Synthesis of chromium(0) and molybdenum(0) bis (eta(6)-arene) derivatives and their monoelectronic oxidation to [M(eta(6)-arene)(2)](+) cations

M(eta(6)-arene)(2) species (M = Cr, arene = 1,3,5-Me(3)C(6)H(3); M = Mo, arene = 1,3,5-Me(3)C(6)H(3), 1,3,5-(i)Pr(3)C(6)H(3)), have been prepared by a modified Fischer-Hafner synthesis or by metal vapour techniques. The reaction of Cr(eta(6)-1,3,5-Me(3)C(6)H(3))(2) with the fulvene derivatives pentacarbomethoxycyclopentadiene (pcmcpH), 1-benzoyl-6-hydroxy-6-phenylfulvene (dbcpH), or 1-benzoyl-3-nitro-6-hydroxy-6-phenylfulvene (dbncpH) proceeds with evolution of dihydrogen and formation of the ionic derivatives [Cr(eta(6)-1,3,5-Me(3)C(6)H(3))(2)][E], where E = pcmcp, dbcp, or dbncp. Mo(eta(6)-arene)(2) derivatives (arene = toluene, 1,3,5-Me(3)C(6)H(3), 1,3,5-(i)Pr(3)C(6)H(3)) are oxidized to [Mo(eta(6)-arene)(2)](+) by pcmcpH. The crystal and molecular structures of [M(eta(6)-1,3,5-R(3)C(6)H(3))(2)][pcmcp] (M = Cr, R = Me; M = Mo, R = Me, (i)Pr) have been solved by X-ray single crystal diffraction.     

A fixed point theorem for a family of mappings in a fuzzy metric space

In this paper we give a common fixed point theorem for a family of mappings of a G-complete fuzzy metric space (X, M, *) into itself. From this result we deduce a common fixed point theorem for a family of mappings of a complete metric space (X, d) into itself. Supported by University of Palermo.     

Three Solvable Many-Body Problems in the Plane

Three solvable many-body problems in the plane are introduced. They are characterized by equations of motion of Newtonian type ('mass times acceleration equals force'), and are solvable for an arbitrary number n of particles. The equations of motion are rotation-invariant. The forces are velocity-dependent. These models give rise to a rich phenomenology of motions, whose detailed exploration is postponed to future papers.     

L'integrale multiplo del Calcolo delle Variazioni ed il problema dell'approssimazione delle funzioni

Sunto Diamo un teorema di approssimazione per l'integrale multiplo del Calcolo delle Variazioni in forma ordinaria, ponendoci nella classe B _λ diC. B. Morrey. Lavaro eseguito nel Seminario di Analisi Matematica della Università di Palermo.     

New C-integrable and S-integrable systems of nonlinear partial differential equations

A technique to identify new C-integrable and S-integrable systems of nonlinear partial differential equations is reported, with two representative examples displayed and tersely discussed.     

Isospectral matrices and classical polynomials

The matrices of order n defined, in terms of the n arbitrary numbers x_j, by the formulae $\text{X=}\text{diag}\text{(x}{}_{\mathrm{j}}\text{)}\text{and}\text{Z}{}_{\mathrm{jk}}\text{=δ}{}_{\mathrm{jk}}\text{∑′}\text{l=1}\text{n}\text{(x}{}_{\mathrm{j}}\text{?x}{}_{\mathrm{l}}\text{)}{}^{\mathrm{?1}}\text{+(1?δ}{}_{\mathrm{jk}}\text{(x}{}_{\mathrm{j}}\text{?x}{}_{\mathrm{k}}\text{)}{}^{\mathrm{?1}}$, are representations of the multiplicative operator ξ and of the differential operator d/dξ in a space spanned by the polynomials in ξ of degree less than n. This elementary fact implies a number of remarkable formulae involving these matrices, including novel representations of the classical polynomials.     

A local notion of absolute continuity in ℝ n

We consider the notion ofp, δ-absolute continuity for functions of several variables introduced in [2] and we investigate the validity of some basic properties that are shared by absolutely continuous functions in the sense of Maly. We introduce the classδ−BV _loc ^p (Ω, ℝ ^m ) and we give a characterization of the functions belonging to this class. Supported by MURST of Italy.