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Correcting thefie sonsitivitv for different N-rax spectra. we found the contribution of the absorbed ACNs is almost comparable o that of less-absorhed ACNs iu the 210 keV source counts at a flux limit of we«10D7.	Correcting the sensitivity for different X-ray spectra, we found the contribution of the absorbed AGNs is almost comparable to that of less-absorbed AGNs in the 2–10 keV source counts at a flux limit of $2\times10^{-13}$.
Figure 10b) shows the correlation between the redshift aud i6 210 keV luninosity ofthe identified AGNs.	Figure 1(b) shows the correlation between the redshift and the 2–10 keV luminosity of the identified AGNs.
The redshitt distribution of the 5 absorbed ACINS is concentrated at 2<0.5. which contrasts to the presence of 15 ess-absorbed ACNs at z>0.5.	The redshift distribution of the 5 absorbed AGNs is concentrated at $z<0.5$, which contrasts to the presence of 15 less-absorbed AGNs at $z>0.5$.
This suegeests a deficiency of AGNs with column ⋅⋅ DalMEME ≺∐∖∐↴∖↴↕⊓↸∖↴∖↴∪↕⊀∖∐∶↓∩−−−⊽⋟⋜↧−∙∶∩∙⋅↱⊐−≻∙∪↥⋅∐↕↑↕∐∖⊸∖≓↥⋅⋜↧⋅↖↽↕∏∐∐∐∪↴∖↴↕↑⋅↖⇁↥⋅⋜⋯∶↴∙⊾↸∖↕⋜∐⋅∶↴∙⊾↸∖↥⋅ . . - ⋅⋅ ∐⋜⋯↕∩⊔↸∖↥⋅∶↴↜∷∖↴⊥∙∪↥⋅↴⋝∪↑∐∙⋀∖⊽∪↑↸∖↑∐⋜↧↑↕↕≯↑↕∐∖⊔⋝	This suggests a deficiency of AGNs with column densities of $N_{\rm H} = 10^{22-23}$ at $z$ = 0.5–2, or in the X-ray luminosity range larger than $10^{44}$, or both.
↥⋅∪⋜⊓↧≓∐∐↸∖⊀≚≼∶⋀∖⊽↴∖↴↖↖↽↕↑∐∐⋜∐⋅≼↧↴∖↴↻↸∖↸⊳⊓⋅⋜↧ ↕⋜↧↖↽↸∖↕∐∏⋅↕∐↴∖↴↕≼⊳⋜∏⋝↴∖↴∪∏≻↑↕∪∐↕∐↴∖↴↑↸∖⋜⊔↧∪↕≯∪↑↕∐∖↥⋅∐⋜∐⋅≼∐∖∐↕∐∶↴⋁∐∐∖↸⊳∐⋜⋯↕↴∖↴⋯↴∖↴⋯⊳∐⋜↧↴∖↴≼⊲∪∐∏≻↑∪∐ reflection. it could complement this deficiency.	Note that if the 4 broad-line AGNs with hard spectra have intrinsic absorption instead of other hardening mechanism such as Compton reflection, it could complement this deficiency.
Deep survevs were performed with oover several fields (Ogasaka 11998). although optical identification is more difficult than the LSS because of faint flux levels aud source confusion problem.	Deep surveys were performed with over several fields (Ogasaka 1998), although optical identification is more difficult than the LSS because of faint flux levels and source confusion problem.
To overcome this difficulty. we have becu conducting a deep survey of the Lockman Tole. where the	To overcome this difficulty, we have been conducting a deep survey of the Lockman Hole, where the
lt can be shown with stability arguments (?) that the effective masses are bounded from below m/m2m/m. where nycny\| or equivalently Since in neutron star core the elfective masses are typically than the bare nucleon mass (therefore AIc0). this inequalityi provides an upper bound for the largestὃν possible strengtho of entreünment cllects between the two fluids.	It can be shown with stability arguments \citep{chamelhaensel-06} that the effective masses are bounded from below $m_\star^q/m>n_q/n_{\rm b}$ , where $n_{\rm b}=n_n+n_p$ or equivalently Since in neutron star core the effective masses are typically than the bare nucleon mass (therefore ${\cal K}^{np}>0$ ), this inequality provides an upper bound for the largest possible strength of entrainment effects between the two fluids.
