Showing papers on "Legendre polynomials published in 1995"

Orientation and Alignment in Reactive Beam Collisions: Recent Progress

Abstract: The concept of directional axis distributions and orientation-dependent reaction cross sections is used to describe the effect of the mutual orientation of reagents on the outcome of reactive beam collisions. The axis distributions and cross sections are expanded in series of Legendre polynomials and real spherical harmonics, respectively, and characterized by the expansion coefficients (moments). The interrelations between the moments of the cross sections and the directionally dependent experimental data (steric effects) on the one hand and the anisotropic properties of the potential energy surfaces on the other hand are presented. Recent progress in the field of dynamical stereochemistry results from the prep­ aration of molecular orientation and alignment by using the novel brute force technique and an optical method, respectively. The theories of both methods are summarized, and typical experimental arrangements are presented. All experimental results based on these techniques are reviewed. Among the...

Spectral methods on arbitrary grids

Abstract: Stable and spectrally accurate numerical methods are constructed on arbitrary grids for partial differential equations. These new methods are equivalent to conventional spectral methods but do not rely on specific grid distributions. Specifically, we show how to implement Legendre Galerkin, Legendre collocation, and Laguerre Galerkin methodology on arbitrary grids.

On the Gibbs phenomenon IV: recovering exponential accuracy in a subinterval from a Gegenbauer partial sum of a piecewise analytic function

Abstract: We continue our investigation of overcoming Gibbs phenomenon, i.e., to obtain exponential accuracy at all points (including at the discontinuities themselves), from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function. We show that if we are given the first N Gegenbauer expansion coefficients, based on the Gegenbauer polynomials C(x) with the weight function (1-x^2)- for any constant 0, of an L_1 function f(x), we can construct an exponentially convergent approximation to the point values of f(x) in any sub-interval in which the function is analytic. The proof covers the cases of Chebyshev or Legendre partial sums, which are most common in applications.

An Improved Two-Dimensional Micromechanical Theory for Brittle Solids with Randomly Located Interacting Microcracks:

Abstract: This paper presents applications of accurate orthogonal function approximation methods to the two-dimensional problems of brittle solids containing many randomly located (and possibly randomly oriented) and strongly interacting microcracks within the framework of micromechanics and ensemble-volume average approach. The randomly located and oriented two-crack interaction problems are solved by using the highly accurate Legendre and Tchebycheff orthogonal polynomials to any desired order. The complex stress potential method is subsequently employed to micromechanically derive microcrack opening displacements under complicated loadings due to microcrack interaction effects-including concentrated loadings, arbitrary loadings and polynomial loadings. Improved local ensemble-averaged and overall effective elastic compliances of brittle solids due to microcracks and their interactions are systematically constructed by using the pairwise microcrack interaction mechanism and the ensemble-volume average approach. A...

Derivatives of addition theorems for Legendre functions

Abstract: Differentiation of the well-known addition theorem for Legendre polynomials produces results for sums over order m of products of various derivatives of associated Legendre functions. The same method is applied to the corresponding addition theorems for vector and tensor spherical harmonics. Results are also given for Chebyshev polynomials of the second kind, corresponding to ‘spin-weighted’ associated Legendre functions, as used in studies of distributions of rotations.

Spectral theory of EM wave scattering by periodic strips

Abstract: A new method is proposed for analyzing electromagnetic (EM) wave scattering and waveguiding by a planar periodic system of thin and perfectly conducting strips. The method exploits some known properties of Fourier series with coefficients expressed by Legendre polynomials. The method can be used to solve problems associated with EM wave propagation and polarization having an arbitrary angle with respect to strips in arbitrary anisotropic media, multiperiodic systems of strips, and layered systems of skewed periodic strips. In the paper the method is presented by an example, namely the scattering of EM waves from a grating consisting of perfectly conducting strips in vacuum. Numerical calculations show that the method converges much faster than do alternative methods. >

Nonlinear Galerkin method using Chebyshev and Legendre polynomials I.: the one-dimensional case

Abstract: A new strategy, stemming from the nonlinear Galerkin method [M. Marion and R. Temam, SIAM J. Numer. Anal., 26 (1989), pp. 1139–1157], for solving linear elliptic and nonlinear dissipative evolution equations by using Chebyshev and Legendre polynomials is presented. The essential idea is to decompose the solution into a low mode part and a high mode part and to treat them separately. The robustness of the method over the classical Galerkin method is substantiated by rigorous error estimates and preliminary numerical experiments.

