When is R the union of an increasing family of null sets?

We study the problem in the title and show that it is equivalent to the fact that every set of reals is an increasing union of measurable sets. We also show the relationship of it with Sierpin’ski set

Main-Sequence and sub-giant stars in the Globular Cluster NGC6397: The complex evolution of the lithium abundance

Thanks to the high multiplex and efficiency of Giraffe at the VLT we have been able for the first time to observe the Li I doublet in the Main Sequence (MS) stars of a Globular Cluster. At the same time we observed Li in a sample of Sub-Giant (SG) stars of the same B-V colour. Our final sample is composed of 84 SG stars and 79 MS stars. In spite of the fact that SG and MS span the same temperature range we find that the equivalent widths of the Li I doublet in SG stars are systematically larger than those in MS stars, suggesting a higher Li content among SG stars. This is confirmed by our quantitative analysis. We derived the effective temperatures, from H$\alpha$ fitting, and NLTE Li abundances of the stars in our the sample, using 3D and 1D models. We find that SG stars have a mean Li abundance higher by 0.1dex than MS stars, using both 1D and 3D models. We also detect a positive slope of Li abundance with effective temperature. These results provide an unambiguous evidence that the Li abundance changes with evolutionary status. The physical mechanisms responsible for this behaviour are not yet clear, and none of the existing models seems to describe accurately these observations. Based on these conclusions, we believe that the cosmological lithium problem still remains an open question.Comment: Proceedings of the contributed talk presented at the IAU Symposium 26

Acontecer latinoamericano

Se presenta lo más importante del acontecer latinoamericano. De México, el Pacto de Solidaridad Económica, la sucesión presidencial y el conflicto en la UNAM. Sobre Centroamérica, los esfuerzos de pacificación en el Itsmo; de Colombia, el cartel de Medellín, las elecciones en Haití y el conflicto Panamá-Estados Unidos.ITESO, A.C

Quasinodal lines in rhombohedral magnetic materials

A well-established result in condensed matter physics states that materials crystallizing in symmetry groups containing glide reflection symmetries possess nodal lines on the energy bands. These nodal lines are topologically protected and appear on the fixed planes of the reflection in reciprocal space. In the presence of inversion symmetry, the energy bands are degenerate and the band crossings on the fixed plane may be one-dimensional, or may intersect in points, including the case of empty intersection. In the latter case, the crossing is partially or totally avoided, thus producing lines on reciprocal space where the energy gap may be small, and in the former, the nodal lines will endure, thus producing Dirac or double nodal lines. In addition, if the material crystallizes in a ferromagnetic phase where the glide reflection symmetry is broken, the nodal lines hybridize, thus defining lines in reciprocal space where the energy gap may be small. In this work, we concentrate our efforts on the study of nodal lines that hybridize due to magnetization; we have coined the term of quasinodal lines for those lines in reciprocal space where the energy gap is small (less than what can be detected experimentally ∼25 meV). We study magnetic trifluorides and trioxides which crystallize in magnetic space groups 167.107 and 161.71 and we show the existence of quasinodal lines on these materials. We furthermore show that whenever the quasinodal lines are located around the Fermi level then interesting charge and spin transport effects are induced and can be used to detect experimentally these lines. Of particular interest are the half-metallic ferromagnetic phases of PdF3 and LiCuF3 where the large signal of the anomalous Hall conductance is due to the presence of the quasinodal lines on the Fermi level

Topological electronic structure and Weyl points in nonsymmorphic hexagonal materials

Using topological band theory analysis we show that the nonsymmorphic symmetry operations in hexagonal lattices enforce Weyl points at the screw-invariant high-symmetry lines of the band structure. The corepresentation theory and connectivity group theory show that Weyl points are generated by band crossings in accordion-like and hourglass-like dispersion relations. These Weyl points are stable against weak perturbations and are protected by the screw rotation symmetry. Based on first-principles calculations we found a complete agreement between the topological predicted energy dispersion relations and real hexagonal materials. Topological charge (chirality) and Berry curvature calculations show the simultaneous formation of Weyl points and nodal-lines in 4d transition-metal trifluorides such as AgF3 and AuF3. Furthermore, a large intrinsic spin-Hall conductivity was found due to the combined strong spin-orbit coupling and multiple Weyl-point crossings in the electronic structure. These materials could be used to the spin/charge conversion in more energy-efficient spintronic devices

Three carbon-enhanced metal-poor dwarf stars from the SDSS - Chemical abundances from CO^5BOLD 3D hydrodynamical model atmospheres

The origin of carbon-enhanced metal-poor stars enriched with both s and r elements is highly debated. Detailed abundances of these types of stars are crucial to understand the nature of their progenitors. The aim of this investigation is to study in detail the abundances of SDSS J1349-0229, SDSS J0912+0216 and SDSS J1036+1212, three dwarf CEMP stars, selected from the Sloan Digital Sky Survey. Using high resolution VLT/UVES spectra (R ~ 30 000) we determine abundances for Li, C, N, O, Na, Mg, Al, Ca, Sc, Ti, Cr, Mn, Fe, Co, Ni and 21 neutron-capture elements. We made use of CO^5BOLD 3D hydrodynamical model atmospheres in the analysis of the carbon, nitrogen and oxygen abundances. NLTE corrections for C I and O I lines were computed using the Kiel code. We classify SDSS J1349-0229 and SDSS J0912+0216 as CEMP-r+s stars. SDSS J1036+1212 belongs to the class CEMP-no/s, with enhanced Ba, but deficient Sr, of which it is the third member discovered to date. Radial-velocity variations have been observed in SDSS J1349-0229, providing evidence that it is a member of a binary system. The chemical composition of the three stars is generally compatible with mass transfer from an AGB companion. However, many details remain difficult to explain. Most notably of those are the abundance of Li at the level of the Spite plateau in SDSS J1036+1212 and the large over-abundance of the pure r-process element Eu in all three stars.Comment: 12 pages, 15 figures. Accepted for publication in A&