text:
\section{Introduction and motivation}

Quasars  are extremely luminous active galaxy nuclei powered by supermassive black holes with mass ranging from millions to tens of billions of Sun mass and consuming the gas from surrounding accretion disks. The effect is powerful, the radiant energy of quasars is giant and quasars have typically luminosities hundreds  of times greater than a galaxy such as the Milky Way. Quasar  surveys (over a million have been observed so far) have demonstrated that quasar activity was more common in the distant past -- the peak epoch was approximately 10 billion years ago. The closest known quasar is observed ca.  600 million light-years away from Earth. Record the farthest quasar is J0313-1806 observed in 2021
 with a 1.6 billion solar-mass black hole at z = 7.64, i.e., 670 million years after the Big Bang. Also in 2021, PSO J172.3556+18.7734 was detected as the most distant known radio-loud quasar discovered. A major concern is the vast amount of energy these objects would have to be radiating, if they were so distant. Commonly accepted explanation of such high power of quasars, that it is due to matter in an accretion disc falling into a supermassive black hole, which was  suggested in 1964 by Salpeter and Zeldovich \cite{salpet,zeld}. Dynamics of the accretion disc has been modelled by Shakura and Sunyaev in terms of classical fluid hydrodynamics and the release of  the gravitational field energy has been assessed due to by viscosity induced transport of angular momentum in opposite direction to accretion flow \cite{sunaev}. The model has been extended onto fully general gravitation approach by Novikov and Thorne \cite{novikov1} and was the subject of further intensive studies and more sophistical developments, e.g., towards rotating black holes in Kerr metric \cite{abramowiczg,reynolds}, however, still within classical hydrodynamic or magnetohydrodynamic approach.

 The energy transferred to the accretion disc matter on the cost of gravitational field of attracting central singularity and controlled by the angular momentum transport in opposite direction versus flow of matter in a flat disc is assumed to be eventually converted into electromagnetic radiation  by the thermal radiation including also the inverse Compton effect and bremsstrahlung. The realistic model of the active black hole luminosity has been formulated by Shapiro et al. \cite{shapiro}, who considered the accretion disc for microquasar Cignus X-1 powered by a 15 Sun mass black hole. By assumption the local temperature in the accretion disc ($10^9$ K for electrons and $10^{11}$ K for ions) the thermal radiation has been assessed as similar as observed, however, to elucidate the giant luminosity in X-ray range the Comptomization mechanism has been invoked, i.e., the increase of energy of soft thermal photons on the cost of energy of hot charge carriers in plasma. To assure the sufficient energy of these charge carriers the extremely high temperature in the accretion disc had to be assumed (just $10^{9-11}$ K). So hot plasma is assumed to occur in the small distance from the central black hole, at $r\sim 6 r_s$ (where $r_s$ is the event horizon radius of the black hole). Though the Shapiro model fits well to observations of microquasars, its generalization onto super-luminous giant quasars powered by $10^9$ Sun mass black holes or larger \cite{supereddington} is problematic, since the temperature $10^9 - 10^{11}$ K  of hot plasma in vast accretion disc of such large black holes  seems to be unrealistic. Moreover, the source for sufficient abundance  of soft photons in a small region where the Comptomization mechanism would be efficient is uncertain and  disputable.  Some numerical simulations done recently by Fragile et al. \cite{6p} allow to match to the observable luminosities of some not distant ($z<0.3$) active binary black hole objects \cite{farahzest}, but rather not to super-luminous remote quasars \cite{supereddington}. Higher radiation efficiency has been modelled within conventional classical magneto-hydrodynamic approach by inclusion of hypothetical giant magnetic component to accretion plasma in the case of a spinning black hole \cite{24p}, assuming, however, unrealistic extremal accretion mass rate to gain sufficiently large luminosity.

Models of the accretion disc  utilize the energy conservation condition and the transport of angular momentum in the flow of matter in gravitational potential of a black hole within a conventional hydrodynamic turbulence picture with some phenomenological assumptions regarding viscosity, inner disc edge and channels of energy dissipation imposed \cite{sunaev,merloni,novikov1,abramowiczg,shapiro}. In particular, the conventional models of the accretion assume the termination of the disc radiation at the inner disc edge located not closer to the singularity than at the innermost stable circular orbit (at $3r_s$, though the active region is not closer than $6r_s$ \cite{shapiro}), neglecting the nearest neighbourhood of the event horizon.  Such classical hydrodynamic or magnetohydrodynamic  models  are not applicable in closer vicinity of the event horizon \cite{ksiazka}. At extreme matter influx rates as in super-luminous quasars, these models have problems with fitting the observations and additional channels of gravitational energy to radiation transfer are still being sought.
 To explain the enormously intensive hard radiation of super-luminous quasars (despite its softening by gravitation relativity effect due to Doppler boosting and gravitational redshift), some other mechanism
 besides thermal radiation or other conventional mechanisms (Comptomization and bremsstrahlung) seems to be contributing.

 In the present paper we propose such a new channel of gravitational energy to radiation transfer close to the event horizon of the central black hole, i.e., in the region neglected in classical hydrodynamic-type previous models. We will demonstrate that the quantum in nature process undergoing on the rim of black hole photon sphere (at distance $1.5r_s$ from the gravitational singularity) can substantially supplement the overall luminosity of quasars and in the case of an uppermost rate of mass consumption by the black hole, it can help to match the observable   radiation energy/mass conversion rate on the level of 30 \% for super-luminous active gravitation centres.
Such high radiation energy/mass conversion rate is actually needed to explain the typical luminosity of giant quasars being of order of $10^{40}$ W and powered by ca. billion Sun mass black holes consuming of order of $10$ Sun mass per year (i.e., $0.1$  Earth mass per second), in extreme case of $1000$ Sun mass per year ($10$ Earth mass per second). Such rates of mass consumption are inferred from the comparison of
central black holes in quasars, which  vary between $ 10^5 - 10^9$ of solar masses, as  have been measured using a reverberation mapping, and cannot increase during the quasar activity period beyond an average  mass of closer non-active super-massive black holes. Several dozen nearby large galaxies, including our own Milky Way galaxy, that do not have an active centre and do not show any activity similar to a quasar, are confirmed to contain a similar supermassive black hole in their  centres. Thus it is now thought that all large galaxies have a giant black hole of this kind, but only a small fraction have sufficient matter in the right kind of orbit at their centre to become active and power radiation in such a way as to be seen as quasars. However, if quasars are active for a long time and their mass cannot increase too much, the efficiency of converting matter into energy must be high to explain the gigantic luminosity of  quasars.

An argument supporting a model of quasar luminosity from the accretion disc close to the Schwarzschild horizon of a central black hole  is the observed variation in time of quasar activity.
Quasars' luminosities vary with time scale that ranges from days to hours. This means that quasars generate and emit their energy from a very small region, since each part of the quasar would have to be in contact with other parts on such a time scale as to allow the coordination of the luminosity variations. This would mean that a quasar cannot be larger than a few light hours or days across -- just as the size of the Schwarzschild horizon radius of the central supermassive black holes, $r_s=\frac{2GM}{c^2}$, where $M$ is the mass of the black hole, $G$ is the gravitational constant, $c$ is light velocity in the vacuum. For exemplary supermassive black holes such radius is 0.08 au for Sigittarius A* in Milky Way with mass $M=8.2 \times 10^{36}$ kg (4.15 million $M_\odot$), or 1300 au for largest known black hole TON 618 with mass $M=1.3 \times 10^{41}$ kg (66 billion of $M_\odot$).  The emission of a large amount of power from a small region requires a power source far more efficient than the nuclear fusion that powers stars. The conversion rate of the gravitational potential energy to the radiation energy at the mass influx to a black hole reaches ca. 30 \%   in quasars, compared to 0.7 \% for the conversion of mass to energy in a star like our Sun. It is the only known process that can produce such high power over a very long term. Stellar explosions such as supernovas and gamma-ray bursts, and direct matter–antimatter annihilation, can also produce very high power output, but supernovas only last for days, and the universe does not appear to have had large amounts of antimatter at the relevant times. A physical mechanism of so effective mass-energy conversion in quasars is still obscure to some extent, as the thermal radiation of hot ionized gas, bremsstrahlung, synchrotron radiation and the inverse Compton effect  cannot be so efficient at case of super-luminous quasars \cite{supereddington} with vast accretion discs probably not so hot as in the case of more concentrated and smaller centres (like Cignus X-1 \cite{shapiro}).
 Probably some additional mechanism of energy conversion  contributes in the vicinity of the Schwarzschild  horizon to produce an additional amount of radiation in  quasars.  We propose such a mechanism taking into account collective quantum effects in dense fermion plasma associated with a phase transition in multiparticle systems passing the rim of the photon sphere of a black hole.

The paper is organized as follows. In the next paragraph we propose the new mechanism of gravitational energy conversion into e-m radiation, which has a quantum character and is effective at passing the photon sphere rim of the black hole (at the distance $1.5 r_s$ from the origin with gravitational singularity). The physical details of the proposed mechanism are next explained. The comparison with observations are placed in the next paragraph where the new model is suggested also to supplement current concept of high energy radiation at collapse of neutron star  merger exceeding the Tolman-Oppenheimer-Volkoff limit and, on the other hand, some short time-lasting luminosity changes of active galactic nuclei (as recently observed for AGN 1ES 1927+654 \cite{ga1flare}). Some numerical estimations and additional comments and mathematical proofs are shifted to Appendices.