On a mapping between hard-core potential models for nematogens and continuous ones

Abstract: Hard-core nematogenic models can be studied using Onsager’s theory, and, on the other hand, continuous interaction potentials can be investigated using the molecular field approach pioneered by Maier and Saupe. Comparison between these treatments shows a certain formal similarity, reflecting their common variational root; on this basis, hard-core potential models can be mapped on separable continuous ones, via their excluded volume. As a specific example, we have, therefore, considered hard spherocylinders and hence the continuous potential model(s) here uj denotes the unit vector defining particle orientation, xj denotes the coordinate vector of its center of mass, and ∊ is a positive quantity setting temperature and energy scales(i.e. T*=kBT/∈). The |sin| function can be expanded in a series of Legendre polynomials of even order, with a dominant P2 term; particles’ centers of mass were associated with a simple-cubic lattice, the function ɸ(r) was truncated at nearest-neighbor separation, and values of the two parameters p and q were chosen so as to make contact with the Lebwohl-Lasher lattice model, i.e. . The resulting pair potential was studied by molecular field theory and by computer simulation; molecular field theory predicts a first-order transition at , whereas the result obtained by simulation was ; comparison with the Lebwohl-Lasher model shows that the higher-order interactions contained in the potential tend to increase the transition temperature towards its molecular field limit.

Radiative transfer in a participating slab with anisotropic scattering and general boundary conditions

Abstract: Radiative transfer has been considered within a participating plane slab bounded by two emitting and reflecting plates. The slab is assumed in radiative equilibrium and scatter radiation anisotropically. The scattering function is expanded into Legendre polynomials and a rigorous solution is developed based on the projection method. The resulting formulae for the partial and total heat fluxes have been numerically processed, for both linearly anisotropic and Rayleigh modes of scattering. For the isotropically scattering case, the computed values completely reproduce those available in the literature. As regards to the anisotropic results it has the same accuracy like that for isotropic.

Method of magnet shimming

Abstract: A method is described which corrects magnetic field inhomogeneities in a magnetic resonance (MR) imaging system that takes into account the effects of a subject on the magnetic fields. A magnetic resonance sequence is applied to the subject to extract MR response signals from the subject indicating the magnetic field inhomogeneity over space. The changes of magnetic field inhomogeneity with a change in radial variable ρ (or r, in a cylindrical coordinate system) are calculated. A derivative with respect to a radial variable is calculated of a generalized 3D infinite series polynomial, such as a Legendre polynomial expressed in spherical coordinates. These inhomogeneity changes are then fit to 3D polynomial derivative to result in coefficients [dI c ]. Similarly coefficients of a derivative shim coil calibration matrix [dM] are determined for each shim coil modeling the effect of each shim coil is on the magnetic field within the imaging volume. An inverse derivative matrix [dM] is calculated to result in the inverse derivative matrix [dM -1 ]. The inverse derivative matrix [dM -1 ] is multiplied by the coefficients [dI c ] to determine current coefficients [C] which define currents to be passed through each shim coil to correct the magnetic field inhomogeneities.

An efficient algorithm for choosing scattering directions in Monte Carlo work with arbitrary phase functions

Abstract: This paper describes an efficient Monte Carlo algorithm for choosing a new direction of a photon after a scattering interaction. The algorithm chooses a scattering angle by linear interpolation in a table of the inverse cumulative scattering probability. A Legendre expansion of the phase function makes it easy to apply Clenshaw's algorithm to build the interpolation table. The points in the table are close enough together that linear interpolation is accurate. With a table of 100,000 entries, we can keep the absolute and relative errors in matching the probability distribution below 10(exp -5).

The evaluation of Legendre functions of the second kind

Abstract: A method of evaluating Legendre functions of the second kind by applying the trapezoidal rule to Heine's integral representation is described. An error analysis is given, and some numerical results are obtained.

