambda d(x,\\hx)}}{\\mathbb{E}_{P}[e^{\\lambda d(X,\\hX)}]} P_{X,\\hX}(x,\\hx), \\quad \\lambda \\in \\mathbb{R}.\n\\label{eq: tilted distribution}\n\\end{equation}\nMoreover, by Theorem 14.3 in \\cite{PolyanskiNotes} we have\n\\begin{align}\nR^l \\left(P_{X,\\hX},f^{-1}_l(t) \\right) &=  \\lambda^*_l f^{-1}_l(t) - \\log \\left(\\mathbb{E}\\left[\\exp(\\lambda^*_l d(X,\\hX))\\right]\\right),\\\\\nR^u \\left(P_{X,\\hX},f^{-1}_u(t) \\right) &= \\lambda^*_u f^{-1}_u(t) - \\log \\left(\\mathbb{E}\\left[\\exp(\\lambda^*_u d(X,\\hX))\\right]\\right),\n\\end{align}\nand therefore\n\\begin{align}\n&\\frac{\\partial}{\\partial P_{X,\\hX}(x,\\hx)} \\left( R^l \\left(P_{X,\\hX},f^{-1}_l(t) \\right) - R^u \\left(P_{X,\\hX},f^{-1}_u(t) \\right) \\right) \\\\\n&=  - \\frac{\\exp \\left(\\lambda^*_l d(x,\\hx)\\right)}{\\mathbb{E}\\left[\\exp\\left(\\lambda^*_l d(X,\\hX)\\right)\\right]} + \\frac{\\exp \\left(\\lambda^*_u d(x,\\hx)\\right)}{\\mathbb{E}\\left[\\exp\\left(\\lambda^*_u d(X,\\hX)\\right)\\right]} \\label{eq: the derivative wrp p},\n\\end{align}\nwhere \\eqref{eq: the derivative wrp p} uses the fact that the derivative of $R^l(P_{X,\\hX},f^{-1}_l(t) )$ and $R^u(P_{X,\\hX},f^{-1}_u(t) )$ with respect to $\\lambda$ is zero at $\\lambda^*_l$ and $\\lambda^*_u$, respectively. Note that the term in \\eqref{eq: the derivative wrp p} cannot be zero for all $(x,\\hx) \\in \\mathcal{X} \\times \\hat{\\mathcal{X}}$, because that contradicts $P^{(\\lambda^*_l)}$ and $P^{(\\lambda^*_u)}$, defined in \\eqref{eq: tilted distribution}, being two separate distributions. Thus, \\eqref{eq: derrivative of difference is nonzero} holds for at least one pair $(x_0,\\hx_0) \\in \\mathcal{X} \\times \\hat{\\mathcal{X}}$. Then, for some $\\delta >0 $ let\n\\begin{equation}\n\\bar{P}_{\\hX|X,Y} (\\hx|x,y) = P_{\\hX|X,Y} (\\hx|x,y) \\pm \\begin{cases}\n\\delta, & \\hx=\\hx_0, x=x_0,\\\\\n-\\frac{\\delta}{|\\hat{\\mathcal{X}}| -1}, & \\hx \\neq \\hx_0, x=x_0,\n\\end{cases}\n\\end{equation}\nwhere the sign of $\\pm$ depends on the sign of \\eqref{eq: derrivative of difference is nonzero}. Since $\\delta$ can be chosen arbitrarily small, $\\bar{P}_{\\hX|X,Y}$ is arbitrarily close to $P_{\\hX|X,Y}$. Therefore, due to \\eqref{eq: derrivative of difference is nonzero} we have $R^l \\left(\\bar{P}_{X,\\hX},f^{-1}_l(t) \\right) < R^u \\left(\\bar{P}_{X,\\hX},f^{-1}_u(t) \\right)$.\n\nThen, \\eqref{eq: equality of LB and UB} holds and \\eqref{eq: UB for L_M t f and special case t 2} together with \\eqref{eq: LB for L_M t f and special case t 2} yield\n\\begin{equation}\n\\begin{array}{rll}\nL^{(M)}(t,f)=&\\min_{P_{\\hX|X,Y}} & I(Y;\\hX)\\\\\n&\\text{subject to:} & {\\substack{ R^l \\left(P_{X,\\hX},f^{-1}_l(t) \\right) \\le R^u \\left(P_{X,\\hX},f^{-1}_u(t) \\right)}}.\n\\end{array}\n\\label{eq: LMTF with inequality constraint}\n\\end{equation}\n\n\nWe now prove that if $L(f^{-1}_u(t)) > 0$, then there exists an optimal mechanism for \\eqref{eq: LMTF with inequality constraint} that satisfies the constraint with equality.\nLet $\\hat{P}_{\\hX|X,Y}$ be an optimal mechanism for $L^{(M)}(t,f)$, which satisfies\n\\begin{equation}\nR^l \\left(\\hat{P}_{X,\\hX},f^{-1}_l(t) \\right) \\le R^u \\left(\\hat{P}_{X,\\hX},f^{-1}_u(t) \\right).\n\\end{equation}\nAlso, let $P^*_{\\hX|X,Y}$ be an optimal mechanism for $L(f^{-1}_u(t))$. Since $L(f^{-1}_u(t)) > 0$, we know that $\\mathbb{E}_{P^*} [d(X^n,\\hX^n)] = f^{-1}_u(t)$. Therefore, we have\n\\begin{equation}\nR^l \\left(P^*_{X,\\hX},f^{-1}_l(t) \\right) \\ge R^u \\left(P^*_{X,\\hX},f^{-1}_u(t) \\right) = 0.\n\\end{equation}\nAccording to \\eqref{eq: LM lower limit}, $P^*_{\\hX|X,Y}$ achieves a lower leakage than that of $\\hat{P}_{\\hX|X,Y}$. Since $D(P||Q)$ is a continuous function in both $P$ and $Q$ and the mutual information is convex in the conditional distribution, there exists a mechanism $\\tilde{P}_{\\hX|X,Y}$ on the line connecting $\\hat{P}_{\\hX|X,Y}$ and $P^*_{\\hX|X,Y}$ that satisfies \n\\begin{equation}\nR^l \\left(\\tilde{P}_{X,\\hX},f^{-1}_l(t) \\right) = R^u \\left(\\tilde{P}_{X,\\hX},f^{-1}_u(t) \\right),\n\\end{equation}\nand achieves a leakage at most equal to that of $\\hat{P}_{\\hX|X,Y}$. Therefore, it suffices to replace the constraint in \\eqref{eq: LMTF with inequality constraint} with equality.\n\n\n\\subsection{Proof of Lemma \\ref{lemma: g bounded asymptotic leakage - simple}} \\label{proof: g bounded asymptotic leakage - simple}\n\\textbf{Achievability:}\nWe build a combination of memoryless mechanisms to show achievability. Specifically, we pick the optimal mechanisms for single letter leakage functions evaluated at approximately $D_1, D_2, \\ldots, D_k$. The reason for not choosing the exact values of $D_i$ is that we need the optimal single letter mechanism to satisfy a slightly smaller average distortion bound so that a tail probability constraint is guaranteed.\n\nRecall that $\\mathcal{P}^*(D)$ is the set of optimal single letter mechanisms for $L(D)$.\nThen, for any $D$ let $P^{*(D)}_{\\hat{X}|X,Y}\\in \\mathcal{P}^*(D)$ and $P^{*(D)}_{\\hat{X}^n|X^n,Y^n}=\\left(P^{*(D)}_{\\hat{X}|X,Y}\\right)^n$. Define $\\epsilon^{(n)}_0 = \\epsilon_0 = 1$, and for any $1\\le i \\le k$ let\n\\begin{align}\nD^{(n)}_i &\\triangleq D_i - D_{\\text{max}} \\sqrt{\\frac{\\log n}{n}} \\label{eq: D^n},\\\\\n\\epsilon^{(n)}_i &\\triangleq \\frac{\\epsilon_i -  \\frac{\\epsilon_{i-1}}{n}  - e^{-n\\delta(i)} }{1- \\frac{1}{n}} \\label{eq: eps^n}.\n\\end{align}\nFor the special case where $D_1=D_{\\text{min}}$, let $D^{(n)}_1 = D_1$ instead.\nNote that for sufficiently large $n$ we have $0 \\le D^{(n)}_i \\le D_i$ and $0 \\le \\epsilon^{(n)}_i \\le \\epsilon_i$, which implies that\n\\begin{equation}\n\\epsilon^{(n)}_i = \\epsilon_i + O\\left(\\frac{1}{n}\\right) \\label{eq: epsilon and epsilon^n}.\n\\end{equation}\nNow let $E$ be a random variable independent from $(X^n,Y^n)$ with alphabet set $\\{1,\\ldots,k+1\\}$, where $P(E=i) = \\epsilon^{(n)}_{i-1} - \\epsilon^{(n)}_{i}$ for $1 \\le i \\le k$, and $P(E=k+1) = \\epsilon^{(n)}_{k}$. Then, consider the following mechanism:\n\\begin{align}\n&P_{\\hX^n|X^n,Y^n}(\\hx^n|x^n,y^n) \\nonumber\\\\\n&= \\begin{cases}\nP^{*(D^{(n)}_{E})}_{\\hX^n|X^n,Y^n}(\\hx^n|x^n,y^n), & \\text{if } \\quad 1\\le E \\le k,\\\\\nP^{*(D^{(n)}_k)}_{\\hX^n} (\\hx^n), & \\text{if } \\quad E=k+1.\n\\end{cases}\n\\label{eq: P achievability}\n\\end{align}\nFirst, we show that it is feasible, i.e. it satisfies $P(d(X^n,\\hX^n) > D_i) \\le \\epsilon_i$ for any $ 1 \\le i \\le k$. Since $D^{(n)}_i \\rightarrow D_i$, and $D_i$ has a distinct value for each $i$, there exists a $\\delta(i) > 0$ and $n_i$ such that $\\delta(i) < e^{- \\frac{\\left(D_i - D^{(n)}_{i-1}\\right)^2}{D^2_{\\text{max}}}}$ for $n \\ge n_i$. Therefore, for any $1 \\le i \\le k$ and $n \\ge n_i$ we can bound the $i$th error probability by\n\\begin{subequations}\n\t\\begingroup\n\t\\allowdisplaybreaks\n\t\\begin{align}\n\t&\\mathbb{P}[d(X^n,\\hX^n) >D_i] \\nonumber\\\\\n\t&= \\epsilon^{(n)}_{k+1} P(d(X^n,\\hX^n) >D_i | E=k+1) \\nonumber\\\\\n\t& \\quad + \\sum_{j=1}^{k}  \\left( \\epsilon^{(n)}_{j-1} - \\epsilon^{(n)}_j  \\right) P(d(X^n,\\hX^n) >D_i | E=j)\\nonumber \\\\\n\t& \\le \\epsilon^{(n)}_i  + \\sum_{j=1}^{i}  \\left( \\epsilon^{(n)}_{j-1} - \\epsilon^{(n)}_j  \\right) P(d(X^n,\\hX^n) >D_i | E=j)\\\\\n\t&\\le \\epsilon^{(n)}_i  + \\sum_{j=1}^{i}  \\left( \\epsilon^{(n)}_{j-1} - \\epsilon^{(n)}_j  \\right) e^{-n \\frac{\\left(D_i - D^{(n)}_j\\right)^2}{D^2_{\\text{max}}}} \\label{eq: Chernoff bound}\\\\\n\t&\\le \\epsilon^{(n)}_i + e^{-n \\frac{\\left(D_i - D^{(n)}_{i-1}\\right)^2}{D^2_{\\text{max}}}}  \\sum_{j=1}^{i-1}  \\left( \\epsilon^{(n)}_{j-1} - \\epsilon^{(n)}_j  \\right) \\nonumber \\\\\n\t& \\quad + \\left( \\epsilon^{(n)}_{i-1} - \\epsilon^{(n)}_i  \\right) e^{-n \\frac{\\left(D_i - D^{(n)}_i\\right)^2}{D^2_{\\text{max}}}}\\\\\n\t& = \\epsilon^{(n)}_i + e^{-n \\frac{\\left(D_i - D^{(n)}_{i-1}\\right)^2}{D^2_{\\text{max}}}}  \\left( 1 - \\epsilon^{(n)}_{i-1}  \\right) + \\frac{1}{n}\\left( \\epsilon^{(n)}_{i-1} - \\epsilon^{(n)}_i  \\right) \\\\\n\t& \\le \\epsilon^{(n)}_i + e^{-n\\delta(i)}   \\left( 1 - \\epsilon^{(n)}_{i-1}  \\right) + \\frac{1}{n}\\left( \\epsilon^{(n)}_{i-1} - \\epsilon^{(n)}_i  \\right)  \\label{eq: delta definition}\\\\\n\t& \\le  \\epsilon^{(n)}_i + \\frac{1}{n}\\left( \\epsilon^{(n)}_{i-1} - \\epsilon^{(n)}_i  \\right)  + e^{-n\\delta(i)} \\\\\n\t& \\le  \\epsilon^{(n)}_i + \\frac{1}{n}\\left( \\epsilon_{i-1} - \\epsilon^{(n)}_i  \\right)  + e^{-n\\delta(i)} \\\\\n\t& \\le \\epsilon_i \\label{eq: total probability of error is less than epsilon},\n\t\\end{align}\n\t\\endgroup\n\\end{subequations}\nwhere \\eqref{eq: Chernoff bound} follows from Lemma \\ref{lemma: Hoeffding}, \\eqref{eq: delta definition} is due to the definition of $\\delta(i)$, and \\eqref{eq: total probability of error is less than epsilon} results from \\eqref{eq: D^n} and \\eqref{eq: eps^n}.\nNote that in the special case where $D_1=D_{\\text{min}}$, we have $D^{(n)}_1 = D_1=D_{\\text{min}}$. Therefore, $\\mathbb{P}[d(X^n,\\hX^n) >D_1] = 0$, because the optimal mechanism achieving $L(D_{\\text{min}})$ has to satisfy $\\mathbb{P}[d(X^n,\\hX^n) = D_1] = 1$.\n\nWe now show that the mechanism introduced in \\eqref{eq: P achievability} achieves \\eqref{eq: multi constraint}. Recalling the definition of $E$ we have\n\\begin{subequations}\n\t\\begingroup\n\t\\allowdisplaybreaks\n\t\\begin{align}\n\t&I(Y^n;\\hX^n)  \\\\\n\t&\\le I(Y^n;\\hX^n|E) \\\\\n\t&= \\sum_{j=1}^{k} (\\epsilon^{(n)}_{j-1}-\\epsilon^{(n)}_{j}) I(Y^n;\\hX^n|E=j) \\nonumber\\\\\n\t& \\quad + \\epsilon^{(n)}_{k} \\; I(Y^n;\\hX^n|E=k+1)\\\\\n\t&=\\sum_{j=1}^{k} (\\epsilon^{(n)}_{j-1}-\\epsilon^{(n)}_{j}) n \\; L(D^{(n)}_j) \\label{eq: lemma 1 UB}\\\\\n\t& = n \\sum_{j=1}^{k} (\\epsilon_{j-1}-\\epsilon_{j}) \\; L(D_j)  + O(\\sqrt{n\\log n}),\n\n\t\\label{eq: lemma 1 UB part2}\n\t\\end{align}\n\t\\endgroup\n\\end{subequations}\nwhere \\eqref{eq: lemma 1 UB} is due to definition of the chosen mechanism in \\eqref{eq: P achievability}, and \\eqref{eq: lemma 1 UB part2} is implied by \\eqref{eq: D^n}. This yields the upper bound in \\eqref{eq: multi constraint LB and UB}.