This volume contains the proceedings of the international workshop Variational Problems in Materials Science. Coverage includes the study of BV vector fields, path functionals over Wasserstein spaces, variational approaches to quasi-static evolution, free-discontinuity problems with applications to fracture and plasticity, systems with hysteresis or with interfacial energies, evolution of interfaces, multi-scale analysis in ferromagnetism and ferroelectricity, and much more.

In recent years many researchers in material science have focused their attention on the study of composite materials, equilibrium of crystals and crack distribution in continua subject to loads. At the same time several new issues in computer vision and image processing have been studied in depth. The understanding of many of these problems has made significant progress thanks to new methods developed in calculus of variations, geometric measure theory and partial differential equations. In particular, new technical tools have been introduced and successfully applied. For example, in order to describe the geometrical complexity of unknown patterns, a new class of problems in calculus of variations has been introduced together with a suitable functional setting: the free-discontinuity problems and the special BV and BH functions. The conference held at Villa Olmo on Lake Como in September 1994 spawned successful discussion of these topics among mathematicians, experts in computer science and material scientists.

This volume contains the Proceedings of the International Workshop "Variational Methods For Discontinuous Structures," held at Villa Erba Antica (Cernobbio) on the Lago di Como, July 4-6, 2001. The workshop was jointly organized by the Dipartimento di Matematica Francesco Brioschi of Milano Politecnico and the International School for Advanced Studies (SISSA) of Trieste. In past years the calculus of variations faced mainly the study of continuous structures, particularly problems with smooth solutions. One of the deepest and more delicate problems was the regularity of weak solutions. More recently, new sophisticated tools have been introduced in order to study discontinuities. In many variational problems solutions develop singularities, and sometimes the most interesting part of a solution is the singularity itself. The conference intended to focus on recent developments in this direction. Some of the talks were devoted to differential or variational modelling of image segmentation, occlusion and textures synthesizing in image analysis, variational description of micro-magnetic materials, dimension reduction and structured deformations in elasticity and plasticity, phase transitions, irrigation and drainage, evolution of crystalline shapes. In most cases theoretical and numerical analysis of these models were provided. Other talks were dedicated to specific problems of the calculus of variations: variational theory of weak or lower-dimensional structures, optimal transport problems with free Dirichlet regions, higher order variational problems, symmetrization in the BV framework. This volume contains contributions by 12 of the 16 speakers invited to deliver lectures in the workshop. Most of the contributions present original results in fields which are rapidly evolving at present.

This volume contains the Proceedings of the International Workshop Variational Methods For Discontinuous Structures, which was jointly organized by the Dipar timento di Matematica Francesco Brioschi of Milano Politecnico and the Interna tional School for Advanced Studies (SISSA) of Trieste. The Conference took place at Villa Erba Antica (Cernobbio) on the Lago di Como on July 4- 6, 2001. In past years the calculus of variations faced mainly the study of continuous structures, say particularly problems with smooth solutions. One of the deepest and more delicate problems was the regularity of weak solutions. More recently, new sophisticated tools have been introduced in order to study discontinuities: in many variational problems solutions develop singularities, and sometimes the most interesting part of a solution is the singularity itself. The conference intended to focus on recent developments in this direction. Some of the talks were devoted to differential or variational modelling of image segmentation, occlusion and textures synthesizing in image analysis, varia tional description of micro-magnetic materials, dimension reduction and structured deformations in elasticity and plasticity, phase transitions, irrigation and drainage, evolution of crystalline shapes; in most cases theoretical and numerical analysis of these models were provided. viii Preface Other talks were dedicated to specific problems of the calculus of variations: variational theory of weak or lower-dimensional structures, optimal transport prob lems with free Dirichlet regions, higher order variational problems, symmetrization in the BV framework."

In recent years many researchers in material science have focused their attention on the study of composite materials, equilibrium of crystals and crack distribution in continua subject to loads. At the same time several new issues in computer vision and image processing have been studied in depth. The understanding of many of these problems has made significant progress thanks to new methods developed in calculus of variations, geometric measure theory and partial differential equations. In particular, new technical tools have been introduced and successfully applied. For example, in order to describe the geometrical complexity of unknown patterns, a new class of problems in calculus of variations has been introduced together with a suitable functional setting: the free-discontinuity problems and the special BV and BH functions. The conference held at Villa Olmo on Lake Como in September 1994 spawned successful discussion of these topics among mathematicians, experts in computer science and material scientists.

Questo volume fornisce una introduzione all analisi dei sistemi dinamici discreti. La materia e presentata mediante un approccio unitario tra il punto di vista modellistico e quello di varie discipline che sviluppano metodi di analisi e tecniche risolutive: Analisi Matematica, Algebra Lineare, Analisi Numerica, Teoria dei Sistemi, Calcolo delle Probabilita. All esame di un ampia serie di esempi, segue la presentazione degli strumenti per lo studio di sistemi dinamici scalari lineari e non lineari, con particolare attenzione all analisi della stabilita. Si studiano in dettaglio le equazioni alle differenze lineari e si fornisce una introduzione elementare alle trasformate Z e DFT. Un capitolo e dedicato allo studio di biforcazioni e dinamiche caotiche. I sistemi dinamici vettoriali ad un passo e le applicazioni alle catene di Markov sono oggetto di tre capitoli. L esposizione e autocontenuta: le appendici tematiche presentano prerequisiti, algoritmi e suggerimenti per simulazioni al computer. Ai numerosi esempi proposti si affianca un gran numero di esercizi, per la maggior parte dei quali si fornisce una soluzione dettagliata. Il volume e indirizzato principalmente agli studenti di Ingegneria, Scienze, Biologia ed Economia. Questa terza edizione comprende l aggiornamento di vari argomenti, l aggiunta di nuovi esercizi e l ampliamento della trattazione relativa alle matrici positive ed alle loro proprieta utili nell analisi di sistemi, reti e motori di ricerca."