Inverse scattering with partial information on the potential

The one-dimensional Schrödinger equation is considered when the potential is real valued and integrable and has a finite first moment. The recovery of such a potential is analyzed in terms of the scattering data consisting of a reflection coefficient, all the bound-state energies, knowledge of the potential on a finite interval, and all of the bound-state norming constants except one. It is shown that a missing norming constant in the data can cause at most a double nonuniqueness in the recovery. In the particular case when the missing norming constant in the data corresponds to the lowest-energy bound state, the necessary and sufficient conditions are obtained for the nonuniqueness, and the two norming constants and the corresponding potentials are determined. Some explicit examples are provided to illustrate the nonuniqueness.     

The motion of a classical spinning particle in a Riemann-Cartan space-time

A consistent set of equations of motion for classical charged particles with spin and magnetic dipole moment in a Riemann-Cartan space-time is generated from a constrained Lagrangian formalism. The equations avoid the spurious free helicoidal solutions and at the same time conserve the canonical condition of normalization of the 4-velocity. The 4-velocity and the mechanical moment are parallel in this theory, where the condition of orthogonality between spin and 4-velocity is treated as a nonholonomic constraint. A generalized BMT precession equation is obtained as one of the results of the formalism.     

Selective preparation of key intermediates for the synthesis of key intermediates for the synthesis of gemini surfactants by phase-transfer catalysis

A comparison of liquid-liquid and solid-liquid PTC for the selective synthesis of diglycidyl ether from protected pentaerythritol and epichlorohydrin is presented. Solid-liquid PTC was found to be more useful than liquid-liquid PTC because the use of water or other solvents can be avoided and higher yields and selectivity are achieved. It was proved that etherification takes place in the solid phase-organic phase system even in the absence of the phase-transfer catalyst. However, the use of tetrabutylammonium bisulfate as catalyst is essential due to its crucial effect on the enhancement of the rate of etherification and on the improvement of the selectivity to diglycidyl ether, which is higher than 98%.     

Schottky type groups and Kleinian groups acting on S3 with the limit set a wild Cantor set

We construct geometrically finite free Kleinian groups acting onS ³ whose limit sets are wild Cantor sets.Partially supported by CNPqSupported by CNPq     

Combined effect of explicit time-stepping and quadrature for curvature driven flows

Summary. The flow of a closed surface of codimension 1 in driven by curvature is first approximated by a singularly perturbed parabolic double obstacle problem with small parameter . Conforming piecewise linear finite elements, with mass lumping, over a quasi-uniform and weakly acute mesh of size are further used for space discretization, and combined with forward differences for time discretization with uniform time-step . The resulting explicit schemes are the basis for an efficient algorithm, the so-called dynamic mesh algorithm, and exhibit finite speed of propagation and discrete ondegeneracy. No iteration is required, not even to handle the obstacle constraints. The zero level set of the fully discrete solution is shown to converge past singularities to the true interface, provided and no fattening occurs. If the more stringent relations are enforced, then an interface rate of convergence is derived in the vicinity of regular points, along with a companion for type I singularities. For smooth flows, an interface rate of convergence of is proven, provided and exact integration is used for the potential term. The analysis is based on constructing fully discrete barriers via an explicit parabolic projection with quadrature, which bears some intrinsic interest, Lipschitz properties of viscosity solutions of the level set approach, and discrete nondegeneracy. These basic ingredients are also discussed. Received June 20, 1995     

Wave propagation phenomena from a spatiotemporal viewpoint: Resonances and bifurcations

By using biorthogonal decompositions, we show how uniformly propagating waves, togehter with their velocity, shape, and amplitude, can be extracted from a spatiotemporal signal consisting of the superposition of various traveling waves. The interaction between the different waves manifests itself in space-time resonances in case of a discrete biorthogonal spectrum and in resonant wavepackets in case of a continuous biorthogonal spectrum. Resonances appear as invariant subspaces under the biorthogonal operator, which leads to closed sets of algebraic equations. The analysis is then extended to superpositions of dispersive waves for which the (Fourier) dispersion relation is no longer linear. We then show how a space-time bifurcation, namely a qualitative change in the spatiotemporal nature of the solution, occurs when the biorthogonal operator is a nonholomorphic function of a parameter. This takes place when two eigenvalues are degenerate in the biorthogonal spectrum and when the spatial and temporal eigenvectors rotate within each eigenspace. Such a scenario applied to the superposition of traveling waves leads to the generation of additional waves propagating at new velocities, which can be computed from the spatial and temporal eigenmodes involved in the process (namely the shape of the propagating waves slightly before the bifurcation). An eigenvalue degeneracy, however, does not necessarily lead to a bifurcation, a situation we refer to as being self-avoiding. We illustrate our theoretical predictions by giving examples of bifurcating and self-avoiding events in propagating phenomena.     

Internal three-dimensional viscous flow solutions using the vorticity-potential method

A numerical solution procedure for internal three-dimensional viscous flow is proposed in this paper. The formulation is based on the non-primitive variables, the vorticity and potentials, on a curvilinear grid. A new upwind difference scheme is introduced to overcome the convective instabilities arising in the central difference scheme for the vorticity transport equations, while keeping false diffusion to a minimum level. Developing flows in both straight and curved square ducts are simulated to validate the procedure. The results are compared with both experimental measurements and analytical solutions.     

Acidity predictions in an electron propagator approach

An alternative way to calculate vertical ionization potentials (VIP) and vertical electron affinity (VEA) is the application of Koopman's theorem, using the electron propagator theory. In the present work, the results of the application of this theorem using the electron propagator formalism have been compared with the experiment in order to validate different basis set. Using the basis set with the best performance, the acidity tendencies in some substituted acetic acid molecules have been analyzed by correlating the proton affinity (PA) with molecular electronegativity (χ) and hardness (η); these last indexes were obtained from the calculated VIP and VEA considering the finite difference approximation. The above correlations were compared with equivalent correlations using the energy of the frontier Hartree–Fock orbitals and the corresponding Kohn–Sham orbitals, which were calculated with the B3LYP‐DFT procedure. The results indicate that the electron propagator theory could be an interesting alternative to evaluate reactivity indexes, since this theory gives reliable values of VIP and VEA. It was also found that (i) the VIP values are very close to experiment, with only a 0.38% of error; (ii) acceptable results are inferred for VEA; (iii) a triple zeta quality function works quite well in these calculations, and particularly the 6‐311G(d,p) basis set is the best, as it had been reported; and (iv) using the depronation energy (DPE), good results were obtained in the correlations δDPE‐VEA and δDPE‐χ. The results tested that P3 approximation in the electron propagator approach can be a new and interesting alternative in predicting VIP, VEA, and some reactivity indexes, such as χ and η, at least for the compounds studied. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006