\n\n\n\\textbf{Converse:}\nAssume a mechanism $P_{\\hat{X}^n|X^n,Y^n}$ satisfying the feasibility constraint of \\eqref{eq: LG definition}. Define the indicator random variable $E$ as\n\\begin{equation}\nE = \\begin{cases}\n1, & \\text{if } d(X^n,\\hX^n) \\le D_1,\\\\\n2, & \\text{if } D_1 < d(X^n,\\hX^n) \\le D_2,\\\\\n\\vdots&\\vdots\\\\\nk+1, & \\text{if } D_k < d(X^n,\\hX^n).\n\\end{cases}\n\\end{equation}\nLet $P_{e_i} = \\mathbb{P}[E\\ge i]$ for $i=1,\\cdots, k+1$ and $\\epsilon_0 = 1$. Clearly, for all feasible $P_{\\hX^n|X^n,Y^n}$ and $1\\le i \\le k+1$, we have $P_{e_i} \\le \\epsilon_{i-1}$. Then:\n\\begin{subequations}\n\t\\begingroup\n\t\\allowdisplaybreaks\n\t\\begin{align}\t\n&I(Y^n;\\hat{X}^n) \\nonumber\\\\\n&= H(Y^n) - H(Y^n | \\hX^n)\\\\\n&=H(Y^n) - H(Y^n|\\hX^n,E) - I(Y^n;E|\\hX^n)\\\\\n&\\ge H(Y^n)-H(Y^n|\\hat{X}^n,E)-H(E) \\\\\n&\\ge n H(Y)-\\sum_i (P_{e_i}-P_{e_{i+1}}) H(Y^n|\\hat{X}^n,E=i)-\\log(k+1). \\label{eq: converse of Lemma 1}\n\t\\end{align}\n\t\\endgroup\n\\end{subequations}\nWe now bound $H(Y^n|\\hX^n, E=i)$ for each $i$. Note that:\n\\begin{subequations}\n\t\\begingroup\n\t\\allowdisplaybreaks\n\t\\begin{align}\n\tH(Y^n|\\hX^n, E=i) &= \\sum_{j=1}^{n} H(Y_j|\\hX_j, Y^{j-1}, E=i)\\\\\n\t& \\le \\sum_{j=1}^{n} H(Y_j|\\hX_j, E=i).\n\t\\end{align}\n\t\\endgroup\n\\end{subequations}\nIf we define $t^{(i)}_j = \\mathbb{E} [d(X_j, \\hX_j) | E=i]$, then we have\n\\begin{subequations}\n\t\\begingroup\n\t\\allowdisplaybreaks\n\t\\begin{align}\n\tL \\left(t^{(i)}_j\\right) &= \\min_{\\substack{P_{\\hX|X,Y}: \\mathbb{E} [d(X,\\hX)] \\le t^{(i)}_j}} I(Y;\\hX) \\\\\n\t&= H(Y) - \\max_{P_{\\hX|X,Y}: \\mathbb{E} [d(X,\\hX)] \\le t^{(i)}_j} H(Y|\\hX).\n\t\\end{align}\n\t\\endgroup\n\\end{subequations}\nTherefore, $H(Y_j|\\hX_j,E=i) \\le H(Y) - L(t^{(i)}_j)$ for any $i=1,\\ldots,n$, and\n\\begin{subequations}\n\t\\begingroup\n\t\\allowdisplaybreaks\n\t\\begin{align}\n\tH(Y^n | \\hX^n,E=i) &\\le \\sum_{j=1}^{n} \\left[ H(Y) - L(t^{(i)}_j) \\right]\\\\\n\t& \\le n H(Y) - n L(D_i)\n\t\\label{eq: upper bound on H(Y|X_hat,E=i)},\n\t\\end{align}\n\t\\endgroup\n\\end{subequations}\nwhere \\eqref{eq: upper bound on H(Y|X_hat,E=i)} is due to convexity of $L(\\cdot)$ and the fact that $\\sum_{j} t^{(i)}_j \\le n D_i$ by definition.\nReplacing \\eqref{eq: upper bound on H(Y|X_hat,E=i)} in \\eqref{eq: converse of Lemma 1} gives\n\\begin{subequations}\n\t\\begingroup\n\t\\allowdisplaybreaks\n\t\\begin{align}\n\t&I(Y^n ; \\hX^n) \\nonumber\\\\\n\t&\\ge \\sum_{i=1}^{k} (P_{e_i}-P_{e_{i+1}})  n L(D_i) -\\log(k+1)\\\\\n\t&\\ge  n \\sum_{i=1}^{k} (\\epsilon_{i-1} - \\epsilon_{i})  L(D_i) -\\log(k+1) \\label{eq: p_i and eps_i},\n\t\\end{align}\n\t\\endgroup\n\\end{subequations}\nwhere \\eqref{eq: p_i and eps_i} is due to the fact that $L(\\cdot)$ is non-increasing, $\\epsilon_0=1$, and for any feasible $P_{\\hX^n|X^n,Y^n}$ and $2\\le i \\le k+1$, we have $P_{e_i} \\le \\epsilon_{i-1}$. This yields the lower bound in \\eqref{eq: multi constraint result}.\n\n\n\\subsection{Proof of Theorem \\ref{theorem: f approximation}} \\label{proof: f approximation}\nWe will need the following lemma in our proof for Theorem \\ref{theorem: f approximation}.\n\\begin{lemma}\n\tFor any given $n$ and $f$, $L^{(G)} (n,t,f)$ is convex in $t$. Consequently, $L^{(G)} (t,f)$ is also convex in $t$, for any $f$.\n\t\\label{lemma: LG is convex in t}\n\\end{lemma}\n\\begin{IEEEproof}\n\tFor any $t_1,t_2$, and some $0\\le\\lambda \\le 1$, let $t_{\\lambda} = \\lambda t_1 + (1-\\lambda) t_2$. We will show that $L(n,t_{\\lambda},f) \\le \\lambda L(n,t_1,f) + (1-\\lambda) L(n,t_2,f)$. Let $P_1$ and $P_2$ be optimal mechanisms for $L^{(G)}(n,t_1,f)$ and $L^{(G)}(n,t_2,f)$ respectively, and $P_{\\lambda} \\triangleq \\lambda P_1 + (1-\\lambda) P_2$. Note that $P_\\lambda$ is feasible for $L^{(G)}(n,t_\\lambda,f)$ because \n\t\\begin{align}\n\t&\\mathbb{E}_{P_{\\lambda}}\\left[f\\left(d(X^n,\\hX^n)\\right)\\right] \\nonumber \\\\\n\t&= \\lambda \\mathbb{E}_{P_{1}}\\left[f\\left(d(X^n,\\hX^n)\\right)\\right] + (1-\\lambda) \\mathbb{E}_{P_2}\\left[f(d(X^n,\\hX^n))\\right] \\nonumber\\\\\n\t&\\le \\lambda t_1 + (1-\\lambda) t_2 = t_{\\lambda}.\n\t\\end{align}\n\tMoreover, since $I(Y^n;\\hX^n)$ is convex in $P_{\\hX^n|X^n,Y^n}$, the leakage achieved by $P_\\lambda$ is at most equal to $\\lambda L^{(G)}(n,t_1,f) + (1-\\lambda) L^{(G)}(n,t_2,f)$ which implies $L^{(G)}(n,t_{\\lambda},f) \\le \\lambda L^{(G)}(n,t_1,f) + (1-\\lambda) L^{(G)}(n,t_2,f)$. Finally we note that the asymptotic leakage $L^{(G)}(t,f)$ is also convex in $t$ because it is the limit of convex functions in $t$.\n\\end{IEEEproof}\n\nWe now present an achievable scheme and a converse for Theorem \\ref{theorem: f approximation}.\n\n\\textbf{Achievability:}\nWe know that $L^{(G)}(t,f) \\le L^{(M)}(t,f) \\le L(f^{-1}_l(t))$, where the latter inequality is due to Theorem \\ref{theorem: f approximation memoryless}. Since by Lemma \\ref{lemma: LG is convex in t}, $L^{(G)}(t,f)$ is a convex function in $t$, the definition of lower convex envelope gives $L^{(G)}(t,f) \\le (L \\circ f^{-1}_l)^{**}(t)$. This in turn gives $L^{(G)}(t,f) \\le (L \\circ f^{-1})^{**} (t)$ due to Remark \\ref{remark: upper and lower g** are equal}.\n\n\\textbf{Converse:}\nWe first focus on the class of piecewise step functions $f$, and then show that the result holds for any function $f$, using piecewise step approximations of $f$.\n\n\n\\textit{Piecewise Step functions $f$:} Let us consider the class of functions $f$ that are of the form\n\\begin{equation}\nf(D) = \\sum_{i=1}^{k} a_i \\boldsymbol{1}(D>D_i),\n\\label{eq: f approx}\n\\end{equation}\nwhere $k$ is finite and each $D_i$ is a distinct distortion level with $f(D_i) < f(D_j)$ for $i<j$. For this class of functions, \\eqref{eq: LG definition} simplifies and can be lower bounded as\n\\begin{subequations}\n\t\\begingroup\n\t\\allowdisplaybreaks\n\t\\begin{align}\n\t&L^{(G)}(n,t,f) = \\min_{\\substack{P_{\\hX^n|X^n,Y^n} :\\\\ \\mathbb{E}[\\sum_{i=1}^{k} a_i \\boldsymbol{1} (d(X^n,\\hX^n) > D_i)]\\le t}} \\frac{1}{n}I(Y^n;\\hat{X}^n) \\nonumber\\\\\n\t& =\\min_{\\substack{P_{\\hX^n|X^n,Y^n} :\\\\ \\sum_{i=1}^{k} a_i \\mathbb{P}[ (d(X^n,\\hX^n) > D_i)]\\le t}} \\frac{1}{n}I(Y^n;\\hat{X}^n)\\\\\n\t&= \\min_{\\substack{0 \\le \\epsilon_k\\le\\ldots\\le\\epsilon_1\\le 1:\\\\ \\sum_{i=1}^{k} a_i \\epsilon_i\\le t}} \\min_{\\substack{P_{\\hX^n|X^n,Y^n} :\\\\ \\mathbb{P}[d(X^n,\\hX^n) > D_i] \\le \\epsilon_i, \\\\ \\forall 1 \\le i  \\le k}} \\frac{1}{n}I(Y^n;\\hat{X}^n)\\\\\n\t&\\ge \\min_{\\substack{0 \\le \\epsilon_k\\le\\ldots\\le\\epsilon_1\\le 1:\\\\ \\sum_{i=1}^{k} a_i \\epsilon_i\\le t}} \\sum_{i=1}^{k} (\\epsilon_{i-1} - \\epsilon_{i})  L(D_i) \\nonumber\\\\\n\t\t& \\qquad + \\epsilon_k L(D_{\\text{max}}) - \\frac{\\log (k+1)}{n} \\label{eq: optimization converse 1}\\\\\n\t&\\ge \\max_{\\lambda \\ge 0} \\min_{0 \\le \\epsilon_k\\le\\ldots\\le\\epsilon_1\\le 1} \\sum_{i=1}^{k} (\\epsilon_{i-1} - \\epsilon_{i}) L(D_i) + \\epsilon_k L(D_{\\text{max}}) \\nonumber\\\\\n\t\t&\\qquad  +\\lambda \\sum_{i=1}^{k} a_i \\epsilon_i - \\lambda t - \\frac{\\log (k+1)}{n}\\label{eq: optimization converse 2}\\\\\n\t&= \\max_{\\lambda \\ge 0} \\min_{\\substack{ \\gamma_1 \\cdots \\gamma_{k+1} :\\\\ \\gamma_i \\ge 0, \\forall i=1,\\ldots,k+1,\\\\\\sum_{i=1}^{k+1} \\gamma_i=1}} \\sum_{i=1}^{k+1} \\gamma_i L(D_i) - \\frac{\\log (k+1)}{n} \\nonumber\\\\\n\t\t& \\qquad +\\lambda \\sum_{i=1}^{k+1} f(D_i)\\gamma_i - \\lambda t \\label{eq: optimization converse 3}\\\\\n\t& = \\max_{\\lambda \\ge 0}  \\min_i L(D_i)+\\lambda f(D_i)-\\lambda t - \\frac{\\log (k+1)}{n} \\label{eq: optimization converse 4}\\\\\n\t& = \\max_{\\lambda \\ge 0}  \\min_i L(f^{-1}_u(t_i))+\\lambda t_i-\\lambda t - \\frac{\\log (k+1)}{n}\\label{eq: optimization converse 5}\n\t\\end{align}\n\t\\endgroup\n\\end{subequations}\nwhere\n\\begin{itemize}\n\t\\item \\eqref{eq: optimization converse 1} follows from Lemma \\ref{lemma: g bounded asymptotic leakage - simple}, and the fact that $L(D_{\\text{max}})=0$,\n\t\\item \\eqref{eq: optimization converse 2} is due to forming the Lagrangian given by incorporating only the last constraint in \\eqref{eq: optimization converse 1}, i.e. $\\sum_{i=1}^{k} a_i \\epsilon_i \\le t$,\n\t\\item \\eqref{eq: optimization converse 3} is derived by letting $\\epsilon_{k+1} = 0$, $D_{k+1} = D_{\\text{max}}$, and $\\gamma_i = \\epsilon_{i-1} - \\epsilon_i$, for $i=1,\\ldots,k+1$.\n\t\\item \\eqref{eq: optimization converse 4} holds because a convex combination of non-negative real numbers is minimized by choosing a $\\boldsymbol{\\gamma}$ with $\\gamma_i=1$ for some $i$ corresponding to the smallest $L(D_i)+\\lambda f(D_i)$, and $\\gamma_j=0$, for all other $j \\neq i$,\n\t\\item and \\eqref{eq: optimization converse 5} is derived by defining $t_i=f(D_i)$, i.e. $D_i = f^{-1}_u(t_i)$.\n\\end{itemize}\nThen, by taking the limit as $n \\rightarrow \\infty$ we have\n\\begin{equation}\nL^{(G)}(t,f) = \\max_{\\lambda}  \\min_i L(f^{-1}_u(t_i))+\\lambda t_i-\\lambda t .\n\\end{equation}\nNote that the $i$th function, $L(f^{-1}_u(t_i))+\\lambda t_i$ is a minimizer for some $\\lambda$, if for all $j \\neq i$ we have\n\\begin{equation}\nL(f^{-1}_u(t_i)) + \\lambda t_i \\le L(f^{-1}_u(t_j)) + \\lambda t_j,\n\\end{equation}\nor equivalently\n\\begin{subequations}\n\t\\begingroup\n\t\\allowdisplaybreaks\n\t\\begin{align}\n\t&\\frac{L(f^{-1}_u(t_i))-L(f^{-1}_u(t_j))}{t_{i}-t_{j}} \\le -\\lambda, &\\text{   for $j < i$}, \\label{eq: on the convex envelope a}\\\\\n\t&\\frac{L(f^{-1}_u(t_i))-L(f^{-1}_u(t_j))}{t_{i}-t_{j}} \\ge -\\lambda, &\\text{   for $j > i$}\\label{eq: on the convex envelope b}.