The current attempt is aimed to
outline the geometrical framework of a well known statistical problem,
concerning the explicit expression of the arithmetic mean standard deviation
distribution. To this respect, after a short exposition, three steps are
performed as 1)
formulation of the arithmetic mean standard deviation, , as a function of the errors, , which, by themselves, are
statistically independent; 2)
formulation of the arithmetic mean standard deviation distribution, , as a function of the errors, ; 3) formulation of the arithmetic mean
standard deviation distribution, , as a function of the
arithmetic mean standard deviation, , and the arithmetic mean rms
error, . The integration domain can
be expressed in canonical form after a change of reference frame in the n-space, which is recognized as an
infinitely thin n-cylindrical corona
where the symmetry axis coincides with a coordinate axis. Finally, the solution
is presented and a number of (well known) related parameters are inferred for
sake of completeness.

Abstract:
The current attempt is aimed to extend previous results, concerning the explicit expression of the arithmetic mean standard deviation distribution, to the general case of the weighted mean standard deviation distribution. To this respect, the integration domain is expressed in canonical form after a change of reference frame in the n-space, which is recognized as an infinitely thin n-cylindrical corona where the axis coincides with a coordinate axis and the orthogonal section is an infinitely thin, homotetic (n-1)-elliptical corona. The semiaxes are formulated in two different ways, namely in terms of (1) eigenvalues, via the eigenvalue equation, and (2) leading principal minors of the matrix of a quadratic form, via the Jacobi formulae. The distribution and related parameters have the same formal expression with respect to their counterparts in the special case where the weighted mean coincides with the arithmetic mean. The reduction of some results to ordinary geometry is also considered.

Abstract:
A theory of collisionless fluids is developed in a unified picture, where nonrotating $(widetilde{Omega_1}=widetilde{Omega_2}= widetilde{Omega_3}=0)$ figures with some given random velocity component distributions, and rotating $(widetilde{Omega_1} ewidetilde{Omega_2} e widetilde{Omega_3}) $ figures with a different random velocity component distributions, make adjoint configurations to the same system. R fluids are defined as ideal, self-gravitating fluids satisfying the virial theorem assumptions, in presence of systematic rotation around each of the principal axes of inertia. To this aim, mean and rms angular velocities and mean and rms tangential velocity components are expressed, by weighting on the moment of inertia and the mass, respectively. The figure rotation is defined as the mean angular velocity, weighted on the moment of inertia, with respectto a selected axis. The generalized tensor virial equations (Caimmi and Marmo 2005) are formulated for R fluidsand further attention is devoted to axisymmetric configurations where, for selected coordinateaxes, a variation in figure rotation has to be counterbalanced by a variation in anisotropy excess and viceversa. A microscopical analysis of systematic and random motions is performed under a fewgeneral hypotheses, by reversing the sign of tangential or axial velocity components of anassigned fraction of particles, leaving the distribution function and other parametersunchanged (Meza 2002). The application of the reversion process to tangential velocitycomponents is found to imply the conversion of random motion rotation kinetic energy intosystematic motion rotation kinetic energy. The application ofthe reversion process to axial velocity components is found to imply the conversionof random motion translation kinetic energy into systematic motion translation kinetic energy, and theloss related to a change of reference frame is expressed in terms of systematic motion (imaginary) rotation kinetic energy. A number of special situations are investigated in greater detail.It is found that an R fluid always admits an adjoint configuration where figure rotationoccurs around only one principal axis of inertia (R3 fluid),which implies that all the results related to R3 fluids (Caimmi 2007) may be extended to R fluids. Finally, a procedure is sketched for deriving the spin parameter distribution (including imaginary rotation) from a sample of observed or simulated large-scale collisionless fluids i.e. galaxies and galaxy clusters.

