Euler Obstruction and Defects of Functions on Singular Varieties

Several authors have proved Lefschetz type formulae for the local Euler obstruction. In particular, a result of this type is proved in [BLS].The formula proved in that paper turns out to be equivalent to saying that the local Euler obstruction, as a constructible function, satisfies the local Euler condition (in bivariant theory) with respect to general linear forms. The purpose of this work is to understand what prevents the local Euler obstruction of satisfying the local Euler condition with respect to functions which are singular at the considered point. This is measured by an invariant (or ``defect'') of such functions that we define below. We give an interpretation of this defect in terms of vanishing cycles, which allows us to calculate it algebraically. When the function has an isolated singularity, our invariant can be defined geometrically, via obstruction theory. We notice that this invariant unifies the usual concepts of {\it the Milnor number} of a function and of the {\it local Euler obstruction} of an analytic set.Comment: 18 page

The Schur-Horn theorem for operators and frames with prescribed norms and frame operator

Let $\mathcal H$ be a Hilbert space. Given a bounded positive definite operator $S$ on $\mathcal H$, and a bounded sequence $\mathbf{c} = \{c_k \}_{k \in \mathbb N}$ of non negative real numbers, the pair $(S, \mathbf{c})$ is frame admissible, if there exists a frame $\{f_k \}_{k \in \mathbb{N}}$ on $\mathcal H$ with frame operator $S$, such that $\|f_k \|^2 = c_k$, $k \in \mathbb {N}$. We relate the existence of such frames with the Schur-Horn theorem of majorization, and give a reformulation of the extended version of Schur-Horn theorem, due to A. Neumann. We use it to get necessary conditions (and to generalize known sufficient conditions) for a pair $(S, \mathbf{c})$, to be frame admissible.Comment: To appear in Illinois Journal of Mat

Hemiparasitic plant impacts animal and plant communities across four trophic levels

1.Understanding the impact of species on community structure is a fundamental question in ecology. There is a growing body of evidence that suggests that both sub-dominant species and parasites can have a disproportionately large impact. 2.Here we report the impacts of an organism that is both subdominant and parasitic, the hemiparasite Rhinanthus minor. Whilst the impact of parasitic angiosperms on their hosts and, to a lesser degree, co-existing plant species, have been well characterized, much less is known about their impacts on higher trophic levels. 3.We experimentally manipulated field densities of the hemiparasite Rhinanthus minor in a species rich grassland, comparing the plant and invertebrate communities in plots where it was removed, at natural densities or at enhanced densities. 4.Plots with natural and enhanced densities of R. minor had lower plant biomass than plots without the hemiparasite, but enhanced densities almost doubled the abundance of invertebrates within the plots across all trophic levels, with effects evident in herbivores, predators and detritivores. 5.The hemiparasite R. minor, despite being a sub-dominant and transient component within plant communities that it inhabits, has profound effects on four different trophic levels. These effects persist beyond the life of the hemiparasite, emphasizing its role as a keystone species in grassland communitie

MERLIN radio detection of an interaction zone within a binary Orion proplyd system

Presented here are high angular resolution MERLIN 5 GHz (6 cm) continuum observations of the binary proplyd system, LV 1 in the Orion nebula, which consists of proplyd 168--326SE and its binary proplyd companion 168--326NW (separation 0.4 arcsec). Accurate astrometric alignment allows a detailed comparison between these data and published HST PC Halpha and [Oiii] images. Thermal radio sources coincide with the two proplyds and originate in the ionized photoevaporating flows seen in the optical emission lines. Flow velocities of approx 50 km/s from the ionized proplyd surfaces and \geq 100 km/s from a possible micro-jet have been detected using the Manchester Echelle spectrometer. A third radio source is found to coincide with a region of extended, high excitation, optical line emission that lies between the binary proplyds 168--326SE/326NW . This is modelled as a bowshock due to the collision of the photoevaporating flows from the two proplyds. Both a thermal and a non-thermal origin for the radio emission in this collision zone are considered.Comment: 23 pages, 9 figures, accepted by Ap

