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).     

Chemistry and the near-sighted nature of the one-electron density matrix

Two different macrospopic pieces of copper have different external potentials and, because of the unique functional relationship between the electron density and the external potential as demanded by density functional theory, should possess different electron density distributions. Experimentally, however, an atom in the bulk exhibits the same electron density in both samples and they possess identical sets of intensive properties. Density functional theory does not account for the fundamental observation underlying the theory of atoms in molecules: that what are apparently identical distributions of charge can be observed for an atom or a grouping of atoms in systems with different external potentials and that these atoms contribute essentially identical amounts to the energies and all other properties of the systems in which they occur. It is shown that, unlike the external potential, the kinetic energy density and the potential energy density, defined by the virial of the Ehrenfest force acting on electron density, are short-range functions. As recorded in the first article on atoms in molecules, they exhibit a local dependence on the electron density that causes them to faithfully mimic the transferability of the atomic charge distributions from one system to another. The electron, the kinetic energy, and the virial densities are all determined directly by the one-electron density matrix, a function termed near-sighted by Professor Kohn. It is this near-sighted property of the one-matrix that underlies the working hypothesis of chemistry—that of a functional group exhibiting a characteristic set of properties. The observations obtained from the theory of atoms in molecules and the atomic theorems it determines demonstrate the existence of a local relationship between the electron density and all properties of a system. © 1995 John Wiley & Sons, Inc.     

Theoretical Definition of a Functional Group and the Molecular Orbital Paradigm

It is the purpose of this review to demonstrate that the empirical classification of the observations of chemistry in terms of the properties assigned to functional groups is a consequence of and is predicted by physics. This is accomplished by showing that the atoms and functional groups of chemistry can be identified with bounded space-filling objects whose properties are defined by quantum mechanics. The quantum mechanical definition of a group is combined with a new pictorial representation of its form to obtain a unified picture which should make it eminently recognizable to chemists. This picture, when combined with the demonstrated ability of these groups to recover the measured properties of atoms in molecules, is offered as one which meets the expectations a chemist associates with the concept of a functional group. The manner in which this physical definition of a group differs fundamentally from models of functional groups based upon molecular orbital theory is discussed.