Bounding the gap between a free group (outer) automorphism and its inverse

For any finitely generated group $G$, two complexity functions $\alpha_G$ and $\beta_G$ are defined to measure the maximal possible gap between the norm of an automorphism (respectively outer automorphism) of $G$ and the norm of its inverse. Restricting attention to free groups $F_r$, the exact asymptotic behaviour of $\alpha_2$ and $\beta_2$ is computed. For rank $r\geqslant 3$, polynomial lower bounds are provided for $\alpha_r$ and $\beta_r$, and the existence of a polynomial upper bound is proved for $\beta_r$.Comment: 24 pages; To appear in Collectanea Mathematic

The puzzle of metallicity and multiple stellar populations in the Globular Clusters in Fornax

We examine the photometric data for Fornax clusters, focussing our attention on their horizontal branch color distribution and, when available, on the RR Lyr variables fraction and period distribution. Based on our understanding of the HB morphology in terms of varying helium content in the context of multiple stellar generations, we show that clusters F2, F3 and F5 must contain substantial fractions of second generation stars (~54-65%). On the basis of a simple chemical evolution model we show that the helium distribution in these clusters can be reproduced by models with cluster initial masses ranging from values equal to ~4 to ~10 times larger than the current masses. Models with a very short second generation star formation episode can also reproduce the observed helium distribution but require larger initial masses up to about twenty times the current mass. While the lower limit of this range of possible initial GC masses is consistent with those suggested by the observations of the low metallicity field stars, we also discuss the possibility that the metallicity scale of field stars (based on CaII triplet spectroscopy) and the metallicities derived for the clusters in Fornax may not be consistent with each other. The reproduction of the HB morphology in F2,F3,F5 requires two interesting hypotheses: 1) the first generation HB stars lie all at "red" colours. According to this interpretation, the low metallicity stars in the field of Fornax, populating the HB at colours bluer than the blue side ((V-I)o<=0.3 or (B-V)o<=0.2) of the RR Lyrs, should be second generation stars born in the clusters;a preliminary analysis of available colour surveys of Fornax field provides a fraction ~20% of blue HB stars, in the low metallicity range; 2) the mass loss from individual second generation red giants is a few percent of a solar mass larger than the mass loss from first generation stars.Comment: 14 pages, 8 figures. Accepted for publication in MNRA

Topological mass mechanism and exact fields mapping

We present a class of mappings between models with topological mass mechanism and purely topological models in arbitrary dimensions. These mappings are established by directly mapping the fields of one model in terms of the fields of the other model in closed expressions. These expressions provide the mappings of their actions as well as the mappings of their propagators. For a general class of models in which the topological model becomes the BF model the mappings present arbitrary functions which otherwise are absent for Chern-Simons like actions. This work generalizes the results of [1] for arbitrary dimensions.Comment: 11 page

Nonrenormalization theorems for N=2 Super Yang-Mills

The BRST algebraic proofs of the the nonrenormalization theorems for the beta functions of N=2 and N=4 Super Yang-Mills theories are reviewed.Comment: 3 pages, contribution to SUSY 2000 Encyclopedi

Integrated methodologies for 3D deformation analysis at Ischia Island (Italy): state of the art, prospectives and modelling.

Ischia Island, located SW of Naples (Italy), has been characterized by both explosive and effusive activity with the last eruption occurred in 1302. Subsequent dynamics, characterized by seismic activity with the strongest events occurred in 1881 and 1883 and a diffuse hydrothermal phenomena, shows a significant subsidence in the S and NW sectors. The presence of the active volcanoes in a very densely area needs continuous monitoring of the dynamics related to the pre-eruptive processes. Ground deformation studies are an important precursor because are linked to magma overpressure and migration. In particular, the geodetic monitoring system is mainly based on GPS and Precise levelling techniques. Here, we present a study of the surface deformation occurring in the island based on Differential Synthetic Aperture Radar Interferometry (DInSAR) referred to as Small BAseline Subset (SBAS) technique. Levelling surveys carried out between 1990 and 2003 on the Mt Epomeo resurgent block record negative dislocations on the northern and southern flanks with a maximum subsidence rate of 1.27 cm/yr. This deformation is not associated with cooling, crystallization or lateral drainage of magma and cannot be explained by a pressure point or prorate ellipsoid source. The data show that between 1990 and 2003 Mt Epomeo has been affected by a subsidence with two maxima located on its northern and southern sectors. Then, the 1992–2003 time interval and SAR data acquired by the European Remote Sensing (ERS) satellites from ascending and descending orbits have been used, thus allowing us to discriminate the vertical and east–west components of the displacements. A validation of the DInSAR results has been carried out first by comparing the vertical deformations estimated from the SAR data with those measured from the spirit levelling network that is present in the area. The deformation is due to the closure of cracks associated with ENE–WSW to E–W preexisting faults along which degassing processes occur. We propose that the recorded dislocations reflect a decrease in the fluid pressure within these cracks