Showing papers on "Legendre polynomials published in 1977"
01 Jan 1977
TL;DR: Spectral Methods Survey of Approximation Theory Review of Convergence Theory Algebraic Stability Spectral Methods Using Fourier Series Applications of algebraic stability analysis Constant Coefficient Hyperbolic Equations Time Differencing Efficient Implementation of Spectral Method as discussed by the authors.
Abstract: Spectral Methods Survey of Approximation Theory Review of Convergence Theory Algebraic Stability Spectral Methods Using Fourier Series Applications of Algebraic Stability Analysis Constant Coefficient Hyperbolic Equations Time Differencing Efficient Implementation of Spectral Methods Numerical Results for Hyperbolic Problems Advection-Diffusion Equation Models of Incompressible Fluid Dynamics Miscellaneous Applications of Spectral Methods Survey of Spectral Methods and Applications Properties of Chebyshev and Legendre Polynomial Expansions.
3,386 citations
TL;DR: In this paper, an approach to numerical convection is presented that exclusively yields upstream-centered schemes, which start from a meshwise approximation of the initial-value distribution by simple basic functions, e.g., Legendre polynomials.
2,206 citations
TL;DR: In this article, the moments of the normalized Legendre polynomials of the cosine of the scattering angle have been used to obtain continuous differential cross sections for molecular collisions.
Abstract: We present a method for obtaining continuous differential cross sections for molecular collisions from trajectories with initial conditions selected by Monte Carlo methods. It is a moment method and we represent the differential scattering cross section in terms of the moments of the normalized Legendre polynomials of the cosine of the scattering angle. The method is applied to five examples, for four of which we evaluate the necessary moments exactly. The fifth is a realistic example of an experimentally obtained differential cross section. For the first four cases we can obtain satisfactory convergence with as few as 400 trajectories with the proper choice of the highest order moment used in the expansion. As a working criterion we select as the highest order coefficient the highest order moment whose absolute value is larger than 0.05. Generally, the method does not converge any more rapidly than does the histogram method. It does provide a simple way of deciding the available angular resolution in the differential cross section for a given number of trajectories.
80 citations
TL;DR: The background of the dispute is sketched, and this little known attack on Gauss in 1820 is presented in translation.
32 citations
TL;DR: In this paper, a general mathematical treatment is given for the problem of eliminating the effect of orientation distributions in fiber diagrams, which involves the inversion of an integral transform the kernel of which is determined by an orientation function already derived in an earlier paper (Ruland andTompa (3)).
Abstract: A general mathematical treatment is given for the problem of eliminating the effect of orientation distributions in fiber diagrams. The method involves the inversion of an integral transform the kernel of which is determined by an orientation function already derived in an earlier paper (Ruland andTompa (3)). The inversion can be obtained by a development of the intensity distributions into a series of Legendre polynomials.
31 citations
TL;DR: In this article, a Legendre transform theory is used to develop two general critical point criteria for a classical thermodynamic system and the problem of indeterminacy is discussed, as is the option of employing mole fractions (rather than mole numbers).
Abstract: Legendre transform theory is used to develop two general critical point criteria for a classical thermodynamic system. The problem of indeterminacy is discussed, as is the option of employing mole fractions (rather than mole numbers).
30 citations
TL;DR: In this paper, the Lagrange expansion was used to obtain a general result for generating polynomials of the type Θ( √ √ n, √ α + β n, β + δ n) for the Hermite, Legendre, Bernoulli, Euler, Jacobi and Bell polynomial.
Abstract: Generating functions of the type $\sum_0^\infty {L_n^{(\alpha + \beta x)} (x)z^n } $, $P_n^{(\alpha + \gamma n,\beta + \delta n)} (x)z^n $ have been obtained recently. In the present paper a general result of this kind is derived making use of the Lagrange expansion. A number of applications are given. These include, in addition to the Laguerre and Jacobi polynomials, the Hermite, Legendre, Bernoulli, Euler and Bell polynomials.
