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.     

A Macroscopic System with Undamped Periodic Compressional Oscillations

A class of macroscopic systems is described which have the remarkable feature that they can sustain undamped compressional radial oscillations. They consist of an arbitrary number of particles confined by a harmonic potential and interacting among themselves through conservative forces scaling as the inverse cube of distances. The radial oscillation leads to a variation of the thermodynamic quantities characterizing the system. The system therefore does not approach equilibrium, since the (macroscopic) amplitude of the oscillation does not decrease as time goes to infinity. The oscillation is harmonic and isochronous, that is, its frequency is fixed and independent of the initial condition. These results hold independently of the dimension of the system and are also valid in the quantal context.     

Generations of Monic Polynomials such that the Coefficients of Each Polynomial of the Next Generation Coincide with the Zeros of a Polynomial of the Current Generation,and New Solvable Many-Body Problems

The notion of generations of monic polynomials such that the coefficients of each polynomial of the next generation coincide with the zeros of a polynomial of the current generation is introduced, and its relevance to the identification of endless sequences of new solvable many-body problems “of goldfish type” is demonstrated.     

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.     

Bianchi cosmologies with anisotropic matter: Locally rotationally symmetric models

The dynamics of cosmological models with isotropic matter sources (perfect fluids) is extensively studied in the literature; in comparison, the dynamics of cosmological models with anisotropic matter sources is not. In this paper we consider spatially homogeneous locally rotationally symmetric solutions of the Einstein equations with a large class of anisotropic matter models including collisionless matter (Vlasov), elastic matter, and magnetic fields. The dynamics of models of Bianchi types I, II, and IX are completely described; the two most striking results are the following. (i) There exist matter models, compatible with the standard energy conditions, such that solutions of Bianchi type IX (closed cosmologies) need not necessarily recollapse; there is an open set of forever expanding solutions. (ii) Generic type IX solutions associated with a matter model like Vlasov matter exhibit oscillatory behavior toward the initial singularity. This behavior differs significantly from that of vacuum/perfect fluid cosmologies; hence “matter matters”. Finally, we indicate that our methods can probably be extended to treat a number of open problems—in particular, the dynamics of Bianchi type VIII and Kantowski-Sachs solutions.     

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.     

Admissible perturbations of α‐ψ‐pseudocontractive operators: convergence theorems

We prove some convergence theorems for α‐ψ‐pseudocontractive operators in real Hilbert spaces, by using the concept of admissible perturbation. Our results extend and complement some theorems in the existing literature. Copyright © 2016 John Wiley & Sons, Ltd.