Composition of generalized derivations as a Lie derivation

Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, F and G non-zero generalized derivations of R. If the composition (FG) acts as a Lie derivation on R, then (FG) is a derivation of R and one of the following holds: there exist α C and a U such that F(x)=[a,x] and G(x)=α x, for all x R; G is an usual derivation of R and there exists α C such that F(x)=α x, for all x R; there exist α, β C and a derivation h of R such that F(x)=α x+h(x), G(x)=β x, for all x R, and α β+h(β)=0. Moreover in this case h is not an inner derivation of R; there exist a\u27, c\u27 U such that F(x)=a\u27x, G(x)=c\u27x, for all x R, with a\u27c\u27=0; there exist b\u27, q\u27 U such that F(x)=xb\u27, G(x)=xq\u27, for all x R, with q\u27b\u27=0; there exist c\u27, q\u27 U, η, γ C such that F(x)=η (xq\u27-c\u27x)+γ x, G(x)=c\u27x+xq\u27, for all x R, with γ c\u27-η c\u272=-γ q\u27-η q\u272

Solar radiation mapping at the micro-urban scale using GIS

The paper describes the solar radiation mapping technique used in the application of ELaR, a methodology and integrated tools to support sustainable design options in rehabilitation projects. In particular the paper explains how Geographic Information Systems can be used to map the solar radiation incident on the roof and facades of existing buildings at the micro-urban scale and how such maps can help to facilitate the diffusion of passive heating and cooling techniques in the energetic/environmental upgrading of existing buildings. ELaR stands for Ecodynamic Land Register, a tool to help designer to evaluate sustainability through energy and material flows analysis (Daly H.E. 1990). The specific information system developed is based on the geometric structure of the land cadastre and allows to evaluate different design options in term of amount of input and output flows activated to sustain the activities related to building uses. From the practical point of view the methodology is based on the use of two specific synergetic tools: - A Geographic Information System, with the main aim to elaborate specific thematic maps, which represent the existing environmental conditions of the local area; such maps include information on potential local renewable resources and on the geometrical configuration of the existing buildings. Particular attention has been paid to the estimation of solar radiation incident on the facades and top of the existing buildings by using digital elevation models (DEM, raster resolution 0,5m / pixel). - Spreadsheets, linked to the thematic maps on the renewable local potential, to define the different design options in terms of input and output flows related to the land parcel where the building stands. These quantities are expressed in terms of specific impact indicators such as, the CO2-equivalent emissions, primary renewable and non-renewable energy

STRONG COMMUTATIVITY PRESERVING GENERALIZED DERIVATIONS ON MULTILINEAR POLYNOMIALS

WOS: 000419972300007Let R be a non-commutative prime ring of characteristic different from 2, with right Utumi quotient ring U and extended centroid C and let F and G be generalized derivations of R such that F(x)G(y) - F(y)G(x) = [x, y], for all x, y is an element of S, where S is a subset of R. Here we will discuss the following cases: (a) S = [R, R]; (b) S = L, where L is a non-central Lie ideal of R; (c) S = f (R), where f (R) is the set of all evaluations of a non-central multilinear polynomial f (x(1), . . . , x(n)) on R. In all cases, if R does not satisfy s4(x(1), . . . , x(1)), the standard polynomial identity on 4 non-commuting variables, then there exist s, c is an element of U such that F (x) = xs, G(x) = cx, for all x is an element of R, and sc = 1(C) (the unit of C). We also study the semiprime case