Showing papers on "Quintic function published in 1997"
TL;DR: Time-localized solitary wave solutions of the one-dimensional complex Ginzburg-Landau equation (CGLE) are analyzed for the case of normal group-velocity dispersion in this article.
Abstract: Time-localized solitary wave solutions of the one-dimensional complex Ginzburg-Landau equation (CGLE) are analyzed for the case of normal group-velocity dispersion Exact soliton solutions are found for both the cubic and the quintic CGLE The stability of these solutions is investigated numerically The regions in the parameter space in which stable pulselike solutions of the quintic CGLE exist are numerically determined These regions contain subspaces where analytical solutions may be found An investigation of the role of group-velocity dispersion changes in magnitude and sign on the spectral and temporal characteristics of the stable pulse solutions is also carried out
161 citations
TL;DR: All approximations constructed in this paper have the optimal order of approximation, twice of the degree of approximate Bezier curves.
59 citations
TL;DR: The additional degree of freedom of PH spirals identified herein relaxes constraints otherwise required to guarantee their existence, and offers the designer precise control over their total length and/or the ability to fine-tune their curvature distributions.
Abstract: Walton and Meek [Walton, D.J. and Meek, D.S., A Pythagorean-hodograph quintic spiral. Computer Aided Design , 1996, 28 , 943–950] have recently advocated the use of Pythagorean-hodograph quintics of monotone curvature, or “PH spirals” for short, as transitional elements that give G 2 connections of linear and circular arcs in applications such as layout of highways and railways—in which context PH curves provide the important advantage of rational offsets and exact rectifications . They construct a PH quintic, interpolating an initial point and tangent and a final tangent, with monotone curvature variation from zero to a given (extremum) final value. We show that using the complex representation for PH curves greatly simplifies this problem and also reveals that the method of Walton and Meek yields a special instance among a one-parameter family of solutions. The additional degree of freedom of PH spirals identified herein relaxes constraints otherwise required to guarantee their existence, and offers the designer precise control over their total length and/or the ability to fine-tune their curvature distributions.
53 citations
TL;DR: In this paper, the effect of quintic nonlinearities in the propagation of optical solitons in cubic nonlinear media was studied, and the decay of higher-order soliton in the presence of self-defocusing quintic perturbations was investigated.
47 citations
14 Sep 1997
TL;DR: In this article, the authors present two algorithms for ne-tuning rational B-spline curves in a projective dual three-space known as the Image Space of Spatial Kinematics.
Abstract: This paper presents two algorithms for ne-tuning rational spatial motions suitable for Computer Aided Design. The rational motions are represented by rational B-spline curves in a projective dual three-space known as the Image Space of Spatial Kinematics. The problem of ne-tuning of rational motions is studied as that of ne-tuning the corresponding rational curves in the Image Space called the image curves. The path-smoothing algorithm automatically detects and smoothes out the third order geometric discontinuities in the path of a cubic rational Bspline image curve. The speed-smoothing algorithm uses a quintic rational spline image curve to obtain a second-order geometric approximation of the path of a cubic rational B-spline image curve while allowing specication of the speed and the rate of change of speed at the key points to obtain a near constant kinetic energy parametrization. The notion of kinetic energy is used in the paper as a natural way of combining the rotational and translational speed of a spatial motion. The results have applications in trajectory generation in robotics, planing of camera movement, spatial navigation in visualization and virtual reality systems, as well as mechanical system simulation.
35 citations
TL;DR: In this paper, the stability of the bright solitary wave solution to the perturbed cubic-quintic Schroedinger equation is considered, and it is shown that in a certain region of parameter space these solutions are unstable, with the instability being manifested as a small positive eigenvalue.
Abstract: The stability of the bright solitary wave solution to the perturbed cubic-quintic Schroedinger equation is considered. It is shown that in a certain region of parameter space these solutions are unstable, with the instability being manifested as a small positive eigenvalue. Furthermore, it is shown that in the complimentary region of parameter space there are no small unstable eigenvalues. The proof involves a novel calculation of the Evans function, which is of interest in its own right. As a consequence of the eigenvalue calculation, it is additionally shown that N-bump bright solitary waves bifurcate from the primary wave.
