Eduardo Pascali

On the semicontinuity and the relaxation for integrals with respect to the lebesgue measure added to integrals with respect to a Radon measure

SummaryWe give conditions on the functionsf, g and the Radon measure μ such that the functional

A dimension reduction technique for two-mode non-convex fuzzy data

F(u) = \int\limits_\Omega {f(x,u,Du) dx} + \int\limits_{\bar \Omega } {g(x,\tilde u)d\mu }

Limits of Obstacle Problems for the Area Functional

is Lp(Ω)-lower semicontinuous on H1,p(Ω) (p⩾1). We study also the relaxation problem.

Relaxation of the non-parametric plateau problem with an obstacle

LR-fuzzy numbers are widely used in Fuzzy Set Theory applications based on the standard definition of convex fuzzy sets. However, in some empirical contexts such as, for example, human decision making and ratings, convex representations might not be capable to capture more complex structures in the data. Moreover, non-convexity seems to arise as a natural property in many applications based on fuzzy systems (e.g., fuzzy scales of measurement). In these contexts, the usage of standard fuzzy statistical techniques could be questionable. A possible way out consists in adopting ad-hoc data manipulation procedures to transform non-convex data into standard convex representations. However, these procedures can artificially mask relevant information carried out by the non-convexity property. To overcome this problem, in this article we introduce a novel computational definition of non-convex fuzzy number which extends the traditional definition of LR-fuzzy number. Moreover, we also present a new fuzzy regression model for crisp input/non-convex fuzzy output data based on the fuzzy least squares approach. In order to better highlight some important characteristics of the model, we applied the fuzzy regression model to some datasets characterized by convex as well as non-convex features. Finally, some critical points are outlined in the final section of the article together with suggestions about future extensions of this work.

Characterization of fuzzy topologies from neighborhoods of fuzzy points

Fuzzy modeling and fuzzy statistics provide useful tools for handling empirical situations affected by vagueness and imprecision in the data. Several fuzzy statistical models and methods (e.g., fuzzy regression, fuzzy principal component analysis, fuzzy clustering) have been developed over the years. Generally the standard LR-fuzzy data representation has been used in these methods. However, several empirical contexts, such as human ratings and decision making, may show more complex fuzzy structures which cannot be successfully modeled by the LR representation. In all these cases another type of fuzzy data representation, the so-called LHIR representation, should be preferred instead. In particular, this novel representation allows to handle with fuzzy data which are characterized by non-convex membership functions. In this paper, we address the problem of summarizing large datasets characterized by two-mode non-convex fuzzy data. We introduce a novel dimension reduction technique (NCFCA) based on the framework of Component Analysis and Least squares programming. Finally, to better highlight some important characteristics of the proposed model, we apply NCFCA to three empirical datasets concerning behavioral and socio-economic issues.

Uniqueness of the one-dimensional bounce problem as a generic property in

SummaryWe prove the L1-convergence of a sequence of boundary value problems for first order P.D.E.s on sets of finite perimeter (in the sense of De Giorgi [13]) to a boundary value problem of transmission type. The previous result is applied to prove also the convergence for the prime integrals equations connected with the one-dimensional bounce problem.This approach allows to deepen previous results on (see [4], [6], [7], [8]) by proving uniqueness for.

Existence theorems for systems of nonlinear integro-differential equations

We study the general form of the limit, in the sense of Γ-convergence, of a sequence of variational problems for the area functional with one-side obstacles.