\n\t\\end{align}\n\t\\endgroup\n\\end{subequations}\nNote that \\eqref{eq: on the convex envelope a} and \\eqref{eq: on the convex envelope a} imply the slope of the line connecting points $\\{(t_i,L(f^{-1}_u(t_i))), (t_j,L(f^{-1}_u(t_j)))\\}$ is not larger than $-\\lambda$, for $j < i$, and not smaller than $-\\lambda$, for $j >i$. This holds if and only if $L(f^{-1}_u(t_i))= (L \\circ f^{-1}_u)^{**}(t_i)$. Since $(L \\circ f^{-1}_u)^{**}(t_i) = (L \\circ f^{-1})^{**}(t_i)$ due to Remark \\ref{remark: upper and lower g** are equal}, the only relevant $i$ in the minimization in \\eqref{eq: optimization converse 4} are those for which $L(f^{-1}_u(t_i))=(L \\circ f^{-1})^{**}(t_i)$. Hence, \\eqref{eq: optimization converse 4} can be rewritten as\n\t\\begin{align}\n\t&L^{(G)}(t,f)\\ge \\nonumber\\\\\n\t&\\max_{\\lambda}  \\min_{i: L(f^{-1}_u(t_i))=(L \\circ f^{-1})^{**}(t_i)} L(f^{-1}_u(t_i))+\\lambda t_i-\\lambda t.\n\t\\end{align}\nFor a chosen $\\lambda$ and $i$, $L(f^{-1}_u(t_i))+\\lambda t_i-\\lambda t$ is the evaluation of a linear function at $t$, which is tangential to $(L \\circ f^{-1})^{**}(\\cdot)$ at $(t_i, (L \\circ f^{-1})^{**}(t_i))$, with slope $-\\lambda$. This value is always smaller than or equal to $(L \\circ f^{-1})^{**}(t)$, and because $(L \\circ f^{-1})^{**}(\\cdot)$ is a convex piecewise linear function, it suffices to optimize over only those values of $\\lambda$ that are equal to the slope of the linear segment of $(L \\circ f^{-1})^{**}(\\cdot)$ that contains $t$. Thus, for an optimal $\\lambda$ we have $\\min_i (L \\circ f^{-1}_u)(t_i)+\\lambda t_i-\\lambda t = (L \\circ f^{-1})^{**}(t)$, resulting in $L^{(G)}(t,f) \\ge (L \\circ f^{-1})^{**}(t)$.\n\t\n\\textit{General functions $f$:} Finally, we now show that $L^{(G)} (t,f) \\ge (L \\circ f^{-1})^{**}(t)$ for the case of general non-decreasing left continuous functions $f$.\nFor any $\\delta > 0$, there exists a lower approximation $f_\\delta$ of $f$ over $[D_{\\text{min}},D_{\\text{max}}]$ that has the form of \\eqref{eq: f approx} with a finite number of step functions, i.e. $f_\\delta(x) = \\sum_{i=1}^{k} a_i 1_{D_i}(x)$, with $a_i = f(D_i)-f(D_{i-1})$ for $1\\le i \\le k$ and $a_{\\text{max}} \\triangleq \\max_{i} a_i \\le \\delta$. Then, we have $f_\\delta(D) < f(D) \\le f_\\delta(D) + \\delta$, and thus\n\\begin{subequations}\n\t\\begingroup\n\t\\allowdisplaybreaks\n\\begin{align}\nL^{(G)}(t,f) &\\ge L^{(G)}(t,f_\\delta)\\label{eq: lower LG(t,f_delta) le LG(t,f)}\\\\\n&\\ge (L \\circ {f_\\delta}^{-1}_u)^{**}(t) \\label{eq: for f_delta LG ge g**}\\\\\n& = (L \\circ {f_\\delta}^{-1})^{**}(t), \\label{eq: g**(bar_f_delta) = g**(f_delta)}\n\\end{align}\n\t\\endgroup\n\\end{subequations}\nwhere \\eqref{eq: lower LG(t,f_delta) le LG(t,f)} holds because we have $L^{(G)}(n,t,f_\\delta) \\le L^{(G)}(n,t,f)$ for any $n$, \\eqref{eq: for f_delta LG ge g**} is based on the result we had earlier on piecewise step functions specifically, and \\eqref{eq: g**(bar_f_delta) = g**(f_delta)} is due to Remark \\ref{remark: upper and lower g** are equal}.\nThen, taking the limit as $\\delta \\rightarrow 0$ and the fact that $\\lim_{\\delta \\rightarrow 0} f_\\delta(D) = f(D)$ gives $L^{(G)}(t,f) \\ge (L \\circ f^{-1})^{**}(t)$.\n\n\n\n\n\\subsection{Proof of Theorem \\ref{theorem: g bounded asymptotic leakage}} \\label{proof: g bounded asymptotic leakage}\n\n\nWe now proceed to proving the result in \\eqref{eq: g bounded asymptotic leakage result} for all non-increasing right-continuous functions $g:[D_{\\text{min}},D_{\\text{max}}] \\rightarrow (0,1]$. Recall that we proved this for simple functions through Lemma \\ref{lemma: g bounded asymptotic leakage - simple}. For any bounded, non-increasing, and right-continuous function $g$, there exist two sequences of simple functions $\\{\\overline{g}_i\\}_{i=1}^{\\infty}$ and $\\{\\underline{g}_i\\}_{i=1}^{\\infty}$ that are bounded away from zero, converge to $g$ uniformly from above and below, respectively, and each of functions $\\overline{g}_i$ and $\\underline{g}_i$ takes $i$ distinct values. Since $\\underline{g}_i(D) \\le g(D) \\le \\overline{g}_i (D)$ for all $i\\ge1$, $D \\in [D_{\\text{min}},D_{\\text{max}}]$, and the asymptotic optimal leakage for simple constraint functions is the integral in \\eqref{eq: multi constraint result}, for each $i \\ge 1$ we have\n\\begin{equation}\n\\int_{D_{\\text{min}}}^{D_{\\text{max}}} L(D) d(\\overline{g}_i(D)) \\le L^{(G)} (g) \\le \\int_{D_{\\text{min}}}^{D_{\\text{max}}} L(D) d(\\underline{g}_i(D)).\n\\label{eq: ub and lb with simple functions}\n\\end{equation}\nSince $L(\\cdot)$ and $g(\\cdot)$ are bounded, the integral $\\int_{D_{\\text{min}}}^{D_{\\text{max}}} L(D) d(g(D))$ exists. Therefore, in order to prove\n\\begin{equation}\nL^{(G)} (g) = \\int_{D_{\\text{min}}}^{D_{\\text{max}}} L(D) d(g(D)),\n\\label{eq: convergence of the leakage to the integral}\n\\end{equation}\nit suffices to show that \n\\begin{equation}\n\\lim_{i \\rightarrow \\infty} \\int_{D_{\\text{min}}}^{D_{\\text{max}}} L(D) d(\\underline{g}_i(D)) = \\int_{D_{\\text{min}}}^{D_{\\text{max}}} L(D) d(g(D)),\n\\end{equation}\nand similarly for the integral with respect to $d(\\overline{g}_i(D))$. In order to do so, we use the uniform convergence of $\\underline{g}_i$ to $g$, and integration by parts. Since $L(\\cdot)$ is a convex, and therefore, continuous function, the Lebesgue\u2013--Stieltjes integral $\\int_{D_{\\text{min}}}^{D_{\\text{max}}} L(D) d(g(D))$ reduces to a Riemann\u2013--Stieltjes integral, and admits integration by parts \\cite{Hille1996}. Thus, we can bound the difference of the two integrals as\n\\begin{subequations}\n\t\\begingroup\n\t\\allowdisplaybreaks\n\\begin{align}\n&\\left|\\int_{D_{\\text{min}}}^{D_{\\text{max}}} L(D) d(\\underline{g}_i(D)) -  \\int_{D_{\\text{min}}}^{D_{\\text{max}}} L(D) d(g(D))\\right|\\\\\n& = \\Bigg | L(D) \\left( \\underline{g}_i(D)\\big|^{D_{\\text{max}}}_{D_{\\text{min}}}  - g(D)\\big|^{D_{\\text{max}}}_{D_{\\text{min}}} \\right) \\nonumber \\\\\n& \\quad + \\int_{D_{\\text{min}}}^{D_{\\text{max}}} \\left(\\underline{g}_i(D)-g(D)\\right) d(L(D))  \\Bigg |\\\\\n& \\le \\left|L(D) \\left( \\underline{g}_i(D)\\big|^{D_{\\text{max}}}_{D_{\\text{min}}}  - g(D)\\big|^{D_{\\text{max}}}_{D_{\\text{min}}} \\right)\\right| \\nonumber\\\\\n& \\quad  + \\left| \\int_{D_{\\text{min}}}^{D_{\\text{max}}} \\left(\\underline{g}_i(D)-g(D)\\right) d(L(D))  \\right |,\n\\end{align}\n\t\\endgroup\n\\end{subequations}\nwhich goes to zero as $i \\rightarrow \\infty$, due to uniform convergence of $\\underline{g}_i$ to $g$. One can also verify the same argument for $d(\\overline{g}_i(D))$. Hence, both of the integrals in \\eqref{eq: ub and lb with simple functions} converge to the same value $\\int_{D_{\\text{min}}}^{D_{\\text{max}}} L(D) d(g(D))$, and therefore \\eqref{eq: convergence of the leakage to the integral} holds.\n\n\n\\subsection{Proof of Lemma \\ref{lemma: single letter leakage for doubly symmetric source}} \\label{proof: single letter leakage for doubly symmetric source}\nDue to the symmetry of the source distribution, and convexity of mutual information in conditional distribution, there exists an optimal mechanism with\n\\begin{align}\nP(\\hX=1|X=0,Y=1)&=P(\\hX=0|X=1,Y=0)&=\\beta_1,\\\\\nP(\\hX=1|X=0,Y=0)&=P(\\hX=0|X=1,Y=1)&= \\beta_2.\n\\end{align}\nTherefore, it suffices to optimize over all feasible values of $\\beta_1$ and $\\beta_2$. Rewriting the joint distribution $P_{Y,\\hX}$ in terms of $\\beta_1$, $\\beta_2$, and $q$ gives\n\\begin{align}\nP(Y=0,\\hX=1) &= P(Y=1,\\hX=0) \\nonumber\\\\\n&= 0.5 \\left[(1-q) \\beta_2 + q (1-\\beta_1) \\right]\\label{eq: Y Xhat joint 1},\\\\\nP(Y=0,\\hX=0) &= P(Y=1,\\hX=1) \\nonumber\\\\\n&= 0.5 \\left[(1-q) (1-\\beta_2) + q \\beta_1 \\right] \\label{eq: Y Xhat joint 2}.\n\\end{align}\nTherefore, we have\n\\begin{subequations}\n\t\\begingroup\n\t\\allowdisplaybreaks\n\\begin{align}\nL(D) &= \\min_{\\substack{0 \\le \\beta_1,\\beta_2 \\le 1: \\\\ (1-q)\\beta_2 + q\\beta_1 \\le D}} H(\\hX) - H(\\hX|Y) \\\\\n&= \\min_{\\substack{0 \\le \\beta_1,\\beta_2 \\le 1: \\\\ (1-q)\\beta_2 + q\\beta_1 \\le D}} 1 - H_b\\left((1-q)\\beta_2 + q(1-\\beta_1)\\right) \\label{eq: lemma: single letter leakage for doubly symmetric source: simplification 1}\\\\\n&=\\min_{\\substack{q- D \\le \\gamma \\le q+D,\\\\ 0 \\le \\gamma \\le 1}} 1 - H_b(\\gamma) \\label{eq: lemma: single letter leakage for doubly symmetric source: simplification 2}\\\\\n&= \\begin{cases}\n1 - H_b(q + D), & D < 0.5 - q,\\\\\n0, &D\\ge 0.5 - q.\n\\end{cases} \\label{eq: lemma: single letter leakage for doubly symmetric source: simplification 3}\n\\end{align}\n\\endgroup\n\\end{subequations}\nwhere \n\\begin{itemize}\n\t\\item \\eqref{eq: lemma: single letter leakage for doubly symmetric source: simplification 1} is due to  \\eqref{eq: Y Xhat joint 1} and \\eqref{eq: Y Xhat joint 2},\n\t\\item \\eqref{eq: lemma: single letter leakage for doubly symmetric source: simplification 2} holds because $q \\le 0.5$ and the minimum and maximum values of $(1-q)\\beta_2 + q(1-\\beta_1)$ subject to $(1-q)\\beta_2 + q\\beta_1 \\le D$ are $\\min\\{q+D, 1\\}$ and $\\max\\{q-D,0\\}$, respectively.\n\t\n\tIf $D < q$, then the extreme values occur at the corner points of the feasible region with $(\\beta_1 = 0, \\beta_2 = \\frac{D}{1-q})$, and $(\\beta_1 = \\frac{D}{q}, \\beta_2 = 0)$. Otherwise, if $ q \\le D \\le 1-q$, then the minimum and maximum values will be $0$ and $q+D$, respectively. Finally, for $D > 1-q$ the extreme values will be $0$ and $1$. The first scenario is depicted in Fig. \\ref{fig: feasible set for LP}.\n\t\\item \\eqref{eq: lemma: single letter leakage for doubly symmetric source: simplification 2} is due to the fact that the binary entropy function $H_b(\\cdot)$ is concave and maximized at $0.5$.\n\\end{itemize}\n\\begin{figure}[htb!]\n\t\\centering\n\t\\includegraphics[width= 0.45 \\columnwidth]{feasible.pdf}\n\t\\caption{The feasible set and extreme values for $(1-q)\\beta_2 + q(1-\\beta_1)$ subject to $(1-q)\\beta_2 + q\\beta_1 \\le D$, if $D < q$.}\n\t\\label{fig: feasible set for LP}\n\\end{figure}\n\n\n\\section{Conclusion}\n\nWe have formulated the tradeoff between privacy and utility as a minimization of mutual information between private and released data subject to two different forms of distortion constraints: the average distortion cost constraint and the complementary CDF bound on distortion. The former allows for taking non-separable distortion measures into account, while the latter enables the data publisher to provide refined guarantees on utility.