A method for computing the coefficients in the product-sum formula of associated Legendre functions

Abstract: The product of two associated Legendre functions can be represented by a finite series in associated Legendre functions with unique coefficients. In this study a method is proposed to compute the coefficients in this product-sum formula. The method is of recursive nature and is based on the straightforward polynomial form of the associated Legendre function's factor. The method is verified through the computation of integrals of products of two associated Legendre functions over a given interval and the computation of integrals of products of two Legendre polynomials over [0,1]. These coefficients are basically constant and can be used in any future related applications. A table containing the coefficients up to degree 5 is given for ready reference.

Nonstationary and Quasi-stationary Treatment of Distribution Anisotropy in the Temporal Relaxation of an Energetic Electron Group

Abstract: A relaxation study of an electron group in collision dominated weakly ionized plasmas has been performed. The study is based on the two-term approximation of the Legendre polynomial expansion of the electron velocity distribution in the nonstationary Boltzmann equation. To overcome the limitation of the conventional quasi-stationary treatment of the distribution anisotropy, a very efficient solution approach of the nonstationary kinetic equation in two-term approximation has been developed which allows for a strict nonstationary treatment of the distribution anisotropy. By using this approach the temporal evolution of the isotropic and anisotropic distribution of the electrons has been investigated for a model plasma, which involves typical features of an inert gas plasma. A comparison of the results with corresponding ones obtained by applying the conventional approach under various parameter conditions clearly indicates a pronounced falsification of the real relaxation course by the latter approach.

Characteristic classes of Lagrangian and Legendre singularities

Abstract: Contents §1. Introduction §2. Preliminaries §3. Lagrangian structures and their characteristic classes §4. Characteristic spectral sequence §5. Characteristic spectral sequence (continuation) §6. Computations in the characteristic spectral sequence of stable right equivalence of functions §7. Characteristic classes of stable V-equivalence §8. Proofs Bibliography

Analytic calculation of the radiance in an anisotropically scattering turbid medium close to a source

Abstract: We present a method to calculate the radiance due to an isotropic point source in an infinite, homogeneous, anisotropically scattering medium. The method is an extension of a well known method for the case of isotropic scattering. Its basic mathematical ingredient is the Fourier transform. Its great advantage is that it also works very close to the source and not just far away from it, as is the case with most other methods. In principle, the method works for any phase function that can be expanded in a finite number of Legendre polynomials. Here, the simple example of linear anisotropic scattering is worked out numerically and the result is compared with Monte Carlo simulation. Good agreement is found between the two.

The derivative expansion of the renormalization group

Abstract: By writing the flow equations for the continuum Legendre effective action (a.k.a. Helmholtz free energy) with respect to a particular form of smooth cutoff, and performing a derivative expansion up to some maximum order, a set of differential equations are obtained which at FPs (Fixed Points) reduce to non-linear eigenvalue equations for the anomalous scaling dimension η . Illustrating this by expanding (single component) scalar field theory, in two, three and four dimensions, up to second order in derivatives, we show that the method is a powerful and robust means of discovering and quantifying non-perturbative continuum limits (continuous phase transitions).

On the legendre coefficients of the moments of the general order derivative of an infinitely differentiable function

Abstract: Formulae for the Legendre coefficients of the moments of the general order derivative of an infinitely differentiable function in terms of its Legendre coefficients are derived. Two numerical applications of how to use these formulae for solving ordinary differential equations with varying coefficients, by reducing them to recurrence relations of lowest order in the Legendre expansion coefficients are discussed. Comparisons with the results obtained by the optimal algorithm of Lewanowicz (1976) are made.

Method for passively shimming an open magnet

Abstract: A method for passively shimming an open magnet having a magnetic field with an inhomogeneity including an amount of positive 2, 0 Legendre polynomial harmonics. The magnetic field is mapped and the amount of positive 2, 0 Legendre polynomial harmonics of the mapped magnetic field is determined. An adjusted magnetic field is defined equal to the mapped magnetic field minus a nonzero portion of the determined amount of positive 2, 0 Legendre polynomial harmonics. A computer shim code is run which calculates adding shims to reduce the inhomogeneity of the adjusted magnetic field. Shims are added to the open magnet as calculated from the running of the computer shim code.