Canonical Noether symmetries and commutativity properties for gauge systems

For a dynamical system defined by a singular Lagrangian, canonical Noether symmetries are characterized in terms of their commutation relations with the evolution operators of Lagrangian and Hamiltonian formalisms. Separate characterizations are given in phase space, in velocity space, and through an evolution operator that links both spaces.Comment: 22 pages; some references updated, an uncited reference deleted, minor style change

Spore germination and development of young mycelia in some Rhizopogon species

Spore germination in Rhizopogon abietis, R. luteolus, R. roseolus and R. villosulus was induced in the presence of Rhodotorula glutinis and activated charcoal, in agar medium (N6:5). In one R. roseolus sample, 51% of spores germinated within 35 days, allowing observation of the course of spore germination and the different developmental patterns of homokaryotic mycelia. In these plates, spores showed two times of germination. The spores that germinated early produced an apical germ tube. Later other spores germinated in proximity to young mycelium, by forming a germ vesicle. One of the hyphal growth patterns obtained (interruption-swelling-ramification) is similar to that reported for other fungi. With this technique for inducing spore germination, it is possible to obtain enough monosporic cultures to perform mating tests. Key words: Basidiomycotina, Hypogeous, Monosporic Cultures

Singular lagrangians: some geometric structures along the Legendre map

New geometric structures that relate the lagrangian and hamiltonian formalisms defined upon a singular lagrangian are presented. Several vector fields are constructed in velocity space that give new and precise answers to several topics like the projectability of a vector field to a hamiltonian vector field, the computation of the kernel of the presymplectic form of lagrangian formalism, the construction of the lagrangian dynamical vector fields, and the characterisation of dynamical symmetries.Comment: 27 pages; minor changes, a reference update

The wave equation for stiff strings and piano tuning

We study the wave equation for a string with stiffness. We solve the equation and provide a uniqueness theorem with suitable boundary conditions. For a pinned string we compute the spectrum, which is slightly inharmonic. Therefore, the widespread scale of 12 equal divisions of the just octave is not the best choice to tune instruments like the piano. Basing on the theory of dissonance, we provide a way to tune the piano in order to improve its consonance. A good solution is obtained by tuning a note and its fifth by minimizing their beats.Peer ReviewedPostprint (published version

Immigració i adquisició de segones llengües: una aproximació als errors en la morfologia verbal

This article presents a descriptive study about the role of transfer in the second language acquisition of Catalan by immigrant learners. The analysis focuses on errors of verbal morphology produced by six learners of Catalan as a second language whose first languages were Arabic, Chinese, and Soninke. The results are analyzed taking into account the development of interlanguage and the influence of the tense-aspect system of the first language