Heegner points,cycles and Maass forms

We derive for Hecke-Maass cusp forms on the full modular group a relation between the sum of the form at Heegner points (and integrals over Heegner cycles) and the product of two Fourier coefficients of a corresponding form of half-integral weight. Specializing to certain cycles we obtain the nonnegativity of theL-function of such a form at the center of the critical strip. These results generalize similar formulae known for holomorphic forms. Partially supported by NSF grant # DMS-9096262. Partially supported by NSF grant # DMS-9102082.     

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.     

Dictator game giving: Rules of fairness versus acts of kindness

In both dictator and impunity games, one player, the dictator, divides a fixed amount of money between himself and one other, the recipient. Recent lab studies of these games have produced seemingly inconsistent results, reporting substantially divergent amounts of dictator giving. Also, one prominent explanation for some of these differences, the impact of experimenter observation, displayed weak explanatory power in a different but related lab game. Data from the new experiment reported here offers some explanations. We find that dictators determine how much they will give on the basis of the total money available for the entire experimental session, not on the basis of what is available per game. This explains the reported differences between impunity and dictator studies. When distributing a gift among several recipients, individual dictators show little tendency towards equal treatment. Also, we find no evidence for the experimenter observation effect. Comparison with earlier experiments suggests that differences in the context of the game, affected by differences in written directions and independent of experimenter observation, account for differences across dictator studies. We propose a hypothetical decision procedure, based on the notion that dictator giving originates with personal and social rules that effectively constrain self-interested behavior. The procedure provides a link between dictator behavior and a broader class of laboratory phenomena. Received August 1993/Final version April 1994