Roughness induced surface acoustic resonances

The vibrational and electronic spectra of a semi-infinite crystal with a planar surface is modified in presence of surface inhomogeneities or roughness such as ridges or grooves, quantum wires or tips… Using a Green's function formalism, we present an exact numerical method for obtaining the variation of the density of states associated with the adsorption of a ridge on a flat surface or with a groove cut into an otherwise planar surface. This general method is applied to the determination of the acoustic resonances of shear horizontal polarization associated with such deterministic surface protuberances or indentations. The positions and widths of the peaks in the total or local densities of states give the frequencies and lifetimes of the resonances, which may be more or less pronounced features depending on the relative parameters of the substrate and ridge materials. We also investigate the modifications of these acoustic surface shape resonances due to the interaction between two such defects. This calculation can also be transposed to the study of electronic structure of a wire near a flat surface, in the framework of an effective mass model.     

Variational particle number approach for rational compound design

Within density functional theory, a variational particle number approach for rational compound design (RCD) is presented. An expression for RCD is obtained in terms of minimization of a suitably defined energy penalty functional whose gradients are the nuclear and the electronic chemical potential. Using combined quantum and molecular mechanics, a nonpeptidic anticancer drug candidate is designed.     

Optimization of effective atom centered potentials for london dispersion forces in density functional theory

We add an effective atom-centered nonlocal term to the exchange-correlation potential in order to cure the lack of London dispersion forces in standard density functional theory. Calibration of this long-range correction is performed using density functional perturbation theory and an arbitrary reference. Without any prior assignment of types and structures of molecular fragments, our corrected generalized gradient approximation density functional theory calculations yield correct equilibrium geometries and dissociation energies of argon-argon, benzene-benzene, graphite-graphite, and argon-benzene complexes.     

Admissible submonoids of Artin-Tits monoids

We show the analogue of Mühlherr’s [B. Mühlherr, Coxeter groups in Coxeter groups, in: Finite Geometry and Combinatorics, Cambridge University Press, 1993, pp. 277-287] for Artin-Tits monoids and for Artin-Tits groups of spherical type. That is, the submonoid (resp. subgroup) of an Artin-Tits monoid (resp. group of spherical type) induced by an admissible partition of the Coxeter graph is an Artin-Tits monoid (resp. group).This generalizes and unifies the situation of the submonoid (resp. subgroup) of fixed elements of an Artin-Tits monoid (resp. group of spherical type) under the action of graph automorphisms, and the notion of LCM-homomorphisms defined by Crisp in [J. Crisp, Injective maps between Artin groups, in: Geom. Group Theory Down Under (Canberra 1996), de Gruyter, Berlin, 1999, pp. 119-137] and generalized by Godelle in [E. Godelle, Morphismes injectifs entre groupes d’Artin-Tits, Algebr. Geom. Topol. 2 (2002) 519-536].We then complete the classification of the admissible partitions for which the Coxeter graphs involved have no infinite label, started by Mühlherr in [B. Mühlherr, Some contributions to the theory of buildings based on the gate property, Dissertation, Tübingen, 1994]. This leads us to the classification of Crisp’s LCM-homomorphisms.     

On the linear search problem

A man in an automobile searches for another man who is located at some point of a certain road. He starts at a given point and knows in advance the probability that the second man is at any given point of the road. Since the man being sought might be in either direction from the starting point, the searcher will, in general, have to turn around many times before finding his target. How does he search so as to minimize the expected distance travelled? When can this minimum expectation actually be achieved? This paper answers the second of these questions.     

The Fried Average Entropy and Slow Entropy for Actions of Higher Rank Abelian Groups

We consider two numerical entropy-type invariants for actions of \({\mathbb{Z}^k}\) , invariant under a choice of generators and well-adapted for smooth actions whose individual elements have positive entropy. We concentrate on the maximal rank case, i.e. \({\mathbb{Z}^k,\,k \geq 2}\) actions on k + 1-dimensional manifolds. In this case we show that for a fixed dimension (or, equivalently, rank) each of the invariants determines the other and their values are closely related to regulators in algebraic number fields. In particular, in contrast with the classical case of \({{\mathbb Z}}\) actions the entropies of ergodic maximal rank actions take only countably many values. Our main result is the dichotomy that is best expressed under the assumption of weak mixing or, equivalently, no periodic factors: either both invariants vanish, or their values are bounded away from zero by universal constants. Furthermore, the lower bounds grow with dimension: for the first invariant (the Fried average entropy) exponentially, and for the second (the slow entropy) linearly.     

The linear search problem rides again

The Linear Search Problem concerns a search for a point in the real line by continuous motion starting at 0. The optimal turning points for such a search under the hypothesis that the location of the target is distributed normally about 0 have been approximated by mechanical calculation, but no proof has been given that there is only a single minimizing strategy or that the numbers calculated do indeed approximate that strategy. Plausible arguments have been made before, both by these authors and others. In this paper, the plausible arguments are supplanted by mathematical proofs. The research of the senior author has been supported by the Wisconsin Alumni Research Foundation. The research of the junior author has been supported by Hewlett Packard, Inc. under a Faculty Development Fellowship at Cornell University.     

Uncountable Homomorphic Images of Polish Groups are not N1-Free Groups

Shelah has recently proved that an uncountable free group cannotbe the automorphism group of a countable structure. In fact,he proved a more general result: an uncountable free group cannotbe a Polish group. A natural question is: can an uncountable₁-free group be a Polish group? A negative answer is given here;indeed, it is proved that an ₁-free group cannot be a homomorphicimage of a Polish group. In fact, a stronger result is proved,involving a non-commutative analogue of the notion of separablegroup. 2000 Mathematics Subject Classification 20E05.