Relative cohomology of polynomial mappings

 Let F be a polynomial mapping from ℂ ⁿ to ℂ ^q with n>q. We study the De Rham cohomology of its fibres and its relative cohomology groups, by introducing a special fibre F ⁻¹(∞) ``at infinity' and its cohomology. Let us fix a weighted homogeneous degree on with strictly positive weights. The fibre at infinity is the zero set of the leading terms of the coordinate functions of F. We introduce the cohomology groups H ^k (F ⁻¹(∞)) of F at infinity. These groups enable us to compute all the other cohomology groups of F. For instance, if the fibre at infinity has an isolated singularity at the origin, we prove that every weighted homogeneous basis of H ^n−q (F ⁻¹ (∞)) is a basis of all the groups H ^n−q (F ⁻¹(y)) and also a basis of the (n−q) ^th relative cohomology group of F. Moreover the dimension of H ^n−q (F ⁻¹(∞)) is given by a global Milnor number of F, which only depends on the leading terms of the coordinate functions of F. Received: 12 February 2002 / Revised version: 25 May 2002 Published online: 3 March 2003     

Alternate dissociation of the screw dislocations in a (0 0 1) buried small-angle twist boundary in silicon

The square dislocation network of a (0 0 1) buried small-angle boundary in silicon was observed by dark-field transmission electron microscopy to examine the structures of more than 100 dissociated dislocation segments. Images were taken with g = (2 2 0), using a many-beam case along the reciprocal lattice row. Dissociation occurs on alternate close-packed planes without systematic rule, although a degree of ordering is taking place. Most of the dislocation segments have lengths equal to half of the square network period. Image simulation studies revealed that their experimental contrasts cannot be explained from the usual assumption of straight dislocations running in an infinite crystal. However, if these dislocations are supposed close and parallel to a nearby free surface, a reasonable agreement is found between the micrographs and the simulated images. A three-dimensional elastic model is proposed to explain the contrasts of the dislocation network.     

A new behaviour of the interstitial element H or D in solution in rare-earth metal

The study of the resistivity of the Lu-(H or D) system makes clear certain characteristics of the α phase and indicates a change for the interstitial element in the transition range 170–400°C. On both sides of this domain an appreciable difference is observed in the properties of the solid solution. The α1 phase of the domain below 170°C has a higher resistivity and a maximum hydrogen concentration independent of the temperature. For the phase α of the domain above 400°C, the expansions both with the concentration growth and with the rise in temperature are isotropic contrary to the α1 phase.The resistivity and X-ray diffraction studies, taking into account the lattice symmetry, allow us to conclude that a change may occur in the type of interstices filled by the hydrogen atoms, tetrahedral for the α phase and octahedral with H atoms associating in pairs for the α1 phase. Among the lanthanide metals, we find that only lutetium and thulium, in uniting with H or D, have the low temperature α1 phase.     

Zigzag lines in a (001)Si low-angle twist boundary

A low angle twist boundary formed by bonding an ultrathin (001) silicon film onto a (001) silicon wafer is investigated using two-beam transmission electron microscopy to identify positively zigzag lines which separate large interfacial regions formed by square networks of 1/2? 110 ? screw misfit dislocations. An approach to the elastic field of a zigzag line is proposed from the repetitive use of angular dislocations added to a ribbon-like uniform distribution of infinitesimal dislocations parallel to a family of pure screw misfit dislocations. Theoretical and experimental images of successive triple nodes are compared to derive the unique set of Burgers vectors attached to a zigzag line. In principle, this approach can be applied to any elongated hexagonal mesh of a dislocation network.     

