Endohedrally confined hydrogen atom with a moving nucleus

We studied the hydrogen atom as a system of two quantum particles in different confinement conditions; a spherical-impenetrable-wall cavity and a fullerene molecule cage. The motion is referred to the center of spherical cavities, and the Schr\"{o}dinger equation solved by means of a Generalized Sturmian Function expansion in spherical coordinates. The solutions present different properties from the ones described by the many models in the literature, where the proton is fixed in space and only the electron is considered as a quantum particle. Our results show that the position of the proton (i.e. the center of mas of the H atom) is very sensitive to the confinement condition, and could vary substantially from one state to another, from being sharply centered to being localized outside the fullerene molecule. Interchange of the localization characteristics between the states when varying the strength of the fullerene cage and mass occurred through crossing phenomena

Modified Frank-Wolfe Algorithm for Enhanced Sparsity in Support Vector Machine Classifiers

This work proposes a new algorithm for training a re-weighted L2 Support Vector Machine (SVM), inspired on the re-weighted Lasso algorithm of Cand\`es et al. and on the equivalence between Lasso and SVM shown recently by Jaggi. In particular, the margin required for each training vector is set independently, defining a new weighted SVM model. These weights are selected to be binary, and they are automatically adapted during the training of the model, resulting in a variation of the Frank-Wolfe optimization algorithm with essentially the same computational complexity as the original algorithm. As shown experimentally, this algorithm is computationally cheaper to apply since it requires less iterations to converge, and it produces models with a sparser representation in terms of support vectors and which are more stable with respect to the selection of the regularization hyper-parameter

Stationary scalar and vector clouds around Kerr-Newman black holes

Massive bosons in the vicinity of Kerr-Newman black holes can form pure bound states when their phase angular velocity fulills the synchronisation condition, i.e. at the threshold of superradiance. The presence of these stationary clouds at the linear level is intimately linked to the existence of Kerr black holes with synchronised hair at the non-linear level. These configurations are very similar to the atomic orbitals of the electron in a hydrogen atom. They can be labeled by four quantum numbers: $n$, the number of nodes in the radial direction; $\ell$, the orbital angular momentum; $j$, the total angular momentum; and $m_j$, the azimuthal total angular momentum. These synchronised configurations are solely allowed for particular values of the black hole's mass, angular momentum and electric charge. Such quantization results in an existence surface in the three-dimensional parameter space of Kerr-Newman black holes. The phenomenology of stationary scalar clouds has been widely addressed over the last years. However, there is a gap in the literature concerning their vector cousins. Following the separability of the Proca equation in Kerr(-Newman) spacetime, this work explores and compares scalar and vector stationary clouds around Kerr and Kerr-Newman black holes, extending previous research.Comment: 17 pages, 6 figures. Contribution to Selected Papers of the Fifth Amazonian Symposium on Physics (accepted in IJMPD