The physical meaning of the dynamical cllective masses defined. by Eq (56)). becomes clear when writing the expressions of the nucleon B-momentum covectors (26)) where ονοSpeyer’.	The physical meaning of the dynamical effective masses defined by Eq \ref{eq:effmass}) ), becomes clear when writing the expressions of the nucleon 3-momentum covectors \ref{eq.3pi}) ) where $v_{_{\rm X}\, \nu}\equiv\eta_{\nu\mu}v_{_{\rm X}}^{\, \mu}$.
This shows that the 3-momoentum and the 3-velocity of a given nucleon species are not. aligned whenever the dynamical cllective masses dilfer from the bare nucleon mass. or equivalenth whenever the non-diagonal cocllicients of the mobility matrix do not vanish.	This shows that the 3-momentum and the 3-velocity of a given nucleon species are not aligned whenever the dynamical effective masses differ from the bare nucleon mass, or equivalently whenever the non-diagonal coefficients of the mobility matrix do not vanish.
By analogy with the definition (56)). let us introduce relativistic nucleon dynamical elective masses by where A7" is the relativistic generalisation of the non-relativistic mobility matrix. AA .	By analogy with the definition \ref{eq:effmass}) ), let us introduce relativistic nucleon dynamical effective masses by where $\widetilde{\cal K}^{q q^\prime}$ is the relativistic generalisation of the non-relativistic mobility matrix ${\cal K}^{qq^\prime}$ .
Using Eq. (46))	Using Eq. \ref{eq.kappa.rel}) )
together with (56)). we find with ;j5,/ the chemical potential defined by Eq.i (23)).	together with \ref{eq:effmass}) ), we find with $\mu_q$ the chemical potential defined by Eq. \ref{eq.muX}) ).
Hc is easily checked that nosm7 in the Newtonian limit.	It is easily checked that $\widetilde{m}_\star^q \rightarrow m_\star^q$ in the Newtonian limit.
With these definitions. the neutron and proton 4-momenta can be explicitly written as Note that the entrainment contributions involve the ellective masses (56)).	With these definitions, the neutron and proton 4-momenta can be explicitly written as Note that the entrainment contributions involve the effective masses \ref{eq:effmass}) ).
Eq. (62))	Eq. \ref{eq:releffmass2}) )
is the generalization to interacting multi-Duid svstems of the elective mass introduced by 2? in the perfect Quid case.	is the generalization to interacting multi-fluid systems of the effective mass introduced by \citet{carter-89} in the perfect fluid case.
Indeed in the absence of entrainment. K=0 so that i7—m while the relativistic ellective masses are given by This equation is identical to Eq.(1.66) in the lectures notes of 2..	Indeed in the absence of entrainment, ${\cal K}^{np}=0$ so that $m_\star^q=m$ while the relativistic effective masses are given by This equation is identical to Eq.(1.66) in the lectures notes of \citet{carter-89}.
Phe physical origin of the dillerence between m7 and m7 is that in rclativity forms of energy contribute to the mass.	The physical origin of the difference between $\widetilde{m}_\star^q$ and $m_\star^q$ is that in relativity forms of energy contribute to the mass.
A remarkable consequence is that even massless particles can have a non-vanishing relativistic effective mass.	A remarkable consequence is that even massless particles can have a non-vanishing relativistic effective mass.
This is for instance the case of leptons for which we have assumed m!=0.	This is for instance the case of leptons for which we have assumed $m^\ell=0$.
Llowever Eq. (47))	However Eq. \ref{eq.pil.rel}) )
show that the dynamical effective lepton mass is not zero but is given by Note that even if leptons were not interacting (which we will actually suppose in Section SN. in order to evaluate the master function A). they would still have an non-zero cllective mass due to the Pauli exclusion principle which prevents all the particles from occupying the lowest energy state with zero momentum (the chemical potential sy is then given by the Fermi energy of the lepton species f£).	show that the dynamical effective lepton mass is not zero but is given by Note that even if leptons were not interacting (which we will actually suppose in Section \ref{sect.micro} in order to evaluate the master function $\Lambda$ ), they would still have an non-zero effective mass due to the Pauli exclusion principle which prevents all the particles from occupying the lowest energy state with zero momentum (the chemical potential $\mu_\ell$ is then given by the Fermi energy of the lepton species $\ell$ ).