\section{Proposition of quantum mechanism driving quasar activity below the innermost unstable circular orbit of a black hole}

According to the conventional models of the accretion disc (\cite{sunaev,novikov1} and developments e.g.,  \cite{abramowiczg,reynolds,merloni,ksiazka}), the radiation of such disc terminates at the inner edge of the disc not closer to the event horizon than ca. $6r_s$ \cite{shapiro}. Such a common theoretical assumption displays the fact that the classical hydrodynamic approach on which  the theory of the accretion disc is based on \cite{sunaev,ksiazka}, does not apply to the region closest to Schwarzschild horizon where the conventional scattering regime of particles in plasma is substituted by another regime  resembling some-kind of laminarization of matter movement induced by the giant gravity, prevailing and marginalizing  local interparticle effects the closer to the horizon.

The timescale of more or less regular changes in the brightness
of  quasars suggest that the responsible radiation  source  must be concentrated close to
the event horizon rather than located in remote regions of the
accretion disk (or in distant jets).
This hypothetical source of some part of quasar radiation located close to the event horizon must be governed by completely different mechanism than that already  described within classical  hydrodynamic accretion  models applicable in more remote regions \cite{sunaev,novikov1,shapiro,ksiazka}.

We propose some supplementation to a model of quasar luminosity source including the region just below the innermost unstable circular orbit (the rim of the photon sphere in Shwarzschild metric), taking advantage of the specific change of homotopy of trajectories of particles which if they enter the photon sphere, they irrevocably spiral onto the Schwarzschild horizon in the laminarized manner governed by the central gravitational singularity no matter how large the initial energy and angular momentum of  particles are  and no matter how strongly they mutually interact.   This is the result of spacetime curvature induced by the gravitational singularity at small distances, what can be  illustrated in standard  Schwarzschild metric. Sharp changing of the homotopy of classical admissible trajectories for particles passing the innermost unstable circular orbit (which coincides with the photon sphere rim for non-rotating and uncharged black hole at $r=1.5 r_s$ distance from the central singularity) causes the quantum phase transition consisting in the local repeal of the Pauli exclusion principle, which results in the rapid collapse of Fermi spheres of collective systems of electrons and protons in infalling plasma in the accretion disk. The local density of fermions in close vicinity of the event horizon attains large values due to the gravitational compression. Fermi liquids of electrons and protons are quantumly degenerated here and accumulate great energy in their Fermi spheres on the cost of the gravitational energy.
The conversion of the gravitational field energy into electromagnetic radiation at rapid collapse of Fermi spheres of fermions  is  very effective for sufficiently high scale of transport of the mass and its local compression close to the event horizon, and can help to explain the observable luminosity of quasars. The consumption of the mass by the black hole cannot exceed in this model the limit imposed by the uppermost density of matter at the photon sphere rim (of order of the density of hadrons in atom nucleus or  the density of neutrons in neutron star at  Tolman-Oppenheimer-Volkoff limit). This constraint regulates the matter consumption rate in dependence of the black hole mass in the case of not limited abundance of matter supply to the accretion disc from surroundings.  The phenomenon of suggested Fermi sphere decay has a topological character sourced in the homotopy of Schwarzschild geodesics in curved spacetime  and displays a unique situation when the energy of fermions accumulated  in Fermi spheres of electrons and protons  in dense highly compressed plasma can be released due to quantum phase transition related to the local decay of quantum statistics triggered  by the  change of the homotopy of trajectories close to central singularity of  the black hole.
At sufficiently high number of particles compressed to a small volume by the  giant gravitation close to the event horizon, just at the innermost unstable circular orbit at $r=1.5r_s$ (coinciding with the photon sphere rim),  the mass to energy  conversion rate at the collapse of the Fermi spheres of electrons and protons reaches the value of 30 \% provided that the density of matter at the rim of the photon sphere attains its uppermost limit. Such a mechanism seems to be responsible for a significant contribution to the luminosity of giant super-luminous quasars for which the matter supply from their surroundings is limited only by the uppermost density limit at the photon sphere rim. The detailed proof of the proposed model
is demonstrated in the following paragraphs.

\subsection{Change of trajectory homotopy below the innermost unstable circular orbit and the collapse of the Fermi sphere as the radiation source}

If identical indistinguishable particles distributed in some  space (mathematically defined as a manifold $M$) can exchange their positions in $M$ (i.e., if the classical trajectories for such exchanges are admissible in the manifold $M$), the quantum statistics of these particles is assigned by a scalar unitary representation of the braid group \cite{mermin1979aaa,lwitt} (as shortly presented in  Appendix \ref{a}). The braid group is defined for $N$ identical indistinguishable particles on the manifold $M$ as the first homotopy group of the multiparticle coordination space of this system, i.e., the braid group is equal to $\pi_1(F_N)$, the first homotopy group of the space $F_N$, where $F_N=(M^N-\Delta)/S_N$ is the classical coordination space of $N$ identical indistinguishable particles in $M$ ($\Delta$ is the diagonal subset of the $N$-fold product of the manifold $M^N=M\times\dots \times M$ and subtracted from $M^N$ to assure the particle number conservation, whereas the division by the permutation group $S_N$ introduces particle indistinguishability -- points in $F_N$ which differ by only numbering of particles are unified). Various scalar unitary representations of $\pi_1(F_N)$ assign different quantum statistics of the same classical particles (more specific explanations are shifted to
 Appendix  \ref{a}).

In 4D spacetime only two scalar unitary representations of the braid group are possible, corresponding to bosons and fermions \cite{mermin1979aaa}. The most prominent consequence of quantum statistics is the Pauli exclusion principle, which   asserts that quantum particles of fermionic type  cannot share any common single-particle quantum state. In particular, fermions cannot approach a spatial region already occupied by another fermion, and thus they mutually repulse themselves. This is called the quantum degeneracy repulsion and the related  pressure is the origin of stopping the collapse of white dwarfs or neutron stars. In the former case the degeneracy pressure of electrons plays the role \cite{chandrasekhar}, whereas in the latter case of neutrons \cite{tolman,volkoff,olandau}.

Pauli exclusion principle leads to the formation of the so-called Fermi sphere in the case of large number of identical fermions in some volume, when the chemical potential $\mu$ (the increase of total system energy in the result of the addition of a single fermion to a multiparticle system) is much greater than the temperature in the system  in energy scale, $k_BT$, where $k_B$ is the Boltzmann constant. In such a case, referred to as quantum degenerated Fermi system,  particles are forced to occupy  consecutive in energy single-particle states one by one (these states can be numbered by the momentum in macroscopic systems, because the momentum  is a good quantum number at translational symmetry, and also for macroscopic finite but large systems with local translational symmetry and conventional periodic Born-Karman boundary conditions imposed \cite{landau1972}). The  occupation one-by-one of ordered in energy single-particle states results in a great accumulation of energy in the  Fermi sphere, and the accumulated energy can achieve a giant value in dense systems. In finite size macroscopic systems with spatial volume $V$, the momentum $\mathbf{p}$ of single-particle states is discrete, and according to periodic Born-Karman boundary conditions the density of quantum states in the phase space is  $\frac{dVd^3\mathbf{p}}{(2\pi\hbar)^3}$ (in agreement with the Bohr-Sommerfeld quasiclassical quantization rule, on the other hand \cite{landau1972}), $\hbar=1.05 \times 10^{-34}$ Js is the reduced Planck constant. Thus, $N$ fermionic particles  fill the Fermi sphere in momentum space up to some radius  called as the Fermi momentum separating discrete filled states inside  the sphere from outside  empty ones (at ground state of the total system, i.e., at $T=0$ K and provided that the single-particle energy spectrum is isotropic in momentum space, as e.g., for free  particles either relativistic with energy $\varepsilon(\mathbf{p})=\sqrt{p^2c^2+m^2c^4}-mc^2$ or classical ones with energy $\varepsilon(\mathbf{p})=\frac{p^2}{2m}$). This picture is maintained even in nonzero temperature, provided that the chemical potential of fermions $\mu\gg k_BT$ (where the chemical potential $\mu$ is the energy increase when a single particle is added to the system). This condition is easy to be satisfied in sufficiently dense systems, for example in astrophysical problems considered in this paper the Fermi sphere is almost the same as at $T=0$ K at temperatures even of order of $10^9$ K (the proof is placed in paragraph \ref{efficiency}).

As an elementary illustrative example let us consider  free electrons in a normal metal with the typical concentration of order of $10^{23}$ (of order of Avogadro number) per cm$^3$. These electrons constitute the large Fermi sphere with Fermi radius in momentum space $p_F \simeq 1.5 \times 10^{-24} $ kg m/s and with accumulated large total energy $\sim 3\times10^{10}$ J/m$^3$. This energy cannot be released because all fermions are blocked in their single-particle stationary states by Pauli exclusion principle, i.e., all lower single-particle states in the Fermi sphere are occupied and thus there is no room for fermions to jump  from the higher energy states to  lower ones. In this example of normal metal the energy of upper electrons in the Fermi sphere, $\frac{p_F^2}{2m}$ ($m=9.1 \times 10^{-31}$ kg is the rest mass of electron) at $T=0$ K reaches $90000$ K  (in thermal scale, i.e., in units for which the Boltzmann constant $k_B =1$). This Fermi energy is the chemical potential at $T=0$ K and it weakly changes with the temperature up to  melting temperature of a metal. So hot electrons on the Fermi surface even at $T=0$ K do not melt the metal by themselves, as they cannot give back their energy blocked in the Fermi sphere by the Pauli exclusion principle.

