Hardy inequalities with optimal constants and remainder terms

We show that the classical Hardy inequalities with optimal constants in the Sobolev spaces and in higher-order Sobolev spaces on a bounded domain can be refined by adding remainder terms which involve norms. In the higher-order case further norms with lower-order singular weights arise. The case being more involved requires a different technique and is developed only in the space .

     

Integral cohomology of the Milnor fibre of the discriminant bundle associated with a finite Coxeter group

Let W be a finite Coxeter group generated by real reflections in a complex vector space. We compute the integral cohomology of the Milnor fibre of the discriminant bundle $\Delta :{\mathbb{C}}^n/W\to \mathbb{C}$, together with the action of the monodromy, for the whole list of exceptional groups. Here Δ is the map induced by the square of the polynomial defining the arrangement of reflection hyperplanes of W. The computation is equivalent to that of the cohomology, with suitable local coefficients, of the corresponding Artin group. These computations complete, for the exceptional cases, those performed by De Concini et al. for rational coefficients. To cite this article: F. Callegaro, M. Salvetti, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

A Sharp Upper Bound for the¶Torsional Rigidity of Rods¶by Means of Web Functions

Using web functions, we approximate the Dirichlet integral which represents the torsional rigidity of a cylindrical rod with planar convex cross-section Ω. To this end, we use a suitably defined piercing function, which enables us to obtain bounds for both the approximate and the exact value of the torsional rigidity. When Ω varies, we show that the ratio between these two values is always larger than ¾; we prove that this is a sharp lower bound and that it is not attained. Several extremal cases are also analyzed and studied by numerical methods.     

Wide Oscillation Finite Time Blow Up for Solutions to Nonlinear Fourth Order Differential Equations

We give sufficient conditions for local solutions to some fourth order semilinear ordinary differential equations to blow up in finite time with wide oscillations, a phenomenon not visible for lower order equations. The result is then applied to several classes of semilinear partial differential equations in order to characterize the blow up of solutions including, in particular, its applications to a suspension bridge model. We also give numerical results which describe this oscillating blow up and allow us to suggest several open problems and to formulate some related conjectures.     

On damage detection by continuous dynamic monitoring in wind-excited suspension bridges

The paper discusses the application of dynamic methods for damage detection in the main cables of suspension bridges, using data continuously recorded under wind excitation through permanent monitoring systems and automated operational modal analysis. A continuum model for predicting the vertical aeroelastic response of wind-excited damaged suspension bridges is formulated and presented at first. The model shows that, for a real sample bridge, typical variations of mean wind speed produce variations of natural frequencies, due to aeroelastic effects, that are more significant than those produced by a small damage. A possible solution to this issue, proposed in the paper, consists of removing the dependence on the excitation source by calculating frequency shifts considering frequencies, in reference and damaged states, associated to approximately the same mean wind speed. This task and the necessary estimation of frequency shifts through a statistical analysis of identified natural frequencies outline the need for a continuous dynamic monitoring. The analytical model is finally employed for generating dynamic wind response data that are successively processed by means of an advanced automated modal identification tool. Although based on the simplifications inherently contained in the analytical model, the results show that frequency shifts caused by a relatively small damage can be accurately estimated from response data recorded under wind excitation with a reasonable number of data sets.     

Fixed points of commuting holomorphic mappings other than the Wolff point

Let be the unit disc of and let be such that . For 1$">, let . We study the behavior of on . In particular, we prove that . As a consequence, besides conditions for , we prove a conjecture of C. Cowen in case and are univalent mappings.

     

Electrolytic Oxidation of Thiocyanate and Selenocyanate Salts and the Photoelectrochemical Effect.

New photosensitive materials can be obtained by electrochemical oxidation of thiocyanate and selenocyanate salts. Using (K,Na)SCN eutectic melt, the formation of an electrodic deposit, with photoelectrochemical properties, has already been reported by us. To improve the photoelectrochemical characteristics of the deposit, futher investigations were carried out in the following sistems: selenocyanate ammoniate, KSCH-Acetamide eutectic mixture and KSCN ethylene carbonate solution:. Attempts to obtain a massive deposit in the ammoniate solution were unsuccessful due to ammonia oxidation. Measurements performed on the other KSCN systems show that temperature is a parameter of remarkable importance. In fact, lowering the temperature one obtains a decrease on formation of parathiocyanogen on behalf of the polytrithiocyanogen, the species that gives rise to the photoeffect.     

X-ray photoelectron spectra of Pd(II) and Pt(II) complexes with 1,3-thiazolidine-2-thione. A quantum mechanics study on the free ligand

Metal complexes of general formula M(ttz)2X2 (with M= Pd(II) or Pt(II); X = Cl or Br; ttz = 1,3-thiazolidine-2-thione) have been synthetized as crystalline compounds and studied by means of X-ray Photoelectron Spectroscopy (XPS). The chemical shift of core level signals showed that ttz is bonded to the metal through the thioketonic sulphur atom and that electronic charge redistribution in the ligand takes place after complexation. No metal-nitrogen bonds are present. This is consistent with the results of all the quantum mechanical models according to which hydrogen is bound to nitrogen, even in the hydrogen bonded complex, making the latter rather unavailable to coordination