Simulation of scanning tunneling microscope images of 1,3-cyclohexadiene bound to a silicon surface

Scanning tunneling microscope (STM) images of 1,3-cyclohexadiene bound to silicon are interpreted using a nonequilibrium Green's function method. The resolution of the carbon-carbon double bond for positive bias voltages but not for negative bias voltages is explained using a quasiprobability density analysis. The asymmetry in the images arises from the system's voltage dependent electronic structure. A pi* orbital is found to be responsible for the empty state STM images of the carbon-carbon double bond, which is observed experimentally. The pi orbital relevant for the opposite bias does not produce an STM image sharply localized in the bond region because the molecule induces a Si-surface dipole dependent on the bias. The dipole voltage dependence arises from molecular charging. This result emphasizes the importance of simulating the molecule as an element in an open quantum system.     

Nonlinear waves in zonostrophic turbulence

The Charney-Hasegawa-Mima equation applies to a broad variety of hydrodynamic systems ranging from the large-scale planetary circulations to small-scale processes in magnetically confined plasma. This equation harbors flow regimes that have not yet been fully understood. One of those is the recently discovered regime of zonostrophic turbulence emerging in the case of small-scale forced, barotropic two-dimensional turbulence on the surface of a rotating sphere or in its beta-plane approximation. The commingling of strong nonlinearity, strong anisotropy and Rossby waves underlying this regime is highlighted by the emergence of stable systems of alternating zonal jets and a new class of nonlinear waves, or zonons. This Letter elucidates the physics of the zonons and their relation to the large-scale coherent structures.     

The Dynamics of Pendulums on Surfaces of Constant Curvature

We define the notion of a pendulum on a surface of constant curvature and study the motion of a mass at a fixed distance from a pivot. We consider some special cases: first a pivot that moves with constant speed along a geodesic, and then a pivot that undergoes acceleration along a fixed geodesic.      

Reactions of 3а,6a-diaza-1,4-diphosphapentalene with activated acetylenes

The reaction of cyclohexane-annulated 3 a,6a-diaza-1,4-diphosphapentalene (DDP) with di-tert-butyl acetylenedicarboxylate (DBAD) affords the 1,3-dipolar cycloaddition product of the acetylene moiety to the phosphorus and sp²-carbon atoms of DDP in 90% yield. No individual products were isolated in the reaction of DDP with dimethyl acetylenedicarboxylate (DMAD). In the three-component DDP–DMAD–carbazole system (1: 2: 1), a product was generated in 73% yield via successive reactions, including the 1(P),3(C)-dipolar cycloaddition of DMAD to DDP, the addition of the second equivalent of DMAD to the three-coordinate phosphorus atom of the intermediate, and the NH addition of carbazole at the unsaturated C=C bond of the second DMAD moiety. The structures of the reaction products were established by X-ray diffraction.     

Billiards with Pockets: A Separation Principle and Bound for the Number of Orbit Types

 We introduce and prove a Separation Principle, similar in form to the familiar Uncertainty Principle of quantum mechanics, which separates the position and direction of any two phase points on distinct unfoldings of (non-parallel) trajectories on a polygonal billiard table with pockets. Applying this principle, we demonstrate that the number of orbit types (that is, classes of trajectories, up to parallelism) on a polygonal billiard table with area A and pockets of area a is strictly bounded above by . More generally, the same bound applies to any compact polyhedral surface with pockets at its vertices. If the boundary is empty (so that billiard trajectories are just geodesics), the bound is reduced by a factor of two to . We believe the Separation Principle will also have fundamental applications to other problems in the theory of billiards and related dynamical systems. Received: 28 December 2001 / Accepted: 9 April 2002 Published online: 4 September 2002     

Non-causal models in long term planning via set contractive optimal control methods

The notion of consistency for optimal plans introduced in [F.E. Kydland, E.C. Prescott, Rules rather than decisions: The inconsistency of optimal plans, J. Polit. Econ. 85 (3) (1977) 473–491] is studied in relation to dynamic programming and to multi-objective optimal control. It is demonstrated that the consistency condition for an optimal sequence of n policies for periods from 1 to n corresponding to a sequence of n economic agents’ decisions that together constitute an optimal plan for an agreed-upon social objective function is equivalent to a multi-objective problem with n+1 criteria. An optimal plan is consistent if and only if the corresponding (n+1)-criteria problem is balanced, and consistent optimal plans do exist in such economic situations. Otherwise, the consistency requirement generates the balance set and a set of Pareto solutions, and both can be computed as illustrated in the inflation–unemployment example extended for two periods with three objectives to optimize. The procedure of “consistent planning for the infinite horizon” considered in [F.E. Kydland, E.C. Prescott, Rules rather than decisions: The inconsistency of optimal plans, J. Polit. Econ. 85 (3) (1977) 473–491] is shown to concur with Bellman’s principle of optimality that may be invalid in many practical cases including the case considered in [F.E. Kydland, E.C. Prescott, Rules rather than decisions: The inconsistency of optimal plans, J. Polit. Econ. 85 (3) (1977) 473–491]. Finitely causal and non-causal models are introduced for long term planning in a multi-objective (Pareto) framework, and it is demonstrated how to deal with non-causal models using new set-contractive methods for multi-objective global optimization. The results are illustrated by examples.     

Nonlinear observation via global optimization methods: Measure theory approach

Nonlinear observation methods developed by Galperin (Refs. 1 and 2) and global optimization methods developed by Zheng (Refs. 3 and 4) are coupled to obtain effective procedures for solution of nonlinear observation and identification problems.The work of this author was supported by the Natural Sciences and Engineering Research Council of Canada under Grant No. A3492.     

Molecular transport junctions: asymmetry in inelastic tunneling processes

Inelastic electron tunneling spectroscopy (IETS) measurements are usually carried out in the low-voltage ("Ohmic", i.e., linear) regime where the elastic conduction/voltage characteristic is symmetric to voltage inversion. Inelastic features, normally observed in the second derivative d(2)I/dV(2) are also symmetric (in fact antisymmetric) in many cases, but asymmetry is sometimes observed. We show that such asymmetry can occur because of different energy dependences of the two contact self-energies. This may be attributed to differences in contact density of states (different contact material) or different energy dependence of the coupling (STM-like geometry or asymmetric positioning of molecular vibrational modes in the junction). The asymmetry scales with the difference between the energy dependence of these self-energies and disappears when this dependence is the same for the two contacts. Our nonequilibrium Green function approach goes beyond proposed WKB scattering theory in properly accounting for Pauli exclusion, as well as providing a path to generalizations, including consideration of phonon dynamics and higher-order perturbation theory.     

On the norms of singular integral operators along certain curves with intersections

It is well known [K] that the norm and essential norm of singular integral operator for contours composed by two rays depend on the angle between the rays. The goal of the paper is both to show that for the contour that is an intersection oftwo straight lines the norm and essential norm of singular integral operatordo not depend on the angle between the lines, andto calculate these norms. Some generalizations for the case when is an intersection of several simple contours are also considered.