Abstract:
Simple closed-box (CB) models of chemical evolution are extended on two respects, namely (i) simple closed-(box+reservoir) (CBR) models allowing gas outflow from the box into the reservoir (Hartwick 1976) or gas inflow into the box from the reservoir (Caimmi 2007) with rate proportional to the star formation rate, and (ii) simple multistage closed-(box+reservoir) (MCBR) models allowing different stages of evolution characterized by different inflow or outflow rates. The theoretical differential oxygen abundance distribution (TDOD) predicted by the model maintains close to a continuous broken straight line. An application is made where a fictitious sample is built up from two distinct samples of halo stars and taken as representative of the inner Galactic halo. The related empirical differential oxygen abundance distribution (EDOD) is represented, to an acceptable extent, as a continuous broken line for two viable [O/H]-[Fe/H] empirical relations. The slopes and the intercepts of the regression lines are determined, and then used as input parameters to MCBR models. Within the errors (-+σ), regression line slopes correspond to a large inflow during the earlier stage of evolution and to low or moderate outflow during the subsequent stages. A possible inner halo - outer (metal-poor) bulge connection is also briefly discussed. Quantitative results cannot be considered for applications to the inner Galactic halo, unless selection effects and disk contamination are removed from halo samples, and discrepancies between different oxygen abundance determination methods are explained.

Abstract:
Macrogases are defined as two-component,large-scale celestial objects where the subsystems interact only via gravitation.The macrogas equation of state is formulated and compared to the van der Waals (VDW)equation of state for ordinary gases.By analogy, it is assumed that real macroisothermal curves in macrogases occur as real isothermal curves in ordinary gases, where a phase transition(vapour-liquid observed in ordinary gases and gas-stars assumed in macrogases) takesplace along a horizontal linein the macrovolume-macropressure{small $({sf O}sX_mathrm{V}sX_mathrm{p})$} plane.The intersections between real and theoretical(deduced from the equation of state) macro isothermalcurves, make two regions of equal surface as for ordinary gases obeying the VDW equation of state.A numerical algorithm is developed for determining the following points of a selected theoretical macroisothermal curve on the {small $({sf O}sX_mathrm{V}sX_mathrm{p})$} plane:the three intersections with the related real macroisothermal curve,and the two extremum points (one maximum and one minimum). Different kinds of macrogases are studied in detail: UU, where U density profiles are flat, to be conceived as a simple guidance case; HH, where H density profiles obey the Hernquist (1990) law, which satisfactorily fits the observed spheroidal components of galaxies; HN/NH, where N density profiles obey the Navarro-Frenk-White (1995,1996, 1997) law, which satisfactorily fits the simulated nonbaryonic dark matter haloes.A different trend is shown by theoretical macroisothermal curves on the{small $({sf O}sX_mathrm{V}sX_mathrm{p})$} plane,according to whether density profiles are sufficiently mild (UU) or sufficiently steep (HH, HN/NH).In the former alternative, no critical macroisothermal curve exists, below or above which the trend is monotonous. In the latter alternative, a critical macroisothermal curve exists, as shown by VDW gases, where the critical point may be defined as the horizontal inflexion point. In any case, by analogy with VDW gases, the first quadrant of the{small $({sf O}sX_mathrm{V}sX_mathrm{p})$} plane may be divided into three parts: (i) The G region, where only gas exists; (ii) The S region,where only stars exist; (iii) The GS region,where both gas and stars, exist. With regard to HH and HN/NH macrogases, an application is made to a subsample ({small $N=16$}) of elliptical galaxies extracted from larger samples {small $(N=25,~N=48)$}of early type galaxies investigated within the SAURON project (Cappellari et al. 2006, 2007).Under the simplifying assumption of universal