The Fermi momentum in the degenerate quantum liquid of arbitrary fermions depends solely  on particle concentration,
\begin{equation}
	\label{fm}
	p_F=\hbar(3\pi^2 \rho)^{1/3},
\end{equation}
where $\hbar$ is the reduced Planck constant and $\rho=\frac{n}{V}$ is the concentration of fermions, i.e., $n$ is the number of particles in the spatial volume $V$.
The Fermi momentum is independent of interaction of fermions according to Luttinger theorem \cite{luttinger}.   This follows from the fact that the phase space volume  $V \frac{4}{3} \pi p_F^3$ with the position space volume $V$ and the volume of momentum-space  of spherical shape $\frac{4}{3}\pi p_F^3$, corresponds to $  n=2(V\frac{4}{3}\pi p_F^3)/ h^3$ quantum states according to the  Bohr-Sommerfeld rule \cite{landau1972} ($h=2\pi\hbar$ is the Planck constant and factor  $2$ accounts here for the additional  spin degree of freedom  of fermions). The formula for Fermi momentum, as quasiclassically derived, is independent of interaction of fermions in Fermi liquid even with arbitrary strong particle interaction. With Fermi momentum the Fermi energy is linked -- the single-particle energy of a particle with Fermi momentum (equal to the chemical potential of fermions in multiparticle system at $T=0$ K). Thus for electrons in a metal with  $p_F\simeq 10^{-24}$ kg m/s the Fermi energy (in gas approximation) $\varepsilon_F=\frac{p_F^2}{2m} \simeq 1.2 \times 10^{-18} $ J (or $\sim 90000$ K in thermal units).

The whole Fermi sphere collects the energy of all fermions from the bottom of this sphere to its surface and this energy  per spatial  volume $V$ (neglecting interaction of fermions) equals to
\begin{equation}
	\label{en}
	\begin{array}{l}
		E=\sum_{\mathbf{p}} \varepsilon(\mathbf{p}) f(\varepsilon(\mathbf{p}))\\
		= \frac{V}{(2\pi \hbar)^3}\int d^3\mathbf{p}\varepsilon(\mathbf{p})f(\varepsilon(\mathbf{p}))\\
		=\int_0^{p_F}dp \int_0^{\pi} d \theta\int_0^{2\pi} d\phi  p^2 sin\theta \varepsilon(\mathbf{p}) \frac{V}{(2\pi \hbar)^3}\\
		=\frac{V}{2 \pi^2 \hbar^3}\int_0^{p_F} dp p^2 \varepsilon(p),\\
	\end{array}
\end{equation}
where the sum runs  over occupied states only, what is guaranteed by Fermi-Dirac distribution function $f(\varepsilon(\mathbf{p}))=\frac{1}{e^{(\varepsilon(\mathbf{p})-\mu)/k_BT}+1}\rightarrow_{T\rightarrow 0}1-\Theta(\varepsilon(\mathbf{p})-\varepsilon_F)$ (here $\Theta(x)$ is the Heaviside step function and $\varepsilon_F=\mu(T=0)$),  $p,\theta, \phi$ are spherical variables in momentum space and $\varepsilon(\mathbf{p})$ is the kinetical energy of a fermion equal to $\frac{p^2}{2m}$ (in nonrealtivistic case), $\sqrt{p^2c^2+m^2c^4}-mc^2$ (in relativistic case) or $cp$ (in ultrarelativistic case). The factor $\frac{V}{(2\pi \hbar)^3}$ is the density of quantum states, i.e., the number of single-particle quantum states in the element of the phase space $Vd^3\mathbf{p}$ -- this is  the density of states $\frac{dVd^3\mathbf{p}}{(2\pi\hbar)^3}$ integrated over the whole spatial volume $V$ of the system, because the energy is independent of spatial position.
We thus see that in the Fermi sphere the energy can be stored up to a giant value (the larger the higher Fermi momentum is; in normal metals of order of $10^{10}$ J/m$^3$).
This giant energy cannot be released as electrons cannot lower their energy, because all preceding states with lower energy are already occupied.

The energy reservoir in the Fermi sphere can be even greater than in metals because it depends on density of matter via the Fermi momentum given by Eq. (\ref{fm}) and the concentration of fermions can be larger than that of free collective  electrons in metals. In another example -- a neutron star with the density of order of $10^{18}$ kg/m$^3$ (i.e., of the order of two Sun mass compressed to the compact neutron star with radius of ca. $10$ km), the concentration of neutrons is of order of $10^{45}$ 1/m$^3$. For such a concentration of neutrons  the neutron Fermi sphere energy in the whole star attains the range of $10^{47}$ J, just as the energy of frequently observed cosmic short giant gamma-ray bursts (assuming the isotropy of their sources) -- for some more particularities of this example cf. Table \ref{tab-kw1}.

\begin{table*}
	\caption{Fermi momentum $p_F$ (acc. to Eq. (\ref{fm})),  Fermi energy $\varepsilon_F=\varepsilon(p_F)$ (for relativistic case of kinetic energy $\varepsilon(\mathbf{p})=\sqrt{c^2p^2+m_n^2c^4}-m_nc^2$, $m_n=1.675\times 10^{-27}$ kg -- the mass of neutron) and total energy of the Fermi sphere $E$ (acc. to Eq. (\ref{en})) released at the decay of the statistics for the example of density $\xi$ and radius $r$ of a neutron star  ($n$ total number of neutrons in the star). The collapse of Fermi sphere in a neutron star caused by the gravitation induced specific homotopy of trajectories near the Schwarzschild radius, which precludes particle exchanges and locally revokes the Pauli exclusion principle, is a good candidate for isotropic source of short giant gamma-ray burst with observable energy $\sim 10^{47}$ J.}
	\label{tab-kw1}
	\centering
	\begin{tabular}{c c c c c c}
		\hline\hline
		$\xi$ [kg/m$^3$]  & $r$ [km] & $n$& $p_F$ [kg m/s]& $\varepsilon_F$ [GeV] & $E$ [J] \\
		\hline
		$5\times10^{18}$& $10$ & $1.13\times 10^{58}$&$4.52 \times 10^{-19}$ &$0.32$ & $1.84 \times 10^{47}$ \\
			\hline
		$2 \times 10^{18}$&10&$4.7 \times 10^{57}$&$3.37 \times 10^{-19}$&$0.2$&$4.8 \times 10^{46} $\\
		\hline
		$1.0 \times 10^{19}$&$8$&$1.08 \times 10^{58}$&$5.57\times 10^{-19}$&$0.58$&$3 \times 10^{47}$\\
		\hline
		$2.5 \times 10^{18}$&$8$&$2.97 \times 10^{57}$&$3.62 \times 10^{-19}$&$0.24$&$3.5 \times 10^{46}$\\
		\hline

	\end{tabular}
\end{table*}

The similar energy estimation can be  performed  for any quantumly degenerated fermion system, i.e., when the chemical potential -- the energy increase caused by the addition of a single particle to multiparticle  system -- is much greater than the actual thermal energy in the system, $k_B T$, $k_B=1.38 \times 10^{-23}$ J/K is the Boltzmann constant, $T$ is the absolute temperature in the system (in normal  metal this chemical potential  is of order of 10 eV, or $10^5$ K [in units with $k_B=1$] and thus $\mu$ is practically equal to the Fermi energy $\varepsilon_F$, the upper electron energy at $T=0$ K, i.e.,  $\mu\simeq \varepsilon_F\gg k_BT$ even at melting temperature.

If one considers an accretion disc of a quasar, then the local density of electrons and protons (assuming accretion of neutral hydrogen) grows with falling of the matter towards the Schwarzschild horizon. The originally diluted neutral gas (let's say of a hydrogen cloud)  ionizes itself due to friction in the accretion disc and eventually becomes a degenerate two-component Fermi liquid of electrons and protons  despite the high temperature. Both these Fermi liquids  attain an ultra-high concentration in an increasingly flattened and compressed region in vicinity of the event horizon. For two component plasma,  both Fermi spheres  of electrons and protons contribute to the energy storage. At the same concentration of electrons and protons (due to the neutrality condition of plasma in the disc) electrons accumulate larger kinetic energy than protons because of lighter mass (the relativistic kinetic energy of the electron and  proton is $\varepsilon(\mathbf{p})=\sqrt{c^2p^2 +m^2c^4}-mc^2$ with $m=m_e=9.1 \times 10^{-31} $ kg and $m=m_p=1.67 \times 10^{-27}$ kg, respectively).

The energy accumulated in the Fermi spheres of electrons and protons can be released in the form of the electromagnetic radiation if the Pauli exclusion principle is locally waived  in close vicinity of the event horizon, due to local decay of quantum statics (as proved in Appendix C). Charged particles couple to an electromagnetic field and the collapse of their Fermi spheres is accompanied with the emission of photons in agreement with Fermi golden rule for quantum transitions between initial states of particles in the Fermi sphere and their ground state.

