The Maximal Positively Invariant Set: Polynomial Setting

This note considers the maximal positively invariant set for polynomial discrete time dynamics subject to constraints specified by a basic semialgebraic set. The note utilizes a relatively direct, but apparently overlooked, fact stating that the related preimage map preserves basic semialgebraic structure. In fact, this property propagates to underlying set--dynamics induced by the associated restricted preimage map in general and to its maximal trajectory in particular. The finite time convergence of the corresponding maximal trajectory to the maximal positively invariant set is verified under reasonably mild conditions. The analysis is complemented with a discussion of computational aspects and a prototype implementation based on existing toolboxes for polynomial optimization

Classical solutions in five dimensional induced matter theory and its relation to an imperfect fluid

We study five dimensional cosmological models with four dimensional hypersufaces of the Bianchi type I and V. In this way the five dimensional vacuum field equations $\rm G_{AB} = 0$, led us to four dimensional matter equations $\rm G_{\mu\nu}=T_{\mu\nu}$ and the matter is interpreted as a purely geometrical property of a fifth dimension. Also, we find that the energy-momentum tensor induced from the fifth dimension has the structure of an imperfect fluid that has dissipative terms.Comment: 16 pages, latex, no figure

Testing a dissipative kinetic k-essence model

In this work, we present a study of a purely kinetic k-essence model, characterized basically by a parameter $\alpha$ in presence of a bulk dissipative term, whose relationship between viscous pressure $\Pi$ and energy density $\rho$ of the background follows a polytropic type law $\Pi \propto \rho^{\lambda+1/2}$, where $\lambda$, in principle, is a parameter without restrictions. Analytical solutions for the energy density of the k-essence field are found in two specific cases: $\lambda=1/2$ and $\lambda=(1-\alpha)/2\alpha$, and then we show that these solutions posses the same functional form than the non-viscous counterpart. Finally, both approach are contrasted with observational data from type Ia supernova, and the most recent Hubble parameter measurements, and therefore, the best values for the parameters of the theory are founds.Comment: 9 pages, 5 figures, accepted in EPJ

VALES I: the molecular gas content in star-forming dusty H-ATLAS galaxies up to z = 0.35

We present an extragalactic survey using observations from the Atacama Large Millimeter/submillimeter Array (ALMA) to characterize galaxy populations up to z = 0.35: the Valparaíso ALMA Line Emission Survey (VALES). We use ALMA Band-3 CO(1–0) observations to study the molecular gas content in a sample of 67 dusty normal star-forming galaxies selected from the Herschel Astrophysical Terahertz Large Area Survey (H-ATLAS). We have spectrally detected 49 galaxies at >5σ significance and 12 others are seen at low significance in stacked spectra. CO luminosities are in the range of (0.03–1.31) × 10^(10) K km s^(−1) pc^2, equivalent to log(M_(gas)/M_⊙)=8.9--10.9 assuming an αCO = 4.6 (K km s^(−1) pc^2)^(−1), which perfectly complements the parameter space previously explored with local and high-z normal galaxies. We compute the optical to CO size ratio for 21 galaxies resolved by ALMA at ∼3.5 arcsec resolution (6.5 kpc), finding that the molecular gas is on average ∼ 0.6 times more compact than the stellar component. We obtain a global Schmidt–Kennicutt relation, given by log[Σ_(SFR)/(M_⊙ yr^(−1)kpc^(−2))]=(1.26±0.02)×log[Σ_(MH2)/(M_⊙ pc^(−2))]−(3.6±0.2). We find a significant fraction of galaxies lying at ‘intermediate efficiencies’ between a long-standing mode of star formation activity and a starburst, specially at L_(IR) = 10^(11–12) L_⊙. Combining our observations with data taken from the literature, we propose that star formation efficiencies can be parametrized by log [SFR/M_(H2)] = 0.19 × (log L_(IR)− 11.45) − 8.26 − 0.41 × arctan [−4.84(logL_(IR) − 11.45)]. Within the redshift range we explore (z < 0.35), we identify a rapid increase of the gas content as a function of redshift