SO2-binding properties of cationic η6,η1-NCN-pincer arene ruthenium platinum complexes: spectroscopic and theoretical studies

The SO(2)-binding properties of a series of η(6),η(1)-NCN-pincer ruthenium platinum complexes (NCN = 2,6-bis[(dimethylamino)methyl]phenyl anion) have been studied by both UV-visible spectroscopy and theoretical calculations. When an electron-withdrawing [Ru(C(5)R(5))](+) fragment (R = H or Me) is η(6)-coordinated to the phenyl ring of the NCN-pincer platinum fragment (cf. [2](+) and [3](+), see Scheme 1), the characteristic orange coloration (pointing to η(1)- SO(2) binding to Pt) of a solution of the parent NCN-pincer platinum complex 1 in dichloromethane upon SO(2)-bubbling is not observed. However, when the ruthenium center is η(6)-coordinated to a phenyl substituent linked in para-position to the carbon-to-platinum bond, i.e. complex [4](+), the SO(2)-binding property of the NCN-platinum center seems to be retained, as bubbling SO(2) into a solution of the latter complex produces the characteristic orange color. We performed theoretical calculations at the MP2 level of approximation and TD-DFT studies, which enabled us to interpret the absence of color change in the case of [2](+) as an absence of coordination of SO(2) to platinum. We analyze this absence or weaker SO(2)-coordination in dichloromethane to be a consequence of the relative electron-poorness of the platinum center in the respective η(6)-ruthenium coordinated NCN-pincer platinum complexes, that leads to a lower binding energy and an elongated calculated Pt-S bond distance. We also discuss the effects of electrostatic interactions in these cationic systems, which also seems to play a destabilizing role for complex [2(SO(2))](+).     

pH‐Responsive Relaxometric Behaviour of Coordination Polymer Nanoparticles Made of a Stable Macrocyclic Gadolinium Chelate

Lanthanide‐containing nanoscale particles have been widely explored for various biomedical purposes, however, they are often prone to metal leaching. Here we have created a new coordination polymer (CP) by applying, for the first time, a stable Gd^III chelate as building block in order to prevent any fortuitous release of free lanthanide(III) ion. The use of the Gd‐DOTA‐4AmP complex as a design element in the CP allows not only for enhanced relaxometric properties (maximum r₁=16.4 mm ^?1 s^?1 at 10 MHz), but also for a pH responsiveness (Δr₁=108 % between pH 4 and 6.5), beyond the values obtained for the low molecular weight Gd‐DOTA‐4AmP itself. The CP can be miniaturised to the nanoscale to form colloids that are stable in physiological saline solution and in cell culture media and does not show cytotoxicity.     

A stabilization process applied to a hidden variables method for evaluating the uncertainties on foundation impedances and their effect on vibrations induced by railways in a building

The paper is devoted to stochastic foundation impedance modeling for buildings submitted to vibrations. The hidden-variables method used in seismic engineering is revisited, due to a larger frequency range used in vibration prediction. Indeed, in this new context, instability of the solution and non-physical nature of mass and stiffness random matrices have been observed. The hidden variable method has been therefore implemented by enforcing explicitly the stability of the solution and the positiveness of mass and stiffness matrices. The effects of numerical parameters used throughout the process are shown and the improved hidden-variables method has been used for predicting the level of vibrations inside a building induced by railway sources. We present here steps of stabilization process, and then discuss on an example of application.     

Implicit BEM formulations for usual and sensitivity problems in elasto-plasticity using the consistent tangent operator concept

This paper presents boundary element method (BEM) formulations for usual and sensitivity problems in (small strain) elasto-plasticity using the concept of the local consistent tangent operator (CTO). “Usual” problems here refer to analysis of nonlinear problems in structural and solid continua, for which Simo and Taylor first proposed the use of the CTO within the context of the finite element method (FEM). A new implicit BEM scheme for such problems, using the CTO, is presented first. A formulation for sensitivity analysis follows. It is shown that the sensitivity of the strain increment, associated with an infinitesimal variation of some design parameter, solves a linear problem which is governed by the (converged value of the) same global CTO as the one that appears in the usual problem. Numerical results for both usual and sensitivity problems are shown for a one-dimensional example. They demonstrate the effectiveness of the present approach. In particular, accurate sensitivities with respect to material parameters (e.g., exponent of the power-type hardening law) are obtained even with few integration cells and for large load increments.