\n\nFor the average distortion cost constraints, we have characterized the asymptotically optimal leakage for both stationary memoryless and general mechanisms as a function of the single letter leakage function $L$ and the distortion cost function $f$. In particular, we have shown that a memoryless mechanism achieves the asymptotically optimal leakage if and only if the information leakage-cost function $L(f^{-1}(\\cdot))$ coincides with its lower convex envelope; otherwise, a mixture of exactly two memoryless mechanisms is sufficient.\n\nFor the complementary CDF bound on distortion, we have derived the asymptotically optimal leakage. We have shown that under general mechanisms the optimal leakage is equal to the integral of the single letter leakage function with respect to the Lebesgue---Stieltjes measure associated with the complementary CDF bound, while for stationary and memoryless mechanisms, it is equal to the single letter leakage function evaluated at the largest value of distortion for which the CDF bound function is equal to one.\n\n\nFor both types of utility constraints, the challenge remains to characterize the second order performance of the leakage as a function of the data size $n$. More generally, the proof techniques developed here for arbitrary cost functions and complementary CDF bounds on distortion are applicable to a broad class of information theoretic problems such as lossy source coding with fidelity constraints and channel coding with input cost constraints.\n\n\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Acknowledgements}\nAuthors express their deep debt of gratitude to Pierre Gaspard for\nstimulating discussions and constructive criticism. S.R.J. is financially\nsupported by the \"Communaute Francaise de Belgique\" under contract no.\nARC-93\/98-166.\n\\label{sectacknow}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Kinematic dependences}\n\n\\paragraph {{\\bf DVCS}}\n\nThe kinematic dependences of DVCS production, presented in\nFig.~\\ref{fig:dvcs}, are well described by models using either GPDs \nor a dipole approach~\\cite{z-dvcs,h1-dvcs}.\n\nThe interference of the DVCS and Bethe-Heitler processes gives access, \nthrough the measurement of beam charge asymmetry, to the ratio $\\rho$ of the\nreal to imaginary parts of the DVCS amplitude.\nThe measurement $\\rho = 0.20 \\pm 0.05 \\pm 0.08$~\\cite{h1-dvcs} is in \nagreement with the value $\\rho = 0.25 \\pm 0.03 \\pm 0.05$ obtained from a \ndispersion relation using the $W$ dependence of the cross section.\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{1.0cm}\n\\begin{picture}(11.,5.)   \n\\put(0.0,0.0){\\epsfig{file=marage_pierre.fig1a.eps,width=6.cm,height=5.cm}}\n\\put(6.,0.8){\\epsfig{file=marage_pierre.fig1b.eps,width=5.0cm}}\n\\end{picture}\n\\vspace{-0.3cm}\n\\caption{(left) \\mbox{$Q^2$}\\ and $W$ dependences of DVCS production, with simple\nfit parameterisations~\\protect\\cite{z-dvcs};\n(right) beam charge asymmetry, $\\cos \\phi$ fit and predictions of a GPD \nmodel~\\protect\\cite{h1-dvcs}.}\n\\label{fig:dvcs}\n\\end{center}\n\\vspace{-0.3cm}\n\\end{figure}\n\n\\paragraph {{\\bf \\mbox{$Q^2$}\\ dependence of light VM production}}\n\nThe cross sections for elastic and proton dissociative \\mbox{$\\rho$}\\ and \\mbox{$\\phi$}\\ \nelectroproduction have been measured with high \nprecision~\\cite{z-rho,h1-rho-hera1,z-phi}.\nThe \\mbox{$Q^2$}\\ dependence, shown for \\mbox{$\\rho$}\\ mesons in Fig.~\\ref{fig:rho_q2}, is reasonably \ndescribed by several models, using either the GPD or the dipole approach.\n\\begin{figure}[h]\n\\begin{minipage}{0.48\\columnwidth}\n\\centerline{\\includegraphics[width=0.80\\columnwidth]{marage_pierre.fig2.eps}}\n\\vspace{-0.3cm}\n\\caption{\\mbox{$Q^2$}\\ dependence of elastic and proton dissociative electroproduction \ncross sections of \\mbox{$\\rho$}\\ mesons, and model predictions~\\protect\\cite{h1-rho-hera1}.}\n\\label{fig:rho_q2}\n\\end{minipage}\n\\hspace{2mm}\n\\begin{minipage}[h]{0.48\\columnwidth}\n\\centerline{\\includegraphics[width=0.80\\columnwidth]{marage_pierre.fig3.eps}}\n\\vspace{-0.3cm}\n\\caption{Elastic $b$ slopes, as a function of $\\mu^2 = \\scaleqsqplmsq$\nfor VM production and $\\mu^2 = \\mbox{$Q^2$}$ for DVCS~\\protect\\cite{h1-rho-hera1}.}\n\\label{fig:b-slopes}\n\\end{minipage}\n\\vspace{-0.3cm}\n\\end{figure}\n\nAlthough the production cross sections for light and heavy VMs differ by several orders \nof magnitude at $\\mbox{$Q^2$} \\simeq 0$,\nit is striking that the ratios are nearly constant when they are studied as a function of the \nscaling variable \\scaleqsqplmsq, with values close to unity when scaled \naccording to the quark charge content of the VMs, $\\mbox{$\\rho$} : \\omega : \\phi : \\mbox{$J\/\\psi$}  = 9 : 1 : 2 : 8$. \nThis confirms the relevance of the dipole size to the cross sections, even though the\nagreement with SU(4) universality is not perfect, indicating that\nwave function effects may need to be taken into account.\n\nThe ratio of the production cross sections with proton dissociative and elastic scattering\nat $\\mbox{$|t|$} = 0$ is found to be independent of \\mbox{$Q^2$}.\nConsistent values around 0.160 are measured for \\mbox{$\\rho$}\\ and \\mbox{$\\phi$}\\ production with \ndissociative mass $M_Y < 5~\\mbox{\\rm GeV}$~\\cite{h1-rho-hera1}. \nThis observation supports the independence of the hard and soft vertex contributions to the \nscattering amplitudes, known as proton vertex or ``Regge\" factorisation.\n\n\n\n\\paragraph {{\\bf $t$ slopes}}\n\nExponentially falling \\mbox{$|t|$}\\ distributions, with $\\rm {d} \\sigma \/ \\rm {d}t \\propto e^{-b |t|}$,\nare measured for DVCS, light and heavy VM production, in both the \nelastic and the proton dissociative channels.\nIn an optical model appraoch, the slope $b$ is given by the convolution of the \ntransverse sizes of the\n$q \\bar q$ dipole, of the diffractively scattered system (which vanishes\nfor proton dissociation) and of the exchange (a contribution which is expected to be small).\nAs shown in Fig.~\\ref{fig:b-slopes}, the elastic slopes for light VMs\nstrongly decrease with increasing \\mbox{$Q^2$}.\nThey reach values of the order of $5~\\mbox{${\\rm GeV}^{-2}$}$, similar to those measured in \n\\mbox{$J\/\\psi$}\\ production, for values of the scaling variable\n$\\scaleqsqplmsq\\ \\gapprox\\ 5~\\mbox{${\\rm GeV}^2$}$.\nThis evolution reflects the shrinkage with \\mbox{$Q^2$}\\ of the light quark colour dipole.\nA similar evolution is observed for DVCS as a function of the variable \\mbox{$Q^2$}.\nThe proton dissociative slopes similarly decrease with the increasing scale, down to values \naround $1.5~\\mbox{${\\rm GeV}^{-2}$}$ for \\mbox{$\\rho$}\\ production and DVCS, and values slightly below $1~\\mbox{${\\rm GeV}^{-2}$}$ \nfor \\mbox{$J\/\\psi$}\\ production.\n\nThe difference between the elastic and proton dissociative slopes,\n$b_{el} - b_{p. diss.}$, provides another test of proton vertex factorisation.\nA value of $3.5 \\pm 0.1~\\mbox{${\\rm GeV}^{-2}$}$ is measured for \n\\mbox{$J\/\\psi$}~\\protect\\cite{z-jpsi-photoprod,h1-jpsi-hera1}, with a similar value for \nDVCS~\\protect\\cite{h1-dvcs}. \nThe difference is higher, around $5.5~\\mbox{${\\rm GeV}^{-2}$}$, for \\mbox{$\\rho$}\\ and \\mbox{$\\phi$}\\ mesons, with\nhowever an indication of a decrease toward the \\mbox{$J\/\\psi$}\\ value with increasing \n$\\scaleqsqplmsq$~\\cite{h1-rho-hera1}.\n\n\\paragraph {{\\bf Energy dependence and effective Regge trajectory}}\n\nThe energy dependence of DVCS and VM production\nis well described by a power law, $\\rm {d} \\sigma \/ \\rm {d}W \\propto W^{\\delta}$.\n\nFigure~\\ref{fig:Regge}~(left) shows that the energy dependence is significantly stronger\nfor heavy quark photoproduction, with $\\delta \\sim 0.8-1.2$, than for (soft) \nhadron--hadron interactions and light VM photoproduction, with $\\delta \\sim 0.2$.\nThis is explained by the fact that the photoproduction of VMs formed of heavy quarks\nis a hard process, characterised by small transverse dipoles which probe the low-$x$ \ngluon density in the proton at a scale where it is quickly increasing with $1\/x$. \n\nFor light VM production, the $W$ dependence is hardening with \\mbox{$Q^2$}, with values of \n$\\delta$ similar to the \\mbox{$J\/\\psi$}\\ values for $\\scaleqsqplmsq\\ \\gapprox\\ 5~\\mbox{${\\rm GeV}^2$}$.\nThis feature is explained by the shrinkage of the colour dipoles at large \\mbox{$Q^2$}\\ \nvalues.\n\n\\begin{figure}[h]\n\\begin{center}\n\\setlength{\\unitlength}{1.0cm}\n\\begin{picture}(14.,5.5)   \n\\put(0.0,0.){\\epsfig{file=marage_pierre.fig4a.eps,width=5.5cm}}\n\\put(6.,0.5){\\epsfig{file=marage_pierre.fig4b.eps,width=4.0cm}}\n\\put(10.,0.5){\\epsfig{file=marage_pierre.fig4c.eps,width=4.0cm}}\n\\end{picture}\n\\vspace{-0.3cm}\n\\caption{(left) $W$ dependence of VM photoproduction~\\protect\\cite{levy}; \nmeasurement of  the intercept $\\mbox{$\\alpha_{I\\!\\!P}$}(0)$ (centre) and of  the slope \\mbox{$\\alpha^\\prime$}\\\n(right) of the effective Regge trajectory, as a function of the scale \n$\\mu^2 =  \\qsqplmsq \/4$ for VM production and $\\mu^2 = \\mbox{$Q^2$}$ \nfor DVCS~\\protect\\cite{h1-rho-hera1}.}\n\\label{fig:Regge}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{figure}\nIn a Regge inspired parameterisation, the energy dependence of the\ncross section and its correlation with $t$ are given by\n$\\delta(t) =  4 \\ ( \\mbox{$\\alpha_{I\\!\\!P}$}(t)  - 1)$, with $\\mbox{$\\alpha_{I\\!\\!P}$}(t) = \\mbox{$\\alpha_{I\\!\\!P}$}(0) + \\mbox{$\\alpha^\\prime$} \\cdot \\ t$,\nwhere \\mbox{$\\alpha^\\prime$}\\ describes the shrinking of the diffractive peak with energy.\nThe hard behaviour of \\mbox{$J\/\\psi$}\\ production and the hardening with $\\scaleqsqplmsq$ of \nthe energy dependence of light VM production for $t = 0$ are shown in \nFig.~\\ref{fig:Regge}~(centre), where values of $1.08$ or $1.11$ for $\\mbox{$\\alpha_{I\\!\\!P}$}(0)$ are typical \nof soft hadron-hadron interactions.\nAs shown in Fig.~\\ref{fig:Regge}~(right), the slope of the effective trajectory for VM production, \nincluding \\mbox{$\\rho$}\\ photoproduction~\\cite{h1-rho-photoprod},\nis smaller than the value $0.25~\\mbox{${\\rm GeV}^{-2}$}$, typical for hadronic interactions.