So(2,1) Lie algebra and the Green's functions for the conditionally exactly solvable potentials

The Green's functions of the recently discovered conditionally exactly solvable potentials are computed. This is done through the use of a second-order differential realization of the so(2,1) Lie algebra. So we present the dynamical symmetry underlying the solvability of such potentials and show that they belong to a general class of solvable and partially solvable potentials. © 1994 The American Physical Society.

Closed expressions for Lie algebra invariants and finite transformations

A simple procedure to obtain complete, closed expressions for Lie algebra invariants is presented. The invariants are ultimately polynomials in the group parameters. The construction of finite group elements requires the use of projectors, whose coefficients are invariant polynomials. The detailed general forms of these projectors are given. Closed expressions for finite Lorentz transformations, both homogeneous and inhomogeneous, as well as for Galilei transformations, are found as examples.

Pure Spinors in AdS and Lie Algebra Cohomology

We show that the BRST cohomology of the massless sector of the Type IIB superstring on AdS(5) x S (5) can be described as the relative cohomology of an infinite-dimensional Lie superalgebra. We explain how the vertex operators of ghost number 1, which correspond to conserved currents, are described in this language. We also give some algebraic description of the ghost number 2 vertices, which appears to be new. We use this algebraic description to clarify the structure of the zero mode sector of the ghost number two states in flat space, and initiate the study of the vertices of the higher ghost number.

Infinite dimensional Lie algebra associated with conformal transformations of the two-point velocity correlation tensor from isotropic turbulence

We deal with homogeneous isotropic turbulence and use the two-point velocity correlation tensor field (parametrized by the time variable t) of the velocity fluctuations to equip an affine space K3 of the correlation vectors by a family of metrics. It was shown in Grebenev and Oberlack (J Nonlinear Math Phys 18:109–120, 2011) that a special form of this tensor field generates the so-called semi-reducible pseudo-Riemannian metrics ds2(t) in K3. This construction presents the template for embedding the couple (K3, ds2(t)) into the Euclidean space R3 with the standard metric. This allows to introduce into the consideration the function of length between the fluid particles, and the accompanying important problem to address is to find out which transformations leave the statistic of length to be invariant that presents a basic interest of the paper. Also we classify the geometry of the particles configuration at least locally for a positive Gaussian curvature of this configuration and comment the case of a negative Gaussian curvature.

Filiera corta: percorsi di innovazione tecnici, organizzativi e sociali nella gestione strategica delle nicchie. Esperienze in Toscana e in Provenza

There is a widening consensus around the fact that, in many developed countries, food production-consumption patterns are in recent years interested by a process of deep change towards diversification and re-localisation practices, as a counter-tendency to the trend to the increasing disconnection between farming and food, producers and consumers. The relevance of these initiatives doesn't certainly lie on their economic dimension, but rather in their intense diffusion and growth rate, their spontaneous and autonomous nature and, especially, their intrinsic innovative potential. These dynamics involve a wide range of actors around local food patterns, embedding short food supply chains initiatives within a more complex and wider process of rural development, based on principles of sustainability, multifunctionality and valorisation of endogenous resources. In this work we have been analysing these features through a multi-level perspective, with reference to the dynamics between niche and regime and the inherent characteristics of the innovation paths. We apply this approach, through a qualitative methodology, to the analysis of the experience of farmers’ markets and Solidarity-Based Consumers Groups (Gruppi di Acquisto Solidale) ongoing in Tuscany, seeking to highlight the dynamics that are affecting the establishment of this alternative food production-consumption model (and its related innovative potential) from within and from without. To verify if and in which conditions they can constitute a niche, a protected space where radical innovations can develop, we make reference to the three interrelated analytic dimensions of socio-technical systems: the actors (i.e. individuals. social groups, organisations), the rules and institutions system, and the artefacts (i.e. the material and immaterial contexts in which the actors move). Through it, we analyse the innovative potential of niches and the level of their structuration and , then, the mechanisms of system transition, focusing on the new dynamics within the niche and between the niche and the policy regime emerging after the growth of interest by mass-media and public institutions and their direct involvement in the initiatives. Following the development of these significant experiences, we explore more deeply social, economic, cultural, political and organisational factors affecting innovations in face-to-face interactions, underpinning the critical aspects (sharing of alternative values, coherence at individual choices level, frictions on organisational aspects, inclusion/exclusion, attitudes towards integration at territorial level), towards uncovering until to the emergence of tensions and the risks of opportunistic behaviours that might arise from their growth. Finally, a comparison with similar experiences abroad is drawn (specifically with Provence), in order to detect food for thought, potentially useful for leading regional initiativestowards transition path.