Eigenfunction expansions for singular Schrodinger operators

We extend Ikebe's theory of eigenfunctions to a class of Schrodinger operators H = - + V on L ² (IR³), where the potential V is replaced by a measure which need not be absolutely continuous with respect to Lebesgue measure. Applications include the proof of the existence and completeness of the wave operators for a free particle which is partly reflected and partly transmitted by a compact 2-dimensional surface.     

On the rigidity of certain surfaces with folds and applications to shell theory

In the asymptotic theory of thin elastic shells the rigidity of the mid-surface with kinematic boundary conditions plays an important role. Rigidity is understood in the sense of infinitesimal (linearized) rigidity, i.e., the displacements vanish provided the variation of the first fundamental form vanishes. In this case the surface is also called “stiff”, as it cannot undergo pure bendings. A stiff surface is imperfectly stiff or perfectly stiff when the origin respectively does or does not belong to the essential spectrum of the boundary-value problem. These questions are investigated in the framework of Douglis-Nirenberg elliptic systems, with boundary conditions and transmission conditions at the folds. The index properties ensures quasi-stiffness, i.e. stiffness up to a finite number of degrees of freedom. The concept of perfect stiffness is linked with estimates for the rigidity system at an appropriate level of regularity for the data and the solution. It is proved that surfaces with folds are never perfectly stiff. It is also shown that the transmission conditions at the folds contain more conditions than those satisfying the Shapiro-Lopatinskii property. This leads to certain rigidity properties of the folds. Some examples are given.     

Parametric expansions in the linear theory of cylindrical shells

Summary Concepts of parametric expansions, leading to univocal definitions of classes of solutions (according to the behaviour of derivatives by vanishing thickness parameter) and, for each class, to a unique classification of terms in equations, are applied to cylindrical shells.In the introduction by Cicala, the fields of application of the various classes of solutions are related to loading and boundary conditions.In the subsequent section, Sassi Perino presents the second approximation formulation for rapid variation solutions for circular cylindrical shells. The first approximation form coincides with Donnell's theory. The second approximation terms take into account effects neglected by the ordinary theory, such as elongation of normal fibers and transverse shear deflections.The next section by Siniscalco contains the second approximation form of the long cylinder theory. The first approximation coincides with the Wagner-Vlasof semimembrane theory. The second approximation includes the deformation of midsurface fibers. For the circular cylinder, the resultant equations are obtained in simple form.

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Sommario Gli sviluppi parametrici, sono applicati al guscio cilindrico.Nell'introduzione Cicala mette in relazione i campi di applicabilità delle varie soluzioni con le situazioni di carico e le condizioni al contorno. Nella parte seguente, Sassi Perino ricava la formulazione di seconda approssimazione per le soluzioni di classe1/2. In prima approssimazione si ha la coincidenza con la teoria di Donnell. I termini di seconda approssimazione tengono conto di effetti che l'ordinaria teoria trascura, come l'allungamento delle fibre normali e le deformazioni dovute al taglio trasversale. Nella sezione seguente Siniscalco ricava la forma di seconda approssimazione della teoria del cilindro lungo. La prima approssimazione coincide con la teoria semimembranale di Wagner-Vlasof. Nella seconda approssimazione intervengono le deformazioni delle fibre della superficie media. Per il cilindro circolare viene ottenuta l'equazione risultante in forma semplice.

     

Superstatistics: theory and applications

Superstatistics is a superposition of two different statistics relevant to driven nonequilibrium systems with a stationary state and intensive parameter fluctuations. It contains Tsallis statistics as a special case. After briefly summarizing some of the theoretical aspects, we describe recent applications of this concept to three different physical problems, namely a) fully developed hydrodynamic turbulence, b) pattern formation in thermal convection states, and c) the statistics of cosmic rays.Received: 13 March 2003, Accepted: 27 May 2003, Published online: 9 December 2003PACS: 05.40.-a     

Nonlinear couple stress theory of elastic dielectrics with applications to dynamic deformations

A general nonlinear couple stress theory of elastic dielectrics in the presence of an electric field is formulated and applied to the problem of nonlinear oscillations of the surface of a spherical cavity in an infinite medium. It is observed that the uniform pressure increases while the azimuthal stress decreases considerably from their corresponding values in the elastic dielectric theory without couples stresses.     

Laminated shells with actuation strains: general theory and applications

Summary A general theory is proposed for laminated shells with integrated actuators. The theory incorporates dynamic effects and satisfies the compatibility condition of transverse shear stress at layer interfaces as well as on the top and bottom surfaces of the shells. The governing equations and the relevant boundary conditions are derived via Hamilton's principle. They contain only five unknown variables, as in the first-order shear-deformable shell theory. As an illustrative example, an infinitely long strip composed of a metallic layer mounted by two piezoelectric actuating layers is analysed. The results are compared with those predicted by some other existing models. Received 4 August 1997; accepted for publication 16 June 1998