For unbound nuclear systems like the liquid core of neutron stars. we have jr,2» mmc.	For unbound nuclear systems like the liquid core of neutron stars, we have $\mu_q> mc^2$ .
Besicles if we assume that the strengthof entrainment ellects decreases withincreasing density. Le. (this is actually the case for the models considered in this work. see Eq. (109))).	Besides if we assume that the strengthof entrainment effects decreases withincreasing density, i.e. (this is actually the case for the models considered in this work, see Eq. \ref{eq.kappanp}) )),
and usingthe Cauchy-Schwartz inequality	and usingthe Cauchy-Schwartz inequality
We thank Alessandra Buonanno. Melvyn Davies. ancl especially the referee. Fred. Rasio [or helpful comments.	We thank Alessandra Buonanno, Melvyn Davies, and especially the referee Fred Rasio for helpful comments.
We also thank the Theoretical Institute for Advanced Research in Astropliysies (Hsinchu. Taiwan) for hospitality during part of this work.	We also thank the Theoretical Institute for Advanced Research in Astrophysics (Hsinchu, Taiwan) for hospitality during part of this work.
This paper was supported in part by NASA erant NAC 5-132290.	This paper was supported in part by NASA grant NAG 5-13229.
for not unreasonable (see discussion in Sect.	for not unreasonable (see discussion in Sect.
7) values of V. the method does the cosmological parameters.	7) values of $N$, the method does the cosmological parameters.
If additional a priori conditions are included (e.g.. if the universe is taken to have O4= 0). parameters can be constrained rather efficiently.	If additional a priori conditions are included (e.g., if the universe is taken to have $\Omega_\Lambda = 0$ ), parameters can be constrained rather efficiently.
In order to check the distribution of (». we have repeated à large number of times the entire process. using each time a different set of source galaxies.	In order to check the distribution of $\ell_2$, we have repeated a large number of times the entire process, using each time a different set of source galaxies.
The measured. probability distribution of ο for different values of Nis shown in Figs.	The measured probability distribution of $\ell_2$ for different values of $N$ is shown in Figs.
7 and 8.	7 and 8.
In these figures. the histograms show the	In these figures, the histograms show the
Soft gamma-ray repeaters (SGRs: seven contirmed members) and anomalous X-ray pulsars (AXPs: twelve confirmed are two classes of X-ray pulsating sources with no evidence for companion stars which share a number of properties.	Soft gamma-ray repeaters (SGRs; seven confirmed members) and anomalous X-ray pulsars (AXPs; twelve confirmed are two classes of X-ray pulsating sources with no evidence for companion stars which share a number of properties.
These include rotation periods of several seconds (P.~2-12 s). rapid. spin down (P~10P ss +), large and variable X-ray luminosities (exceeding the rate of rotational energy loss). and the emission of flares and short bursts (see Woods&Thompson2006:Mereghetti2008 for reviews).	These include rotation periods of several seconds $P\sim2$ $12$ s), rapid spin down $\dot{P}\sim10^{-11}$ s $^{-1}$ ), large and variable X-ray luminosities (exceeding the rate of rotational energy loss), and the emission of flares and short bursts (see \citealt{woods06,mereghetti08} for reviews).
SGRs and AXPs (note that the distinction is becoming increasingly blurred) are currently interpreted as observational manifestations of magnetars. namely neutron stars powered by their huge magnetic field (e.g. Paczynski1992:Duncan&Thompson1992:Duncan1995. 1996:: Thompson.Lyutikov&Kulkarni 20025).	SGRs and AXPs (note that the distinction is becoming increasingly blurred) are currently interpreted as observational manifestations of magnetars, namely neutron stars powered by their huge magnetic field (e.g. \citealt{paczynski92,duncan92,thompson95,thompson96}; \citealt*{tlk02}) ).
This picture is supported by the fact that the dipole magnetic fields inferred for SGRs and AXPs from their period and period are above. or at the high end of. those of the radio pulsars.	This picture is supported by the fact that the dipole magnetic fields inferred for SGRs and AXPs from their period and period are above, or at the high end of, those of the radio pulsars.