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Lt is seen that the esu(8=2.37) −⋅∕∕⋅from the Euclidean⊲⋠ counts is. a .factor of 2 hisher than from the shallow counts. in direct. contrast with the expectation from source extraction. simulation and surface density analysis above. | It is seen that the $\sigma_{\rm conf} (\theta = 2.3 \arcsec)$ from the Euclidean counts is a factor of 2 higher than from the shallow counts, in direct contrast with the expectation from source extraction simulation and surface density analysis above. |
For the steeper counts one can assign 5 pty as the confusion limit. which would. be equivalent to 4.5o,,,; or 32 beams/source. | For the steeper counts one can assign 5 $\umu$ Jy as the confusion limit, which would be equivalent to $4.5\sigma_{\rm conf}$ or 32 beams/source. |
The corresponding numbers at TO per cent completeness (10μ.ν) for shallow curve are 190,4! and 17 beams/source. | The corresponding numbers at 70 per cent completeness $10\umu$ Jy) for shallow curve are $19\sigma_{\rm conf}$ and 17 beams/source. |
Lt thus turns out that the slope of the counts determines which sources dominate the confusion. | It thus turns out that the slope of the counts determines which sources dominate the confusion. |
If the source count slope is shallow. it is the bright. sources which dominate. reducing the completeness of detection of an object at a eiven lux level. | If the source count slope is shallow, it is the bright sources which dominate, reducing the completeness of detection of an object at a given flux level. |
In this case the rule-of-thumb taken at κ beams/source (lower for shallow counts and higher [or steeper ones) gives a good. idea of the practical confusion imit. | In this case the rule-of-thumb taken at $\ga 20$ beams/source (lower for shallow counts and higher for steeper ones) gives a good idea of the practical confusion limit. |
If the slope is steep (5> 2) the confusion noise due to inler sources starts reducing the completeness faster than he effect of bright neighbouring sources (see also discussion in Helou Beichman 1990). | If the slope is steep $\gamma > 2$ ) the confusion noise due to fainter sources starts reducing the completeness faster than the effect of bright neighbouring sources (see also discussion in Helou Beichman 1990). |
In this case the traditional way of caleulating 4 from Equation 5. ancl using &I0 or the confusion limit is more appropriate. | In this case the traditional way of calculating $\sigma_{\rm conf}$ from Equation 5, and using $\approx 10\sigma_{\rm conf}$ for the confusion limit is more appropriate. |
Figures 11.. 12.. and 13. show the expected. completeness levels using different integration times for observations with IRAC’s 3.6 and S qun filters with the realistic PSE's. | Figures \ref{completeness3d}, \ref{completeness8d}, and \ref{completeness8i} show the expected completeness levels using different integration times for observations with IRAC's 3.6 and 8 $\umu$ m filters with the realistic PSF's. |
Lere we used a comprehensive Monte Carlo simulation: rather than extracting the source count once from an simulated image (as was done for Figs. | Here we used a comprehensive Monte Carlo simulation: rather than extracting the source count once from an simulated image (as was done for Figs. |
S and 9)). test sources of a given Hux were placed one-bv-one on the simulated. [ramoes and then extracted. | \ref{cumcounts3} and \ref{cumcounts8}) ), test sources of a given flux were placed one-by-one on the simulated frames and then extracted. |
In cach flux bin for every simulation with clifferent noise Door. the test was performed 2000 times. | In each flux bin for every simulation with different noise floor, the test was performed 2000 times. |
‘The integration times corresponding to the different La noise Iloors are printed in&secs. | The integration times corresponding to the different $1\sigma$ noise floors are printed in. |
We show the baseline moclels ane also the ISO-fitted model for the δ qun band. | We show the baseline models and also the ISO-fitted model for the 8 $\umu$ m band. |
A cdetection® can be defined in various wavs. | A `detection' can be defined in various ways. |
Here i was defined as finding a source (the closest one) within a ΕΛΛΝΤΕ radius ancl within a 4e flux interval determine from the observed spread of Huxes at the given [ux density. | Here it was defined as finding a source (the closest one) within a FWHM radius and within a $4\sigma$ flux interval determined from the observed spread of fluxes at the given flux density. |
Alaking the selection in this way we. first. of all. avoid a serious knowledge bias which would be introduce if the exact same input source were searched for from he simulated. image. | Making the selection in this way we, first of all, avoid a serious knowledge bias which would be introduced if the exact same input source were searched for from the simulated image. |
