Showing papers on "Quintic function published in 1994"

Self-trapped propagation in the nonlinear cubic-quintic Schrodinger equation: a variational approach

Abstract: In the framework of a variational approach, we determine approximate analytical expressions for the self-trapped solutions in the cubic-quintic nonlinear Schrodinger equation (NLCQSE) and compare our solutions with the available exact ones. Through our approach, we also discuss the stability of the solutions and give simple explanations to the physical meaning of the bistable soliton regime, which can be obtained for some parameter values. >

Symmetric Daubechies' wavelets and numerical solution of NLS equations

Abstract: Recent work has shown that wavelet-based numerical schemes are at least as effective and accurate as standard methods and may allow an 'easy' implementation of a spacetime adaptive grid Up to now, wavelets which have been used for such studies are the 'classical' ones (real Daubechies' wavelets, splines, Shannon and Meyer wavelets, etc) and were applied to diffusion-type equations The present work differs in two points Firstly, for the first time we use a new set of complex symmetric wavelets which have been found recently The advantage of this set is that, unlike classical wavelets, they are simultaneously orthogonal, compactly supported and symmetric Secondly, we apply these wavelets to the physically meaningful cubic and quintic nonlinear Schrodinger equations The most common method to simulate these models numerically is the symmetrized split-step Fourier method For the first time, we propose and study a new way of implementing a global spacetime adaptive discretization in this numerical scheme, based on the interpolation properties of complex-symmetric scaling functions Second, we propose a locally adaptive 'split-step wavelet' method

Positivity conditions for quartic polynomials

Abstract: Simple necessary and sufficient conditions that a quartic polynomial f(z) be non-negative for z greater than or equal to 0 or a is less than or equal to z and z is less than or equal to b are derived, and illustrated geometrically. The geometry provides considerable insight and suggests various approximations and computational simplifications. The theory is applied to monotone quintic spline interpolations, giving necessary and sufficient conditions and an algorithm for monotone Hermite quintic interpolation.

Explicit error estimates for Quintic and biquintic spline interpolation II

Abstract: The purpose of this paper is to provide explicit error estimates between a given function x ( t ∈ PC ( n ) [ a, b ], 2 ≤ n ≤ 6 and its quintic spline interpolate S Δ x ( t ). The results obtained are sharp and improve on, or supplement, those established by Hall [1] and Schultz [2]. These results are then used to acquire precise error bounds for the approximated and biquintic spline interpolates. Sufficient numerical illustration which dwells upon the importance of the obtained results is also included.

Generalized Gans—Gill Method for Smoothing and Differentiation of Composite Profiles in Practice

Abstract: A comparison has been made of the effectiveness of smoothing and differentiating Savitzky-Golay filters in terms of the minimization of total error criterion. The advantages of multipassing filters based on quartic/quintic (for smoothing) and cubic/quartic (for differentiation) polynomials have been proved. A generalized method is proposed for the choice of polynomial length, for both smoothing and differentiating filters, in the fine-structure analysis of composite spectra. This method follows the suggestion of P. Gans and G. B. Gill to locate an inflection point on the curve of variance of residuals between observed and smoothed spectra with respect to the polynomial length of the filter applied. A block diagram of the computer program is shown for optimum choice of polynomial lengths, and an example of its application in the analysis of the fine structure of an experimental multiplet band system is demonstrated.

The quadratic formula made hard: A less radical approach to solving equations

Abstract: It appears that, along with many of my friends and colleagues, I had been brainwashed by the great and tragic lives of Abel and Galois to believe that no general formulas are possible for roots of equations higher than quartic. This seemed to be confirmed by the brilliant and arduous solution of the general quintic by Hermite. Yet, below we find a formula giving a root to any algebraic equation of degree 2-5 and any reduced equation (see below) of higher degree. This algorithm, which must have been familiar to Lagrange, resulted when I was working on a paper on the asymptotics of hypergeometric functions where Gauss' multiplication formula for the gamma function is used to reduce certain infinite series, and by a happy accident my copy of Whittaker and Watson opened at p. 133.

Modeling scattered function data on curved surfaces

Abstract: We present efficient algorithms to model a collection of scattered function data defined on a given smooth domain surface D in three dimensional real space (lR3 ), by a C1 cubic or a C2 quintic piecewise trivariate polynomial approximation F (a mapping from D into lR ). The smooth polynomial pieces or finite elements of F are defined on a three dimensional triangulation called the simplicial hull and defined over the domain surface D. Our smooth polynomial approximations allows one to additionally control the local geometry of the modeled function F. We also present two different techniques for visualizing the graph of the function F.

Sharp error bounds for the derivatives of Lidstone-spline interpolation II

Abstract: In this paper, we shall derive explicit error estimates in L ∞ norm between a given function f ( x ) ∈ PC ( n ) [ a , b ], 4 ≤ n ≤ 6 and its quintic Lidstone-spline interpolate. The results obtained are then used to establish precise error bounds for the approximated and biquintic Lidstone-spline interpolates. We also include applications to integral equations and boundary value problems as well as sufficient numerical examples which dwell upon the sharpness of the obtained results.

On the resolution of index form equations in dihedral quartic number fields

Abstract: We describe a new algorithm, basedon sieving procedures, for determining the minimal index and all elements with minimal index in a class of totally real quartic fields with Galois group D 8. It is not universally applicable, but its applicability is easily checked for any particular example, and it is very fast when applicable. We include several tables demonstrating the potential of the method. (A more general approach for quartic fields, described in [Gaal et al.], requires much more computation time for each field.) Finally, we present a family of totally real quartic fields with Galois group D 8 and having minimal index 1 (that is, a power integral basis).

Smooth Low Degree Approximations of Polyhedra

Abstract: We present efficient algorithms to construct both C1 and C 2 smooth meshes of cubic and quintic A-patches to approximate a given polyhedron P in three dimensions. The A~patch is a smooth and single-sheeted zero-contour patch of a trivariate polynomial in Bernstein-Bezier (BB) form defined within a tetrahedron. The smooth mesh constructions rely on a novel scheme to build an inner simplicial hull E consisting of tetrahedra and defined by the faces of the given polyhedron p, A single cubic or quintic A-patch is then constructed within each tetrahedron of the simplicial hull :E with the resulting surface being C l or C 2 smooth, respectively. The free parameters of each individual A-patch can be independently controlled to achieve both local and globa1shape deformations and a family of C 1 or C 2 smooth approximations of the original polyhedron.