non-Cartesian Coordinates Research Articles

The finite basis representation as the primary space in multidimensional pseudospectral schemes

We emphasize the merits and the superiority of the most complete nondirect product representation in non-Cartesian coordinates. Beyond the proper choice of basis set we show how to further optimize the spectral range in multidimensional calculations. The combined use of a fully pseudospectral scheme and the finite basis representation (FBR) as the primary space ensures the smallest prefactor in the semilinear scaling law of the Hamiltonian evaluation with respect to the FBR size. In the context of scattering simulations we present a simplified asymptotic treatment which increases the efficiency of the FBR-based pseudospectral approach. An optimal 6D pseudospectral scheme is proposed to treat the vibrational excitation and/or dissociation of a diatomic molecule scattering from a rigid, corrugated surface, and serves to illustrate our discussion. A 5D numerical demonstration is made for the rotationally inelastic scattering of N2 from a model LiF surface.

Application of Distributed Approximating Functionals for Atom-Rigid Rotor Inelastic Scattering: Body Frame Close-Coupling Time-Dependent and Time-Independent Wavepacket Approaches

We provide the first example of the application of distributed approximating functionals (DAFs) for 3D molecular scattering using non-Cartesian coordinates (atom-rigid rotor inelastic scattering), in both time- dependent and time-independent wavepacket schemes. With the aid of the body frame close-coupling wavepacket technique, the 3D problem is expressed as a set of coupled

Configuration ray representations in non-Abelian quantum kinematics and dynamics

Ray extensions of the regular representation of noncompact non-Abelian Lie groups are examined as generalizations of the Cartesian coordinate representation of ordinary quantum mechanics to the case of generalized non-Cartesian coordinates and generalized noncommuting momenta. (The momenta are in fact the generators of the representation, and so they satisfy the Lie algebra of the group.) The concept of configuration ray representation is introduced within this new kinematic formalism as subrepresentations of the regular representation which are embossed with the “relativity theory” of a given system. The main features of the mathematical formalism leading to these representations in configuration spacetime are discussed, and their importance for non-Abelian quantum kinematics and dynamics is emphasized. Two miscellaneous examples on the calculus of phase functions for configuration ray representations are given.

Stochastic imaging of the Wilmington clastic sequence : Journel, A G; Gomez-Hernandez, J J SPE Form EvalV8, N1, March 1993, P33–40

The geometric architecture of the sand-shale sequence in two layers of the Wilmington field is characterized by indicator variograms produced with noncartesian coordinates ensuring stratigraphic conformity. The sequential indicator simulation (SIS) algorithm produces several alternative equiprobably three-dimensional models of the sand-shale sequence, all of them honoring the data at their locations and reproducing the indicator variogram model. These alternative models provide a direct visualization of spatial uncertainty that can be used to assess the need for additional data.

Stochastic Imaging of the Wilmington Clastic Sequence

Summary Correct modeling of the geometric architecture of the sand/shale geometry within shaly layers of the Wilmington field is critical to predict performance recovery accurately. Stochastic images of that clastic sequence are obtained with sequential indicator simulation with non-Cartesian coordinates to ensure stratigraphic conformity.

On a Tensorial Study of the Energy Density Equation of a Momentless Elastic Solid

The absolute tensorial energy density equation of a momentless elastic solid is studied. The energy density concept is developed in arbitrary non-cartesian curvilinear coordinates and the final equation is applicable to theories such as those developed for beams, plates or shells. The use of the Trefftz’s method in connection with the absolute energy density equation will allow us to obtain the following: the non-linear equations of equilibrium, the stability condition, the post-buckling equation and a fourth order functional related to the fourth variation of the energy density equation for an elastic solid.

Errors in Fixed and Moving Frame of References: Applications for Conventional and Doppler Radar Analysis

Techniques for the correction of errors in Doppler radar scans due to advection effects are presented. A moving frame of reference is shown to be useful in least-squares estimates with stationary observations expressed in scalars or Cartesian coordinates. For non-Cartesian coordinates, such as the deduction of radial velocities from triple Doppler radar data, an integral is defined for accounting for advection effects and consequent coordinate transformations. The multiple radars are necessary for unambiguous characterization of the horizontal wind velocity. A scale analysis is employed to estimate errors with and without the error correction procedure. Improvements in correlations between scans are demonstrated when the error correction method is used.

The concept of a collective path and its range of validity

The space of Slater determinants is described in terms of certain non-cartesian coordinates in which the time-dependent Hartree-Fock theory (TDHF) appears to be fully equivalent to a set of classical canonical equations. In this framework the concept of a collective path is developed. This consists in selecting a subspace of Slater determinants in which the collective channel is decoupled from the non-collective excitations. The adiabatic time-dependent Hartree-Fock theory (ATDHF) and the local harmonic approach (LHA) are discussed, which both provide equations to determine the collective path. The main task is to study the limits of validity of either method and the conditions under which the decoupling of collective and non-collective channels is achieved. There the curvatures of the determinantal space appear to be crucial quantities. The LHA aiming at a dynamical path | q, p〉 has the stronger restriction than the ATDHF, being merely concerned with a static path | q, p = 0〉. The condition for the validity of ATDHF consists in requiring the collective kinetic energy per particle to be small compared with an average single-particle energy divided by a simple matrix element representing the coupling between collective and non-collective degrees of freedom. Finally, this condition is cast into a tractable form involving only RPA operations.

A direct method for the solution of poisson's equation with neumann boundary conditions on a staggered grid of arbitrary size

A method based on cyclic reduction is described for the solution of the discrete Poisson equation on a rectangular two-dimensional staggered grid with an arbitrary number of grid points in each direction. Neumann boundary conditions are assumed in one direction and any boundary condition may be used in the other direction. The coefficients of the equation can be functions of the latter direction so that, e.g., non-equidistant grid spacings or non-Cartesian coordinates can be used. Poisson's equation with these boundary conditions describes, e.g., the pressure field of an incompressible fluid flow within rigid boundaries. Numerical results are reported for a FORTRAN subroutine using the method. For an M × N grid the operation count is proportional to MN log 2 N, and about MN storage locations are required.

Search for a Universal Symmetry Group in Two Dimensions

Recently several authors have proposed a universal symmetry group and demonstrated the classical validity of the concept. Supposedly, an SU(n) symmetry group could be constructed for whatsoever system of n degrees of freedom. The claim is assuredly valid for all classically degenerate systems, but is in contradiction with most of the well-known and widely accepted solutions of Schrödinger's equation. We examine the reasons for this discrepancy on the quantum-mechanical level. The construction of the universal symmetry group requires ladder operators, which in most cases are the ladder operators of Infeld and Hill. Complications which owe their origin entirely to numerical relationships imposed by quantization prevent these operators from forming a von Neumann algebra and, in turn, an SU(n) group of constants of the motion. Two important effects are those imposed by anisotropy, wherein not all quanta have the same size, and by non-Cartesian coordinates, wherein quanta in some dimensions are restricted in size by those in other dimensions. These effects can be seen quite clearly in two-dimensional systems: anisotropy in the Cartesian anisotropic harmonic oscillator, and conflict between dimensions in the polar form of the isotropic oscillator. Further complications arise when the two effects are combined, as in the harmonic oscillator or hydrogen atom with ``excess'' angular momentum. Enough residue of the universal symmetry concept remains that many similarities between the hydrogen atom and harmonic oscillator may be understood, including the fact that some levels of the hydrogen atom, which ordinarily transform according to an orthogonal group, may form irreducible representations of the unitary group.