Secondly. we chose the mentione ας cut to be as little as possible dependent. on the usec photometric technique. | Secondly, we chose the mentioned flux cut to be as little as possible dependent on the used photometric technique. |
LE for instance. we hack required a etection to be within X30 per cent of the input [lux (this is used in Lloge 2000) vere would have been more non-detections due to the typical brightening of sources when approaching the confusion limit. | If, for instance, we had required a detection to be within $\pm 30$ per cent of the input flux (this is used in Hogg 2000) there would have been more non-detections due to the typical brightening of sources when approaching the confusion limit. |
In this case the resulting no-noise completeness curves in Figs. 11-13. | In this case the resulting no-noise completeness curves in Figs. \ref{completeness3d}- – |
would be lower by à factor of ~1.5 at a few ply level. | \ref{completeness8i}
would be lower by a factor of $\sim 1.5$ at a few $\umu$ Jy level. |
Figures 11. to 13. can be usedto estimate the longest useful integration times for the URAC instrument in deep | Figures \ref{completeness3d} to \ref{completeness8i} can be usedto estimate the longest useful integration times for the IRAC instrument in deep |
significant mass mixing throught the gas disce. then (he gas and cust will share à common scale height 77;=I£,. | significant mass mixing throught the gas disc, then the gas and dust will share a common scale height $H_d=H_g$. |
In addition. very small grains will have the same bulk velocity as any turbulent eddy in which they reside. and will interact with collisional velocities corresponding to the dust thermal speed. i.e. ος=cj. | In addition, very small grains will have the same bulk velocity as any turbulent eddy in which they reside, and will interact with collisional velocities corresponding to the dust thermal speed, i.e. $v_c=v_{th}$. |
The gas sound speed ος is much greater than the thermal speed of even small dust grains ey, so the dust growth rate would be prohibatively low if such conditions held for large ranges of ry. | The gas sound speed $c_s$ is much greater than the thermal speed of even small dust grains $v_{th}$ so the dust growth rate would be prohibatively low if such conditions held for large ranges of $r_d$. |
We can. however. imagine (hat as the cust grains grow ancl couple less stronely with the gas. the (turbulence can collide dust grains together with collisional speeds Όση! that are much greater than ey,. | We can, however, imagine that as the dust grains grow and couple less strongly with the gas, the turbulence can collide dust grains together with collisional speeds $v_c \sim v_{turbulent}$ that are much greater than $v_{th}$. |
Eventually the growing dust. grains. will settle out of the eas disc. so that fy will decrease. | Eventually the growing dust grains will settle out of the gas disc, so that $H_d$ will decrease. |
For very large dust exains. wilh mean tree paths (with respect (o each other) longer than the largest scale edcdies. the effects of the edcdies will be uncorrelated. causing vc. to equal Off. drastically increasing the erowth rate (Eq. 2)) | For very large dust grains, with mean free paths (with respect to each other) longer than the largest scale eddies, the effects of the eddies will be uncorrelated, causing $v_c$ to equal $\Omega H_d$, drastically increasing the growth rate (Eq. \ref{2}) ) |
above the value for very small grains ancl speeding the onset of gravitational instability. | above the value for very small grains and speeding the onset of gravitational instability. |
To facilitate deriving the Formulae for v. and £7; in different regimes. we define ó=prj. which is proportional to a single dust grains surface density. | To facilitate deriving the formulae for $v_c$ and $H_d$ in different regimes, we define $\phi \equiv \rho_d r_d$, which is proportional to a single dust grain's surface density. |
Then. presuming a constanl pa we can combine (1)) and (2)) to obtain: the form of the growth equation we will use for the rest of the paper. | Then, presuming a constant $\rho_d$, we can combine \ref{1}) ) and \ref{2}) ) to obtain: the form of the growth equation we will use for the rest of the paper. |
The variable o will later help clarify the effects of different dust grain densities on (ime scales and critical grain sizes. | The variable $\phi$ will later help clarify the effects of different dust grain densities on time scales and critical grain sizes. |