\subsection{Energy efficiency of the collapse of Fermi spheres of electrons and protons in the accretion disc near the  event horizon of a quasar}

For concreteness of the estimation let us assume that the central black hole in quasar consumes 5.6 $M_{\odot}$ per year, i.e., ca.  $0.06$ Earth mass per second. Let us assume the stable uniform in time process of matter accretion. The transport of matter across the disk is steady,  thus we can perform calculation e.g., per a single second. Using Eqs (\ref{fm}) and (\ref{en}) one can assess the energy stored in the Fermi spheres for electrons and protons, if all the electrons and protons from the gas mass equalled to $0.06 $ Earth mass, are compressed to the spatial volume $V$  per second (this volume depends on the distance from the event horizon). The local Fermi momentum
\begin{equation}
	\label{fmm}
	p_F(r)=\hbar(3\pi^2 \rho(r))^{1/3}=\hbar\left(3\pi^2 \frac{n}{V(r)}\right)^{1/3},
\end{equation}
where $r$ is the distance from the centre. $p_F(r)$  is constant in time and grows across the disk with increasing local concentration $\rho(r)=\frac{dn}{dV}=\frac{n}{V(r)}$, the same for electrons and protons. The latter equality holds for steady accretion and $n$ is the total number of electrons (or protons) per second, compressed in total to the volume $V(r)$ at the distance $r$ from the origin with central gravitational singularity. This means that  portions $dn$ of electrons and protons in infinitely small consecutive periods $dt$ incoming in radial direction towards the central singularity compressed to $dV(r)$ at radius $r$ add up in due of a single second time period to the total constant flow of mass (in the example, of $0.06 \times M_Z$ kg/s, the Earth mass $M_Z=5.97 \times 10^{24}$ kg) and as the whole are compressed locally at $r$ to $V(r)$. The locally accumulated energy in the Fermi spheres of electrons and protons  grows with lowering $r$ due to the increase of the compression caused by the gravitational field. This energy is proportional to $V(r)$ and, moreover, depends on $V(r)$ via the local  Fermi momentum  (\ref{fmm}) and in accordance  with  Eq. (\ref{en})) can be expressed as,
\begin{equation}
	\label{enn}
	\begin{array}{l}
		E(r)=E_e(r)+E_p(r),\\
		E_e(r)=\frac{V(r)}{2 \pi^2 \hbar^3}\int_o^{p_F(r)}dp p^2 \left( \sqrt{p^2 c^2 +m_e^2 c^4}-m_e c^2\right),\\
		E_p(r)=\frac{V(r)}{2 \pi^2 \hbar^3}\int_o^{p_F(r)}dp p^2 \left(\sqrt{p^2 c^2 +m_p^2 c^4}-m_p c^2\right),\\
	\end{array}
\end{equation}
where the energy $E_{e(p)}$ refers to electrons (protons).

At  the critical radius $r^*$ close to Schwarzschild zone (we argue that $r^*=1.5r_s$ as detailed in Appendix \ref{b})  the decay of  quantum statistics takes place due to  the topology reason (cf. Appendices \ref{a} and \ref{b}) and both Fermi spheres of electrons and protons collapse. The amount of energy given by Eq. (\ref{enn}) per one second can be thus released in the vicinity of the Schwarzschild horizon (at the rim of the photon sphere). Those energies continuously released add up per single second to ca. $10^{40}$ J and can contribute in large part to the observed luminosity of quasar (of order of just $10^{40}$ W). This process  undergoes by portions $dn$ of particle flow incoming to $r^*$ region in infinite small time periods $dt$, adding up   to  $0.06 \times  M_Z$ kg per second in total. The value  of the  released energy depends on local  Fermi momentum and  attains  $10^{40}$ J at sufficiently high level of compression, i.e., at sufficiently small $V(r^*)$ determined from the self-consistent system of Eqs (\ref{fmm}) and (\ref{enn}) if one assumes $E(r^*)=10^{40}$ J. Obtained in this way $V(r^*)$ allows for the assessment of local matter density -- it occurs of order of the uppermost limit for matter density as in neutron star at Tolman-Oppenheimer-Volkoff limit (i.e., of order of density of atomic nuclei), which evidences self-consistency of the model.

To the initial mass of a gas (assuming to be composed of hydrogen $H$)
contribute mostly protons (ca. 2000 times more massive than electrons), thus the total number of electrons, the same as the number of protons falling onto the considered black hole, equals to, $n\simeq 0.06 M_Z /m_p\simeq 2.14 \times 10^{50}$ per second. Simultaneously solving Eqs (\ref{fmm}) and (\ref{enn}), assuming $n=2.14 \times  10^{50}$  in volume $V(r^*)$ and released energy $E(r^*)=10^{40}$ J,  we find the volume of plasma compression $V(r^*)=0.5 \times 10^5$ m$^3$ and electron or proton Fermi sphere radius $p_F(r^*)=5.4 \times 10^{-19}$ kg m/s. Electrons and protons (their amount per second) are compressed to the same volume $V(r^*)$ (due to the neutrality of plasma), hence,  their  concentration at $r^*$, $\rho(r^*)=4.3 \times 10^{45}$ 1/m$^3$. The mass density at $r^*$ (including mass equivalent to the energy stored up in Fermi spheres of electrons and protons) is thus  $\xi(r^*)=\frac{0.06 M_Z}{V(r^*)}+\frac{E(r^*)}{c^2 V(r^*)}\simeq 9 \times 10^{18}$ kg/m$^3$ (similar to mass density in neutron stars). The  released energy of $E(r^*)=10^{40}$ J is equivalent to 30\% of the falling mass of $0.06$ Earth mass (per second). It means that the compressed plasma with degenerate Fermi liquid of electrons (and also of protons) is at $r=r^*$ by 30\% more massive than initial remote diluted gas. This mass increase is due to the gravitational field of the central black hole that compresses both fermion systems, and the accumulation of energy in the Fermi spheres of electrons and protons is at the expense of gravitational energy.

The energy of the gravitational field  accumulates itself in Fermi spheres of electrons and protons in a continuous way during matter compression and then is suddenly released at passing the rim of the photon sphere.  The ratio of total Fermi sphere energies of electrons and protons is $\frac{E_n(r^*)}{E_p(r^*)}\simeq 1.4$. The Fermi energy of electrons with Fermi momentum $p_F(r^*)=5.48 \times  10^{-19}$ kg m/s equals to $\varepsilon_F= 1 $ GeV (it is the upper possible energy of emitted photons), which in thermal scale (in units at $k_B=1$) is of order of $9 \times 10^{12}$ K -- this makes the electron liquid quantumly degenerated at lower temperatures (quasars are not source of thermal gamma radiation \cite{shapiro}, thus their actual temperatures are much lower). The Fermi energy of protons with Fermi momentum $p_F(r^*)=5.48 \times  10^{-19}$ kg m/s equals to $\varepsilon_F= 0.4 $ GeV (it is the upper possible energy of emitted photons by jumping of protons), which in thermal scale (in units at $k_B=1$) is of order of $4 \times 10^{12}$ K.

The release  of energy  due to the  collapse of the  Fermi sphere of charged particles
undergoes according to the Fermi golden rule scheme for quantum transitions \cite{landau1972}, when such transitions are admitted by the local revoking of Pauli exclusion principle (cf. Appendix \ref{b}).  Charged carriers (electrons and protons) couple to the electromagnetic field and the matrix element of this coupling between an individual particle state in the Fermi sphere and its ground state is the kernel of the Fermi golden rule formula for transition  probability per time unit for this particle.
This interaction depends also on electromagnetic field strength (it arises from the single-particle kinetic energy with momentum $\mathbf{p}$ substituted by the kinematic momentum $\mathbf{p}\pm e\mathbf{A}(\mathbf{r},t)$ and developed to linear term with respect to $\mathbf{A}$, which is the vector potential of the electromagnetic wave at gauge that $div\mathbf{A}=0$, $\pm$ corresponds to proton and electrons, respectively), thus the increasing number of excited photons strengthens the coupling (via the increase of  $\mathbf{A}$) in the similar manner as at stimulated emission (known from e.g.,  laser action) and accelerates quantum transition of the Fermi sphere collapse.

Note that the above energy estimation has been done in conventional rigid coordinates, time and spherical coordinates  $t,r,\theta,\phi$ of the remote observer. The Schwarzschild metric written in these coordinates has the form \cite{schwarzschild},
\begin{equation}
	\label{metryka1}
	-c^2d\tau^2=-\left(1-\frac{r_s}{r}\right)         	c^2dt^2+\left(1-\frac{r_s}{r}\right)^{-1}dr^2+r^2(d\theta^2+sin^2\theta d\phi^2),
\end{equation}
where $\tau$ is the proper time, $t$ is the time measured infinitely far of the massive body, $r_s=\frac{2GM}{c^2}$ is the Schwarzschild radius ($G$ is the gravitational constant, $c$ is the light velocity in the vacuum). The Schwarzschild metric has a singularity at $r=0$, which is an intrinsic curvature singularity. It also  has a singularity on the event horizon $r=r_{s}$ due to the second term in (\ref{metryka1}). The metric (\ref{metryka1}) is therefore defined  on the exterior region $r>r_{s}$ or on the interior region $r<r_{s}$.  However, the singularity on the event horizon disappears, as one sees in other  coordinates.  At passing the event horizon  time-type and space-type intervals mutually change their role, which is also an artifact related to the choice of ordinary remote observer coordinates in Scwarzschild metric. For $r\gg r_{s}$, the Schwarzschild metric is asymptotic to the standard Lorentz metric on Minkowski space. The Schwarzschild metric is a solution of Einstein field equations in empty space for a non-rotating and uncharged spherical body, meaning that it is valid only outside the gravitating body with the mass $M$. That is, for a spherical body of radius $R$ the solution is valid for $ r>R$. To describe the gravitational field both inside and outside the gravitating body the Schwarzschild solution must be matched with some suitable interior solution at $r=R$ such as the interior Schwarzschild metric \cite{frolov}. In the case of a classical concept of a black hole, $R=0$ and the above described problem disappears.

