List pricing and pure strategy outcomes in a Bertrand-Edgeworth duopoly

Non-existence of a pure strategy equilibrium in a Bertrand-Edgeworth duopoly model is analyzed. The standard model is modified to include a list pricing stage and a subsequent price discounting stage. Both firms first simultaneously choose a maximum list price and then decide to lower the price, or not, in a subsequent discounting stage. List pricing works as a credible commitment device that induces the pure strategy outcome. It is shown that for a general class of rationing rules there exists a sub-game perfect equilibrium that involves both firms playing pure strategies. This equilibrium payoff dominates any other sub-game perfect equilibrium of the game. Further unlike the dominant firm interpretation of a price leader, we show that the small firm may have incentives to commit to a low price and in this sense assume the role of a leader

Non-detection of magnetic fields in the central stars of the planetary nebulae NGC 1360 and LSS 1362

The presence of magnetic fields is an attractive hypothesis for shaping PNe. We report on observations of the central star of the two Planetary Nebulae NGC1360 and LSS1326. We performed spectroscopy on circularly polarized light with the FOcal Reducer and low dispersion Spectrograph at the Very Large Telescope of the European Southern Observatory. Contrary to previous reports (Jordan et al. 2005, A&A, 432, 273), we find that the effective magnetic field, that is the average over the visible stellar disk of longitudinal components of the magnetic fields, is null within errors for both stars. We conclude that a direct evidence of magnetic fields on the central stars of PNe is still missing --- either the magnetic field is much weaker (< 600 G) than previously reported, or more complex (thus leading to cancellations), or both. Certainly, indirect evidences (e.g., MASER emission) fully justify further efforts to study the strength and morphology of such magnetic fields.Comment: Published in ApJ letters: Leone et al. 2011, ApJ, 731, 33

Singular random matrix decompositions: distributions.

Assuming that Y has a singular matrix variate elliptically contoured distribution with respect to the Hausdorff measure, the distributions of several matrices associated to QR, modified QR, SV and Polar decompositions of matrix Y are determined, for central and non-central, non-singular and singular cases, as well as their relationship to the Wishart and Pseudo-Wishart generalized singular and non-singular distributions. We present a particular example for the Karhunen-Lòeve decomposition. Some of these results are also applied to two particular subfamilies of elliptical distributions, the singular matrix variate normal distribution and the singular matrix variate symmetric Pearson type VII distribution

HCO3- enrichment causes cytosolic NO3- efflux in Posidonia oceanica leaf cells

Posidonia oceanica is a seagrass, the only group of vascular plants to colonize the marine environment. Seawater is an extreme yet stable environment, characterized by high salinity, alkaline pH and low availability of essential nutrients, such as nitrate or phosphate. In addition, in aquatic environments the supply of CO2 for the photosynthesis is limited by diffusion and therefore many aquatic plants use HCO3- as the inorganic carbon source for photosynthesis. Previous results have shown that Na+ -dependent transport systems operate on the plasma membrane of P. oceanica mesophyll leaf cells for the high-affinity NO3-, Pi or amino acids uptake. Also, a direct transport of HCO3- driven by H+ has been found in this species that provides inorganic carbon for photosynthesis and could be a significant component of a carbon concentrating mechanism in this species. Interestingly, this HCO3- direct uptake caused the efflux of chloride from the cytosol, probably through S-type anion channels, pointing that other anions could also be removed from the cytosol. This hypothesis could be relevant in the case of NO3-, since the decrease of cytosolic NO3- in response to HCO3- enrichment could limit N-assimilation. Here we analyse the effect of HCO3- increase on NO3- uptake and cytosolic homeostasis in P. oceanica. Enrichment of natural seawater with 3 mM HCO3- evokes the on-going decrease of cytosolic NO3-, from 5.7 ± 0.2 to 4.8 ± 0.7 mM after 40 min of treatment. The incubation of P. oceanica leaf pieces in 3 mM HCO3- NSW causes an initial increase of NO3- concentration in the medium. Maximum efflux (21 nmol NO3- gFM-1 min-1) occurs within the first minute of incubation. Then, external NO3- is depleted from the medium at lower net uptake rate than the value observed in non HCO3- -enriched natural seawater. These results fit the hypothesis that HCO3- enrichment causes the nitrogen loose and could impair nitrogen assimilation promoting N biomass impoverishment.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech. Spanish MINECO, projects BFU2017-85117-R and BIO2016-81957-RED