Abstract:
Simple multistage closed-(box+reservoir) (MCBR) models of chemical evolution, formulated in an earlier attempt, are extended to the limit of dominant gas inflow or outflow with respect to gas locked up into long-lived stars and remnants. For an assigned empirical differential oxygen abundance distribution (EDOD), which can be linearly fitted, a family of theoretical differential oxygen abundance distribution (TDOD) curves is built up with the following prescriptions: (i) the initial and the ending points of the linear fit are common to all curves; (ii) the flow parameter k ranges from an extremum point to ± ∞, where negative and positive k correspond to inflow and outflow, respectively; (iii) the cut parameter ζO ranges from an extremum point (which cannot be negative) to the limit (ζO) ∞ related to |k|→ + ∞. For curves with increasing ζO, the gas mass fraction locked up into long-lived stars and remnants is found to attain a maximum and then decrease towards zero as |k|→ + ∞ while the remaining parameters show a monotonic trend. The theoretical integral oxygen abundance distribution (TIOD) is also expressed. An application is made to the EDOD deduced from two different samples of disk stars, for both the thin and the thick disk. The constraints on formation and evolution are discussed in the light of the model. The evolution is tentatively subdivided into four stages, namely: assembling (A), formation (F), contraction (C), equilibrium (E). The EDOD related to any stage is fitted by all curves where 0 ≤ ζO ≤ (ζO) ∞ for inflowing gas and (ζO) ∞ ≤ ζO ≤ 1.2 for outflowing gas, with a single exception related to the thin disk (A stage), where the range of fitting curves is restricted to 0.35 ≤ ζO ≤ (ζO) ∞. The F stage may safely be described by a steady inflow regime (k= -1), implying a flat TDOD, in agreement with the results of hydrodynamical simulations. Finally, (1) the change of fractional mass due to the extension of the linear fit to the EDOD, towards both the (undetected) low-metallicity and high-metallicity tail, is evaluated and (2) the idea of a thick disk - thin disk collapse is discussed, in the light of the model.

Abstract:
On the basis of earlier investigations onhomeoidally striated Mac Laurin spheroids and Jacobi ellipsoids (Caimmi and Marmo2005, Caimmi 2006a, 2007), different sequences of configurations are defined and represented in the ellipticity-rotation plane, $({sf O}hat{e}chi_v^2)$. The rotation parameter, $chi_v^2$, is defined as the ratio, $E_mathrm{rot}/E_mathrm{res}$, of kinetic energy related to the mean tangential equatorial velocity component, $M(overline{v_phi})^2/2$, to kineticenergy related to tangential equatorial component velocity dispersion, $Msigma_{phiphi}^2/2$, andresidual motions, $M(sigma_{ww}^2+sigma_{33}^2)/2$.Without loss of generality (above a thresholdin ellipticity values), the analysis is restricted to systems with isotropic stress tensor, whichmay be considered as adjoint configurationsto any assigned homeoidally striated density profile with anisotropic stress tensor, different angular momentum, and equal remaining parameters.The description of configurations in the$({sf O}hat{e}chi_v^2)$ plane is extendedin two respects, namely (a) from equilibriumto nonequilibrium figures, where the virialequations hold with additional kinetic energy,and (b) from real to imaginary rotation, wherethe effect is elongating instead of flattening,with respect to the rotation axis.An application is made toa subsample $(N=16)$ of elliptical galaxies extracted from richer samples $(N=25,~N=48)$of early type galaxies investigated within theSAURON project (Cappellari et al. 2006, 2007).Sample objects are idealized as homeoidallystriated MacLaurinspheroids and Jacobi ellipsoids, and theirposition in the $({sf O}hat{e}chi_v^2)$plane is inferred from observations followinga procedure outlined in an earlier paper(Caimmi 2009b). The position of related adjoint configurations with isotropic stresstensor is also determined. With a singleexception (NGC 3379), slow rotators arecharacterized by low ellipticities $(0lehat{e}<0.2)$, low anisotropy parameters$(0ledelta<0.15)$, and low rotationparameters $(0lechi_v^2<0.15)$, while fastrotators show large ellipticities $(0.2lehat{e}<0.65)$, large anisotropy parameters$(0.15ledelta<0.35)$, and large rotationparameters $(0.15lechi_v^2<0.5)$. Analternative kinematic classification withrespect to earlier attempts (Emsellem etal. 2007) requires larger samples for providingadditional support to the above mentionedresults. A possible interpretation of slowrotators as nonrotating at all and elongated due to negative anisotropy parameters,instead of flattened due to positiveanisotropy parameters, is exploited.Finally, the elliptical