\nFor DVCS $\\mbox{$\\alpha^\\prime$} = 0.03 \\pm 0.09 \\pm 0.11~\\mbox{${\\rm GeV}^{-2}$}$~\\cite{h1-dvcs}; \nfor \\mbox{$J\/\\psi$}\\ photoproduction at high \\mbox{$|t|$}, combining H1~\\cite{h1-jpsi-large-t}\nand ZEUS~\\cite{z-jpsi-large-t}\nmeasurements, $\\mbox{$\\alpha^\\prime$} = -0.02 \\pm 0.01 \\pm 0.01~\\mbox{${\\rm GeV}^{-2}$}$.\n\n\n\\paragraph {{\\bf Remarks on the interaction scales}}\n\nThe energy dependence of the total $ep$ cross section at fixed values of \\mbox{$Q^2$}\\ can be \nparameterised as $F_2 \\propto x^{- \\lambda}$, with values of $\\lambda$ increasing with \\mbox{$Q^2$},\na feature which is attributed to the increase with \\mbox{$Q^2$}\\ of the parton density at small $x$.\nThe prediction that for VM production $\\delta = 2 \\lambda$, when taken at the same scale, \ncan thus provide information on the relevant effective scale for the reaction.\nThe present results clearly indicate that the variable \\scaleqsqplmsq\\ is a better candidate\nthan \\mbox{$Q^2$}\\ for such a unified scale, but high precision measurements of the energy \ndependence of \\mbox{$\\rho$}\\ and \\mbox{$J\/\\psi$}\\ electroproduction remain necessary to settle the scale \nissue~\\cite{levy}.\n\nFor the DVCS process, where both LO and NLO (dipole-type) diagrams contribute, the \npresent high energy data seem to favour an effective scale $\\approx \\mbox{$Q^2$}$ rather than \n$\\approx \\mbox{$Q^2$}\/4$ in order to ensure diffraction universality, but here also more precise data \nare required.\n\n\n\\section {Spin dynamics}\nThe VM production and decay angular distributions allow the measurement of\nfifteen spin density matrix elements, which are bilinear combinations of helicity\namplitudes.\nUnder natural parity exchange, five $T_{\\lambda_V \\lambda_{\\gamma}}$ amplitudes \nare independent: two $s$-channel helicity conserving (SCHC) amplitudes \n($T_{00}$ and $T_{11}$), \ntwo single helicity flip amplitudes ($T_{01}$ and $T_{10}$) \nand one double flip amplitude ($ T_{-11}$).\n\nThe \\mbox{$Q^2$}\\ dependence of the matrix elements for \\mbox{$\\rho$}\\ and \\mbox{$\\phi$}\\ production \nindicates that the five elements which contain products of the SCHC amplitudes \nare non-zero, whereas those formed with the helicity violating amplitudes are\ngenerally consistent with 0.\nA notable exception is the element \\mbox{$r_{00}^{5}$}, which\ninvolves the product of the dominant $T_{00}$ SCHC \namplitude with $T_{01}$,\nwhich describes the transition from a transversely polarised photon to a longitudinal\n\\mbox{$\\rho$}\\ meson~\\cite{z-rho,h1-rho-hera1}.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{1.0cm}\n\\begin{picture}(14.8,3.5)   \n\\put(0.0,0.){\\epsfig{file=marage_pierre.fig5a.eps,width=3.5cm}}\n\\put(3.7,0.){\\epsfig{file=marage_pierre.fig5c.eps,width=3.5cm}}\n\\put(7.5,0.2){\\epsfig{file=marage_pierre.fig5b.eps,width=3.3cm}}\n\\put(11.1,0.2){\\epsfig{file=marage_pierre.fig5d.eps,width=3.3cm}}\n\\end{picture}\n\\vspace{-0.3cm}\n\\caption{Measurement of $R = \\sigma_L \/ \\sigma_T$, as a function of \n\\mbox{$Q^2$}\\ and\n\\mbox{$|t|$}~\\protect\\cite{h1-rho-hera1},\n$W$, \nand the invariant mass, $M_{\\pi\\pi}$, of the two decay pions~\\protect\\cite{z-rho}, \nfor \\mbox{$\\rho$}\\ electroproduction.}\n\\label{fig:R}\n\\end{center}\n\\vspace{-0.3cm}\n\\end{figure}\nThe ratio $R = \\sigma_L \/ \\sigma_T$ of the longitudinal to transverse cross sections\nfor \\mbox{$\\rho$}\\ production is shown in Fig.~\\ref{fig:R} as a function of \\mbox{$Q^2$}, $W$, $t$, and the \ninvariant mass of the two decay pions, for several domains in \\mbox{$Q^2$}.\n\nA strong increase of $R$ with \\mbox{$Q^2$}\\ is observed, which is tamed at large \\mbox{$Q^2$}. \nThese features are relatively well described by GPD and dipole models.\nThe \\mbox{$Q^2$}\\ dependence of $R$ for \\mbox{$\\rho$}, \\mbox{$\\phi$}\\  and \\mbox{$J\/\\psi$}\\ production\nfollows a universal trend when plotted as a function of \n$Q^2  \/ M_{V}^2$~\\cite{h1-rho-hera1}.\nWith the present data, no $W$ dependence is observed, but it should be stressed that\nthe lever arms in $W$ for fixed \\mbox{$Q^2$}\\ values are relatively limited.\n\nNo \\mbox{$|t|$}\\ dependence is observed for $R$ by ZEUS with $\\mbox{$|t|$} \\leq 1~\\mbox{${\\rm GeV}^2$}$~\\cite{z-rho},\nwhereas an increase of $R$ with \\mbox{$|t|$}\\ is observed by H1 for $\\mbox{$Q^2$} > 5~\\mbox{${\\rm GeV}^2$}$, \n$\\mbox{$|t|$} \\leq 3~\\mbox{${\\rm GeV}^2$}$~\\cite{h1-rho-hera1}.\nThis increase can be translated into a measurement of the \ndifference between the longitudinal and transverse $t$ slopes, through the relation\n$R(t) = \\sigma_L(t) \/ \\sigma_T(t) \\propto e^{- (b_L - b_T) |t|}$.\nA slight indication ($1.5 \\sigma$) is thus found for a negative value of $b_L - b_T$\n($-0.65 \\pm 0.14_{-0.51}^{+0.41}$), suggesting that the\naverage transverse size of dipoles for transverse amplitudes is larger than for\nlongitudinal amplitudes\n\nThe strong dependence of $R$ with the dipion mass, observed by both\nexperiments~\\cite{z-rho,h1-rho-hera1}, cannot be attributed solely to the interference of \nresonant \\mbox{$\\rho$}\\ and non-resonant $\\pi \\pi$ production, and \nindicates that the spin dynamics of \\mbox{$\\rho$}\\ production depends of the effective\n$q \\bar q$ mass.    \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.19\\columnwidth]{marage_pierre.fig6a.eps}\n\\hspace {-0.3cm}\n\\includegraphics[width=0.19\\columnwidth]{marage_pierre.fig6b.eps}\n\\hspace {-0.3cm}\n\\includegraphics[width=0.19\\columnwidth]{marage_pierre.fig6c.eps}\n\\hspace {-0.3cm}\n\\includegraphics[width=0.19\\columnwidth]{marage_pierre.fig6d.eps}\n\\hspace {0.5cm}\n\\includegraphics[width=0.19\\columnwidth]{marage_pierre.fig6e.eps}\n\\vspace{-0.3cm}\n\\caption{(a-d) Helicity amplitude ratios, as a function of $t$; (right plot)\nphase difference between the two SCHC amplitudes, $T_{00}$ and \n$T_{11}$~\\protect\\cite{h1-rho-hera1}.}\n\\label{fig:ampl_ratios_t}\n\\end{center}\n\\vspace{-0.3cm}\n\\end{figure}\n\nHelicity amplitude ratios are measured, under the approximation that they are\nin phase, through fits to the 15 matrix elements.\nThe four ratios to the dominant $T_{00}$ amplitude are presented in \nFig.~\\ref{fig:ampl_ratios_t} as a function of $t$, for two domains in \\mbox{$Q^2$}.\nAt large \\mbox{$Q^2$}, a $t$ dependence compatible with the expected $\\sqrt{ \\mbox{$|t|$} }$\nlaw is observed for both single helicity flip amplitudes.\nA significant double-flip amplitude $T_{-11}$ is observed, which may be related\nto gluon polarisation in the proton.\nThe $t$ dependence of $T_{11}\/T_{00}$ at large \\mbox{$Q^2$}, a $3 \\sigma$ effect, is related to the\n$t$ dependence of $R$ and supports the indication of a difference between \nthe transverse sizes of dipoles in transversely and longitudinally polarised photons.\n\nA small non-zero phase difference between the two SCHC amplitudes, which decreases with\nincreasing \\mbox{$Q^2$}, is visible in Fig.~\\ref{fig:ampl_ratios_t}.\nThrough dispersion relations, this non-zero value is suggestive of different $W$ \ndependences of the longitudinal and transverse amplitudes. \n\n\\section {Large \\boldmath {\\mbox{$|t|$}} VM production}\n\nIn exclusive real photon and VM production at high energy and \nlarge \\mbox{$|t|$}, \na hard scale is present at both ends of the gluon ladder which extends over a large\nrapidity range, between the struck parton in the proton (mostly gluons at small $x$) and \nthe quark or antiquark from the photon fluctuation.\nThese processes thus offer a unique testing ground for the BFKL evolution, since no strong\n$k_T$ ordering along the ladder is expected.\nThis is at variance with high \\mbox{$Q^2$}\\ VM production at low \\mbox{$|t|$}, where a large scale is present\nat the photon end of the ladder and a small scale at the proton end, a configuration which\nis described by the DGLAP evolution. \nFor real photon and \\mbox{$J\/\\psi$}\\ production, there is little uncertainty related to\nthe wave functions.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\setlength{\\unitlength}{1.0cm}\n\\begin{picture}(14.,4.0)   \n\\put(0.2,0.1){\\epsfig{file=marage_pierre.fig7a.eps,height=3.6cm,width=3.6cm}}\n\\put(4.1,-0.3){\\epsfig{file=marage_pierre.fig7b.eps,height=4.5cm,width=5.0cm}}\n\\put(9.3,3.4){\\epsfig{file=marage_pierre.fig7c.eps,height=4.3cm,width=3.cm,angle=270}}\n\\end{picture}\n\\vspace{-0.3cm}\n\\caption{Large \\mbox{$|t|$}\\ photoproduction measurements: \n$W$ dependence of real photon~\\protect\\cite{h1-gamma-larget} (left) \nand \\mbox{$J\/\\psi$}~\\protect\\cite{z-jpsi-large-t} production (centre); \nspin density matrix elements for \\mbox{$\\rho$}\\ \nproduction~\\protect\\cite{h1-rho-photoprod-large-t}.}\n\\label{fig:hight-t}\n\\end{center}\n\\vspace{-0.3cm}\n\\end{figure}\nA specific QCD prediction for large \\mbox{$|t|$}\\ production is the power-law dependence of the\n\\mbox{$|t|$}\\ distribution, at variance with the exponential dependence for $|t| \\ \\lapprox\\ $ \na few~\\mbox{${\\rm GeV}^2$}.\nThe $t$ dependences for $\\mbox{$|t|$} \\geq 4~\\mbox{${\\rm GeV}^2$}$ of $\\gamma$ and \\mbox{$J\/\\psi$}\\ production \nare indeed well described by power laws with exponents \n$n = 2.60 \\pm 0.19 ^{+0.03}_{-0.08}$~\\cite{h1-gamma-larget},\nand $n = 3.0 \\pm 0.1$~\\cite{z-jpsi-large-t}, respectively.\n\nFigures~\\ref{fig:hight-t} (left) and (centre) present the $W$ evolutions of high \\mbox{$|t|$}\\ \nreal photon and \\mbox{$J\/\\psi$}\\ production, respectively.\nA strong $W$ dependence is observed, compatible with calculations based on\nthe BFKL approach, whereas the DGLAP evolution (valid for $\\mbox{$|t|$} \\leq m^2_{\\psi}$),\npredicts a significantly weaker dependence.  \n\nThe spin density matrix elements for \\mbox{$J\/\\psi$}\\ production are in agreement with \nSCHC~\\cite{h1-jpsi-large-t,z-jpsi-large-t},\nwhereas substantial helicity flip contributions are observed in Fig.