. As m is. (he dust. density. in. the disc.. MayBdl is the surface density of the dust sheet that a cust erain travels through in time d/. | As $\frac {\Sigma_d }{H_d}$ is the dust density in the disc, $\Sigma_d \frac {v_c}{H_d} dt$ is the surface density of the dust sheet that a dust grain travels through in time $dt$. |
Hence J) (3)) states that the stickine-parameter mocdified surface density. of that sheet can be added to the penetrating dust grains surface density. much like a ball moving through cling wrap. | Hence \ref{main}) ) states that the sticking-parameter modified surface density of that sheet can be added to the penetrating dust grain's surface density, much like a ball moving through cling wrap. |
The basic question we seek (o answer is whether the grain growth is sullicientlv rapid that the eravitationallv unstable stage of planet lormation can be reached long belore the total tme scale available for planet formation (~109 vr. D'Alessioetal. (2005))). | The basic question we seek to answer is whether the grain growth is sufficiently rapid that the gravitationally unstable stage of planet formation can be reached long before the total time scale available for planet formation $\sim 10^6$ yr, \citet{d'Alessio05}) ). |
In our studs we need to consider (hat very rapid collisional velocities could be too high for sticking to occur and instead result in the destruction of the grains. | In our study, we need to consider that very rapid collisional velocities could be too high for sticking to occur and instead result in the destruction of the grains. |
We also need to determine how large the grains must become for them to settle out. ancl allow gravitational instabilities to occur. | We also need to determine how large the grains must become for them to settle out and allow gravitational instabilities to occur. |
While there exists a dust scale above which the effect of turbulence on the cust velocity dispersion becomes sulliciently weak Lor the dust cise to be gravitationally unstable. that size scale will be far greater (han the initial size of the dust. as well as that of the eritical size in (he absence of turbulence. | While there exists a dust scale above which the effect of turbulence on the dust velocity dispersion becomes sufficiently weak for the dust disc to be gravitationally unstable, that size scale will be far greater than the initial size of the dust, as well as that of the critical size in the absence of turbulence. |
A last consideration is our requirement that the growth occurs belore significant orbital migration. | A last consideration is our requirement that the growth occurs before significant orbital migration. |
since we expect that the condition E«5£F,, will be broken at some point. | since we expect that the condition $E<5 E_{\mathrm{roll}}$ will be broken at some point. |
Clearly, a more realistic collision model that include compaction and fragmentation of dust aggregates, is then needed. | Clearly, a more realistic collision model that include compaction and fragmentation of dust aggregates, is then needed. |
Next, we will investigate the consequences of including these physical regimes, first by using a simplified prescription. | Next, we will investigate the consequences of including these physical regimes, first by using a simplified prescription. |
maser. | maser. |
The reason for the apparent inner edge of the maser disc is not vet. known. | The reason for the apparent inner edge of the maser disc is not yet known. |
The high-velocity masers and are not colinear with the svstemic masers and have negligible accelerations (Cireenhilletal. 1995). | The high-velocity masers and are not colinear with the systemic masers and have negligible accelerations \citep{G95}. |
. The radial dependence of ceclination with respect to the svstemic velocity. of NGC 4258 is. symmetric in the the red ancl blue-shifted masers (Mivoshietal. 1995). | The radial dependence of declination with respect to the systemic velocity of NGC 4258 is anti-symmetric in the the red and blue-shifted masers \citep{M95}. |
.. Lt is suggested that the rotation axis of the disc varies with radius by an angle of up to 0.2 radians (Llerrnstein.Greenhill&Moran1996). ancl so the clise is warped. | It is suggested that the rotation axis of the disc varies with radius by an angle of up to $0.2$ radians \citep{HGM96} and so the disc is warped. |
Herrnsteinetal.(2005) found that the maser spots show a deviation from Weplerian rotation of about Okms | \cite{H05} found that the maser spots show a deviation from Keplerian rotation of about $9\,\rm km\,s^{-1}$. |
They modelled this with a warped Keplerian accretion disc with a racial gracient in its inclination of 0.034mas | They modelled this with a warped Keplerian accretion disc with a radial gradient in its inclination of $0.034\,\rm mas^{-1}$. |
There are several suggested explanations for the origin of the warp in the disc. | There are several suggested explanations for the origin of the warp in the disc. |