Abstract:
Macrogases are defined as two-component, large-scale celestial objects where the subsystems interact only via gravitation. The macrogas equation of state is formulated and compared to the van der Waals equation of state for ordinary gases. By analogy, it is assumed that real macroisothermal curves in macrogases occur as real isothermal curves in ordinary gases, where a phase transition takes place along a horisontal line in the macrovolume-macropressure (Mv-Mp) plane. A simple guidance case and two density profiles which satisfactorily fit to observations or simulations, are studied in detail. For sufficiently steep density profiles, a critical macroisothermal curve exists as shown by ordinary gases, where the critical point coincides with the horisontal inflexion point. By analogy with ordinary gases, the first quadrant of the (Mv-Mp) plane may be divided into three parts, namely (i) the G region, where only gas exists; (ii) the S region, where only stars exist; (iii) the GS region, where both gas and stars exist. An application is made to a subsample of elliptical galaxies investigated within the SAURON project. Different models characterized by equal subsystem mass ratio and different scaled truncation radii, are considered and the related position of sample objects on the (Mv-Mp) plane is determined. Tipically, fast rotators are found to lie within the S region, while slow rotators are close (from both sides) to the boundary between the S and the GS region. The net effect of the uncertainty affecting observed quantities, on the position of sample objects on the (Mv-Mp) plane, is also investigated. Finally, a principle of corresponding states is formulated for macrogases with assigned density profiles and scaled truncation radii.

Abstract:
Concerning bivariate least squares linear regression, the classical results obtained for extreme structural models in earlier attempts are reviewed using a new formalism in terms of deviation (matrix) traces which, for homoscedastic data, reduce to usual quantities leaving aside an unessential (but dimensional) multiplicative factor. Within the framework of classical error models, the dependent variable relates to the independent variable according to the usual additive model. The classes of linear models considered are regression lines in the limit of uncorrelated errors in X and in Y. For homoscedastic data, the results are taken from earlier attempts and rewritten using a more compact notation. For heteroscedastic data, the results are inferred from a procedure related to functional models. An example of astronomical application is considered, concerning the [O/H]-[Fe/H] empirical relations deduced from five samples related to different stars and/or different methods of oxygen abundance determination. For low-dispersion samples and assigned methods, different regression models yield results which are in agreement within the errors for both heteroscedastic and homoscedastic data, while the contrary holds for large-dispersion samples. In any case, samples related to different methods produce discrepant results, due to the presence of (still undetected) systematic errors, which implies no definitive statement can be made at present. Asymptotic expressions approximate regression line slope and intercept variance estimators, for normal residuals, to a better extent with respect to earlier attempts. Related fractional discrepancies are not exceeding a few percent for low-dispersion data, which grows up to about 10% for large-dispersion data. An extension of the formalism to generic structural models is left to a forthcoming paper.

Abstract:
Different sequences of ellipsoids are represented on the ellipticity-rotation plane. The rotation parameter is defined as the ratio of kinetic energy related to the mean tangential equatorial velocity component to kinetic energy related to tangential equatorial component velocity dispersion and residual motions. Systems with isotropic stress tensor are considered as adjoint configurations to their counterparts with anisotropic stress tensor, different angular momentum, and equal remaining parameters. Both nonequilibrium figures and figures elongated by imaginary rotation are represented on the ellipticity-rotation plane. An application is made to a reduced sample of elliptical galaxies. The position on the ellipticity-rotation plane of both sample objects and related adjoint configurations with isotropic stress tensor is inferred from existing observations within the SAURON project. With a single exception, slow rotators are characterized by low ellipticities, low anisotropy parameters, and low rotation parameters, while the contrary holds for fast rotators. A possible interpretation of slow rotators as nonrotating at all and elongated due to negative anisotropy parameters, is exploited. Finally, the elliptical side of the Hubble sequence is interpreted as a sequence of equilibrium (adjoint) configurations where the ellipticity is an increasing function of the rotation parameter, slow rotators correspond to early classes and fast rotators to late classes. In this view, boundaries are rotationally distorted regardless of angular momentum and stress tensor, where rotation has to be intended as due to additional kinetic energy of tangential equatorial velocity components, with respect to spherical configurations with isotropic stress tensor.