Non-unitary representations of the SU(2) algebra in the Dirac equation with a Coulomb potential

A novel realization of the classical SU(2) algebra is introduced for the Dirac relativistic hydrogen atom defining a set of operators that, besides, allow the factorization of the problem. An extra phase is needed as a new variable in order to define the algebra. We take advantage of the operators to solve the Dirac equation using algebraic methods. To acomplish this, a similar path to the one used in the angular momentum case is employed; hence, the radial eigenfuntions calculated comprise non unitary representations of the algebra. One of the interesting properties of such non unitary representations is that they are not labeled by integer nor by half-integer numbers as happens in the usual angular momentum representation.Comment: 20 pages 1 eps figure in a single zipped file, submitted to J. Math. Phy

Review of Cephalanthus (Rubiaceae), with the segregation of the new African genus Sylvainia

The genus Cephalanthus belongs to the tribe Naucleeae along with 26 other genera. Members o f Cephalanthus grow mainly in températe and subtropical wetlands in the Americas and Asia, but there is a single African species that is associated with forest maigins and rocky outcrops in humid températe grasslands. The genus comprises six species with a markedly disjunct distribution: three species are from the Americas: C. glabratus, C. occidentalis, and C. salicifolius-, two species are from tropical Asia: C. angustifolius and C. tetrandrus, and one species is from tropical Africa: C. natalensis. Recent molecular phylogenetic studies have supported the monophyly of the genus, however, most morphological synapomorphies are absent in the single African species, which is sister to the remaining species o f the genus. The aim o f this paper is to provide a brief taxonomic revisión o f the genus Cephalanthus, considering previously undervalued or unconsidered morphological aspects, and to describe a new monospecific genus based on comparative morphological analyses. Vegetative and reproductive material o f all Cephalanthus species was analysed following conventional taxonomic techniques. We transferred C. natalensis to a new genus, Sylvainia, taking into account the morphological similarities and diíferences between Cephalanthus and the new genus. Cephalanthus glabrifolius, a little-known species previously considered synonymous with C. tetrandrus, is resurrected as a valid species, bringing the number of Cephalanthus species back to six. The subtribe Cephalanthinae now has two genera, its type Cephalanthus with six species, and a new monospecific genus Sylvainia endemic to southeastem Africa. Based on the modified description of Cephalanthus, now absent from Alfica, the generic concept is substantially modified, being now represented by three American and three Asian species

From circular paths to elliptic orbits: A geometric approach to Kepler's motion

The hodograph, i.e. the path traced by a body in velocity space, was introduced by Hamilton in 1846 as an alternative for studying certain dynamical problems. The hodograph of the Kepler problem was then investigated and shown to be a circle, it was next used to investigate some other properties of the motion. We here propose a new method for tracing the hodograph and the corresponding configuration space orbit in Kepler's problem starting from the initial conditions given and trying to use no more than the methods of synthetic geometry in a sort of Newtonian approach. All of our geometric constructions require straight edge and compass only.Comment: 9 pages, 4 figure

A useful form of the recurrence relation between relativistic atomic matrix elements of radial powers

Recently obtained recurrence formulae for relativistic hydrogenic radial matrix elements are cast in a simpler and perhaps more useful form. This is achieved with the help of a new relation between the $r^a$ and the $\beta r^b$ terms ($\beta$ is a $4\times 4$ Dirac matrix and $a, b$ are constants) in the atomic matrix elements.Comment: 7 pages, no figure

New non-unitary representations in a Dirac hydrogen atom

New non-unitary representations of the SU(2) algebra are introduced for the case of the Dirac equation with a Coulomb potential; an extra phase, needed to close the algebra, is also introduced. The new representations does not require integer or half integer labels. The set of operators defined are used to span the complete space of bound state eigenstates of the problem thus solving it in an essentially algebraic way

Efectos de la ablación telencefálica sobre el aprendizaje apetitivo instrumental en la carpa dorada (Carassius auratus)

Efectos de la ablación telencefálica sobre el aprendizaje apetitivo instrumental en la carpa dorada (Carassius auratus). En este experimento se emplearon carpas doradas (Carassius auratus) con ablación bilateral del telencéfalo o con operación de control en un procedimiento de entrenamiento instrumental con distribución de ensayos ampliamente espaciados (un ensayo por día). Los animales fueron entrenados a nadar en un corredor bajo condiciones de reforzamiento continuo o reforzamiento parcial al 50%. Se realizaron 32 sesiones diarias de adquisición seguidas de 32 sesiones diarias de extinción en las que la comida fue suprimida. La lesión telencefálica afectó a la adquisición durante los primeros ensayos, aunque, al final, los animales lesionados alcanzaron una ejecución asintótica similar a la de los animales controles. La extinción se vio facilitada por las lesiones, independientemente del programa de reforzamiento usado durante la adquisición. Los resultados se interpretan en relación a los estudios comparativos de aprendizaje instrumental apetitivo con ensayos espaciados realizados en otros vertebrados.Goldfish received either bilateral ablation of the telencephalon or a sham operation, followed by instrumental training under widely-spaced conditions (one trial per day). Animals were trained to swim in a runway for either continuous reinforcement or 50% partial reinforcement. Thirty-two daily acquisition sessions were followed by 32 daily extinction sessions during which food was withheld. Telencephalic lesions disrupted early acquisition performance, but lessoned animals achieved an asymptotic performance similar to that of sham controls. Extinction was facilitated by the lesions, independently of the schedule of reinforcement used during acquisition. The results are interpreted in relation to comparative research on spaced-trial appetitive instrumental learning in vertebrates.Ministerio de Educación y Cultura PB96-1334Junta de Andalucía CVI-24

Recurrence relation for relativistic atomic matrix elements

Recurrence formulae for arbitrary hydrogenic radial matrix elements are obtained in the Dirac form of relativistic quantum mechanics. Our approach is inspired on the relativistic extension of the second hypervirial method that has been succesfully employed to deduce an analogous relationship in non relativistic quantum mechanics. We obtain first the relativistic extension of the second hypervirial and then the relativistic recurrence relation. Furthermore, we use such relation to deduce relativistic versions of the Pasternack-Sternheimer rule and of the virial theorem.Comment: 10 pages, no figure