Capronictal.(2006). considere some mechanisms for the warping and precession of galactic accretion cdises. | \cite{C06} considered some mechanisms for the warping and precession of galactic accretion discs. |
Papaloizou.Terquem&Lin(1998). showec that it could be produced by a binary companion orbiting outside the maser disc. | \cite{PTL98} showed that it could be produced by a binary companion orbiting outside the maser disc. |
Such a companion would nee a mass comparable to that of the cise but there is no observational evidence for it. | Such a companion would need a mass comparable to that of the disc but there is no observational evidence for it. |
A second. suggestion is tha radiation pressure from the central black hole. produces torques on a slightly warped disc and the warp grows (Pringle 1996).. | A second suggestion is that radiation pressure from the central black hole produces torques on a slightly warped disc and the warp grows \citep{P96}. . |
However the masing dise is stable agains this racliation instability if ay:0.2 (Capronictal.2007).. where à, is the viscosity parameter (Shakura& 973). | However the masing disc is stable against this radiation instability if $\alpha_1 \le 0.2$ \citep{C07}, where $\alpha_1$ is the viscosity parameter \citep{SS73}. |
.. Alternatively. in the absence of other torques. Capronietal.(2007). concluded: that the warping in the disc of GC 4258 is due to the Bardeen-Petterson οσοι. | Alternatively, in the absence of other torques, \cite{C07} concluded that the warping in the disc of NGC 4258 is due to the Bardeen-Petterson effect. |
I£ we lave an accretion disc around a misaligned. spinning black ole. Lense-Phirring precession drives a warp in the disc. | If we have an accretion disc around a misaligned spinning black hole, Lense-Thirring precession drives a warp in the disc. |
The inner parts of the disc are aligned with the black hole (Dardeen&Petterson1975). | The inner parts of the disc are aligned with the black hole \citep{BP}. |
. Capronietal.(2007). found that the warp radius in he disc is comparable to or smaller than the radius of the inner masers. | \cite{C07} found that the warp radius in the disc is comparable to or smaller than the radius of the inner masers. |
They find the timescale of alignment of the system to be a few billion vears. | They find the timescale of alignment of the system to be a few billion years. |
They. based their work on he results of Scheuer&Feiler(1996) who assumed that the surface density is constant. | They based their work on the results of \cite{SF} who assumed that the surface density is constant. |
We use the more realistic power aw surface density and viscosities described by Martin.-rinele&Tout (2007). | We use the more realistic power law surface density and viscosities described by \cite{MPT07}. |
. We fit these analytical disc moclels. warped by the Lense-Phirring cllect to the shape of the observed maser distribution. | We fit these analytical disc models, warped by the Lense-Thirring effect to the shape of the observed maser distribution. |
In this Section we consider some properties of accretion discs. | In this Section we consider some properties of accretion discs. |
We assume that we have a steady state disc where m=const and that the surface density is a power law (Shakura&SunvaevLOT3). | We assume that we have a steady state disc where $\nu_1 \Sigma=\,\rm
const$ and that the surface density is a power law \citep{SS73}. |
. To be in steady state. the viscosity must obev where 3. Ny and vp) are constants and fy is some fixed raclius. | To be in steady state, the viscosity must obey where $\beta$, $\Sigma_0$ and $\nu_{10}$ are constants and $R_0$ is some fixed radius. |
According to Shakura&Sunvaev(1973). the thickness of the disc in the gas pressure dominatedregion has where r—A/BAR. and Ro=206Albuοἳ. | According to \cite{SS73} the thickness of the disc in the gas pressure dominatedregion has where $r=R/3R_{\rm g}$ and $R_{\rm g}=2GM_{\rm BH}/c^2$. |
This is well approximated by {1Η=const lor ARa. | This is well approximated by $H/R=\,$ const for $R\gg R_{\rm g}$. |
For NGC 425s Ro=1.0510mas which is much smaller that the inner οσο of the- disc at 2.8mas and so we assume that in our work. | For NGC 4258 $R_{\rm g}=1.05\times 10^{-3}\,\rm mas$ which is much smaller that the inner edge of the disc at $2.8\,\rm mas$ and so we assume that $H/R=\,\rm const$ in our work. |
There are two viscosities: £j corresponds to the aziniuthal shear (the viscosity normally associated with accretion discs) ancl νο corresponds to the vertical shear in the disc which smoothes out the twist. | There are two viscosities: $\nu_1$ corresponds to the azimuthal shear (the viscosity normally associated with accretion discs) and $\nu_2$ corresponds to the vertical shear in the disc which smoothes out the twist. |