~\\ref{fig:hight-t} (right)\nfor \\mbox{$\\rho$}\\ production with \n$1.5\\ \\mathrel{\\rlap{\\lower4pt\\hbox{\\hskip1pt$\\sim$}\\ \\mbox{$|t|$}\\ \\mathrel{\\rlap{\\lower4pt\\hbox{\\hskip1pt$\\sim$} 10~\\mbox{${\\rm GeV}^2$}$~\\cite{z-high-t,h1-rho-photoprod-large-t}, \nwhich can be understood in a BFKL approach with a chiral-odd component of the \nphoton wave function.\n\n\n\n\n\\section*{Acknowledgments}\nIt is a pleasure to thank numerous colleagues from the ZEUS and H1 collaborations \nas well as theorists for enlightening discussions, \nand the workshop organisers for the lively discussions and the pleasant atmosphere of the \nmeeting.\n\n\\begin{footnotesize}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe concept of fuzzy sets along with various operations has been introduced\nby Lofti A. Zadeh in 1965 \\cite{Zad}. Due to the diverse applications\nranging from engineering, computer science and social behavior studies,\nthe researchers have taken keen interest in the subject in its related\nfields. The study of fuzzy algebraic structures was started by introducing\nthe concept of fuzzy subgroups by A. Rosenfeld \\cite{Ros}. He formulated\nthe concept of fuzzy subgroup and extend the main idea of group theory\nto develop the theory of fuzzy groups. Anthony and Sherwood further\nredefined fuzzy groups \\cite{Anto}. Many other papers on fuzzy subgroups\nhave also appeared which generalize various concepts of group theory\nsuch as normal subgroups, quotient groups and cosets \\cite{Meng,Wu,Wan}.\n\n\\noindent In the forty years history of AG-groupoids, though it was\nexplored slowly, yet in the last couple of years abundant research\nwas carried out in this area which attracted the attention of many\nnew researchers. An AG-groupoid is a generalization of commutative\nsemigroup. It is a nonassociative groupoid in general, in which the\nleft invertive law $(ab)c=(cb)a$ holds. In general, an AG-group is\na nonassociative structure in which commutativity and associativity\nimply each other, and thus it becomes abelian group if any one of\nthem is allowed. An AG-groupoid $(G,\\cdot)$ is called an AG-group\nor left almost group (LA-group), if there exists a unique left identity\n$e\\in G$ (that is $ea=a$ for all $a\\in G$ ), and for all $a\\in G$\nthere exists $a^{^{-1}}\\in G$ such that $a^{^{-1}}a=aa^{-1}=e$.\nM. Kamran extended the notion of AG-groupoid to an AG-group and defined\ncosets of an AG-subgroup $H$ of an AG-group $G$ and proved that\nquotient $G\/H$ is defined for every AG-subgroup $H$. He also proved\nthat Lagrange's Theorem holds for AG-group \\cite{Kamran}. The third\nauthor of this article has discussed various basic properties of AG-groups\nand explored new results such as: complexes and cosets decomposition,\nconjugacy relations in AG-groups, normality, normalizers and many\nmore \\cite{Sha Thes,MshahT}. For the first time in 2003, Q. Mushtaq\nand M. Khan introduced ideals in AG-groupoid and fuzzified these concepts\n\\cite{idealsinAG}. This attracted the attention of various other\nresearchers to the field of AG-groupoids and AG-groups, as a result\nsince then we can see lots of papers in this area. It is also worth\nmentioning that various new classes of AG-groupoids have been recently\nintroduced \\cite{SIA1,intro2,key-1-13,RAAS4,AGst5,AGst-6} and some\nare just have been arXived \\cite{mod,mod m} and their fuzzification\nis suggested as an interesting future work.\n\nIn this paper we extend the concepts of normal fuzzy AG-subgroup \\cite{I. Ahmad,key-1}.\nWe further define fuzzy cosets, quotient AG-subgroups and quotient\nfuzzy AG-subgroups, which will provide new direction to the researchers\nin this area. We also introduce a fuzzy version of the famous Lagrange's\nTheorem for finite AG-groups.\n\n\n\\section{Preliminaries}\n\nIn this section we list some basic definitions that will frequently\nbe used in the subsequent sections of the paper.\n\n\\noindent A \\emph{fuzzy subset} $\\mu$ is a mapping $\\mu:X\\rightarrow[0,1]$.\nThe set of all fuzzy subsets of $X$ is called the \\emph{fuzzy power\nset} of $X$ and is denoted by $FP(X)$. Let $\\mu\\in FP(X)$, then\nthe the image of $\\mu$ is a set $\\{\\mu(x)\\,:\\, x\\in X\\}$ and is\ndenoted by $\\mu(X)$ or $Im(\\mu).$ \n\n\\noindent In the rest of this paper $G$ will denote an AG-group otherwise\nstated and $e$ will denote the left identity of $G$.\n\n\\noindent \\begin {definition} \\cite{I. Ahmad} Let $\\mu\\in FP(G)$,\nthen $\\mu$ is called a fuzzy AG-subgroup of $G$ if for all $x,y\\in G$;\n\n\\begin{enumerate}[(i)]\n\n\\item $\\mu(xy)\\geq\\mu(x)\\wedge\\mu(y)$;\n\n\\item $\\mu(x^{-1})\\geq\\mu(x)$.\n\n\\end{enumerate}\n\n\\noindent The set of all fuzzy AG-subgroups of $G$ is denoted by\n$F(G)$.\n\n\\noindent If $\\mu\\in F(G)$, then\n\\begin{equation}\n\\mu_{\\ast}=\\{x\\in G\\,|\\text{\\,}\\mu(x)=\\mu(e)\\}.\\label{a1}\n\\end{equation}\n\\end {definition}\n\n\\noindent \\begin {lemma}\\emph{\\label{ad}\\cite{I. Ahmad}} Let $\\mu$\nbe any fuzzy AG-subgroup of $G$ i.e $\\mu\\in F(G)$, then for all\n$x\\in G$,\n\n\\begin{enumerate}[(i)]\n\n\\item$\\mu(e)\\geq\\mu(x)$;\n\n\\item $\\mu(x)=\\mu(x^{-1})$.\n\n\\end{enumerate}\n\n\\noindent \\end {lemma}\n\n\\noindent \\begin {definition}\\cite{I. Ahmad} Let $\\mu\\in F(G)$.\nThen $\\mu$ is called a \\emph{normal fuzzy AG-subgroup}\\textbf{ }of\n$G$ if\n\\[\n\\mu(xy\\cdot x^{-1})=\\mu(y)\\text{ }\\forall x,y\\in G.\n\\]\nThe set of all normal fuzzy AG-subgroups of $G$ is denoted by $NF(G)$.\n\\end {definition}\n\n\\noindent \\begin {example} Consider an AG-group of order $4$:\n\n\\noindent \\begin{center}\n\\begin{tabular}{l|llll}\n$\\cdot$  & $0$  & $1$  & $2$  & $3$\\tabularnewline\n\\hline \n$0$  & $0$ & $1$ & $2$ & $3$\\tabularnewline\n$1$  & $3$ & $0$ & $1$ & $2$\\tabularnewline\n$2$  & $2$ & $3$ & $0$ & $1$\\tabularnewline\n$3$ & $1$ & $2$ & $3$ & $0$\\tabularnewline\n\\end{tabular}\n\\par\\end{center}\n\n\\noindent \\begin{flushleft}\ndefine fuzzy subset $\\mu$ by $\\mu(0)=t_{0}$ and $\\mu(x)=t_{1}$\notherwise; where $t_{0},t_{1}\\in[0,1]$ and $t_{0}>t_{1}$. Then $\\mu$\nis fuzzy AG-subgroup. However, $\\mu$ is not normal fuzzy AG-subgroup.\nHere $\\mu_{*}=\\{0\\}$ which is normal fuzzy AG-subgroup.\n\\par\\end{flushleft}\n\n\\noindent \\end {example}\n\n\\noindent \\begin {example} Consider an AG-group $G=<a,b:\\, a^{2}=b^{2}=(ab)^{2}=e>$,\ndefine fuzzy subset $\\mu:G\\rightarrow[0,1]$ by $\\mu(e)=t_{0},\\,\\mu(a)=t_{1}$\nand $\\mu(x)=t_{2}$ otherwise; where $t_{0},t_{1},t_{2}\\in[0,1]$\nand $t_{0}>t_{1}>t_{2}$. Then $\\mu$ is normal fuzzy AG-subgroup\nof $G$.\n\n\\noindent \\end {example}\n\n\n\\section{Main Results}\n\n\\begin {proposition} \\label{b-1}Let $\\mu\\in F(G)$. Then $\\mu(xy)=\\mu(yx)$\n$\\forall x,y\\in G$.\\end {proposition}\n\n\\noindent \\begin {proof} Let $\\mu\\in F(G)$, then\n\\begin{eqnarray*}\n\\mu(xy) & = & \\mu(ex\\cdot y)=\\mu(yx\\cdot e)\\qquad[\\mbox{using left invertive law}]\\\\\n & \\geq & \\mu(yx)\\wedge\\mu(e)=\\mu(yx)\\,\\,\\quad[\\mu(e)\\geq\\mu(yx)\\,\\,\\forall\\, x,y\\in G]\\\\\n\\Rightarrow\\mu(xy) & \\geq & \\mu(yx)\\quad\\forall\\, x,y\\in G.\n\\end{eqnarray*}\nSimilarly, we can show that $\\mu(yx)\\geq\\mu(xy)\\,\\,\\forall\\, x,y\\in G$.\nThus $\\mu(xy)\\geq\\mu(yx)\\geq\\mu(xy)\\,\\,\\forall\\, x,y\\in G$. Hence\n$\\mu(yx)=\\mu(xy)$. \\end {proof}\n\n\\noindent \\begin {lemma}\\label{a}\\cite[Lemma 17]{I. Ahmad} Let\n$\\mu$ be a fuzzy AG-subgroup of $G$. Let $x\\in G$ then $\\mu(xy)=\\mu(y)$\n$\\forall\\, y\\in G$ if and only if $\\mu(x)=\\mu(e)$. \\end {lemma}\n\n\\noindent \\begin {theorem} Let $f$ be a homomorphism on AG-group\n$G$ and $\\mu$ is any normal fuzzy AG-subgroup of $f(G)$. Then $\\mu\\circ f\\in NF(G)$.\n\\end {theorem}\n\n\\noindent \\begin {proof} First we show that $\\mu\\circ f\\in F(G)$.\nSince\n\\begin{eqnarray*}\n\\mu\\circ f(xy) & = & \\mu(f(xy))\\\\\n & = & \\mu(f(x)\\cdot f(y))\\\\\n & \\geq & \\mu(f(x))\\wedge\\mu(f(y))\\\\\n & = & \\mu\\circ f(x)\\wedge\\mu\\circ f(y)\n\\end{eqnarray*}\n$\\forall\\, x,y\\in G$, and\n\\begin{eqnarray*}\n\\mu\\circ f(x^{-1}) & = & \\mu(f(x^{-1}))\\\\\n & = & \\mu((f(x))^{-1})\\,\\qquad\\quad\\\\\n & = & \\mu(f(x))\\\\\n & = & \\mu\\circ f(x).\n\\end{eqnarray*}\n$\\forall\\, x\\in G$. Hence $\\mu\\circ f\\in F(G)$.\n\n\\noindent Next we show that $\\mu\\circ f\\in NF(G)$, since\n\\begin{eqnarray*}\n\\mu\\circ f(xy\\cdot x^{-1}) & = & \\mu(f(xy\\cdot x^{-1}))\\\\\n & = & \\mu(f(xy)\\cdot f(x^{-1}))\\\\\n & = & \\mu(\\{f(x)f(y)\\}\\cdot(f(x))^{-1})\\\\\n & = & \\mu(f(y))\\,\\,[\\mu\\in NF(f(G))]\\\\\n & = & \\mu\\circ f(y).\n\\end{eqnarray*}\n$\\forall\\, x,y\\in G$. Hence $\\mu\\circ f\\in NF(G)$. \\end {proof}\n\n\\noindent \\begin {definition} \\label{b}Let $\\mu\\in F(G)$, for any\n$x\\in G$ define a mapping\n\\begin{eqnarray*}\n\\mu_{x} & : & G\\rightarrow[0,1],\\,\\,\\mbox{by}\n\\end{eqnarray*}\n\\begin{eqnarray}\n\\mu_{x}(g) & = & \\mu(gx^{-1})\\,\\,\\forall g\\in G.\\label{eq:1a-1}\n\\end{eqnarray}\nThen $\\mu_{x}$ is called fuzzy coset of $G$ determined by $x$ and\n$\\mu$, and the collection of all fuzzy cosets of $\\mu$ is represented\nby \\emph{$\\mathsf{\\mathit{\\mathtt{\\mathbb{\\mathfrak{\\mathscr{F}}}}}}$.}\\end {definition}\n\n\\noindent In AG-groups we can define quotient AG-group by any AG-subgroup\nwithout normality. Therefore, make use of this we can define quotient\nAG-groups or factor AG-group as follows: \n\n\\noindent \\begin {theorem} \\label{s}Let $\\mu\\in NF(G)$ and $G\\diagup\\mu=\\{\\mu_{x}\\,:\\, x\\in G\\}$.\nThen $G\\diagup\\mu$ form an AG-group under the usual composition of\nmappings define by \\emph{$\\mu_{x}\\circ\\mu_{y}=\\mu_{xy}\\,\\,\\forall\\, x,y\\in G$}.\n\\end {theorem}\n\n\\noindent \\begin {proof} First we show that the composition of cosets\nis well defined. Let $x,y,x_{\\circ},y_{\\circ}\\in G$ such that $\\mu_{x}=\\mu_{x_{\\circ}}$and\n$\\mu_{y}=\\mu_{y_{\\circ}}$.\n\n\\noindent We show that $\\mu_{x}\\circ\\mu_{y}=\\mu_{x_{\\circ}}\\circ\\mu_{y_{\\circ}}$,\ni.e. $\\mu_{xy}=\\mu_{x_{\\circ}y_{\\circ}}$. Thus by (\\ref{eq:1a-1})\n\\begin{eqnarray*}\n\\mu_{xy}(g) & = & \\mu(g(xy)^{-1})=\\mu(g\\cdot x^{-1}y^{-1})\\quad\\forall\\, g\\in G;\\,\\mbox{and}\\\\\n\\mu_{x_{\\circ}y_{\\circ}}(g) & = & \\mu(g(x_{\\circ}y_{\\circ})^{-1})=\\mu(g\\cdot x_{\\circ}^{-1}y_{\\circ}^{-1})\\quad\\forall\\, g\\in G.