Finite Mechanical Proxies for a Class of Reducible Continuum Systems

We present the exact finite reduction of a class of nonlinearly perturbed wave equations, based on the Amann-Conley-Zehnder paradigm. By solving an inverse eigenvalue problem, we establish an equivalence between the spectral finite description derived from A-C-Z and a discrete mechanical model, a well definite finite spring-mass system. By doing so, we decrypt the abstract information encoded in the finite reduction and obtain a physically sound proxy for the continuous problem.Comment: 15 pages, 3 figure

Singular Continuation: Generating Piece-wise Linear Approximations to Pareto Sets via Global Analysis

We propose a strategy for approximating Pareto optimal sets based on the global analysis framework proposed by Smale (Dynamical systems, New York, 1973, pp. 531-544). The method highlights and exploits the underlying manifold structure of the Pareto sets, approximating Pareto optima by means of simplicial complexes. The method distinguishes the hierarchy between singular set, Pareto critical set and stable Pareto critical set, and can handle the problem of superposition of local Pareto fronts, occurring in the general nonconvex case. Furthermore, a quadratic convergence result in a suitable set-wise sense is proven and tested in a number of numerical examples.Comment: 29 pages, 12 figure

New activity pattern in human interactive dynamics

We investigate the response function of human agents as demonstrated by written correspondence, uncovering a new universal pattern for how the reactive dynamics of individuals is distributed across the set of each agent's contacts. In long-term empirical data on email, we find that the set of response times considered separately for the messages to each different correspondent of a given writer, generate a family of heavy-tailed distributions, which have largely the same features for all agents, and whose characteristic times grow exponentially with the rank of each correspondent. We furthermore show that this universal behavioral pattern emerges robustly by considering weighted moving averages of the priority-conditioned response-time probabilities generated by a basic prioritization model. Our findings clarify how the range of priorities in the inputs from one's environment underpin and shape the dynamics of agents embedded in a net of reactive relations. These newly revealed activity patterns might be present in other general interactive environments, and constrain future models of communication and interaction networks, affecting their architecture and evolution.Comment: 15 pages, 7 figure

A Polynomial Chaos Approach to Robust Multiobjective Optimization

Robust design optimization is a modeling methodology, combined with a suite of computational tools, which is aimed to solve problems where some kind of uncertainty occurs in the data or in the model. This paper explores robust optimization complexity in the multiobjective case, describing a new approach by means of Polynomial Chaos expansions (PCE). The aim of this paper is to demonstrate that the use of PCE may help and speed up the optimization process if compared to standard approaches such as Monte Carlo and Latin Hypercube sampling

Globalizzazione dell'ottica geometrica e di Fresnel: singolarità di proiezione e caustiche.

In questa tesi si affronta uno studio sistematico di rivisitazione dell'ottica ondulatoria basato sulla equazione di Helmholtz con particolari enfasi su taluni aspetti legati alla ricerca di soluzioni globalmente definite e alle ostruzioni all'esistenza di esse: le caustiche. Le strutture simplettiche si sono rilevate il contesto adeguato nel quale inserire questo studio

Global search perspectives for multiobjective optimization

Extending the notion of global search to multiobjective optimization is far than straightforward, mainly for the reason that one almost always has to deal with infinite Pareto optima and correspondingly infinite optimal values. Adopting Stephen Smale's global analysis framework, we highlight the geometrical features of the set of Pareto optima and we are led to consistent notions of global convergence. We formulate then a multiobjective version of a celebrated result by Stephens and Baritompa, about the necessity of generating everywhere dense sample sequences, and describe a globally convergent algorithm in case the Lipschitz constant of the determinant of the Jacobian is known

Lack of critical phase points and exponentially faint illumination

The Stationary Phase Principle (SPP) states that in the computation of oscillatory integrals, the contributions of non-stationary points of the phase are smaller than any power n of 1/k, for k\u2192 1e. Unfortunately, SPP says nothing about the possible growth in the constants in the estimates with respect to the powers n. A quantitative estimate of oscillatory integrals with amplitude and phase in the Gevrey classes of functions shows that these contributions are asymptotically negligible, like exp( 12akb ), a, b > 0. An example in Optics is given

A Pareto–Pontryagin Maximum Principle for Optimal Control

In this paper, an attempt to unify two important lines of thought in applied optimization is proposed. We wish to integrate the well-known (dynamic) theory of Pontryagin optimal control with the Pareto optimization (of the static type), involving the maximization/minimization of a non-trivial number of functions or functionals, Pontryagin optimal control offers the definitive theoretical device for the dynamic realization of the objectives to be optimized. The Pareto theory is undoubtedly less known in mathematical literature, even if it was studied in topological and variational details (Morse theory) by Stephen Smale. This reunification, obviously partial, presents new conceptual problems; therefore, a basic review is necessary and desirable. After this review, we define and unify the two theories. Finally, we propose a Pontryagin extension of a recent multiobjective optimization application to the evolution of trees and the related anatomy of the xylems. This work is intended as the first contribution to a series to be developed by the authors on this subject