Showing papers on "Quintic function published in 2003"
19 Feb 2003
TL;DR: An online method for obtaining smooth, jerk-bounded trajectories has been developed and implemented and a method for blending these straight-line trajectories over a series of way points is also discussed.
Abstract: An online method for obtaining smooth, jerk-bounded trajectories has been developed and implemented. Jerk limitation is important in industrial robot applications, since it results in improved path tracking and reduced wear on the robot. The method described herein uses a concatenation of fifth-order polynomials to provide a smooth trajectory between two way points. The trajectory approximates a linear segment with parabolic blends trajectory. A sine wave template is used to calculate the end conditions (control points) for ramps from zero acceleration to nonzero acceleration. Joining these control points with quintic polynomials results in a controlled quintic trajectory that does not oscillate, and is near time optimal for the jerk and acceleration limits specified. The method requires only the computation of the quintic control points, up to a maximum of eight points per trajectory way point. This provides hard bounds for online motion algorithm computation time. A method for blending these straight-line trajectories over a series of way points is also discussed. Simulations and experimental results on an industrial robot are presented.
400 citations
TL;DR: In this article, the nonlinear properties of two distinct chalcogenide glasses were investigated experimentally and theoretically through a spatially resolved Mach-Zehnder interferometer, and it was shown that the resulting nonlinear index coefficient cannot be correctly described with the usual cubic model.
183 citations
TL;DR: In this paper, an extension of the time-splitting sine-spectral (TSSP) method for solving damped focusing nonlinear Schrodinger equations (NLSs) is presented.
Abstract: This paper introduces an extension of the time-splitting sine-spectral (TSSP) method for solving damped focusing nonlinear Schrodinger equations (NLSs). The method is explicit, unconditionally stable, and time transversal invariant. Moreover, it preserves the exact decay rate for the normalization of the wave function if linear damping terms are added to the NLS. Extensive numerical tests are presented for cubic focusing NLSs in two dimensions with a linear, cubic, or quintic damping term. Our numerical results show that quintic or cubic damping always arrests blowup, while linear damping can arrest blowup only when the damping parameter ${\delta}$ is larger than a threshold value ${\delta}_{\rm th}$. We note that our method can also be applied to solve the three-dimensional Gross-Pitaevskii equation with a quintic damping term to model the dynamics of a collapsing and exploding Bose-Einstein condensate (BEC).
97 citations
TL;DR: Two methods for simultaneously approximating both branches of a two-branched function using Chebyshev polynomials are introduced, which remove the pernicious, convergence-wrecking effects of the square root singularity at the limit point where the two branches meet.
85 citations
Posted Content•
TL;DR: In this paper, an extension of the time-splitting sine-spectral (TSSP) method for solving damped focusing nonlinear Schrodinger equations (NLS) is presented.
Abstract: This paper introduces an extension of the time-splitting sine-spectral (TSSP) method for solving damped focusing nonlinear Schrodinger equations (NLS). The method is explicit, unconditionally stable and time transversal invariant. Moreover, it preserves the exact decay rate for the normalization of the wave function if linear damping terms are added to the NLS. Extensive numerical tests are presented for cubic focusing nonlinear Schrodinger equations in 2d with a linear, cubic or a quintic damping term. Our numerical results show that quintic or cubic damping always arrests blowup, while linear damping can arrest blowup only when the damping parameter $\dt$ is larger than a threshold value $\dt_{\rm th}$. We note that our method can also be applied to solve the 3d Gross-Pitaevskii equation with a quintic damping term to model the dynamics of a collapsing and exploding Bose-Einstein condensate (BEC).
55 citations
TL;DR: A fourth-order method based on quintic splines for the solution of third-order linear and non-linear boundary-value problems (BVPs) of the form y^'^'=f(x,y),a= is presented.
53 citations
Posted Content•
TL;DR: The periodic non-linear Schrodinger equations with odd integer power nonlinearities were studied in this article for initial data which are assumed to be small in some negative order Sobolev space, but which may have large L 2 mass.
Abstract: We study the periodic non-linear Schrodinger equations with odd integer power nonlinearities, for initial data which are assumed to be small in some negative order Sobolev space, but which may have large L^2 mass. These equations are known to be illposed in H^s for all negative s, in the sense that the solution map fails to be uniformly continuous from H^s to itself, even for short times and small norms. Here we show that these equations are even more unstable. For the cubic equation, the solution map is discontinuous from H^s to even the space of distributions. For the quintic and higher order nonlinearities, there exist pairs of solutions which are uniformly bounded in H^s, are arbitrarily close in any C^N norm at time zero, and fail to be close in the distribution topology at an arbitrarily small positive time.
