Why define atoms in real space?

The physics of a system is determined by a variation of the action integral, i.e., by a variation of the space–time volume integral of the Lagrange function. If one demands that the properties of an atom in a molecule be derived from physics, the atom must generate its own space–time volume, requiring that its boundaries be defined in real space. The variations in the action are related to the actions of generators of infinitesimal unitary transformations. In the general case, the action integral is altered by generators acting in both the spacelike and timelike surface bounding the space–time volume, whereas for a total isolated system, the physics is totally determined by their action in just the spacelike surfaces at the two time endpoints. It is shown and illustrated for a one-dimensional system that the definition of an atom corresponds to the possibility of choosing a subsystem in such a way that the contributions to the change in action resulting from the evolution in time of its spatial boundaries vanishes identically. The properties of these subsystems and of the total system of which they are a part are, therefore, determined by one and the same action principle. This choice of subsystem corresponds to the possibility of augmenting the Lagrange function by the divergence of the gradient of the electron density a step that, while leaving the equations of motion unchanged, modifies the generating operators in the required manner. © 1994 John Wiley & Sons, Inc.     

Property (<Emphasis Type="Italic">T</Emphasis>) and rigidity for actions on Banach spaces

We study property (T) and the fixed-point property for actions on L ^p and other Banach spaces. We show that property (T) holds when L ² is replaced by L ^p (and even a subspace/quotient of L ^p ), and that in fact it is independent of 1≤p<∞. We show that the fixed-point property for L ^p follows from property (T) when 1<p< 2+ε. For simple Lie groups and their lattices, we prove that the fixed-point property for L ^p holds for any 1< p<∞ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive spaces. Bader partially supported by ISF grant 100146; Furman partially supported by NSF grants DMS-0094245 and DMS-0604611; Gelander partially supported by NSF grant DMS-0404557 and BSF grant 2004010; Monod partially supported by FNS (CH) and NSF (US).     

Symmetries of BLT-Sets

We do the tentative beginnings of a study of BLT-sets of generalised quadrangles via their symmetries. In particular, the study of whorls about a line leads us to hyperbolic reflections preserving a BLT-set of Q(4, q).     

Theoretical framework and experimental procedure for modelling mesoscopic volume fraction stochastic fluctuations in fiber reinforced composites

Many engineering materials exhibit fluctuations and uncertainties on their macroscopic mechanical properties. This randomness results from random fluctuations observed at a lower scale, especially at the meso-scale where microstructural uncertainties generally occur. In the present paper, we first propose a complete theoretical stochastic framework (that is, a relevant probabilistic model as well as a non-intrusive stochastic solver) in which the volume fraction at the microscale is modelled as a random field whose statistical reduction is performed using a Karhunen–Loeve expansion. Then, an experimental procedure dedicated to the identification of the parameters involved in the probabilistic model is presented and relies on a non-destructive ultrasonic method. The combination of the experimental results with a micromechanical analysis provides realizations of the volume fraction random field. In particular, it is shown that the volume fraction can be modelled by a homogeneous random field whose spatial correlation lengths are determined and may provide conditions on the size of the meso-volumes to be considered.