Massive runaway stars in the Small Magellanic Cloud

Using archival Spitzer Space Telescope data, we identified for the first time a dozen runaway OB stars in the Small Magellanic Cloud (SMC) through the detection of their bow shocks. The geometry of detected bow shocks allows us to infer the direction of motion of the associated stars and to determine their possible parent clusters and associations. One of the identified runaway stars, AzV 471, was already known as a high-velocity star on the basis of its high peculiar radial velocity, which is offset by ~40 km/s from the local systemic velocity. We discuss implications of our findings for the problem of the origin of field OB stars. Several of the bow shock-producing stars are found in the confines of associations, suggesting that these may be "alien" stars contributing to the age spread observed for some young stellar systems. We also report the discovery of a kidney-shaped nebula attached to the early WN-type star SMC-WR3 (AzV 60a). We interpreted this nebula as an interstellar structure created owing to the interaction between the stellar wind and the ambient interstellar medium.Comment: Accepted by A&

The impact of mass-loss on the evolution and pre-supernova properties of red supergiants

The post main-sequence evolution of massive stars is very sensitive to many parameters of the stellar models. Key parameters are the mixing processes, the metallicity, the mass-loss rate and the effect of a close companion. We study how the red supergiant lifetimes, the tracks in the Hertzsprung-Russel diagram (HRD), the positions in this diagram of the pre-supernova progenitor as well as the structure of the stars at that time change for various mass-loss rates during the red supergiant phase (RSG), and for two different initial rotation velocities. The surface abundances of RSGs are much more sensitive to rotation than to the mass-loss rates during that phase. A change of the RSG mass-loss rate has a strong impact on the RSG lifetimes and therefore on the luminosity function of RSGs. At solar metallicity, the enhanced mass-loss rate models do produce significant changes on the populations of blue, yellow and red supergiants. When extended blue loops or blue ward excursions are produced by enhanced mass-loss, the models predict that a majority of blue (yellow) supergiants are post RSG objects. These post RSG stars are predicted to show much smaller surface rotational velocities than similar blue supergiants on their first crossing of the HR gap. The position in the HRD of the end point of the evolution depends on the mass of the hydrogen envelope. More precisely, whenever, at the pre-supernova stage, the H-rich envelope contains more than about 5\% of the initial mass, the star is a red supergiant, and whenever the H-rich envelope contains less than 1\% of the total mass the star is a blue supergiant. For intermediate situations, intermediate colors/effective temperatures are obtained. Yellow progenitors for core collapse supernovae can be explained by the enhanced mass-loss rate models, while the red progenitors are better fitted by the standard mass-loss rate models.Comment: 19 pages, 11 figures, 6 tables, accepted for publication in Astronomy and Astrophysic

Overcoming Algorithm Aversion: People will Use Imperfect Algorithms If They Can (Even Slightly) Modify Them

Although evidence-based algorithms consistently outperform human forecasters, people often fail to use them after learning that they are imperfect, a phenomenon known as algorithm aversion. In this paper, we present three studies investigating how to reduce algorithm aversion. In incentivized forecasting tasks, participants chose between using their own forecasts or those of an algorithm that was built by experts. Participants were considerably more likely to choose to use an imperfect algorithm when they could modify its forecasts, and they performed better as a result. Notably, the preference for modifiable algorithms held even when participants were severely restricted in the modifications they could make (Stuides 1-3). In fact, our results suggest that participants\u27 preference for modifiable algorithms was indicative of a desire for some control over the forecasting outcome, and not for a desire for greater control over the forecasting outcome, as participants\u27 preference for modifiable was relatively insensitive to the magnitude of the modifications they were able to make (Study 2). Additionally, we found that giving participants the freedom to modify an imperfect algorithm made them feel more satisfied with the forecasting process, more likely to believe that the algorithm was superior, and more likely to choose to use an algorithm to make subsequent forecasts (Study 3). This research suggests that one can reduce algorithm aversion by giving people some control—even a slight amount—over an imperfect algorithm\u27s forecast