29 citations
Book•
31 Jan 1997
TL;DR: The complex sphere finite automorphism groups of the sphere invariant functions inverses of the invariant function functions are inversed by the complex sphere automorphisms as discussed by the authors.
Abstract: The complex sphere finite automorphism groups of the sphere invariant functions inverses of the invariant functions.
29 citations
TL;DR: In this paper, the authors proposed a rational mapping of the Riemann sphere with the icosahedral symmetry of a general quintic, which is the same as the one presented in this paper.
Abstract: Recently, Peter Doyle and Curt McMullen devised an iterative solution to the fifth degree polynomial At the method's core is a rational mapping of the Riemann sphere with the icosahedral symmetry of a general quintic Moreover, this map posseses "reliable" dynamics: for almost any initial point, the its trajectory converges to one of the periodic cycles that comprise an icosahedral orbit This symmetry-breaking provides for a reliable or "generally-convergent" quintic-solving algorithm: with almost any fifth-degree equation, associate a rational mapping that has reliable dynamics and whose attractor consists of points from which one computes a root
An algorithm that solves the sixth-degree equation requires a dynamical system with the symmetry of the alternating group on six things This group does not act on the Riemmann sphere, but does act on the complex projective plane--this is the Valentiner group The present work exploits the resulting 2-dimensional geometry in finding a Valentiner-symmetric rational mapping whose elegant dynamics experimentally appear to be reliable in the above sense---transferred to the 2-dimensional setting This map provides the central feature of a conjecturally-reliable sextic-solving algorithm analogous to that employed in the quintic case
10 citations
23 Jun 1997
TL;DR: In this article, an integral interpolating subspline curves for given data points and the knot vector are constructed, which are connected in an affinely invariant way with the control points.
Abstract: We construct (integral) interpolating subspline curves for given data points and the knot vector. The algorithm is very close to B spline approximation. The idea is to blend interpolating Lagrangian splines using B spline techniques. Everything is connected in an affinely invariant way with the control points and the knot vector. We are able to show that our scheme produces high quality subsplines, which include known procedures like Overhauser or quintic interpolation schemes. In addition we may sweep to B splines and return in a very lucid way. Examples show the power of the method. The given procedure allows generalisations to rational subsplines and to tensor product interpolating surfaces.
5 citations
TL;DR: The focus of this paper is the construction of a new tetrahedral patch, which allows an explicit representation but does not require a split of the domain tetrahedra.
2 citations
Posted Content•
TL;DR: Decker et al. as discussed by the authors showed that the union of two quintic elliptic scrolls in P^4 intersecting transversally along an elliptic normal quintic curve is a singular surface Z which behaves numerically like a bielliptic surface.
Abstract: The union of two quintic elliptic scrolls in P^4 intersecting transversally along an elliptic normal quintic curve is a singular surface Z which behaves numerically like a bielliptic surface. In the appendix to the paper [W. Decker et al.: Syzygies of abelian and bielliptic surfaces in P^4, alg-geom/9606013] where the equations of this singular surface were computed, we proved that Z defines a smooth point in the appropriate Hilbert scheme and that Z cannot be smoothed in P^4. Here we consider the analogous situation in higher dimensional projective spaces P^{n-1}, where, to our surprise, the answer depends on the dimension n-1. If n is odd the union of two scrolls cannot be smoothed, whereas it can be smoothed if n is even. We construct an explicit smoothing.
01 Jan 1997
TL;DR: In this paper, it was shown that every quadratic field K is a subfield of the splitting field of a dihedral quintic trinomial x5 + ax + b if and only if the discriminant of K is of the form -49 or -89, where q is the product of distinct primes congruent to 1 modulo 4.
Abstract: It is known that every quadratic field K is a subfield of the splitting field of a dihedral quintic polynomial. In this paper it is shown that K is a subfield of the splitting field of a dihedral quintic trinomial x5 + ax + b if and only if the discriminant of K is of the form -49 or -89, where q is the (possibly empty) product of distinct primes congruent to 1 modulo 4.