The singularity at $r = r_s$ divides the Schwarzschild coordinates in two disconnected patches. The exterior Schwarzschild solution with $r > r_s$ is the one that is related to the gravitational fields of stars and planets. The interior Schwarzschild solution with $0 \leq r < r_s$, which contains the singularity at r = 0, is completely separated from the outer patch by the singularity at $r = r_s$. The Schwarzschild coordinates therefore give no physical connection between the two patches, which may be viewed as separate solutions. The singularity at $r = r_s$ is an illusion however; it is an instance of what is called a coordinate singularity. As the name implies, the singularity arises from a choice of coordinates or coordinate conditions. When one changes  to a different coordinate system (for example Lemaitre, Eddington–Finkelstein, Kruskal–Szekeres, Novikov or Gullstrand–Painlevé coordinates \cite{novikov,kruskal,szekeres}) the metric becomes regular at $r = r_s$ and can extend the external patch to values of $r$ smaller than $r_s$. Using a different coordinate transformation one can then relate the extended external patch to the inner patch. The queer property of Schwarzschild metric is related with the fact that it is impossible to describe the outside and inside of a black hole's event horizon simultaneously in the same rigid and stationary (time independent) metric \cite{lanfield}. The Schwarzschild metric belongs to such a class and it describes the outside of the event horizon using an ordinary coordinate system of the remote observer. In this metric the inside of the horizon is inaccessible (its volume is zero in this metric) and each matter movement terminates for $t\rightarrow \infty$ at the event horizon. If to change, however, to other coordinates like Kruskal-Szekeres or Novikov and to exchange the time $t$ by the proper time $\tau$, then matter smoothly passes the event horizon within a finite proper time period and terminates any movement (also within a proper time period) in central singularity. Different properties of various metrics for the same folded spacetime illustrate different slicing of the same four-dimensional curvature into its space and time components possible to be done in various ways,  with emphasizing of Kruskal-Szekeres metric, being the maximally extended solution of the Einstein equations (analytic in the whole accessible domain) \cite{kruskal,szekeres}.

Nevertheless, for the purpose of the present paper we consider only the upper vicinity of the event horizon in which the Schwarzschild metric well describes geodesics in terms of conventional rigid and stationary  coordinates. They are certainly convenient at the critical innermost unstable circular radius $r=1.5 r_s$. Hence, even if in  Eq. (\ref{fmm}) one substitutes $dV(r)$ by the proper volume at the distance $r$ from the central singularity, which  according to the Schwarzschild metric (\ref{metryka1}) is
\begin{equation}
	\label{correction}
	d{\cal{V}}=\left(1-\frac{r_s}{r}\right)^{-1/2}drr^2 sin\theta d\theta d\phi,
\end{equation}
the change  is only by the factor $\left(1-\frac{r}{r_s}\right)^{-1/2}$. $d{\cal{V}}$ is by this factor  greater than $dV=drr^2sin\theta d\theta d\phi$ observed by the remote observer. At $r^*=1.5r_s$ this factor is ca. 1.7, which gives the reduction  of $p_F$ caused by the gravitational curvature by factor ca. $1.7^{-1/3}\simeq0.84$, which does not change orders in the  estimations presented in this paragraph. The change of $p_F$ by one order of the magnitude would need the closer approaching the Schwarzschild horizon, at $r\simeq 1.000001 r_s$, i.e., rather distant from the $r^*=1.5 r_s$. Hence, for the rough estimation of the effect of Fermi sphere collapse  the correction (\ref{correction})
is unimportant and can be included as the factor $0.84$ to the right-hand side of Eq. (\ref{fmm}), which does not change the orders in the energy estimation.

\label{efficiency}

\section{Supplement to the conventional model of accretion disc and comparison with observations}

The collapse of Fermi spheres in  degenerate Fermi systems  at passing  the innermost unstable circular orbit in Schwarzschild metric  (photon sphere rim of a nonrotating  black hole) does not conflict with the conventional models of the accretion disc of quasars \cite{sunaev,novikov}. The latters base on Shakura-Sunyaev classical hydrodynamic approach to plasma in accretion disc \cite{sunaev} where the inverse transfer of the orbital momentum is modelled by friction and turbulence factors allowing  to increase plasma internal energy on the cost of the black hole gravitational energy \cite{salpet,zeld,ksiazka}.  Then the hot plasma irradiates energy by the thermal radiation \cite{sunaev,merloni}. This gives, however, only soft photons, and to elucidate giant luminosity of quasars (or micro-quasars) in X-ray range, the mechanism of inverse Compton scattering of soft photons on hot electrons or ions is invoked \cite{shapiro}. The model of hot accretion disc, originally proposed for micro-quasar Cignus X-1 with 15 Sun mass black hole \cite{shapiro} assumes extremely high temperature of the inner part of the accretion disc ($10^9$ K for electrons and $10^{11}$ K for ions) to assure sufficient energy of charge carriers needed to Comptonization of soft photons (the identification of a sufficiently abundant soft photon source is not clear, however). Though the Comptomization mechanism in this hot accretion disc model is adjusted to X-ray radiation luminosity of Cignus X-1, the generalization of the model to extremely luminous giant quasars with supermassive black holes ($\sim 10^9$ Sun mass or larger) \cite{supereddington} is problematic, since the temperature $10^9 - 10^{11}$ K  of hot plasma in vast disc seems to be unrealistic. Nevertheless, some numerical simulations of developments of  Shakura-Sunyaev and Thorne-Novikov  model \cite{sunaev,novikov}  done recently by Fragile et al. \cite{6p} allow to match to observable luminosities of some not distant ($z<0.3$) active binary black hole objects \cite{farahzest}, but rather not to super-luminous remote quasars \cite{supereddington}. Higher radiation efficiency has been modelled within conventional classical magneto-hydrodynamic approach by inclusion of hypothetical giant magnetic component to accretion plasma in the case of a spinning black hole \cite{24p}, assuming, however, unrealistic extremal accretion mass rate to gain sufficiently large luminosity.

All hydrodynamic or magnetohydrodynamic models of matter accretion onto a black hole are applicable, however, only relatively far from the event horizon of the black hole. In all such models the inner edge of the accretion disc is assumed to be located well above the photon sphere of the black hole (the latter coincides with radius of the innermost unstable circular orbit in Schwawrzschild metric, $1.5 r_s$, $r_s= \frac{2GM}{c^2}$), i.e., even more distant than the innermost stable circular orbit with the radius $3r_s$ (conventionally assumed the inner edge  of the accretion disc is at $\sim 6 r_s$ \cite{shapiro}).

The collapse of the Fermi spheres described in the present paper takes place at the rim of the photon sphere at $r=1.5 r_s$, in the region completely neglected in conventional hydrodynamic models of matter accretion \cite{sunaev,novikov,6p,24p}, thus this mechanism does not interfere with thermal (including bremsstrahlung) and Comptonization mechanisms for radiation emission from more distant parts of the accretion disc (typically for $r>6 r_s$). Inclusion of the quantum effect of Fermi sphere collapse at the photon sphere rim ($r=1.5 r_s$) can, however, considerably  supplement developments of Shakura-Sunyaev-Thorne-Novikov  classical hydrodynamics approach \cite{sunaev,novikov} not applicable in close vicinity of the event horizon. The release of high energy photons (depending on the accretion mass rate and the black hole mass and governed by the Fermi momentum in compressed plasma) due to Fermi sphere collapse can add to the total luminosity of quasars and micro-quasars, allowing for the avoidance of some parameter-fitting  problems and shortages of conventional classical models \cite{6p,24p}.

The rejection of the quantum statistics in a system of indistinguishable identical particles at passing the rim of the photon sphere is a general property of any black hole, regardless of its mass.
The energy per unit volume stored  in the Fermi sphere of a degenerate Fermi system is a monotonic function of the density of matter. Its maximum is attained for the uppermost possible density of compressed electron-hadron plasma, as in extremely bright quasars or in the case of neutron star mergers exceeding the Tolman-Oppenheimer-Volkoff  stability limit (the uppermost density of compressed Fermi liquid is of order of the density of hadrons in atom nuclei).
In the case of quasars the electron and proton Fermi sphere collapse in the stream of ionized matter in accretion disc  gives the steady luminosity $\sim 10^{40}$ W for  $\sim 10^9$ Sun mass supermassive black holes consuming ca. 10 Sun mass per year ($0.1$ Earth mass per second) for a long time. In the  case of a neutron star merger which exceeds Tolman-Oppenheimer-Volkoff limit of ca. $2.3$ Sun mass compressed to the uppermost density of hadrons, the merger rapidly collapses  due to the relief of internal pressure caused by local recall of the Pauli exclusion principle. The released energy due to the collapse of neutron Fermi sphere in the neutron star merger reaches  $10^{47}$ J (this energy  partly escapes from the photon sphere of a rising black hole in the form of a short giant burst of gamma-rays). In both these extremal cases of the matter compression (in super-luminous quasars and at collapse of neutron star mergers)  the efficiency of mass to radiation conversion is ca. 30 \%, not achievable in any other physical mechanism except for matter-antimatter annihilation (the efficiency of nuclear fusion in stars is only of order $0.7$ \% for mass to energy conversion rate).