The second. viscosity acts when the disc is non-planar. | The second viscosity acts when the disc is non-planar. |
We assume that the second Viscosity obeys the same power law as £j so and voy ds a constant. | We assume that the second viscosity obeys the same power law as $\nu_1$ so and $\nu_{20}$ is a constant. |
We use the a-prescription to describe the viscosities in the disc so that where /4 is the scale height in the disc and © is the angular velocity and similarly where with ayy. Gey and ο Constant (see below) and. 2) is some fixed radius which we define in Section 3.1.. | We use the $\alpha$ -prescription to describe the viscosities in the disc so that where $H$ is the scale height in the disc and $\Omega$ is the angular velocity and similarly where with $\alpha_{10}$, $\alpha_{20}$ and $x$ constant (see below) and $R_0$ is some fixed radius which we define in Section \ref{sec:rwarp}. |
We generally take ayy=0.2 and oso=2 (Lodato&Pringle2007). | We generally take $\alpha_{10}=0.2$ and $\alpha_{20}=2$ \citep{LP07}. |
. Phe angular velocity in thedisc is Ixeplerian so that This can be written in the form of equation (2)) with eedi which is a constant and so We can find a similar equation for the vertical shear viscosity. f». | The angular velocity in thedisc is Keplerian so that Because $c_{\rm s}=H\Omega$ the azimuthal shear viscosity is This can be written in the form of equation \ref{viscs}) ) with $x=\beta-\frac{1}{2}$ which is a constant and so We can find a similar equation for the vertical shear viscosity, $\nu_2$ . |
We let. the surface. density. be the power law given. by | We let the surface density be the power law given by |
the rans. | the r.m.s. |
turbulent velocity to the thermal velocity σον. the ratio of the cloud thickness to the correlation length L/7. as well as one realization of the velocity field. clistribution along the line of sight. | turbulent velocity to the thermal velocity $\sigma_t/v_{th}$, the ratio of the cloud thickness to the correlation length $L/l$, as well as one realization of the velocity field distribution along the line of sight. |
Because of the very. large. time scale for changes in the hvdrodynamic flows the randonr structure of the velocity Ποια along a given line of sight has to be considered as "frozen" over the exposure time. | Because of the very large time scale for changes in the hydrodynamic flows the random structure of the velocity field along a given line of sight has to be considered as `frozen' over the exposure time. |
It follows that the observed. absorption spectrum in the light of a point-like source (star. QSO) corresponds to only one realization of the velocity field. | It follows that the observed absorption spectrum in the light of a point-like source (star, QSO) corresponds to only one realization of the velocity field. |
Therefore. the intervening absorption spectrum cannot be considered. as à time or space average in the statistical sense. | Therefore, the intervening absorption spectrum cannot be considered as a time or space average in the statistical sense. |
Ελπίς means that the actual clistribution of the hvdrodynamic velocities at a given instant of time corresponds to an sample and. ius. may deviate from the average distribution function usstuned to be Gaussian. | This means that the actual distribution of the hydrodynamic velocities at a given instant of time corresponds to an sample and, thus, may deviate from the average distribution function assumed to be Gaussian. |
Accordingly. significant deviations from the expected. average intensity (Z4? can oceur. | Accordingly, significant deviations from the expected average intensity $\langle I_\lambda \rangle$ can occur. – |
For us reason we used in Paper HL. à. Monte Carlo technique to calculate spectra corresponding to the absorption along individual lines of sight. | For this reason we used in Paper II a Monte Carlo technique to calculate spectra corresponding to the absorption along individual lines of sight. |
In Paper LE we considered the direct. problem. ic. we specified the physical parameters and. generated. individual random realizations of the velocity distribution more exactly the distribution of the velocity component parallel to the line of sight e(3)] with which we then caleulated individual spectra. | In Paper II we considered the direct problem, i.e. we specified the physical parameters and generated individual random realizations of the velocity distribution [more exactly – the distribution of the velocity component parallel to the line of sight $v(s)$ ] with which we then calculated individual spectra. |
