A non-isothermal Ginzburg–Landau model in superconductivity: Existence,uniqueness and asymptotic behaviour

In this paper we study a Ginzburg–Landau model which describes the behaviour of a superconducting material including thermal effects. We extend the traditional formulation of the problem, by introducing the temperature as an additional state variable. Accordingly, together with the Gor’kov–Eliashberg system, we introduce an evolution equation for the absolute temperature. We examine in detail the case which allows only variations of the concentration of superconducting electrons and of the temperature, neglecting the electromagnetic field. For this problem existence and uniqueness of the solution are shown. Finally we analyze the asymptotic behaviour of the solutions, proving that the system possesses a global attractor.     

Extension results for slice regular functions of a quaternionic variable

In this paper we prove a new Representation Formula for slice regular functions, which shows that the value of a slice regular function f at a point q=x+yI can be recovered by the values of f at the points q+yJ and q+yK for any choice of imaginary units I,J,K. This result allows us to extend the known properties of slice regular functions defined on balls centered on the real axis to a much larger class of domains, called axially symmetric domains. We show, in particular, that axially symmetric domains play, for slice regular functions, the role played by domains of holomorphy for holomorphic functions.     

Overconvergent modular sheaves and modular forms for GL 2/F

Given a totally real field F and a prime integer p which is unramified in F, we construct p-adic families of overconvergent Hilbert modular forms (of non-necessarily parallel weight) as sections of, so called, overconvergent Hilbert modular sheaves. We prove that the classical Hilbert modular forms of integral weights are overconvergent in our sense. We compare our notion with Katz’s definition of p-adic Hilbert modular forms. For F = ?, we prove that our notion of (families of) overconvergent elliptic modular forms coincides with those of R. Coleman and V. Pilloni.     

Design of a Composite Membrane with Patches

This paper is concerned with minimization and maximization problems of eigenvalues. The principal eigenvalue of a differential operator is minimized or maximized over a set which is formed by intersecting a rearrangement class with an affine subspace of finite co-dimension. A solution represents an optimal design of a 2-dimensional composite membrane Ω, fixed at the boundary, built out of two different materials, where certain prescribed regions (patches) in Ω are occupied by both materials. We prove existence results, and present some features of optimal solutions. The special case of one patch is treated in detail.     

Experimental evidence of bifurcating nonlinear normal modes in piecewise linear systems

A system with piecewise linear restoring forces, typical of damaged beams with a breathing crack, exhibits bifurcations characterized by the onset of superabundant modes in internal resonance with a significantly different shape than that of modes on a fundamental branch. A 2-DOF frame with piecewise linear stiffness is analyzed by means of an experimental investigation; the frame is forced by an harmonic base excitation and the operative modal shapes as well as the response amplitude are directly measured; the results are compared with numerical outcomes for different damping values. This study shows that the shapes and the frequencies of certain nonlinear normal modes (NNMs) of the related autonomous system strongly affect the forced response, in both the numerical and the experimental environments. Therefore, it is possible to match the NNM with the forced response of the system, leading to the prospect of identifying the severity and position of the damage from experimental tests.     

Active Longitudinal Control of Wind-Induced Oscillations of a Suspended Cable

Abstract. Active control of oscillations of a suspended cable under transversal wind, described by a constant term and a fluctuating turbulent component, is dealt with. Control of transversal in-plane and out-of-plane oscillations is performed by imposing a longitudinal motion to one support; the control law depends, linearly or quadratically, on measures of displacement and velocity of a selected point. First, the steady-state oscillations around the planar static configuration, determined by the wind constant component, under sinusoidal in-plane and out-of-plane loading are studied, showing the nonlinear behaviour in the regions of primary and parametric resonance and the effects of the control action. Afterwards, attention is focused on the response of the cable to transversal wind turbulence, whose fluctuating component is described by a random process with an assigned spectral density. The effectiveness of longitudinal control is analysed by means of numerical investigations.Sommario. Si analizza il controllo attivo delle oscillazioni di un cavo sospeso, sottoposto all'azione di un vento agente in direzione ortogonale al piano del cavo e descritto mediante un termine di carico costante ed una componente variabile di turbolenza. Il controllo delle oscillazioni piane e spaziali viene realizzato mediante spostamento longitudinale di uno degli appoggi; la legge di controllo dipende linearmente o quadraticamente da misure di spostamento e velocità di un punto prescelto. In una prima fase, vengono studiate le oscillazioni stazionarie sotto carichi sinusoidali agenti rispettivamente nel piano e fuori del piano del cavo, intorno alla configurazione di equilibrio statico determinata dalla presenza della componente costante del vento, mostrando il comportamento nonlineare nelle regioni di risonanza primaria e parametrica e gli effetti dell'azione di controllo. Successivamente, si focalizza l'attenzione sulla risposta del cavo alla componente trasversale di turbolenza del vento, la quale è descritta da un processo stocastico a media nulla con densità spettrale assegnata. L'efficacia del controllo viene analizzata attraverso un'indagine numerica.     

Spatial harmonics modelling of planar periodic segmented waveguides

A theoretical approach based on Bloch theorem and spatial harmonics has been used to investigate the propagation characteristics in planar periodic segmented waveguides. This analytical method allows to evaluate the field distribution, the effective index and the propagation losses of these structures, taking into account both the propagating and the counterpropagating field components, thus evidencing all the phenomena which can take place in: Bragg reflections, leaky waves, etc. Results for TE fields are presented and compared with those obtained using a paraxial 2D FD-BPM method and a Padé based one, showing that no more than (1, 1) order approximants are needed to provide good estimates of the device characteristics.