\n\\end{eqnarray*}\nNow $\\forall\\, x,y\\in G$,\n\\begin{eqnarray*}\n\\mu(g\\cdot x^{-1}y^{-1}) & = & \\mu[e(g\\cdot x^{-1}y^{-1})]\\\\\n & = & \\mu[((x_{\\circ}y_{\\circ})^{-1}(x_{\\circ}y_{\\circ}))(g\\cdot x^{-1}y^{-1})]\\\\\n & = & \\mu[((x_{\\circ}^{-1}y_{\\circ}^{-1})(x_{\\circ}y_{\\circ}))(g\\cdot x^{-1}y^{-1})]\\\\\n & = & \\mu[g(((x_{\\circ}^{-1}y_{\\circ}^{-1})(x_{\\circ}y_{\\circ}))(x^{-1}y^{-1}))]\\mbox{\\,\\,[in \\ensuremath{G}; \\ensuremath{a(bc)=b(ac)\\,}[8]]}\\\\\n & = & \\mu[g(((x^{-1}y^{-1})(x_{\\circ}y_{\\circ}))(x_{\\circ}^{-1}y_{\\circ}^{-1}))]\\,\\,[\\mbox{using left invertive law]}\\\\\n & = & \\mu[((x^{-1}y^{-1})(x_{\\circ}y_{\\circ}))(g(x_{\\circ}^{-1}y_{\\circ}^{-1}))]\\mbox{\\,\\,[in \\ensuremath{G;}\\ensuremath{\\, a(bc)=b(ac)\\,}[8]]}\\\\\n & \\geq & \\mu((x^{-1}y^{-1})(x_{\\circ}y_{\\circ}))\\wedge\\mu(g(x_{\\circ}^{-1}y_{\\circ}^{-1}))\n\\end{eqnarray*}\n\\begin{eqnarray}\n\\Rightarrow\\mu(g\\cdot x^{-1}y^{-1}) & \\geq & \\mu((x^{-1}y^{-1})(x_{\\circ}y_{\\circ}))\\wedge\\mu(g(x_{\\circ}^{-1}y_{\\circ}^{-1}))\\label{p}\n\\end{eqnarray}\n\n\n\\noindent Now we show that $\\mu((x^{-1}y^{-1})(x_{\\circ}y_{\\circ}))=\\mu(e)$\nin (\\ref{p}) . Let $\\mu_{x}=\\mu_{x_{\\circ}}\\Rightarrow\\mu_{x}(g)=\\mu_{x_{\\circ}}(g)\\,\\,\\forall\\, g\\in G$,\n\\begin{eqnarray}\n\\Rightarrow\\mu(gx^{-1}) & = & \\mu(gx_{\\circ}^{-1}).\\qquad[\\mbox{by }(\\ref{eq:1a-1})]\\label{eq:2a}\n\\end{eqnarray}\nSimilarly, since $\\mu_{y}=\\mu_{y_{\\circ}}\\Rightarrow\\mu_{y}(g)=\\mu_{y_{\\circ}}(g)\\,\\,\\forall\\, g\\in G,$\n\\begin{eqnarray}\n\\Rightarrow\\mu(gy^{-1}) & = & \\mu(gy_{\\circ}^{-1}).\\qquad[\\mbox{by }(\\ref{eq:1a-1})]\\label{eq:3a}\n\\end{eqnarray}\nNow,\n\\begin{eqnarray*}\n\\mu((x^{-1}y^{-1})(x_{\\circ}y_{\\circ})) & = & \\mu((x_{\\circ}y_{\\circ}\\cdot y^{-1})x^{-1})\\,\\,[\\mbox{using left invertive law]}\\\\\n & = & \\mu((x_{\\circ}y_{\\circ}\\cdot y^{-1})x_{\\circ}^{-1})\\\\\n &  & [\\mbox{substituting \\,\\ensuremath{g}\\,\\ by }(x_{\\circ}y_{\\circ}\\cdot y^{-1})\\mbox{ in }(\\ref{eq:2a})]\\\\\n & = & \\mu(x_{\\circ}(y_{\\circ}y^{-1}\\cdot x_{\\circ}^{-1}))\\,\\,\\mbox{[in \\ensuremath{G}; \\ensuremath{(ab\\cdot c)d=a(bc\\cdot d)\\,}[8]]}\\\\\n & = & \\mu((y_{\\circ}y^{-1})(x_{\\circ}x_{\\circ}^{-1}))\\mbox{\\,\\,[in \\ensuremath{G}; \\ensuremath{\\, a(bc)=b(ac)\\,}[8]]}\\\\\n & = & \\mu(y_{\\circ}y^{-1}\\cdot e)=\\mu(ey^{-1}\\cdot y_{\\circ})\\,\\,[\\mbox{using left invertive law]}\\\\\n & = & \\mu(y^{-1}y_{\\circ})\\\\\n & = & \\mu(y_{\\circ}y^{-1})\\qquad\\qquad\\,\\,[\\mbox{by Proposition }\\ref{b-1}]\\\\\n & = & \\mu(y_{\\circ}y_{\\circ}^{-1})\\qquad\\qquad\\,\\,[\\mbox{substituting \\ensuremath{\\, g\\,}by }y_{\\circ}\\mbox{ in }(\\ref{eq:3a})]\\\\\n & = & \\mu(e).\n\\end{eqnarray*}\nTherefore, (\\ref{p}) implies that $\\mu(g\\cdot x^{-1}y^{-1})\\geq\\mu(g\\cdot x_{\\circ}^{-1}y_{\\circ}^{-1})$.\nSimilarly one can prove that $\\mu(g\\cdot x_{\\circ}^{-1}y_{\\circ}^{-1})\\geq\\mu(g\\cdot x^{-1}y^{-1})$.\nConsequently,\n\\begin{eqnarray*}\n\\mu(g\\cdot x^{-1}y^{-1}) & = & \\mu(g\\cdot x_{\\circ}^{-1}y_{\\circ}^{-1})\\\\\n\\Rightarrow\\mu(g\\cdot(xy)^{-1}) & = & \\mu(g\\cdot(x_{\\circ}y_{\\circ})^{-1})\\\\\n\\Rightarrow\\mu_{xy}(g) & = & \\mu_{x_{\\circ}y_{\\circ}}(g)\\,\\,\\forall\\, g\\in G\\\\\n\\Rightarrow\\mu_{xy} & = & \\mu_{x_{\\circ}y_{\\circ}}.\n\\end{eqnarray*}\nHence the product of cosets is well-defined. Now we show that $G\\diagup\\mu$\nform an AG-group under the operation $\\circ$.\n\n\\noindent $G\\diagup\\mu$ is closed under the operation $\\circ$. Also\n$G\\diagup\\mu$ satisfies left invertive law under $\\circ$ ; since\n$(\\mu_{x}\\circ\\mu_{y})\\circ\\mu_{z}=\\mu_{xy}\\circ\\mu_{z}=\\mu_{xy\\cdot z}=\\mu_{zy\\cdot x}=\\mu_{zy}\\circ\\mu_{x}=(\\mu_{z}\\circ\\mu_{y})\\circ\\mu_{x}$\n$\\forall\\, x,y,z\\in G$. Now for any $x\\in G$, $(\\mu_{e}\\circ\\mu_{x})(g)=(\\mu_{ex})(g)=(\\mu_{x})(g)\\Rightarrow(\\mu_{e}\\circ\\mu_{x})=\\mu_{x}\\,\\,\\forall\\, g\\,\\in G$,\nbut $(\\mu_{x}\\circ\\mu_{e})(g)=(\\mu_{xe})(g)\\neq(\\mu_{x})(g)\\Rightarrow(\\mu_{x}\\circ\\mu_{e})\\neq\\mu_{x}\\,\\,\\forall g\\,\\in G$.\nThis implies that $\\mu_{e}$ is the left identity of $G\\diagup\\mu$.\nAs an AG-group $G$ is non associative therefore, $(\\mu_{x}\\circ\\mu_{y})\\circ\\mu_{z}\\neq\\mu_{x}\\circ(\\mu_{y}\\circ\\mu_{z})$.\nFinally, $\\forall\\, x\\in G$, once $(\\mu_{x}\\circ\\mu_{x^{-1}})(g)=(\\mu_{xx^{-1}})(g)=(\\mu_{e})(g)\\Rightarrow\\mu_{x}\\circ\\mu_{x^{-1}}=\\mu_{e}\\,\\,\\forall\\, g\\in G$,\nand $(\\mu_{x^{-1}}\\circ\\mu_{x})(g)=(\\mu_{x^{-1}x})(g)=(\\mu_{e})(g)\\Rightarrow\\mu_{x^{-1}}\\circ\\mu_{x}=\\mu_{e}\\,\\,\\forall\\, g\\in G$\nthe inverse of each $\\mu_{x}$ exists and is $\\mu_{x^{-1}}$. Hence\nit follows that $G\\diagup\\mu$\\emph{ }is an AG-subgroup. \\end {proof}\n\n\\noindent \\begin {remark} The AG-group $G\\diagup\\mu$ defined in\nTheorem \\ref{s} is called quotient AG-group of $G$ relative to the\nnormal fuzzy AG-subgroup $\\mu$.\n\n\\noindent \\end {remark}\n\n\\noindent \\begin {theorem} \\label{FC}Let $\\nu\\in F(G)$ and $H$\nbe any AG-subgroup of $G$. Define $\\xi\\in FP(G\\diagup H)$ as follows:\n\\begin{eqnarray*}\n\\xi(Hx) & = & \\vee\\{\\nu(z)\\,:\\, z\\in Hx\\}\\,\\,\\,\\forall x\\in G.\n\\end{eqnarray*}\nThen $\\xi\\in F(G\\diagup H)$. \\end {theorem}\n\n\\noindent \\begin {proof} Since $\\forall\\, x,y\\in G$,\n\\begin{eqnarray*}\n\\xi(HxHy)=\\xi(H(xy)) & = & \\vee\\{\\nu(z)\\,:\\, z\\in H(xy)\\}\\\\\n & = & \\vee\\{\\nu(uv)\\,:\\, u\\in Hx,\\, v\\in Hy\\}\\\\\n & \\geq & \\vee\\{\\nu(u)\\wedge\\nu(v)\\,:\\, u\\in Hx,\\, v\\in Hy\\}\\\\\n & = & (\\vee\\{\\nu(u)\\,:\\, u\\in Hx\\})\\wedge(\\vee\\{\\nu(v)\\,:\\, v\\in Hy\\})\\\\\n & = & \\xi(Hx)\\wedge\\xi(Hy)\n\\end{eqnarray*}\nand $\\forall\\, x\\in G$,\n\\begin{eqnarray*}\n\\xi(Hx)^{-1}=\\xi(Hx^{-1}) & = & \\vee\\{\\nu(z)\\,:\\, z\\in Hx^{-1}\\}\\qquad\\qquad\\qquad\\qquad\\\\\n & = & \\vee\\{\\nu(w^{-1})\\,:\\, w^{-1}\\in Hx^{-1}\\}\\qquad\\qquad\\qquad\\\\\n & \\geq & \\vee\\{\\nu(w)\\,:\\, w\\in Hx\\}\\qquad\\qquad\\qquad\\\\\n & = & \\xi(Hx).\n\\end{eqnarray*}\nHence $\\xi\\in F(G\\diagup H)$. \\end {proof}\n\n\\noindent \\begin {remark} The fuzzy AG-subgroup defined in Theorem\n\\ref{FC} is called Quotient fuzzy AG-subgroup or factor fuzzy AG-subgroup\nof $G$, and is denoted by $\\nu\\diagup H$.\n\n\\noindent \\end {remark}\n\n\\noindent \\begin {theorem} \\label{c}Let $\\mu\\in\\mathcal{F}(G)$.\nThen $\\mu_{x}=\\mu_{y}\\Leftrightarrow\\underset{x}{\\mu_{*}}=\\underset{y}{\\mu_{*}}\\,\\,\\forall\\, x,y\\in G$.\\end {theorem}\n\n\\noindent \\begin {proof} Let $\\mu_{x}=\\mu_{y}$, then\n\\begin{eqnarray}\n\\mu_{x}(g) & = & \\mu_{y}(g)\\,\\,\\forall\\, g\\in G\\nonumber \\\\\n\\Rightarrow\\mu(gx^{-1}) & = & \\mu(gy^{-1})\\,\\,\\forall\\, g\\in G\\quad[\\mbox{using (\\ref{eq:1a-1})}]\\label{eq:1a}\n\\end{eqnarray}\nput $g=y$, in (\\ref{eq:1a}) we get $\\mu(yx^{-1})=\\mu(e)$ $\\Rightarrow yx^{-1}\\in\\mu_{*}.\\,\\,\\,[\\mbox{by (\\ref{a1})}]$\n$\\Rightarrow$$(yx^{-1})x\\in\\underset{x}{\\mu_{*}}\\Rightarrow(xx^{-1})y\\in\\underset{x}{\\mu_{*}}$\n$(\\mbox{using left invertive law})$ $\\Rightarrow y\\in\\underset{x}{\\mu_{*}}$,\nbut $y\\in\\underset{y}{\\mu_{*}}$, therefore, $\\underset{y}{\\mu_{*}}\\subseteq\\underset{x}{\\mu_{*}}$.\n\n\\noindent Again put $g=x$, in (\\ref{eq:1a}) we get $\\mu(xx^{-1})=\\mu(xy^{-1})\\Rightarrow\\mu(xy^{-1})=\\mu(e)\\Rightarrow xy^{-1}\\in\\mu_{*}.\\,\\,\\,[\\mbox{by (\\ref{a1})}]\\Rightarrow(xy^{-1})y\\in\\underset{y}{\\mu_{*}}\\Rightarrow x\\in\\underset{y}{\\mu_{*}}$,\nbut $x\\in\\underset{x}{\\mu_{*}}$. This implies that, $\\underset{x}{\\mu_{*}}\\subseteq\\underset{y}{\\mu_{*}}$.\nThus $\\underset{x}{\\mu_{*}}\\subseteq\\underset{y}{\\mu_{*}}\\subseteq\\underset{x}{\\mu_{*}}$.\nHence $\\underset{x}{\\mu_{*}}=\\underset{y}{\\mu_{*}}.$\n\n\\noindent Conversely; let $\\underset{x}{\\mu_{*}}=\\underset{y}{\\mu_{*}}\\Rightarrow\\underset{x}{\\mu_{*}}\\circ\\underset{y^{-1}}{\\mu_{*}}=\\underset{y}{\\mu_{*}}\\circ\\underset{y^{-1}}{\\mu_{*}}\\Rightarrow\\underset{xy^{-1}}{\\mu_{*}}=\\underset{yy^{-1}}{\\mu_{*}}\\Rightarrow\\underset{xy^{-1}}{\\mu_{*}}=\\underset{e}{\\mu_{*}}=\\mu_{*}\\Rightarrow xy^{-1}\\in\\mu_{*}$.\nNow for any $x,y\\in G$ it follows that\n\\begin{eqnarray*}\n\\mu(gx^{-1}) & = & \\mu(g((y^{-1}y)x^{-1}))\\\\\n & = & \\mu(g((x^{-1}y)y^{-1}))\\qquad[\\mbox{using left invertive law]}\\\\\n & = & \\mu((x^{-1}y)(gy^{-1}))\\qquad\\mbox{[in \\ensuremath{G}; \\ensuremath{a(bc)=b(ac)}\\,[8]]}\\\\\n & \\geq & \\mu(x^{-1}y)\\wedge\\mu(gy^{-1})\\quad[\\mu\\in\\mathcal{F}(G)]\\\\\n & = & \\mu((xy^{-1})^{-1})\\wedge\\mu(gy^{-1})\\\\\n & = & \\mu(xy^{-1})\\wedge\\mu(gy^{-1})\\quad\\,[\\mbox{by Lemma \\ref{ad}-(ii)}]\\\\\n & = & \\mu(e)\\wedge\\mu(gy^{-1})\\qquad\\quad[\\mbox{by }(\\ref{a1});\\mbox{ as }xy^{-1}\\in\\mu_{*}]\\\\\n & = & \\mu(gy^{-1})\\qquad\\qquad\\quad\\quad[\\mbox{by Lemma \\ref{ad}-(i)}]\n\\end{eqnarray*}\nThis implies that $\\mu(gx^{-1})\\geq\\mu(gy^{-1})$.\n\n\\noindent By similar arrangements we can show that, $\\mu(gy^{-1})\\geq\\mu(gx^{-1})$.\nConsequently $\\mu(gx^{-1})=\\mu(gy^{-1})\\Rightarrow\\mu_{x}(g)=\\mu_{y}(g)\\,\\,\\forall\\, g\\in G$\nby (\\ref{eq:1a-1}). Hence $\\mu_{x}=\\mu_{y}$. \\end {proof} \n\n\\noindent \\begin {theorem} \\label{aa}Let $\\mu\\in NF(G)$ and $\\mu_{x}=\\mu_{y}$,\nthen $\\mu(x)=\\mu(y)$ $\\forall\\, x,y\\in G$. \\end {theorem}\n\n\\noindent \\begin {proof} Let $x,y\\in G$, then $\\mu_{x}=\\mu_{y}\\Leftrightarrow\\underset{x}{\\mu_{*}}=\\underset{y}{\\mu_{*}}\\Rightarrow\\underset{x}{\\mu_{*}}\\circ\\underset{y^{-1}}{\\mu_{*}}=\\underset{y}{\\mu_{*}}\\circ\\underset{y^{-1}}{\\mu_{*}}\\Rightarrow\\underset{xy^{-1}}{\\mu_{*}}=\\underset{yy^{-1}}{\\mu_{*}}\\Rightarrow\\underset{xy^{-1}}{\\mu_{*}}=\\underset{e}{\\mu_{*}}=\\mu_{*}\\Rightarrow xy^{-1}\\in\\mu_{*}$;\n(using Theorem \\ref{c} and the definition of fuzzy cosets). Therefore,\n\\begin{eqnarray*}\n\\mu(y)=\\mu(y^{-1}) & = & \\mu(x^{-1}y^{-1}\\cdot(x^{-1})^{-1})\\quad[\\mu\\in NF(G)]\\\\\n & = & \\mu(x^{-1}y^{-1}\\cdot x)\\\\\n & = & \\mu(xy^{-1}\\cdot x^{-1})\\qquad[\\mbox{using left invertive law]}\\\\\n & \\geq & \\mu(xy^{-1})\\wedge\\mu(x^{-1})\\\\\n & = & \\mu(e)\\wedge\\mu(x)\\qquad\\,\\,\\,\\,[\\mbox{ by }(\\ref{a1}),\\mbox{ as }xy^{-1}\\in\\mu_{*}]\\\\\n & = & \\mu(x)\\qquad\\qquad\\quad\\,\\,\\,\\,[\\mbox{by Lemma \\ref{ad}-(i)}]\\\\\n\\Rightarrow\\mu(y) & \\geq & \\mu(x).