Marco Laudato

Acoustic Metamaterials Based on Local Resonances: Homogenization, Optimization and Applications

The aim of this review is to give an overview of techniques and methods used in the modeling of acoustic and elastic metamaterials. Acoustic and elastic metamaterials are man-made materials which present exotic properties capable to modify and drive wave propagation. In particular in this work we will focus on locally resonant microstructures. Such metamaterials are based on local resonances of the internal structure, the dimensions of which are much smaller than the wavelengths of the waves under analysis. We will consider the seminal papers in the fields to grasp the most important ideas used to develop locally resonant metamaterials, such as homogenization techniques and optimization topology. Finally, we will discuss some interesting application to clarify the aforementioned methods.

Dynamical Vector Fields on the Manifold of Quantum States

In this paper we shall consider the stratified manifold of quantum states and the vector fields which act on it. In particular, we show that the infinitesimal generator of the GKLS evolution is com...

Differential calculus on manifolds with boundary applications

This paper contains a set of lecture notes on manifolds with boundary and corners, with particular attention to the space of quantum states. A geometrically inspired way of dealing with these kind of manifolds is presented, and explicit examples are given in order to clearly illustrate the main ideas.

A 1D Continuum Model for Beams with Pantographic Microstructure: Asymptotic Micro-Macro Identification and Numerical Results

In the standard asymptotic micro-macro identification theory, starting from a De Saint-Venant cylinder, it is possible to prove that, in the asymptotic limit, only flexible, inextensible, beams can be obtained at the macro-level. In the present paper we address the following problem: is it possible to find a microstructure producing in the limit, after an asymptotic micro-macro identification procedure, a continuum macro-model of a beam which can be both extensible and flexible? We prove that under certain hypotheses, exploiting the peculiar features of a pantographic microstructure, this is possible. Among the most remarkable features of the resulting model we find that the deformation energy is not of second gradient type only because it depends, like in the Euler beam model, upon the Lagrangian curvature, i.e. the projection of the second gradient of the placement function upon the normal vector to the deformed line, but also because it depends upon the projection of the second gradient of the placement on the tangent vector to the deformed line, which is the elongation gradient. Thus, a richer set of boundary conditions can be prescribed for the pantographic beam model. Phase transition and elastic softening are exhibited as well. Using the resulting planar 1D continuum limit homogenized macro-model, by means of FEM analyses, we show some equilibrium shapes exhibiting highly non-standard features. Finally, we conceive that pantographic beams may be used as basic elements in double scale metamaterials to be designed in future.

A pedagogical intrinsic approach to relative entropies as potential functions of quantum metrics: The q–z family

Abstract The so-called q -z-Renyi Relative Entropies provide a huge two parameter family of relative entropies which include almost all well-known examples of quantum relative entropies for suitable values of the parameters. In this paper we consider a log-regularized version of this family and use it as a family of potential functions to generate covariant ( 0 , 2 ) symmetric tensors on the space of invertible quantum states in finite dimensions. The geometric formalism developed here allows us to obtain the explicit expressions of such tensor fields in terms of a basis of globally defined differential forms on a suitable unfolding space without the need to introduce a specific set of coordinates. To make the reader acquainted with the intrinsic formalism introduced, we first perform the computation for the qubit case, and then, we extend the computation of the metric-like tensors to a generic n -level system. By suitably varying the parameters q and z , we are able to recover well-known examples of quantum metric tensors that, in our treatment, appear written in terms of globally defined geometrical objects that do not depend on the coordinates system used. In particular, we obtain a coordinate-free expression for the von Neumann–Umegaki metric, for the Bures metric and for the Wigner–Yanase metric in the arbitrary n -level case.

Hamilton Principle in Piola’s work published in 1825

In this paper we assess how a version of Hamilton Principle is formulated by Piola in his Memoir composed between 1822 and 1824 and published in 1825. The Italian title being Sull’applicazione de’ principj della Meccanica Analitica del Lagrange ai principali problemi, in this volume its English translation was chosen to be: On the application of the principles of Analytical Mechanics by Lagrange to the main problems. After a detailed discussion, we observe that also in this case the Stigler’s Law of Eponymy is verified: Hamilton Principle was not formulated at first by Hamilton. Also, some conclusions about the epic vision of science (that is the belief that science is developed by solitary heroes) are drawn and the due tribute to the great contributions by Lagrange, Piola and Hamilton is paid. In particular one of the first appearances of matrix notation, apparently used by Piola, is described and an observation about the first appearance of Piola stress in Lagrange works due to Piola himself is made.

On the Applications of Principles of Analytical Mechanics by Lagrange to the Principal Problems

In this chapter it is presented the translation of the opera “Sull’applicazione de’ principj della Meccanica Analitica del Lagrange ai principali problemi” by Gabrio Piola. It was presented for the contest for the prize and crowned by the I.R. Istituto di Scienze the 4th of October 1824.