Nilpotent normal form for divergence-free vector fields and volume-preserving maps

We study the normal forms for incompressible flows and maps in the neighborhood of an equilibrium or fixed point with a triple eigenvalue. We prove that when a divergence free vector field in $\mathbb{R}^3$ has nilpotent linearization with maximal Jordan block then, to arbitrary degree, coordinates can be chosen so that the nonlinear terms occur as a single function of two variables in the third component. The analogue for volume-preserving diffeomorphisms gives an optimal normal form in which the truncation of the normal form at any degree gives an exactly volume-preserving map whose inverse is also polynomial inverse with the same degree.Comment: laTeX, 20 pages, 1 figur

The stellar and sub-stellar IMF of simple and composite populations

The current knowledge on the stellar IMF is documented. It appears to become top-heavy when the star-formation rate density surpasses about 0.1Msun/(yr pc^3) on a pc scale and it may become increasingly bottom-heavy with increasing metallicity and in increasingly massive early-type galaxies. It declines quite steeply below about 0.07Msun with brown dwarfs (BDs) and very low mass stars having their own IMF. The most massive star of mass mmax formed in an embedded cluster with stellar mass Mecl correlates strongly with Mecl being a result of gravitation-driven but resource-limited growth and fragmentation induced starvation. There is no convincing evidence whatsoever that massive stars do form in isolation. Various methods of discretising a stellar population are introduced: optimal sampling leads to a mass distribution that perfectly represents the exact form of the desired IMF and the mmax-to-Mecl relation, while random sampling results in statistical variations of the shape of the IMF. The observed mmax-to-Mecl correlation and the small spread of IMF power-law indices together suggest that optimally sampling the IMF may be the more realistic description of star formation than random sampling from a universal IMF with a constant upper mass limit. Composite populations on galaxy scales, which are formed from many pc scale star formation events, need to be described by the integrated galactic IMF. This IGIMF varies systematically from top-light to top-heavy in dependence of galaxy type and star formation rate, with dramatic implications for theories of galaxy formation and evolution.Comment: 167 pages, 37 figures, 3 tables, published in Stellar Systems and Galactic Structure, Vol.5, Springer. This revised version is consistent with the published version and includes additional references and minor additions to the text as well as a recomputed Table 1. ISBN 978-90-481-8817-

Opaque spiral galaxies

The photographic images of a carefully selected sample of about 16,000 galaxies have been digitized, and an extensive set of photometric parameters has been determined. These data allow a reanalysis of the effective transparency using the measured surface brightness profiles for much larger, more strictly defined samples. Several tests provide evidence that the major parts of many spiral disks are opaque and that in many cases perhaps only the outer layer of stars is observable. This invalidates most determinations of mass-to-light ratios. Various photometric properties point to an obscuring component with a larger exponential scale length than obscuring component with a larger exponential scale length than that of the stars, possibly composed of cool compact opaque clouds. This result diminishes the evidence for weak haloes around spirals, as inferred from rotation curves

Opaque spiral disks - Some empirical facts and consequences

Results for the Sb and Sc galaxies, as obtained from the analysis of the optical ESO-LV data, are reviewed, and the implied constraints for the properties of the absorbing components in spiral disks are discussed. An alternative interpretation of flat rotation curves and a revised extinction model of the Galaxy is presented. It is argued that the presently available data are best understood when in addition to a cirruslike dust component that causes extinction (i.e., absorption plus scattering) a second component is causing complete obscuration (occultation). This second component could be identified with compact opaque clouds that have a temperature well below the typical IRAS temperature of about 60 K and a spatial distribution described by an exponential with a scale length larger than that of the stars

Modulation-Specific and Laminar-Dependent Effects of Acetylcholine on Visual Responses in the Rat Primary Visual Cortex

Acetylcholine (ACh) is secreted from cholinergic neurons in the basal forebrain to regions throughout the cerebral cortex, including the primary visual cortex (V1), and influences neuronal activities across all six layers via a form of diffuse extrasynaptic modulation termed volume transmission. To understand this effect in V1, we performed extracellular multi-point recordings of neuronal responses to drifting sinusoidal grating stimuli from the cortical layers of V1 in anesthetized rats and examined the modulatory effects of topically administered ACh. ACh facilitated or suppressed the visual responses of individual cells with a laminar bias: response suppression prevailed in layers 2/3, whereas response facilitation prevailed in layer 5. ACh effects on the stimulus contrast-response function showed that ACh changes the response gain upward or downward in facilitated or suppressed cells, respectively. Next, ACh effects on the signal-to-noise (S/N) ratio and the grating-phase information were tested. The grating-phase information was calculated as the F1/F0 ratio, which represents the amount of temporal response modulation at the fundamental frequency (F1) of a drifting grating relative to the mean evoked response (F0). In facilitated cells, ACh improved the S/N ratio, while in suppressed cells it enhanced the F1/F0 ratio without any concurrent reduction in the S/N ratio. These effects were predominantly observed in regular-spiking cells, but not in fast-spiking cells. Electrophysiological and histological findings suggest that ACh promotes the signaling of grating-phase information to higher-order areas by a suppressive effect on supragranular layers and enhances feedback signals with a high S/N ratio to subcortical areas by a facilitatory effect on infragranular layers. Thus, ACh distinctly and finely controls visual information processing in a manner that is specific for the modulation and cell type and is also laminar dependent