Fabio Raciti

Remarks on infinite dimensional duality

We present an improvement of a recent duality theorem and a new result which stresses the fact that the strong duality, without assumptions on the interior of the ordering cone, is related to the normal cone.

On a Class of Random Variational Inequalities on Random Sets

We study a class of random variational inequalities on random sets and give measurability, existence, and uniqueness results in a Hilbert space setting. In the special case where the random and the deterministic variables are separated, we present a discretization technique based on averaging and truncation, prove a Mosco convergence result for the feasible random set, and establish norm convergence of the approximation procedure.

Spin and electron structure

The recent literature shows a renewed interest, with various independent approaches, in the classical theories for spin. Considering the possible interest of those results, at least for the electron case, we purpose in this paper to explore their physical and mathematical meaning, by the natural and powerful language of Clifford algebras (which, incidentally, will allow us to unify those different approaches). In such theories, the ordinary electron is in general associated to the mean motion of a point–like “constituent” Q, whose trajectory is a cylindrical helix. We find, in particular, that the object Q obeys a new, non-linear Dirac–like equation, such that —when averaging over an internal cycle (which corresponds to linearization)— it transforms into the ordinary Dirac equation (valid for the electron as a whole). PACS nos.: 03.70.+k ; 11.10.Qr ; 14.60.Cd . 0 † Work partially supported by INFN, CNR, MURST; by CAPES, CNPq, FAPESP; and by the Slovenian Ministry of Science and Technology. 0∗ On leave from the J.Stefan Institute; University of Ljubljana; 61111–Ljubljana; Slovenia. 0∗∗ Also: Facolta di Ingegneria, Universita statale di Bergamo, 24044 Dalmine (BG), Italy; and C.C.S., State University at Campinas, 13083-970 Campinas, S.P.; Brazil. 0∗∗∗ On leave from Departamento de Matematica Aplicada — Imecc; UNICAMP; 13084–Campinas, S.P.; Brazil.Abstract The recent literature shows a renewed interest, with various independent approaches, in the classical models for spin. Considering the possible interest of those results, at least for the electron case, we purpose in this paper to explore their physical and mathematical meaning, by the natural and powerful language of Clifford algebras (which, incidentally, will allow us to unify those different approaches). In such models, the ordinary electron is in general associated to the mean motion of a point-like “constituent” Q , whose trajectory is a cylindrical helix. We find, in particular, that the object Q obeys a new, non-linear Dirac-like equation, such that — when averaging over an internal cycle (which corresponds to linearization) — it transforms into the ordinary Dirac equation (valid, of course, for the electron as a whole).

Random equilibrium problems on networks

We apply the theory of random variational inequalities to study a class of random equilibrium problems on networks. By means of two classical test problems we treat the case of random demand and random cost and compute mean values and variances for two special probability distributions.

On general infinite dimensional complementarity problems

The aim of this paper is to present a way to study directly generalized complementarity problems in normed spaces. By means of new results on infinite dimensional Lagrange theory we show some optimality conditions which reduce the study of the problems to the one of suitable systems of equalities and inequalities.

More about Tunnelling Times, the Dwell Time and the “Hartman Effect”

In a recent review paper [Phys. Reports 214 (1992) 339] we proposed, within conventional quantum mechanics, new definitions for the sub-barrier tunnelling and reflection times. Aims of the present paper are : i) presenting and analysing the results of various numerical calculations (based on our equations) on the penetration and return times , , during tunnelling inside a rectangular potential barrier, for various penetration depths x f ; ii) putting forth and discussing suitable definitions, besides of the mean values, also of the variances (or dispersions) Dτ T and Dτ R for the time durations of transmission and reflection processes ; iii) mentioning, moreover, that our definition for the average transmission time results to constitute an improvement of the ordinary dwell-time τ -Dw formula : iv) commenting, at last, on the basis of our new numerical results, upon some recent criticism by C.R. Leavens. We stress that our numerical evaluations confirm that our approach implied, and implies, the existence of the Hartman effect : an effect that in these days (due to the theoretical connections between tunnelling and evanescent-wave propagation) is receiving -at Cologne, Berkeley, Florence and Vienna- indirect, but quite interesting, experimental verifications. Eventully, we briefly analyze some other definitions of tunnelling times.

Equilibrium conditions and vector variational inequalities: a complex relation

Since a look to the present literature shows that the notion of vector equilibrium is not univocally defined, the relationship between vector equilibrium problems and vector variational inequalities deserves a specific analysis, deeper than in the scalar case. We analyse and compare various definitions and clarify how they depend on the modelling of the problem under consideration. Furthermore, we show that at least one vector variational inequality formulation of the problem is possible, which is equivalent to a Wardrop-type equilibrium condition. The analysis is illustrated through the concrete example of a multiclass and multiobjective supply demand network.

On the inverse problem of identifying Lamé coefficients in linear elasticity

An output least-squares type functional is employed to identify the Lame parameters in linear elasticity. To be able to identify even the discontinuous Lame parameters the regularization is performed by the BV-seminorm. Finite element discretization is used and convergence analysis is given. Numerical examples are given to show the feasibility of the approach.

Some equilibrium problems under uncertainty and random variational inequalities

In this paper we describe some nonlinear equilibrium problems under uncertainty arising from economics and operations research. In particular we treat Wardrop equilibria in traffic networks. We show how the theory of monotone random variational inequalities, where random variables occur both in the operator and the constraint set, can be applied to model these problems.Therefore in this contribution we introduce the topic of random variational inequalities and present some of our recent results in this field. In particular, we treat the more structured case where a finite Karhunen-Loève expansion leads to a separation of the random and the deterministic variables. Here we describe a norm convergent approximation procedure based on averaging and truncation. We illustrate this procedure by means of some small sized numerical examples.

Time-dependent Variational Inequalities and Applications to Equilibrium Problems

We present two different applications of time-dependent variational inequalities. First we propose a new model of time-dependent distributed markets networks which includes delay effects. Afterwards, we deal with time-dependent and elastic models of transportation networks.