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We prove that if there are $\mathfrak c$ incomparable selective ultrafilters then, for every infinite cardinal $\kappa$ such that $\kappa^\omega=\kappa$, there exists a group topology on the free Abelian group of cardinality $\kappa$ without nontrivial convergent sequences and such that every finite power is countably compact. In particular, there are arbitrarily large countably compact groups. This answers a 1992 question of D. Dikranjan and D. Shakhmatov.

In his survey "Beyond graph energy: Norms of graphs and matrices" (2016), Nikiforov proposed two problems concerning characterizing the graphs that attain equality in a lower bound and in a upper bound for the energy of a graph, respectively. We show that these graphs have at most two nonzero distinct absolute eigenvalues and investigate the proposed problems organizing our study according to the type of spectrum they can have. In most cases all graphs are characterized. Infinite families of graphs are given otherwise. We also show that all graphs satifying the properties required in the problems are integral, except for complete bipartite graphs $K_{p,q}$ and disconnected graphs with a connected component $K_{p,q}$, where $pq$ is not a perfect square., Comment: 17 pages, 5 figures, submitted

Assuming the existence of $\mathfrak c$ incomparable selective ultrafilters, we classify the non-torsion Abelian groups of cardinality $\mathfrak c$ that admit a countably compact group topology. We show that for each $\kappa \in [\mathfrak c, 2^\mathfrak c]$ each of these groups has a countably compact group topology of weight $\kappa$ without non-trivial convergent sequences and another that has convergent sequences. Assuming the existence of $2^\mathfrak c$ selective ultrafilters, there are at least $2^\mathfrak c$ non homeomorphic such topologies in each case and we also show that every Abelian group of cardinality at most $2^\mathfrak c$ is algebraically countably compact. We also show that it is consistent that every Abelian group of cardinality $\mathfrak c$ that admits a countably compact group topology admits a countably compact group topology without non-trivial convergent sequences whose weight has countable cofinality., Comment: new revised and expanded version that includes new co-authors

We study the relations between a generalization of pseudocompactness, named $(\kappa, M)$-pseudocompactness, the countably compactness of subspaces of $\beta \omega$ and the pseudocompactness of their hyperspaces. We show, by assuming the existence of $\mathfrak c$-many selective ultrafilters, that there exists a subspace of $\beta \omega$ that is $(\kappa, \omega^*)$-pseudocompact for all $\kappa<\mathfrak c$, but $\text{CL}(X)$ isn't pseudocompact. We prove in ZFC that if $\omega\subseteq X\subseteq \beta\omega$ is such that $X$ is $(\mathfrak c, \omega^*)$-pseudocompact, then $\text{CL}(X)$ is pseudocompact, and we further explore this relation by replacing $\mathfrak c$ for some small cardinals. We provide an example of a subspace of $\beta \omega$ for which all powers below $\mathfrak h$ are countably compact whose hyperspace is not pseudocompact, we show that if $\omega \subseteq X$, the pseudocompactness of $\text{CL}(X)$ implies that $X$ is $(\kappa, \omega^*)$-pseudocompact for all $\kappa<\mathfrak h$, and provide an example of such an $X$ that is not $(\mathfrak b, \omega^*)$-pseudocompact., Comment: 15 pages

We report a high-pressure study of orthorhombic rare-earth manganites AMnO3 using Raman scattering (for A = Pr, Nd, Sm, Eu, Tb and Dy) and synchrotron X-ray diffraction (for A = Pr, Sm, Eu, and Dy). In all cases, a structural and insulator-to-metal transition was evidenced, with a critical pressure that depends on the A-cation size. We analyze the compression mechanisms at work in the different manganites via the pressure dependence of the lattice parameters, the shear strain in the a-c plane, and the Raman bands associated with out-of-phase MnO6 rotations and in-plane O2 symmetric stretching modes. Our data show a crossover across the rare-earth series between two different kinds of behavior. For the smallest A-cations, the compression is nearly isotropic in the ac plane, with presumably only very slight changes of tilt angles and Jahn-Teller distortion. As the radius of the A-cation increases, the pressure-induced reduction of Jahn-Teller distortion becomes more pronounced and increasingly significant as a compression mechanism, while the pressure-induced bending of octahedra chains becomes conversely less pronounced. We finally discuss our results in the light of the notion of chemical pressure, and show that the analogy with hydrostatic pressure works quite well for manganites with small A-cations but can be misleading with large A-cations., Comment: 23 pages, 11 figures

In this work we studied the synthesis of BSCCO-2212 superconducting phase associating a quite similar method developed by Pechini (PM) with the microwave-assisted hydrothermal method, MAH. To study the influence of MAH on the properties of BSCCO system, we synthesized two samples by such method. For one sample we used carbonates (CMAH) and for the other one we used nitrates (NMAH) as chemical reagents. We also produced a reference sample (REF) just using carbonates by PM to compare their morphological and superconducting properties. The structural properties of the samples were analyzed by scanning electron microscopy and X-ray diffraction. It can be noted that the Bi-2212 phase is predominant in all samples; MAH samples present larger grains than the REF one. However, we can observe that the magnetic behaviour of NMAH is closer to that presented by REF., Comment: 4 pages, 5 figures, work presented in EUCAS 2013

We present a study of the effect of very high pressure on the orthorhombic perovskite GdMnO3 by Raman spectroscopy and synchrotron x-ray diffraction up to 53.2 GPa. The experimental results yield a structural and insulator-to-metal phase transition close to 50 GPa, from an orthorhombic to a metrically cubic structure. The phase transition is of first order with a pressure hysteresis of about 6 GPa. The observed behavior under very high pressure might well be a general feature in rare-earth manganites., Comment: 4 pages, 3 figures and 2 tables