29 citations
TL;DR: In this paper, an expression for FPl(ŝ1ŝ2) h (12) dΩ1dΩ2dr in terms of atom-atom correlation functions, where h(12) is the full molecular pair correlation function, and Pl is the Legendre polynomial.
Abstract: We find an expression for FPl(ŝ1ŝ2) h (12) dΩ1dΩ2dr in terms of atom–atom correlation functions, where h (12) is the full molecular pair correlation function, ŝi is a unit vector that describes the orientation of the ith molecule, and Pl is the Legendre polynomial. For l=1 the result yields our earlier expression for the dielectric constant of a polar (nonpolarizable) fluid. For l=2, the result yields the Kerr constant for a nonpolar fluid of linear molecules, and is also relevant to the theory of dipolarized light scattering and quadratic field effects in NMR theory.
24 citations
TL;DR: In this paper, the authors extended the work of de Boor and Swartz (SIAM J. Numer. Anal., 10 (1973), pp. 582-606) on the solution of two-point boundary value problems by collocation.
Abstract: In this note we extend the work of de Boor and Swartz (SIAM J. Numer. Anal., 10 (1973), pp. 582-606) on the solution of two-point boundary value problems by collocation. In particular, we are concerned with boundary value problems described by integro-differential equations involving derivatives of order up to and including m with m boundary conditions. We study the approximation of (isolated) solutions by means of piecewise polynomial functions of degree less than m + k possessing m -1 continuous derivatives. If the problem is sufficiently smooth and the solution has m +2k continuous derivatives, then one can achieve O(IAlk+m) global convergence by collocating at the zeros of the kth Legendre polynomial relative to each subinterval. At the knots, the approximation and its first m - 1 derivatives are O(IA12k) accurate.
21 citations
TL;DR: In this article, the emission and absorption coefficients for Langmuir waves due to anisotropic unmagnetized particles are reduced to two complementary forms: one involving integrals over momentum p and pitch angle ex; the other involving an integral over p and a sum over Legendre polynomials.
Abstract: The emission and absorption coefficients for Langmuir waves due to anisotropic unmagnetized particles are reduced to two complementary forms: one involving integrals over momentum p and pitch angle ex; the other involving an integral over p and a sum over Legendre polynomials. The quasilinear diffusion coefficients are reduced to the former. It is also shown how the absorption coefficient may be reduced to forms involving neither a p derivative nor an ex derivative. The absorption coefficient is evaluated explicitly for five idealized anisotropic distributions, called a 'forwardcone' anisotropy, a 'semi-cos2 ex' anisotropy, a loss-cone anisotropy, a P, anisotropy and a P2 anisotropy respectively. All except the P2 anisotropy can lead to growth of Langmuir waves, but only if the distribution function is an increasing function of p at the resonant phase speed, e.g. only for gap distributions. The results have important implications in connection with the theory of solar radio bursts.
20 citations
01 Jul 1977
TL;DR: In this article, a precise equivalence is established between the multi-valued Legendre transforms of certain polynomials, and the bifurcation sets of the cuspoid catastrophes.
Abstract: : A precise equivalence is established between the multi-valued Legendre transforms of certain polynomials, and the bifurcation sets of the cuspoid catastrophes. Examples are explored in detail, including a graphical representation and a physical interpretation. (Author)
TL;DR: The results were fitted with Legendre polynomials and the energy dependence of the resulting coefficients indicated resonance-like behaviour at several energies corresponding to excitation between 22 and 32 MeV in 8Be as mentioned in this paper.
TL;DR: In this article, a quasi-static solution for the circular microstrip disk is studied using a GaIerkin solution to the Fredholm integral equation of the first kind derived by using the Green's function approach.
Abstract: The quasi-static solution for the circular microstrip disk is studied using a GaIerkin solution to the Fredholm integral equation of the first kind derived by using the Green's function approach. The basis functions are modified Legendre polynomials combined with a reciprocal square root to provide the correct singularity in charge density at the edge of the disk. The integrals involving the singular part of the Green's function are evaluated exactly, the remainder by using Gaussian quadrature. The method is compared in computational efficiency with recent methods based either on a Galerkin approach in the spectral domain, or the use of dual integral equations. Numerical results are given for charge distribution and capacitance; they are compared to exact results and those obtained by others, and the limitations of those methods are discussed. Closed form expressions are given for the capacitance of a disk based on two simple charge distributions.
