A geometric approach to time evolution operators of Lie quantum systems

Lie systems in Quantum Mechanics are studied from a geometric point of view. In particular, we develop methods to obtain time evolution operators of time-dependent Schrodinger equations of Lie type and we show how these methods explain certain ad hoc methods used in previous papers in order to obtain exact solutions. Finally, several instances of time-dependent quadratic Hamiltonian are solved.Comment: Accepted for publication in the International Journal of Theoretical Physic

Karyotype and NOR-banding of mitotic chromosomes of some Vitis L. species

Chromosome studies were performed in V. champinii, V. cinerea, V. girdiana, V. labrusca, V. rotundifolia, V. rupestris and V. vinifera with the purpose of species characterization using chromosome morphometric data and NOR banding. A median ideogram was obtained for each species. The karyotype formula obtained varied from 7m + 12sm to 9m + 11sm. The species showed moderate chromosome asymmetry values according to TF% form, Stebbins, Romero Zarco and Paszko indices. V. champinii and V. girdiana were apart from the other species by CVcl and CVci graphic representation and also formed a group apart in the dendrogram based on Euclidian distances. The chromosome pair number 3 harbors the secondary constriction and a satellite segment in all species analyzed with Giemsa staining and it may be the same observed after NOR banding technique. It seems that the process of speciation in the North American Euvitis species studied involved some discrete changes in chromosome morphometry which have been reflected in the asymmetry index

Chloroplast microsatellites to investigate the origin of grapevine

The origin of the grapevine, Vitis vinifera L. (ssp.) sativa has been investigated with archaeobotanical–archaeological, cultural and historical data indicating a unique domestication centre located in the Caucasianand Middel-East regions about 6–7000 years ago, but, events leading to the domestication of thisspecies are still an open issue. In this work, eight universal chloroplast microsatellites are used to assessgenetic relationships among varieties selected as representatives of four distinct geographical groups fromMiddle-East to Western European regions. Results show that two out of the eight analysed chloroplast lociare polymorphic within the 142 individuals. Allele variants of the cpSSR loci combine in a total of sixdifferent haplotypes. The analysis of haplotypes distribution and haplotype diversity (HD) suggest that onlythree out of the six haplotypes are represented in the Caucasian and Middle-East samples, with 90% ofindividuals sharing the same haplotype. Moreover, the presence of all six haplotypes in the Europeanaccessions, with a high level of haplotype diversity, suggests varietal influx in these areas. Concerning theWestern European varieties, especially in Spanish accessions, half of the individuals share haplotype VIwhich is completely absent in the Caucasian and Middle-East cultivars. This result opens the discussionabout the existence of a unique and common domestication centre, located in the Caucasian and Middle-East area, for all the European cultivars.This work suggests the usefulness of chloroplast genome markers to provide information on haplotypedistributions that could help to identify further geographical areas for grapevine varietal evolution