\n\\end{eqnarray*}\n\n\n\\noindent Similarly, we can show that $\\mu(x)\\geq\\mu(y)$. This implies\nthat $\\mu(x)\\geq\\mu(y)\\geq\\mu(x).$ Hence $\\mu(x)=\\mu(y)$. \\end {proof}\n\n\\noindent \\begin {proposition} Let $\\mu\\in NF(G)$. Then $\\mu_{x}(xg)=\\mu_{x}(gx)=\\mu(g)\\,\\,\\forall\\, g\\in G$.\\end {proposition}\n\n\\noindent \\begin {proof} Using definition of cosets of fuzzy AG-subgroup,\nit follows that for $g\\in G$; $\\mu_{x}(xg)=\\mu(xg\\cdot x^{-1})=\\mu(g)$.\nAnd $\\mu_{x}(gx)=\\mu(gx\\cdot x^{-1})=\\mu(x^{-1}x\\cdot g)=\\mu(eg)=\\mu(g)$.\nHence $\\mu_{x}(xg)=\\mu_{x}(gx)=\\mu(g)\\,\\,\\forall\\, g\\in G$. \\end {proof}\n\n\\noindent \\begin {theorem}\\label{d}Let $\\mu\\in NF(G)$. Then the\nfollowing assertions hold:\n\n\\begin{enumerate}[(i)]\n\n\\item $G\\diagup\\mu\\cong G\\diagup\\mu_{*}$;\n\n\\item If\\emph{ }$\\nu\\in FP(G\\diagup\\mu)$; defined by $\\nu(\\mu_{x})=\\mu(x)\\,\\,\\forall\\, x\\in G$.\nThen $\\nu\\in NF(G\\diagup\\mu)$.\n\n\\end{enumerate}\n\n\\noindent \\end {theorem}\n\n\\noindent \\begin {proof}  As both $G\\diagup\\mu$ and $G\\diagup\\mu_{*}$\nare AG-groups by Theorem \\ref{s} and $\\phi:G\\diagup\\mu\\rightarrow G\\diagup\\mu_{*}$\ngiven by $\\phi(\\mu_{x})=\\underset{x}{\\mu_{*}}\\,\\,\\forall\\, x\\in G$\nis an isomorphism by Theorem \\ref{c} and the fact that $\\mu_{x}\\circ\\mu_{y}=\\mu_{xy}$\nand $\\underset{x}{\\mu_{*}}\\circ\\underset{y}{\\mu_{*}}=\\underset{xy}{\\mu_{*}}$.\n\n(ii) Let\\emph{ }$\\nu\\in FP(G\\diagup\\mu)$, be defined by $\\nu(\\mu_{x})=\\mu(x)\\,\\,\\forall\\, x\\in G$.\nWe show that $\\nu\\in NF(G\\diagup\\mu)$. Since\n\\begin{eqnarray*}\n\\nu(\\mu_{x}\\circ\\mu_{y}) & = & \\nu(\\mu_{xy})\\\\\n & = & \\mu(xy)\\qquad\\,\\,\\qquad[\\mbox{by definition of \\ensuremath{\\nu}}]\\\\\n & \\geq & \\mu(x)\\wedge\\mu(y)\\,\\,\\,\\qquad\\,\\,[\\mu\\in NF(G)]\\\\\n & = & \\nu(\\mu_{x})\\wedge\\nu(\\mu_{y})\\quad\\,[\\mbox{by definition of \\ensuremath{\\nu}}]\n\\end{eqnarray*}\n$\\forall\\, x,y\\in G$, and\n\\begin{eqnarray*}\n\\nu((\\mu_{x})^{-1}) & = & \\nu(\\mu_{x^{-1}})=\\mu(x^{-1})\\geq\\mu(x)=\\nu(\\mu_{x})\\qquad\\,\\,\n\\end{eqnarray*}\n$\\forall\\, x\\in G$. Hence $\\nu\\in F(G\\diagup\\mu)$. Further, since\n\\begin{eqnarray*}\n\\nu((\\mu_{x}\\circ\\mu_{y})\\circ(\\mu_{x})^{-1}) & = & \\nu(\\mu_{xy}\\circ\\mu_{x^{-1}})\\\\\n & = & \\nu(\\mu_{xy\\cdot x^{-1}})\\\\\n & = & \\mu(xy\\cdot x^{-1})\\,\\,\\quad\\,[\\mbox{by definition of \\ensuremath{\\nu}}]\\\\\n & = & \\mu(y)\\qquad\\,\\,\\,\\,\\,\\,\\qquad[\\mu\\in NF(G)]\\\\\n & = & \\nu(\\mu_{y}).\\qquad\\qquad[\\mbox{by definition of \\ensuremath{\\nu}}]\n\\end{eqnarray*}\n$\\forall\\, x,y\\in G$. Hence $\\nu\\in NF(G\\diagup\\mu)$. \\end {proof}\n\n\\noindent \\begin {theorem}\\label{e}Let $\\mu\\in NF(G)$. Define a\nmapping $\\theta:G\\rightarrow G\\diagup\\mu$ as follows:\n\\begin{eqnarray}\n\\theta(x) & = & \\mu_{x}\\,\\,\\forall\\, x\\in G.\\label{eq:4a}\n\\end{eqnarray}\nThen $\\theta$ is homomorphism with kernel $\\mu_{*}$. \\end {theorem}\n\n\\noindent \\begin {proof} Since \n\\begin{eqnarray*}\n\\theta(xy) & = & \\mu_{xy}=\\mu_{x}\\circ\\mu_{y}=\\theta(x)\\theta(y)\\,\\,\\,\\forall\\, x,y\\in G.\n\\end{eqnarray*}\nHence $\\theta$ is homomorphism. Further, the kernel of $\\theta$\nconsists of all $x\\in G$ for which $\\mu_{x}=\\mu_{e}\\Leftrightarrow\\mu(x)=\\mu(e)$,\n(by Theorem \\ref{aa}) $\\Leftrightarrow x\\in\\mu_{*}$. Thus $Ker\\theta=\\mu_{*}$.\n\\end {proof}\n\n\\noindent \\begin {theorem}\\emph{ }Let $\\mu\\in NF(G)$, and $G\\diagup\\mu$\nis an AG-group. Then each $\\zeta\\in NF(G\\diagup\\mu)$ corresponds\nin a natural way to $\\nu\\in NF(G)$.\\end {theorem}\n\n\\noindent \\begin {proof} Let $\\zeta\\in NF(G\\diagup\\mu)$. Define\na mapping $\\nu:G\\rightarrow[0,1]$ as follows:\n\\begin{eqnarray*}\n\\nu(x) & = & \\zeta(\\mu_{x})\\,\\,\\forall\\, x\\in G.\n\\end{eqnarray*}\nFirst we show that $\\nu\\in F(G)$. Since $\\forall\\, x,y\\in G$,\n\\begin{eqnarray*}\n\\nu(xy) & = & \\zeta(\\mu_{xy})\\\\\n & = & \\zeta(\\mu_{x}\\circ\\mu_{y})\\\\\n & \\geq & \\zeta(\\mu_{x})\\wedge\\zeta(\\mu_{y})\\qquad[\\zeta\\in NF(G\\diagup\\mu)]\\\\\n & = & \\nu(x)\\wedge\\nu(y)\n\\end{eqnarray*}\nand $\\forall\\, x\\in G$,\n\\begin{eqnarray*}\n\\nu(x^{-1}) & = & \\zeta(\\mu_{x^{-1}})=\\zeta(\\mu_{x})^{-1}\\geq\\zeta(\\mu_{x})=\\nu(x)\\qquad\\qquad\\qquad\n\\end{eqnarray*}\nThus $\\nu\\in F(G)$.\n\n\\noindent Further, since $\\forall\\, x,y\\in G$,\n\\begin{eqnarray*}\n\\nu(xy\\cdot x^{-1}) & = & \\zeta(\\mu_{xy\\cdot x^{-1}})\\\\\n & = & \\zeta(\\mu_{y})\\qquad[\\mu\\in NF(G)]\\\\\n & = & \\nu(y).\n\\end{eqnarray*}\nHence $\\nu\\in NF(G)$. \\end {proof}\n\n\\noindent In the following we introduce fuzzy Lagrange's Theorem for\nAG-group of finite order. We start with the following definition.\n\n\\noindent \\begin {definition} Let $G$ be a finite AG-group, $\\mu\\in F(G)$\nand $G\\diagup\\mu$ is an AG-group. Then the cardinality of $G\\diagup\\mu$\nis called the index of fuzzy AG-subgroup of $\\mu$ in $G$ written\nas $[G:\\mu]$.\\end {definition}\n\n\\noindent \\begin {theorem} (Fuzzy Lagrange's Theorem for AG-subgroup).\nLet $G$ be a finite AG-group, $\\mu\\in F(G)$. Then the index of fuzzy\nAG-subgroup of $\\mu$ divides the order of $G$.\\end {theorem}\\begin {proof}\nIt follows from Theorem \\ref{e}, that there is homomorphism $\\theta$\nfrom $G$ into $G\\diagup\\mu$, the set of all fuzzy cosets of $\\mu$,\ndefined in (\\ref{eq:4a}). Let $H$ be an AG-subgroup of $G$ defined\nby $H=\\{h\\in G\\,:\\,\\mu_{h}=\\mu_{e}\\}$. Let $h\\in H$, then $\\mu_{h}=\\mu_{e}\\Leftrightarrow\\underset{h}{\\mu_{*}}=\\underset{e}{\\mu_{*}}$\nusing Theorem \\ref{c}. Therefore, $H=\\{h\\in G\\,:\\,\\underset{h}{\\mu_{*}}=\\underset{e}{\\mu_{*}}\\}$.\nNow decomposing $G$ as a disjoint union of the cosets of $G$ with\nrespect to $H$ i.e.\n\\begin{eqnarray}\nG & = & (H=Hx_{1})\\cup Hx_{2}\\cup\\cdots\\cup Hx_{k},\\label{eq:5a}\n\\end{eqnarray}\nwhere $x_{1}\\in H$ and $x_{i}\\in G;\\,1\\leq i\\leq k$. Now, we show\nthat corresponding to each cost $Hx_{i};\\,\\,1\\leq i\\leq k$, given\nin (\\ref{eq:5a}) there is a fuzzy coset belonging to $G\\diagup\\mu$,\nand further this correspondence is one-one. To see this, consider\nany coset $Hx_{i}$ for any $h\\in H$, we have that; $\\theta(hx_{i})=\\mu_{hx_{i}}=\\mu_{h}\\circ\\mu_{x_{i}}=\\mu_{e}\\cdot\\mu_{x_{i}}=\\mu_{ex_{i}}=\\mu_{x_{i}}$.\nThus $\\theta$ maps each element of $Hx_{i}$ into the fuzzy cosets\n$\\mu_{x_{i}}$.\n\n\\noindent Now we show that $\\theta$ is well-defined. Let $Hx_{i}=Hx_{j}$$\\ensuremath{\\mbox{ , for each \\ensuremath{i,j:}\\,}1\\leq i\\leq k}\\,\\ \\textrm{and }\\ensuremath{1\\leq j\\leq k}.$\nThen\n\\begin{eqnarray*}\nx_{j}^{-1}x_{i}\\in H &  & \\qquad[\\mbox{cosets in AG-groups}]\\\\\n\\Rightarrow\\mu_{x_{j}^{-1}x_{i}} & = & \\mu_{e}\\\\\n\\Rightarrow\\underset{(x_{j}^{-1}x_{i})}{\\mu_{*}} & = & \\underset{e}{\\mu_{*}}\\qquad[\\mbox{by Theorem \\ref{c}]}\\\\\n\\Rightarrow\\underset{(x_{j}^{-1}x_{i})}{\\mu_{*}}\\circ\\underset{x_{i}^{-1}}{\\mu_{*}} & = & \\underset{e}{\\mu_{*}}\\circ\\underset{x_{i}^{-1}}{\\mu_{*}}\\\\\n\\Rightarrow\\underset{(x_{j}^{-1}x_{i})x_{i}^{-1}}{\\mu_{*}} & = & \\underset{ex_{i}^{-1}}{\\mu_{*}}\n\\end{eqnarray*}\n\n\n\\noindent \n\\begin{eqnarray*}\n\\Rightarrow\\underset{(x_{i}^{-1}x_{i})x_{j}^{-1}}{\\mu_{*}} & = & \\underset{ex_{i}^{-1}}{\\mu_{*}}\\qquad[\\mbox{using left invertive law}]\\\\\n\\Rightarrow\\underset{x_{i}^{-1}}{\\mu_{*}} & = & \\underset{x_{j}^{-1}}{\\mu_{*}}\\\\\n\\Rightarrow\\underset{x_{i}}{\\mu_{*}} & = & \\underset{x_{j}}{\\mu_{*}}\\\\\n\\Rightarrow\\mu_{x_{i}} & = & \\mu_{x_{j}}\\,\\qquad[\\mbox{by Theorem \\ref{c}]}\\\\\n\\Rightarrow\\theta(Hx_{i}) & = & \\theta(Hx_{j}).\n\\end{eqnarray*}\nThus $\\theta$ is well-defined.\n\n\\noindent Further, we show that $\\theta$ is one-one; for each $i,\\, j$\nwhere $1\\leq i\\leq k$ and $1\\leq j\\leq k$; assume that\n\\begin{eqnarray*}\n\\theta(Hx_{i}) & = & \\theta(Hx_{j})\\\\\n\\Rightarrow\\mu_{x_{i}} & = & \\mu_{x_{j}}\\\\\n\\Rightarrow\\underset{x_{i}}{\\mu_{*}} & = & \\underset{x_{j}}{\\mu_{*}}\\qquad\\qquad[\\mbox{ Theorem}\\ref{c}]\\\\\n\\Rightarrow\\underset{x_{i}^{-1}}{\\mu_{*}} & = & \\underset{x_{j}^{-1}}{\\mu_{*}}\\qquad[\\mbox{cosets in AG-groups }]\\\\\n\\Rightarrow\\underset{e}{\\mu_{*}}\\circ\\underset{x_{i}^{-1}}{\\mu_{*}} & = & \\underset{e}{\\mu_{*}}\\circ\\underset{x_{j}^{-1}}{\\mu_{*}}\n\\end{eqnarray*}\n\\begin{eqnarray*}\n\\Rightarrow\\underset{ex_{i}^{-1}}{\\mu_{*}} & = & \\underset{ex_{j}^{-1}\\textrm{}}{\\mu_{*}}\\textrm{\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad}\\\\\n\\Rightarrow\\underset{ex_{i}^{-1}}{\\mu_{*}} & = & \\underset{(x_{i}^{-1}x_{i}\\cdot x_{j}^{-1})}{\\mu_{*}}\\\\\n\\Rightarrow\\underset{ex_{i}^{-1}}{\\mu_{*}} & = & \\underset{(x_{j}^{-1}x_{i}\\cdot x_{i}^{-1})}{\\mu_{*}}\\\\\n\\Rightarrow\\underset{(x_{j}^{-1}x_{i})}{\\mu_{*}}\\circ\\underset{x_{i}^{-1}}{\\mu_{*}} & = & \\underset{e}{\\mu_{*}}\\circ\\underset{x_{i}^{-1}}{\\mu_{*}}\n\\end{eqnarray*}\n\\begin{eqnarray*}\n\\Rightarrow\\underset{(x_{j}^{-1}x_{i})}{\\mu_{*}} & = & \\underset{e}{\\mu_{*}}\\\\\n\\Rightarrow\\mu_{x_{j}^{-1}x_{i}} & = & \\mu_{e}\\qquad[\\mbox{by Theorem }\\ref{c}]\\\\\n\\Rightarrow x_{j}^{-1}x_{i}\\in H\n\\end{eqnarray*}\n\\begin{eqnarray*}\n\\Leftrightarrow Hx_{i} & = & Hx_{j}\\qquad\\mbox{\\ensuremath{\\mbox{[for each \\ensuremath{i\\,}and \\ensuremath{j\\,}where \\,\\,}1\\leq i\\leq k\\,}and \\ensuremath{1\\leq j\\leq k.}]}\n\\end{eqnarray*}\nFrom above discussion it is now clear that the number of distinct\ncosets of $H$ (index) in $G$ equals the number of fuzzy cosets of\n$\\mu$, which is a divisor of the order of $G$. Hence we conclude\nthat the index of $\\mu$ also divides the order of $G$. \\end {proof}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}