In the case of not extremal matter compression in the accretion disc of a black holes (as in many active galactic nuclei or in micro-quasars with lower rate of the matter influx) the radiation emitted due to Fermi sphere collapse is less intensive (the mass to energy conversion rate is lower than 30 \%) and softer,  but still contributes to the total luminosity and can help to explain radiation properties of observed binary black hole systems \cite{farahzest,supereddington}. In particular, the Fermi sphere collapse can help to elucidate the relatively short-lasting  brightening of active galactic nuclei, like the recently observed for AGN 1ES 1927+654 \cite{ga1flare}. The 100-fold increase of its luminosity within a few months period would be associated with accidental increase of the matter consumption rate during the corresponding time. If this accidental matter influx is not extremal, then photons emitted due to the Fermi sphere collapse  may not reach over-MeV energy (cf. Table \ref{tab-kw1}) and are not able to produce electron-positron pairs in the ergosphere of this spinning black hole. However, the massive isotropic flux of lower energy photons caused by the Fermi sphere collapse  of electrons and protons may push electron-positron pairs created in the ergosphere according to Blandford-Znajek electromagnetic mechanism \cite{znajek} towards the event horizon, lowering in this way their evaporation to jets across ergosphere nodes. The model by Blandford-Znajek \cite{znajek} gives the theory of jet formation for  spinning Kerr-like black holes where due to dragging  of reference frame in Kerr metric the magnetic field carried with the accretion matter  rotates. The rotation of the ergosphere causes the magnetosphere inside it to rotate, the outgoing flux of angular momentum results in extraction of energy from the black hole. The magnetic field beams in the form of jets and electrons and positrons diffuse across nodes of the ergosphere and next are  highly accelerated in a magnetic field beam in jets producing intensive X-ray radiation. The hypothetical source of electron-positron pairs is a strong electrical field created by the rotating magnetic field frozen in the ergosphere.  The sufficient intensity of an electric field to generate particle-antiparticle pairs in the ergosphere is, however, speculative. A collapse of Fermi sphere in plasma approaching the Kerr black hole would be helpful here, as supplying the abundance of over-MeV photons in the case of extreme matter influx, when the energy of these released photos can reach GeV level and can produce large number of electron-positron pairs, which are able to power up jets besides the Blandford-Znajek mechanism. Nevertheless, in the case when the released photons are sub-MeV at smaller rate of matter consumption by a black hole, then they cannot produce  additional electron-positron pairs inside the ergosphere but can push towards the event horizon those created according to Blandford-Znajek mechanism, reducing their supply to jets.    	This could explain   the temporal change in the radiation spectrum during brightening episode consisting  in optical 100 fold increase of the luminosity and simultaneous lowering of X-ray radiation  (the latter probably due to reduction of the amount of electrons and positrons in the jet, the source of X-ray radiation in jets), without need to speculate on remagnetisation  of the AGN and quenching of its jets by oppositely magnetized gas cloud  during this episode \cite{ga1flare}. When the Fermi energy in accreting plasma does not exceed MeV, then the collapse of electron and proton Fermi spheres can produce an increase of the luminosity with lower frequency (just as has been observed for AGN 1ES 1927+654) and, simultaneously, can temporarily reduce intensity of X-ray radiation form jets, also in agreement with observations.

\section{Conclusions}

We proposed a supplementary model for conversion of gravitational energy  close to a giant black hole of quasar into electromagnetic radiation with the efficiency reaching ca. 30\% of mass to energy conversion rate as observed in super-luminous quasars.  This so effective energy engine is quantum in the nature and is based on	the collapse of the Fermi spheres of electrons and protons in dense plasma flow approaching in a spiral way the Schwarzschild horizon near the photon sphere of a giant black hole of the quasar. The collapse of electron and proton Fermi spheres takes place due to the restriction imposed on particle classical trajectories close to the Schwarzschild zone, beneath  the innermost unstable circular orbit (the rim of the photon sphere of non-rotating and uncharged black hole). In this region the homotopy of trajectories qualitatively changes. Below the photon sphere rim (coinciding with sphere with radius of the innermost unstable circular orbit, $r=1.5r_s$ where $r_s$ is the Schwarzschild radius) only existing trajectories unavoidably one-way spiral onto Schwarzschild horizon for particles passing inward the photon sphere. This precludes particle exchanges in this region and thus locally revokes Pauli exclusion principle, which results in the collapse of Fermi spheres of electrons and protons.  The related release of the energy stored in Fermi spheres of  degenerate quantum liquids of electrons and protons has a continuous character provided a steady supply of galactic gas to the accretion disc, and can contribute substantially to the luminosity of quasars for a long term.

The described mechanism of Fermi sphere collapse below the innermost unstable circular orbit of a black hole seems to be universal, but the energy efficiency depends on  the supply of the matter to be consumed by the black hole and is limited by the uppermost density of matter compressed by the gravitational singularity at the rim of the photon sphere. The uppermost density agrees with the limiting density of fermions like in atom nuclei or in neutron stars achieving the Tolman-Oppenheimer-Volkoff stability limit. The collapse  of a neutron star merger, which has exceeded the limit of stability, may be also accompanied by giant emission of photons released during the rapid decay of the Fermi sphere of neutrons, when neutrons not compressed by quantum degeneracy pressure decay into protons and electrons interacting with the electromagnetic field. The energy stored in the neutron Fermi sphere of neutron star achieving Tolman-Oppenheimer-Volkoff limit is of order of $ 10^{47}$ J,  just  the same as the typical energy of frequently observed  short giant gamma-ray bursts, which would be sourced by neutron Fermi sphere decays. Another support for the presented model may be associated with the mechanism of creation of jets of rotating quasars. The model by Blandford and  Znajek of jet formation would be supplemented with new mechanism of the creation of electron-positron pairs in the ergosphere of a Kerr black hole by GeV$-$MeV photons released here in large amount in due of collapse of Fermi spheres of electrons and protons in analogy to the process described for nonrotating Schwarzschild black hole. In the case of smaller portion of mass consumption by a black hole, the decay of Fermi spheres in falling matter does not achieve the highest efficiency but can be still helpful in elucidation of temporal  changes in luminosity and spectral changes of less intensive radiation sources like recently observed incident in AGN 1ES 1927+654.

redpajama_set_name: RedPajamaArXiv


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Home / Mac / News
Google Analytics subscriptions suspended
By Juan Carlos Perez and IDG News Service
Macworld NOV 20, 2005 4:00 pm PST
Google Inc. has temporarily shut the door of its Google Analytics hosted service to new users because the company has been unable to keep up with demand after it began offering service free last week.
Visitors to the Google Analytics sign-up page are met with the following message: "Thank you for your interest in Google Analytics! Google Analytics has experienced extremely strong demand, and as a result, we have temporarily limited the number of new signups as we increase capacity." The company had previously charged a fee for the service.
Users are asked to submit their name and e-mail address so that the Mountain View, California, company can notify them once it begins accepting new accounts again.
Google didn't immediately reply to a call seeking comment.
This is the latest snafu to hit Google Analytics since last Monday, when Google began to offer it for free. That same day, Google Analytics, which used to cost US$199 per month, began to experience serious performance problems as new and existing users alike struggled to access the service.
The availability problems lasted through Tuesday, but things seemed to stabilize on Wednesday, according to several users contacted at the time by IDG News Service. "The demand for Google Analytics was much higher than we expected. The service is now completely restored and full service is available to everyone," a Google spokesman wrote via e-mail when asked for comment on Wednesday. But at some point between then and today, Google decided to stop accepting new users for the service.
Formerly known as Urchin on Demand, Google Analytics lets users monitor visits to their Web sites to track, for example, the effectiveness of online marketing campaigns and to determine how to modify Web pages to improve sales conversions.


Pictures have emerged of an overturned caravan on the M5 that caused traffic delays earlier today (July 12).
The caravan overturned on the M5 northbound between Junction 26 Wellington (A38) and Junction 25 Taunton (A358) at around 10.40am.
Delays of over an hour and a half were reported as two lanes were closed following the crash, with traffic backing up behind the blockage.
Police were called to the incident at 10.52am, and pictures show the caravan left on its side, straddling lanes two and three.
An Avon and Somerset Constabulary spokeswoman said: "We were called to an overturned caravan and car. Nobody was injured and Highways England are dealing with the incident."
The caravan overturned near the Taunton Deane services, and traffic southbound was also moving slowly due to drivers slowing down to look at the accident on the northbound carriageway.
By 1pm, recovery had taken place, and all lanes were re-opened.


A Mexican man accused of hurling rocks at cars and police officers was shot at a total of 17 times, Washington state police revealed Wednesday. He was hit by five or six of the shots during a controversial confrontation that spawned protests in the area.
According to Reuters, Kennewick Police Sergeant Ken Lattin confirmed that all three officers responding to the scene fired their weapons at 35-year-old Antonio Zambrano-Montes, an unemployed orchard worker and Mexican national who was reportedly throwing rocks at vehicles at a busy intersection when law enforcement arrived. He had spent the last 10 years in Pasco, Washington, and also had two daughters.
Lattin added that it's unclear whether Zambrano-Montes, whose record indicated drug use in the past, suffered from any physical or mental issues, though investigators are looking into the situation.
"Did he have some sort of injury? Did he have some mental health situations that he was dealing with in the days and hours (before the incident)? Or was he under the influence of drugs? We need to know," Lattin said.
Caught on tape via a smartphone recording, the shooting unfolded earlier this month after police made their way to an intersection in downtown Pasco, where reports were coming in about a man tossing rocks at passing vehicles. Zambrano-Montes then reportedly threw rocks at the three responding officers – two of whom were injured and treated at the scene – before he was chased across the street.
As he ran away from police, the video shows Zambrano-Montes with his hands up and in front of him. Police said he did not respond to their orders to surrender. At one point in the video, he turns around to face the officers, who then fatally shoot him on the sidewalk. Officials said stun guns were also used prior to the shooting, to no effect.
Zambrano-Montes turned out to be unarmed, and several witnesses said they believed police overreacted.
"I could not believe they were shooting guns. There were cars and people everywhere," Pasco resident Benjamin Patrick said at the time.
All three officers have been placed on administrative leave until officials conclude their investigation.
According to local KNDO, this wasn't the first run-in between police and Zambrano-Montes. In 2014, officers responded to complaints that he was striking cars with a broom. When police arrived to deal with the situation, Zambrano-Montes allegedly threw items at the officers, including a rocking chair, before attempting to take one of their guns. He was sent to a hospital and treated for methamphetamine use.
The fatal shooting sparked a wave of protests from residents in the area – composed primarily of Hispanics – who view the event as a case of police brutality. Some have even compared it to the controversial police shootings of Michael Brown in Ferguson, Missouri and Eric Garner in New York City.
It also drew criticism from the Mexican Foreign Affairs Ministry.