The aim of the present paper is the inverse. problem. ie. the problem. to. deduce. physical parameters from an observed. spectrum. | The aim of the present paper is the inverse problem, i.e. the problem to deduce physical parameters from an observed spectrum. |
“Lo reproduce a given (observed) spectrum. within the framework of our model. one has to find the proper physical parametersamd an appropriate realization of the velocity field. | To reproduce a given (observed) spectrum within the framework of our model, one has to find the proper physical parameters an appropriate realization of the velocity field. |
In. general. o(s) isa continuous random function of the coordinate s. bu in the numerical procedure it is sampled at evenly. spacec intervals As. | In general, $v(s)$ is a continuous random function of the coordinate $s$, but in the numerical procedure it is sampled at evenly spaced intervals $\Delta s$. |
The necessary number of intervals depends on the values of a,fen, and Lf. being typically ~LOO or hwdrogen absorption lines. | The necessary number of intervals depends on the values of $\sigma_t/v_{th}$ and $L/l$, being typically $\sim 100$ for hydrogen absorption lines. |
Thus. to estimate niocle xwameters [rom the observed spectrum one has to solve an optimization problem in a parameter space of very large aic variable (depending on afry. £/f) dimension. | Thus, to estimate model parameters from the observed spectrum one has to solve an optimization problem in a parameter space of very large and variable (depending on $\sigma_t/v_{th}$, $L/l$ ) dimension. |
It is known hat such kind of problems may be solved. using stochastic optimization methods. | It is known that such kind of problems may be solved using stochastic optimization methods. |
The Reverse Monte. Carlo MC echnique based on the computational scheme invented. by Metropolis (1953) proved to be adequate in our case. | The Reverse Monte Carlo [RMC] technique based on the computational scheme invented by Metropolis (1953) proved to be adequate in our case. |
The method is successfully used in many applications where he state space of a physical system is huge (see e.g. Press 1992. or Lolfmann 1995). | The method is successfully used in many applications where the state space of a physical system is huge (see e.g. Press 1992, or Hoffmann 1995). |
We apply the RAIC method to the problem. of determining the primordial deuterium abundance at. high redshift. | We apply the RMC method to the problem of determining the primordial deuterium abundance at high redshift. |
In. particular. the LE}D Lye profile observed. by 3urles "Tytler (1996. BATT hereinafter) in the spectrum of the quasar Q 1009|2956 is considered. | In particular, the H+D $\alpha$ profile observed by Burles Tytler (1996, T hereinafter) in the spectrum of the quasar Q 1009+2956 is considered. |
TFhis spectrum was selected since (a) ib shows a pronounced DI absorptions at the redshift z,=2.504. (6) it. was obtained with hieh signal-to-noise ratio and spectral resolution. and (ο) the total hydrogen. column density. estimated. by D&TT from the intensity level bevond the Lyman limit in this svstem is consistent with the normalized intensities in the LIL Lya wings. | This spectrum was selected since $a$ ) it shows a pronounced DI absorptions at the redshift $z_a = 2.504$, $b$ ) it was obtained with high signal-to-noise ratio and spectral resolution, and $c$ ) the total hydrogen column density estimated by T from the intensity level beyond the Lyman limit in this system is consistent with the normalized intensities in the HI $\alpha$ wings. |
In the present paper we extend our study of the accuracy of the D/LIE ratio determinations from our Galaxy (Paper HE) to the very distant Lya-svstems (putative intervening galaxies). | In the present paper we extend our study of the accuracy of the D/H ratio determinations from our Galaxy (Paper II) to the very distant $\alpha$ -systems (putative intervening galaxies). |
The background: source is supposed o be point-like anc even in the case of distant. QSOs we consider absorption along one line of sight only. | The background source is supposed to be point-like and even in the case of distant QSOs we consider absorption along one line of sight only. |
This means hat any gravitational focusing which may. in principle. lead o an additional spatial averaging over dilferent lines of sight (an example may be found in Frye 1997) is not taken into account. ( | This means that any gravitational focusing which may, in principle, lead to an additional spatial averaging over different lines of sight (an example may be found in Frye 1997) is not taken into account. ( |
Lhe average mesoturbulent HE]D Lye spectra ave been considered by Levshakov Lakahara 1996a.) | The average mesoturbulent H+D $\alpha$ spectra have been considered by Levshakov Takahara 1996a.) |
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