TL;DR: In this article, a solution to the motion of a satellite in an eccentric orbit and in resonance with the second-degree sectorial harmonic of the potential field is developed, which parallels the well known von Zeipel method of general perturbations.
Abstract: The solution to the motion of a satellite in an eccentric orbit and in resonance with the second-degree sectorial harmonic of the potential field is developed. The method of solution used parallels the well known von Zeipel method of general perturbations. The solution consists of expressions for the variations of the Delaunay variables. These expressions are composed of the perturbations developed by Brouwer in 1959 for the motion of an artificial satellite plus first-order perturbations due to the second-degree sectorial harmonic (in terms of the Legendre normal elliptic integrals of the first and second kind).
TL;DR: In this article, the mean field equation for liquid crystals is reduced to a set of coupled integrodifferential equations by means of expanding the distribution function f (r, δ) in Legendre polynomials and the reciprocal lattice space.
Abstract: Starting with a pairwise, spatially and orientationally dependent intermolecular potential of the Kobayashi-McMillan form, we carry out a systematic solution of the mean field equation for liquid crystals. The mean field equation, presented as first of a hierarchy of BBGKY equations, is first reduced to a set of coupled integrodifferential equations by means of expanding the distribution function f (r, δ) and In f (r, δ) in Legendre polynomials and the reciprocal lattice space. In the first level of approximation, the expansion retains only the lowest-order coefficients, permitting a complete decoupling of the equations. In the second level of approximation, the leading coefficient which couples spatial order to orientational order is included. In the third level of approximation two more higher order coefficients are included. At each level, the free energy functional is evaluated to determine the equilibrium phase at given temperatures and chainlengths of a homologous series. It is shown that t...
TL;DR: In this paper, a method to compute differential cross sections from classical trajectory studies is presented, where the cross section is expanded exactly in a series of Legendre polynomials, and the series coefficients are computed directly from the deflection angles determined from each trajectory.
TL;DR: In this paper, an experiment is described to measure the differential cross section for the charge exchange reaction K 0 p → K + n from 0.6 to 1.5 GeV/c incident kaon momentum in the angular range −0.65 cos θ ∗ ∗.
Journal Article•
TL;DR: In this article, the authors present a legal analysis of the agreement between the Scuola Normale Superiore di Pisa (SNSN) and the National Archives of Italy.
Abstract: © Scuola Normale Superiore, Pisa, 1977, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
TL;DR: In this article, a new formalism has been developed giving the solutions in explicit form of multiterm linear homogeneous recursion relations with nonconstant coefficients, where the Legendre polynomials are viewed as special cases of combinatorics functions, based on the partitions of an interval (j,m), 0⩽j ⩽m, into parts of lengths 1 and 2.
Abstract: Recently a new formalism has been developed giving the solutions in explicit form of multiterm linear homogeneous recursion relations with nonconstant coefficients. The basic idea was to relate this problem to the one of partitioning an interval into parts of given lengths. This idea is extensively used here to obtain the solutions of linear inhomogeneous difference equations. The resulting method has the advantage of being general: It does not rely on any special device and does not assume any special form for the recursion relation. Applications of these techniques to physical problems are presented elsewhere. Here we show how the method works for first and second order equations. The three‐term Legendre polynomial recursion relation with an arbitrary inhomogeneous term is discussed in detail. The Legendre polynomials are then viewed as special cases of combinatorics functions, P (j,m;z), based on the partitions of an interval (j,m), 0⩽j⩽m, into parts of lengths 1 and 2. P (j,m;z) reduces to the Legendr...