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When you are organizing on acquiring a lifestyle insurance policy, select an independent broker. Impartial brokers can generally offer you far more variety in conditions of policy and value than a broker who works completely for a certain insurance coverage organization. Firm brokers are minimal to the products their business sells, and might also be pushed by the business to suggest a particular product.
When picking lifestyle insurance coverage, be certain to have a great thought of how considerably income your cherished kinds will demand. This is critical due to the fact of the sum of financial debt that they may possibly be left with the load of paying out off right after you die. Think about vehicle payments, house loan, credit rating card debts, funeral and burial charges, and the price of elevating children.
An insurance prepare is not a buy-it and neglect-it purchase. You will want to revise your insurance coverage plan as things change in your lifestyle and you get more mature. Alterations in marital status, possessing kids, or reaching retirement age are all reasons to review your program and make adjustments.
When calculating the acceptable quantity of protection to buy with your lifestyle insurance plan, consider all the costs that will want to be coated. However, the proceeds from a life insurance policy payout are also utilized for paying that takes place only as soon as. For instance, it is typically utilised to pay out the fees of possessing a funeral, and getting an estate attorney. These costs can be very pricey.
Assess your group life insurance policy against other insurance policies. Group lifestyle policies, usually provided by your employer, might not often be the the very least costly selection. The costs are established based mostly on averages -- common age, average wellness and other crucial danger variables that might not apply to you. If you have exceptional overall health, store around to see if you can get much better prices in other places.
Don't acquire more life insurance coverage than you really need. You want to get just adequate for your family's demands. Acquiring too a lot existence insurance means that you are going to conclude up having to pay a good deal more in premiums. Be confident, even though, that you will not skimp on your insurance coverage and stop up with a coverage that will not likely supply for your family.
Attempt to acquire your existence insurance policies plan as soon as you want it. It is undoubtedly greatest to do this when you are younger and healthier because the policy's rates have a tendency to be considerably less expensive. If you wait around until later when you are older or when you are in inadequate health, the rates can be extremely pricey.
If you happen to be a smoker who was addicted and using tobacco cigarettes at the time of your life insurance plan, you can actually give up smoking cigarettes and save income. Most companies will permit you reapply for a plan as a non-smoker after a specified volume of time has passed. So there is some incentive in it if you can stop.
From knowing how to help save funds on lifestyle insurance policies to figuring out which variety of program you should go with, this write-up has described some of the finer factors of the insurance policies market, so that you can discover the ideal offer feasible on a high quality insurance policies prepare. Act prior to it's way too late. Select up that strategy.


Title: 100 Organic Skin Care recipesAuthor: Jessica RessSubtitle: Revitalize your natural glow with Facial masks, Body scrubs Home Spa treatments and morePages: 224Language: English..
Natural thickener used to stabilise emulsions (creams, lotions) & improve texture.Natural alternative to chemical binders.May also be used as a resin and carrier for essential oils.Appearance: Fin..
Easily absorbed nourishing carrier oil used for massage and skincare.Excellent moisturising oil when applied directly to chapped, rough or skin.Blend with essential oils to create your own moisturiser..
Versatile ingredient that can help to soothe, cool and protect the skin when incorporated into skin care formulations.Excellent for dry, damaged and ageing skin (usage level is between 0.1% - 10%)Appe..
15 mL Amber glass bottle. (Bottle Only) Approx. External Measurements = Diameter 28 mm / Height 44 mm (glass) 63 mm ..
Spring blooms of bursting Apple blossoms that are warm, yet light and delicate, soft , floral fragrance with a subtle touch of apple. Useful for sachets, powders, pomanders, draw liners, coat hangers..
Light textured, nourishing and easily absorbed carrier oil which contains Vitamin E. May provide soothing relief when applied directly to inflammed, sun damaged or chapped skin.Blend with essenti..
Title: Aromatherapy for the Healthy ChildAuthor: Valerie Ann WorwoodSubtitle: More than 300 natural, non-toxic and fragrant essential oil blendsPages: 320Language: English..
Title: Aromatherapy Pocket Reference BookAuthor: Rebekah HolmesSubtitle: A pocket sized reference to the healing science covering 67 essential oilsPages: 108Language: English..
Thick moisturising and soothing carrier oil when directly applied.May relieve dry, damaged or mild skin conditions. Also very beneficial when incorporated into skin and hair care formulations.Aur..
Thick moisturising and soothing carrier oil when directly applied.May relieve dry, damaged or mild skin conditions. Also very beneficial when incorporated into skin and hair care formulations.Org..


And I Quote offers elegant calligraphy options to complement your custom invitation order. We also offer custom printing services for place cards, table numbers, menus, programs, and anything else you may need printed to coordinate.
ONLY $1.00 per Guest Addressed Envelope, plus a $15 one time set-up fee. If you need additional envelopes printed after initial order, just send us the names/addresses and drop off the envelopes….we will get them printed ASAP for you and you will just pay the $1.00 each envelope. We try to make your event less stressful than it needs to be for our wonderful customers!
For the ultimate in elegance and tradition, Pen & Ink Hand Calligraphy is also available. Please call in advance to be sure our calligrapher is in the store to speak with you. A non-refundable deposit is required since this a hand-made item. Turn around time for Hand Calligraphy varies, depending on quantity and style.
ONLY $1.60-$2.00 per Pen & Ink Guest Addressed Envelope!


...the ideas in this book might give even the most ardent anti-dungeon hater a second thought about using this venerable institution in their D&D game.
D&D ENCOUNTERS SUPPORT: This product supports the D&D Encounters in-store play program by providing Dungeons & Dragons players with exciting new options for their D&D Encounters characters. The player material in this book is also suitable for home game use.
CHARACTER THEMES: This book presents several new character themes tied to dungeon exploration. Themes are popular with Dungeons & Dragons players, as they provide more depth to each character's back story and provide intriguing in-game benefits.
This book also includes advice for Dungeon Masters on how to integrate character themes into an ongoing campaign that features dungeons and other subterranean environments, as well as dungeon-building tips.
This product has tangential ties to the drow-themed marketing campaign.
...if you turn to supplements like this for more than the numbers – for things like quest ideas, character motivations and new settings – then Into the Unknown deserves a spot on your D&D shelf.
While I can't exactly call Into the Unknown: The Dungeon Survival Handbook a disappointment - because there is quite a bit of cool material here - I nevertheless came away from reading the book, well, disappointed as a Dungeon Master.
The supplement is a useful guide for players and Dungeon Masters alike as well as even being an interesting read for those of us not firmly familiar with the 4th Edition of the rules.
...if you or your group is new to D&D, Into the Unknown would be a great addition to your library. For everyone else, you're not throwing your money away but you'll get most of the benefit this book has to offer you in the character builder.
...if you're a 4e player, this book once again gives you plenty of new material that can be used in the current Encounters season, which is a great benefit... For DM's I'd put this book up there with Neverwinter as being an essential purchase for its advice on how to build better adventures.
While the player section is a blast, I think the DM's section is the best part of the book...There's a sense of completion here that I just don't think they're going to be able to beat.


Equal Time with Martha Burk originates weekly from KSFR public radio in Santa Fe. Burk's commentary explores many topics on past progress, future change, and what it means for women and men -- as citizens, as voters, and as inhabitants of the world we live in and help to shape. For those who are curious and who care about what's going on the the public debate (with no shouting) and issues/ what's at stake now and in the future. Each commentary is two and one half minutes (02:30) Host Martha Burk is a nationally recognized journalist and women's leader. She is Money Editor for Ms. Magazine, and her blog is frequently on the front page of The Huffington Post. Burk is also immediate past Chair of the National Council of Women's Organizations, the nation's oldest and largest coalition of women's groups.


Sydney D. Edwards enlisted in Civil War, Jan., 1862, in Co. I, 5th. Me. Regt. Inf.; was discharged; re-enlisted Nov., 1863 in Co. B. 30th. Me. Regt. Inf. being mustered into service Dec. 28; was transferred to Co. C, of same Regt. 1864; and was finally discharged Aug. 5, 1865. He participated in Red River Expedition.
He was for several years member of Board of Selectmen of Oxford, and also Supt. of Schools of the same town. He was Trial Justice for over fourteen years.
Member of AF&AM of Maine; and T. A. Roberts Post, GAR of Maine.
For more information about the ancestors of Annie (Joanna) Files Wardwell, see this section of Samuel Wardwell of Andover.


text:
The ambitious Julie does it her way
november 6, 2016 Geen reacties
The 25-year old Julie Pallesen is an ambitious model and blogger from Denmark. I met her at a shoot for Stradivarius in Barcelona. I noticed how open minded the easy talker was and got inspired by her personality.
We meet at a cute restaurant in Barcelona called Flax and Kale. 'I am already there', says her message. 'Me too, I am inside', I write back. 'I am outside, in front of the church.' As I walk towards the church I see a smiling girl on a terrace with her hand waving in the air.
Travel addiction
"I see so many things of the world. It inspires me. Travelling makes me strong", tells the 25-year old when I ask her what she likes about her life as a model. "I see a lot of different cultures and meet so many different people. It makes you open your eyes." She adds that she does not get this in her hometown Copenhagen. "Of course, my friends and family live there. But after I lived in Paris for one year, I got a little travel addicted."
Open mindedness
"It is good to see different places. The way we grow up is a cultural thing, which is normal, but I see a huge difference between people who travel a lot and people who just stay in their own comfort zone. People who stay in their own narrow lifes, do not accept different facts and values that fast."
With her blog This Way, Julie wants to inspire people by showing her readers all the places she visits and her way of living. She wants to show them her open mindedness. "I also want to expose people that if you work hard for things, you can reach things in life." Working hard and staying positive, that is her way of living.
For a long time, the travel addict has been thinking about starting a blog. "I was looking for the right purpose to start, I did not want to force it", she says after taking a sip from her macchiato with coconut milk. After working for a Danish magazine for six months, Julie suggested them to start a blog page on their website. "That was a good platform to start and after a year with them I went solo."
This Way's beauty products
Having a blog is a stimulating process for the Danish girl. "I am surprised how well it is going. I have readers from all over the world", she pauses for a moment and then continues talking. "I actually want to grow more and I want to make This Way becoming its own label, like a brand." Her ambitions become even more clear when she starts talking about her latest project.
The diligent model wants to sell her own beauty products on her blog. Julie wants to keep the products very natural, without using perfume. "I already started developing the products", she explains. "It is a lot of work, but I am ready for it", she says while eating her vegan burger. "I have finished university, I am ready for the next step."
barcelonabloginterviewjournalismstradivariustravelwork
Me for Vein Magazine
'Perfect beauty bores me'
Model talk – part two
− = twee
Staring at my display I'm waiting until the inspiration comes falling down from the sky. While studying journalism last year, one of my teachers told me that the key to writing is simply beginning - the rest will come by itself.
'Clothes are something you stand for'
Couture in water


text:
package signalmap

import (
	"fmt"
	"unsafe"

