Morphisms of Projective Varieties from the viewpoint of Minimal Model Theory

In this Lecture Notes we present, in a sufficiently self contained way, our contributions and interests in the field of Minimal Model Theory. We study Fano-Mori spaces, both from the biregular and the birational point of view. For the former we recall and develop Kawamata's Base Point Free technique and some of Mori's deformation arguments. For the latter we lean on Sarkisov and #-Minimal Model Programs. In writing these notes we want to give our point of view on this area of research. We are not trying to give a treatment of the whole subject. These notes collect some topics we presented in three mini-courses which were held in Wykno (Pl) (1999), Recife (Br) (2000) and Ferrara (It) (2000), respectively.Comment: 78 pages, AMSLaTeX2e, to appear on Dissertationes Mathematicae, Polish Ac. Sc. Warsa

How are cities working in the Spanish territorial development? ( 1985-2000)

The economic regional convergence process has attracted much attention through the last decade by regional scientists. This paper tries to deep in this topic, taking into account as a unit of analysis the role that cities have in regional convergence. Sigma and beta convergence are measured following a method based on a production function of the main spanish cities( 124 cities bigger than 50.000 inhabitants.

Quantum Monte Carlo study of the H- impurity in small helium clusters

We report ground state energies and structural properties for small helium clusters (4He) containing an H- impurity computed by means of variational and diffusion Monte Carlo methods. Except for 4He_2H- that has a noticeable contribution from collinear geometries where the H- impurity lies between the two 4He atoms, our results show that our 4He_NH- clusters have a compact 4He_N subsystem that binds the H- impurity on its surface. The results for $N\geq 3$ can be interpreted invoking the different features of the minima of the He-He and He-H- interaction potentials.Comment: 12 pages, 7 Ps figure

On possible simplifications in the theoretical description of gas phase atomic cluster dissociation

In this work, we investigate the possibility of describing gas phase atomic cluster dissociation by means of variational transition state theory (vTST) in the microcanonical ensemble. A particular emphasis is placed on benchmarking the accuracy of vTST in predicting the dissociation rate and kinetic energy release of a fragmentation event as a function of the cluster size and internal energy. The results for three Lennard-Jones clusters (LJ(n), n=8,14,19) indicate that variational transition state theory is capable of providing results of accuracy comparable to molecular dynamics simulations at a reduced computational cost. Possible simplifications of the master equation formalism used to model a dissociation cascade are also suggested starting from molecular dynamics results. In particular, it is found that the dissociation rate is only weakly dependent on the cluster total angular momentum J for the three cluster sizes considered. This would allow one to partially neglect the J-dependency of the kinetic coefficients, leading to a substantial decrease in the computational effort needed for the complete description of the cascade process. The impact of this investigation on the modeling of the nucleation process is discussed

Effect of the cluster angular momentum J and the projectile orbital momentum L on capture probability and postcollision dynamics

In this work, collisions between rotating atomic clusters composed of Lennard-Jones (LJ(n)) particles and an identical projectile have been investigated by means of trajectory simulations as a function of the cluster angular momentum J and internal energy E, and for different values of the projectile impact parameter b and relative velocity v(p). As expected, the collision (P(c)(b)) and capture [or sticking P(s)(b)] probabilities are found to decay below unity for values of b larger than the average surface radius of the cluster, with dP/db being strongly dependent on v(p). Both P(c)(b) and P(s)(b), however, appear to be largely insensitive to the modulus of the cluster angular momentum vertical bar J vertical bar and only weakly dependent on E for collisions involving target clusters with a lifetime tau > 100 ps. The latter findings are interpreted as indicating the absence of strong changes in the structure of the target as a function of vertical bar J vertical bar and E. The comparison between the dissociation lifetime (tau(dyn)) of the postcapture complexes (LJ*(n+1))obtained continuing trajectories after monomer capture and the one computed from the fragmentation of statistically prepared clusters (tau(stat)) supports the validity of a two-step capture-dissociation model; similarly, the comparison between the average amount of energy exchanged during trajectories (Delta E(dyn)) in the process LJ(n) + LJ --> LJ*(n+ 1) --> LJ(n) + LJ and the one predicted by statistical simulations (Delta E(stat)) suggests a fast statistical energy redistribution in the collisional complex even for very short tau(dyn) (e.g., 40 ps). In the case of projectiles aimed at the edge of the cluster [(grazing collisions, P(c)(b) < 1]; however, the time elapsed between formal collision and dissociation, tau(coll), is such that tau(coll) < tau(stat) and the trajectories indicate the presence of ballistic dynamics and of a weak energy exchange (Delta E(coll) < Delta E(dyn), with Delta E(coll) being the average energy exchanged during collisions). The relevance of these results to the study of gas phase nucleation and to the possibility of building a fully microcanonical framework for its description is discussed. (C) 2009 American Institute of Physics. [doi:10.1063/1.3239476

