Against inertia

Revised version added 12 March 2012In this paper I challenge the Inertial Theory of language change put forward by Longobardi (2001), which claims that syntactic change does not arise unless caused and that any such change must originate as an ‘interface phenomenon’. It is shown that these two claims and the resulting contention that ‘syntax, by itself, is diachronically completely inert’ (Longobardi 2001: 278), if construed as a substantive, falsifiable theory of diachrony, make predictions that are too strong, and that they cannot be reduced (as seems desirable) to properties of language acquisition. I also express doubt as to the utility and necessity of a methodological/heuristic principle of Inertia, broadly following Lass’s (1980) view of causality.This work was supported by AHRC doctoral award AH/H026924/1

The outer regions of the giant Virgo galaxy M87. Kinematic separation of stellar halo and intracluster light

We present a spectroscopic study of 287 Planetary Nebulas (PNs) in a total area of ~0.4 deg^2 around the BCG M87 in Virgo A. With these data we can distinguish the stellar halo from the co-spatial intracluster light (ICL). PNs were identified from their narrow and symmetric redshifted lambda 5007\4959 Angstrom [OIII] emission lines, and the absence of significant continuum. We implement a robust technique to measure the halo velocity dispersion from the projected phase-space to identify PNs associated with the M87 halo and ICL. The velocity distribution of the spectroscopically confirmed PNs is bimodal, containing a narrow component centred on the systemic velocity of the BCG and an off-centred broader component, that we identify as halo and ICL, respectively. Halo and ICPN have different spatial distributions: the halo PNs follow the galaxy's light, whereas the ICPNs are characterised by a shallower power-law profile. The composite PN number density profile shows the superposition of different PN populations associated with the M87 halo and the ICL, characterised by different PN alpha-parameters, the ICL contributing ~3 times more PNs per unit light. Down to m_5007=28.8, the M87 halo PN luminosity function (PNLF) has a steeper slope towards faint magnitudes than the IC PNLF, and both are steeper than the standard PNLF for the M31 bulge. Moreover, the IC PNLF has a dip at ~1-1.5 mag fainter than the bright cutoff, reminiscent of the PNLFs of systems with extended star formation history. The M87 halo and the Virgo ICL are dynamically distinct components with different density profiles and velocity distribution. The different alpha values and PNLF shapes of the halo and ICL indicate distinct parent stellar populations, consistent with the existence of a gradient towards bluer colours at large radii. These results reflect the hierarchical build-up of the Virgo cluster.Comment: 16 pages, 13 figures, 3 tables, A&A, in pres

Planetary Nebulae and their parent stellar populations. Tracing the mass assembly of M87 and Intracluster light in the Virgo cluster core

The diffuse extended outer regions of galaxies are hard to study because they are faint, with typical surface brightness of 1% of the dark night sky. We can tackle this problem by using resolved star tracers which remain visible at large distances from the galaxy centres. This article describes the use of Planetary Nebulae as tracers and the calibration of their properties as indicators of the star formation history, mean age and metallicity of the parent stars in the Milky Way and Local Group galaxies . We then report on the results from a deep, extended, planetary nebulae survey in a 0.5 sqdeg region centred on the brightest cluster galaxy NGC 4486 (M87) in the Virgo cluster core, carried out with SuprimeCam@Subaru and FLAMES-GIRAFFE@VLT. Two PN populations are identified out to 150 kpc distance from the centre of M87. One population is associated with the M87 halo and the second one with the intracluster light in the Virgo cluster core. They have different line-of-sight velocity and spatial distributions, as well as different planetary nebulae specific frequencies and luminosity functions. The intracluster planetary nebulae in the surveyed region correspond to a luminosity of four times the luminosity of the Large Magellanic Cloud. The M87 halo planetary nebulae trace an older, more metal-rich, parent stellar population. A substructure detected in the projected phase-space of the line-of-sight velocity vs. major axis distance for the M87 halo planetary nebulae provides evidence for the recent accretion event of a satellite galaxy with luminosity twice that of M33. The satellite stars were tidally stripped about 1 Gyr ago, and reached apocenter at a major axis distance of 60-90 kpc from the centre of M87. The M87 halo is still growing significantly at the distances where the substructure is detected.Comment: 8 pages, 3 figures, To appear in the proceedings of the IAU Symposium 317 "The General Assembly of Galaxy Halos: Structure, Origin and Evolution'', A. Bragaglia, M. Arnaboldi, M. Rejkuba & D. Romano, ed

The Planetary Nebulae Populations in the Local Group

Planetary nebulae have been used as tracers of light and kinematics for the stellar populations in early-type galaxies since more than twenty years. Several empirical properties have surfaced: for example the invariant bright cut-off of the planetary nebulae luminosity function and correlations of the luminosity specific PN number with the integrated properties of the parent stellar populations. These observed properties are poorly understood in terms of a simple model of a ionized nebula expanding around a non-evolving central star. In order to make further steps, we need to study self-contained systems at know distances whose PN populations are sufficiently nearby to permit investigation into their physical properties. The galaxies in the Local Group represent a valid proxies to study these late phases of evolved stellar populations with a spread of metallicities, $\alpha$-element enhancements, and star forming histories.Comment: 10 pages, 5 figures. To appear in "Lessons from the Local Group - A Conference in Honour of David Block and Bruce Elmegreen" (eds. Freeman, K.C., Elmegreen, B.G., Block, D.L. \& Woolway, M., SPRINGER: NEW YORK

Imbedding estimates and elliptic equations with discontinuous coefficients in unbounded domains

In this paper we deal with the multiplication operator u ∈ W^{k,p} (Ω) → gu ∈ L^q (Ω), with g belonging to a space of Morrey type. We apply our results in order to establish an a-priori bound for the solutions of the Dirichlet problem concerning elliptic equations with discontinuous coefficients

A Dual Measure of Uncertainty: The Deng Extropy

The extropy has recently been introduced as the dual concept of entropy. Moreover, in the context of the Dempster–Shafer evidence theory, Deng studied a new measure of discrimination, named the Deng entropy. In this paper, we define the Deng extropy and study its relation with Deng entropy, and examples are proposed in order to compare them. The behaviour of Deng extropy is studied under changes of focal elements. A characterization result is given for the maximum Deng extropy and, finally, a numerical example in pattern recognition is discussed in order to highlight the relevance of the new measure