	"github.com/cilium/cilium/pkg/bpf"
)

var (
	MaxEntries int
)

const (
	// MapName is the BPF map name.
	MapName = "cilium_signals"
)

// Key is the index into the prog array map.
// +k8s:deepcopy-gen=true
// +k8s:deepcopy-gen:interfaces=github.com/cilium/cilium/pkg/bpf.MapKey
type Key struct {
	index uint32
}

// Value is the program ID in the prog array map.
// +k8s:deepcopy-gen=true
// +k8s:deepcopy-gen:interfaces=github.com/cilium/cilium/pkg/bpf.MapValue
type Value struct {
	progID uint32
}

// GetKeyPtr returns the unsafe pointer to the BPF key.
func (k *Key) GetKeyPtr() unsafe.Pointer { return unsafe.Pointer(k) }

// GetValuePtr returns the unsafe pointer to the BPF value.
func (v *Value) GetValuePtr() unsafe.Pointer { return unsafe.Pointer(v) }

// String converts the key into a human readable string format.
func (k *Key) String() string { return fmt.Sprintf("%d", k.index) }

// String converts the value into a human readable string format.
func (v *Value) String() string { return fmt.Sprintf("%d", v.progID) }

// NewValue returns a new empty instance of the structure representing the BPF
// map value.
func (k Key) NewValue() bpf.MapValue { return &Value{} }

// InitMap creates the signal map in the kernel.
func InitMap(maxEntries int) error {
	MaxEntries = maxEntries
	signalMap := bpf.NewMap(MapName,
		bpf.MapTypePerfEventArray,
		&Key{},
		int(unsafe.Sizeof(Key{})),
		&Value{},
		int(unsafe.Sizeof(Value{})),
		MaxEntries,
		0,
		0,
		bpf.ConvertKeyValue,
	)
	_, err := signalMap.Create()
	return err
}

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text:
MEN'S SOCCER: Bulldogs play to 1–1 tie against national powerhouse New Hampshire
In its penultimate non-conference matchup, the Yale men's soccer team played to a tie against No. 3 New Hampshire.
Drew Beckmen 12:35 am, Oct 15, 2021
muscosportsphotos.com
On Tuesday night, the Yale men's soccer team fought to a 1–1 draw against No. 4 New Hampshire, asserting itself as a team capable of contesting the nation's top programs.
Yale (4–3–3, 1–0–1 Ivy) traveled to Durham to face off against the undefeated Wildcats (12–0–1, 4–0–0 America East). New Hampshire — ranked fourth nationally at the time — entered the contest without a single loss or tie on the season. However, after 110 minutes of soccer, the Bulldogs tarnished the Wildcats' record by playing to a 1–1 tie. After a scoreless first half, the Wildcats struck first as Linus Fallberg scored on a rebound after a New Hampshire shot rattled off the crossbar. Yale equalized with less than four minutes remaining in the game as the team's leading scorer, Paolo Carroll '23, got on the end of a cross from midfielder Jules Oberg '23. The Elis created a number of potentially game-winning chances in overtime, but the contest ultimately ended in Yale's third tie of the season.
"We are really happy with the performance against a very good team on the road," head coach Kylie Stannard said in an email to the News. "The team has a lot of belief in each other no matter the result, but a performance like that can only help build positive momentum and a desire to want more."
According to Stannard, a number of starters missed the game due to "load management." As a result, Yale began the game with an extremely young roster, with just three upperclassmen among its 11 starters. Despite their youth, the Elis competed throughout the first half, holding their opponent scoreless through the first 45 minutes of play. The Wildcats scored 30 goals in their first 12 games alone.
15 minutes into the game, a New Hampshire defender lofted a long ball forward over the Elis' back line. Goalkeeper Chris Edwards '24 — who made his first start of the season — came off his line to punch the ball away before the Wildcat forward could make an attempt on goal.
The Wildcats finally broke through for the first tally of the game nine minutes into the second half. A pass from the right midfielder split Yale defenders Jake Schaffer '24 and Yasin Aly '25, and an eager Wildcat forward rocketed a left-footed shot off of the crossbar. The rebound fell to Linus Fallberg, who found the back of the net before the Yale keeper could get back in goal.
As the clock approached the 90 minute mark, the Bulldogs continued searching for an equalizer. An opportunity arose on a Yale free kick with just over four minutes remaining on the clock. Forward Eric Lagos '24 headed the ball on target, forcing a save from the New Hampshire keeper. The equalizer quickly followed when midfielder Thomas Toney '23 passed the ball back to Oberg on the right side of the pitch. Using his right foot, Oberg whipped an inswinging cross into the box, finding Carroll, who was unmarked on the back post. Carroll blasted a one time finish into the back of the net to record his seventh goal of the season and send the game into overtime.
During the two 10-minute periods, the Elis had the better chances. Outshooting New Hampshire 4–3, Yale had numerous opportunities to score the golden goal that would have secured the win. In the second overtime period, midfielder Quanah Brayboy '25 forced the Wildcat keeper to make a save from close range. Brayboy's shot appeared to have deflected off of the arm of a New Hampshire defender, and the Bulldogs appealed for a handball. The referee waved away the protests, however, and did not grant the penalty kick.
With three ties in its last four games, Yale now returns to Ivy play as it prepares to face off against No. 19 Cornell (8–1–1, 2–0–0 Ivy) on Saturday.
"The team has been learning and trying to get better each game," Carroll said. "The game against UNH showed we can compete with anyone right now. Saturday against Cornell is going to be a tough game, and we are excited to play."
The Big Red, who currently sits atop the Ivy League standings, will challenge the Blue and White with its offensive firepower. Cornell currently ranks second nationally in points per game and has scored 10 goals in its last three games alone. The Big Red also ranks second in the United Soccer Coaches Northeast Region, trailing the Bulldogs' most recent opponent New Hampshire.
"We are focusing on ourselves — being disciplined, detail-oriented, and stubborn," Schaffer said when asked about the team's preparation for the matchup against Cornell. "We have obviously scouted them, but we are more concerned with the things that we can control."
The Blue and White kick off against Cornell at 4 p.m on Saturday at Reese Stadium. The game will stream live on ESPN+.
DREW BECKMEN
Tweets by YDNSports


text:
<?php

/**
 * Navigation form base class.
 *
 * @method Navigation getObject() Returns the current form's model object
 *
 * @package    OpenPNE
 * @subpackage form
 * @author     Your name here
 * @version    SVN: $Id: sfDoctrineFormGeneratedTemplate.php 24171 2009-11-19 16:37:50Z Kris.Wallsmith $
 */
abstract class BaseNavigationForm extends BaseFormDoctrine
{
  public function setup()
  {
    $this->setWidgets(array(
      'id'         => new sfWidgetFormInputHidden(),
      'type'       => new sfWidgetFormInputText(),
      'uri'        => new sfWidgetFormTextarea(),
      'sort_order' => new sfWidgetFormInputText(),
      'created_at' => new sfWidgetFormDateTime(),
      'updated_at' => new sfWidgetFormDateTime(),
    ));

    $this->setValidators(array(
      'id'         => new sfValidatorDoctrineChoice(array('model' => $this->getModelName(), 'column' => 'id', 'required' => false)),
      'type'       => new sfValidatorString(array('max_length' => 64, 'required' => false)),
      'uri'        => new sfValidatorString(array('max_length' => 2147483647)),
      'sort_order' => new sfValidatorInteger(array('required' => false)),
      'created_at' => new sfValidatorDateTime(),
      'updated_at' => new sfValidatorDateTime(),
    ));

    $this->widgetSchema->setNameFormat('navigation[%s]');

    $this->errorSchema = new sfValidatorErrorSchema($this->validatorSchema);

    $this->setupInheritance();

    parent::setup();
  }

  public function getModelName()
  {
    return 'Navigation';
  }

}


text:
Q: How do you say (adjective) you are wearing cold clothes? In portuguese there is the word:

Agasalhado

which is used for people who are wearing clothes for cold seasons.
I would like to know if there is an equivalent word.
"Clothed" seemed to be right, although not sure about meaning it is wearing cold clothes or just opposite of naked.

A: There are many expressions for this. In the UK, we often use the phrase wrap (sb) up:

wrap (sb) up — phrasal verb with wrap

to dress in warm clothes, or to dress someone in warm clothes:

*

*Wrap up well - it's cold outside.

Cambridge Dictionary
You can also use:

*

*Make sure that you are wrapped up well.

*Make sure that you are well wrapped up.

*Stand still while I wrap you up properly.

Needless to say, it is children who are usually "wrapped up". Husbands come a poor second. Wives have to do it themselves.

redpajama_set_name: RedPajamaStackExchange