TL;DR: In this paper, the Boltzmann equation for the distribution of Landau quasiparticles on the surface of the Fermi sphere is solved by an expansion in Legendre polynomials having symmetry appropriate to longitudinal sound.
Abstract: The Boltzmann equation for the distribution of Landau quasiparticles on the surface of the Fermi sphere is solved by an expansion in Legendre polynomials having symmetry appropriate to longitudinal sound. It is assumed that the collision operator can be replaced by three finite relaxation times and that the quasiparticle interaction includes a nonzero Landau parameterF
. From the solution, the heat flux and temperature gradient in a zero-sound wave can be computed, and comparison with a corresponding phenomenological generalization of Fourier's law yields an expression relating thermal conductivity, λ,F
, and the parameter ξ-c
0
/V
F
, wherec
0
is the longitudinal zero-sound velocity andv
F
the Fermi velocity. This expression should hold simultaneously with a second equation expressing the condition that zero sound should propagate undamped atO K, and thus we can solve for ξ andF
. We obtain the valueF
=−2.99 (v
F
=56.62 m/sec), which depends hardly at all on the experimental value of λ, specific heat, and sound absorption used. These estimates agree with earlier ones from transverse zero sound as to sign ofF
, but it appears that complete quantitative consistency may necessitate invoking Landau parameters of third and higher order.
TL;DR: In this paper, a general analytical method is presented for calculating the flow field about bodies of revolution in incompressible potential flow as a sequence of elementary analytic functions (Fourier, Chebyshev and Legendre).
TL;DR: In this paper, the Hooke, Prandtl-Reuss equations of elastic, plastic deformations are derived from the constitutive equations for elastic-plastic shells, which do not embody an initial yield condition, but admit immediate, though gradual, evolution of inelastic strain.
TL;DR: In this paper, an an initio computed potential surface for the H3+ system at various geometries is expanded in Legendre polynomials and the ensuing radial coefficients are employed to yield the S −matrix elements from the close coupling expansion of the scattering equations, at the collision energy of recently performed molecular beam experiments.
Abstract: An an initio computed potential surface for the H3+ system at various geometries is expanded in Legendre polynomials and the ensuing radial coefficients are employed to yield the S‐matrix elements from the close coupling expansion of the scattering equations, at the collision energy of recently performed molecular beam experiments. The rotationally inelastic cross sections are obtained for an energy corrected, rigid rotor H2 target and the influence of increasing the basis set size is discussed. The dominant contribution of the charge–quadrupole interaction to the computed observables is examined and an interesting effect is observed on the impact parameter dependence of the 0→2 excitation cross section. All these features are explained in terms of the special nature of the proton interaction with the molecular system and of the dynamics that stem from it.
TL;DR: In this paper, the authors measured differential cross sections at 14 momenta between 610 and 940 MeV/c, over the c.m. angular range −0.7 cos θ ∗.
TL;DR: In this article, the polarization parameter has been measured in K−p elastic scattering at eight incident beam momenta between 650 MeV/c and 1071 MeV /c throughout a center of mass angular range of −0.75 cos θ ∗.
01 Sep 1977
TL;DR: In this paper, it was shown that the nonexistence of a limit does not imply the non-existence of eigenvalues, and the question of what happens if no limit exists.
Abstract: Given an infinite system of linear equationswhere the aij depend on a parameter λ, the eigenvalue problem is to determine values of λ for which xj (j = 1, 2, …) are not all zero. This problem (Taylor (3) and Vaughan (4)) can arise in the vibration of rectangular plates. Little theoretical work, however, appears to have been done concerning the existence and determination of the eigenvalues. The usual procedure (see (3) and (4)) is to consider a truncated or reduced system of N equations and find the values of λ for which the determinant of the N × N matrix [aij] vanishes. If a particular λ tends to a constant value as N is increased then this value is assumed to be an eigenvalue. The question therefore arises as to what happens if no limit exists. Can we assert that there are no eigenvalues? By constructing an appropriate example we show that the non-existence of a limit does not imply the non-existence of eigenvalues. In order to construct our example we first establish a result concerning the Legendre polynomials.