Macroscopic evidences for non-Rice-Ramsperger-Kassel effects in the reaction between H3O+ and D2O: The occurrence of nonstatistical isotopic branching ratio

The dynamics of the isotopic scrambling in the energized and metastable complex D2O-H3O+ has been studied using classical molecular dynamics (MD) trajectories starting from regions of phase space corresponding to an already formed collisional complex. The simulations cover the range of internal energies spanned by gas phase collision experiments. Rate constants for the isotopic exchange and the complex dissociation have been computed; the isotopic branching ratio R=[HD2O+]/[H2DO+] has also been obtained from MD simulations and found to deviate substantially from an equivalent prediction based on a previously proposed kinetic scheme. This finding suggests the possibility that details of the reaction dynamics play a role in defining the isotopic branching ratio. The analysis of trajectory results indicated a relatively long lifetime for the collisional complex and the presence of multiple time scales for the exchange process, with a large fraction of the exchange events being separated only by a single oxygen-oxygen vibration or half of it. The occurrence of these fast consecutive jumps and their different probabilities as a function of the relative direction between first and second jumps suggest the presence of ballistic motion in the complex following each reactive event. This can be explained on the basis of overlapping regions in phase space and it is used to provide an explanation of the difference between kinetic and MD branching ratios. (c) 2007 American Institute of Physics

Discretization error-free estimate of low temperature statistical dissociation rates in gas phase: Applications to Lennard-Jones clusters X(13-n)Y(n) (n=0-3)

In this work, an improved approach for computing cluster dissociation rates using Monte Carlo (MC) simulations is proposed and a discussion is provided on its applicability as a function of environmental variables (e.g., temperature). With an analytical transformation of the integrals required to compute variational transition state theory (vTST) dissociation rates, MC estimates of the expectation value for the Dirac delta delta(q(rc)-q(c)) have been made free of the discretization error that is present when a prelimit form for delta is used. As a by-product of this transformation, the statistical error associated with is reduced making this step in the calculation of vTST rates substantially more efficient (by a factor of 4-2500, roughly). The improved MC procedure is subsequently employed to compute the dissociation rate for Lennard-Jones clusters X(13-n)Y(n) (n=0-3) as a function of temperature (T), composition, and X-Y interaction strength. The X(13-n)Y(n) family has been previously studied as prototypical set of systems for which it may be possible to select and stabilize structures different from the icosahedral global minimum of X(13). It was found that both the dissociation rate and the dissociation mechanism, as suggested by the statistical simulations, present a marked dependence on n, T, and the nature of Y. In particular, it was found that a vacancy is preferentially formed close to a surface impurity when the X-Y interaction is weaker than the X-X one whatever the temperature. Differently, the mechanism was found to depend on T for stronger X-Y interactions, with vacancies being formed opposite to surface impurities at higher temperature. These behaviors are a reflex of the important role played by the surface fluctuations in defining the properties of clusters. (c) 2008 American Institute of Physics