52 citations
TL;DR: A high accuracy of the approximation is achieved by giving a proper parametrization of the curve, and the approximation order of the height function along the helix axis is 9 provided that the screw angle of the Helix is fixed.
51 citations
TL;DR: In this paper, the authors used Painleve analysis, the Hirota multi-linear method and a direct ansatz technique to study analytic solutions of the (1+1)-dimensional complex cubic and quintic Swift-Hohenberg equations.
37 citations
TL;DR: In this paper, the shape-preservation property is secured by adjusting 'tension' parameters that arise upon relaxing parametric continuity to geometric continuity, and a simpler and cheaper alternative is also introduced.
Abstract: The interpolation of a planar sequence of points p 0 ,..., p N by shape-preserving G 1 or G 2 PH quintic splines with specified end conditions is considered. The shape-preservation property is secured by adjusting 'tension' parameters that arise upon relaxing parametric continuity to geometric continuity. In the G 2 case, the PH spline construction is based on applying Newton-Raphson iterations to a global system of equations, commencing with a suitable initialization strategy-this generalizes the construction described previously in Numerical Algorithms 27, 35-60 (2001). As a simpler and cheaper alternative, a shape-preserving G 1 PH quintic spline scheme is also introduced. Although the order of continuity is lower, this has the advantage of allowing construction through purely local equations.
TL;DR: These transformations for the elimination of some of the intermediate terms in a polynomial are described in modern notation and their possible utility for polynomials solving is discussed, particularly with respect to the Mathematica poster on the solution of the quintic.
Abstract: Tschirnhaus gave transformations for the elimination of some of the intermediate terms in a polynomial. His transformations were developed further by Bring and Jerrard, and here we describe all these transformations in modern notation. We also discuss their possible utility for polynomial solving, particularly with respect to the Mathematica poster on the solution of the quintic.
TL;DR: In this paper, the authors introduce a new method to construct a C3 function which interpolates a given set of data in a shape-preserving way, and the resulting function is a parametric quintic curve whose shape is controlled by tension parameters.
TL;DR: In this article, a new quintic discrete nonlinear Schrodinger (QDNLS) equation is studied and exact localized solutions for integrable cases are presented for certain sets of parameters.
TL;DR: In this paper, the authors investigated low-degree points on the Fermat curve of degree 13, the Snyder quintic curve and the Klein quartic curve, and used Coleman's effective Chabauty method to obtain bounds for the number of cubic points on each of the former two curves.
Abstract: We investigate low-degree points on the Fermat curve of degree 13, the Snyder quintic curve and the Klein quartic curve. We compute all quadratic points on these curves and use Coleman's effective Chabauty method to obtain bounds for the number of cubic points on each of the former two curves.
TL;DR: In this paper, a number of periodic zero-velocity analytic solutions of the complex quintic Swift-Hohenberg equation (CSHE) have been found using a direct ansatz approach.
Journal Article•
TL;DR: In this article, the stability and bifurcations of limit cycles of the equator in a quintic polynomial system are discussed, and some results obtained in this paper present an intersting contrast with the related results on cubic systems.
Abstract: In this paper, the stability and bifurcations of limit cycles of the equator in a quintic polynomial system are discussed. Some results obtained in this paper present an intersting contrast with the related results on cubic systems.
Posted Content•
TL;DR: In this paper, the relative Hilbert scheme of lines in the Dwork pencil of quintic threefolds is described and the corresponding relative Hilbert schemes associated to the mirror family of quintics are described.
Abstract: We give a description of the relative Hilbert scheme of lines in the Dwork pencil of quintic threefolds. We describe the corresponding relative Hilbert scheme associated to the mirror family of quintic threefolds.
TL;DR: In this paper, it was shown that the vector bundles obtained by Serre's construction from smooth elliptic quintic curves on X form an open part of an irreducible component M' of M(2;0,3) of rank-2 stable vector bundles with Chern classes c_1=0, c_2=3 on the Fano threefold X, the double solid of index two.
Abstract: Let M(2;0,3) be the moduli space of rank-2 stable vector bundles with Chern classes c_1=0, c_2=3 on the Fano threefold X, the double solid of index two. We prove that the vector bundles obtained by Serre's construction from smooth elliptic quintic curves on X form an open part of an irreducible component M' of M(2;0,3) and that the Abel-Jacobi map F:M'-->J(X) into the intermediate Jacobian J(X) defined by the second Chern class is generically finite of degree 84 onto a translate of the theta divisor. We also prove that the family of elliptic quintics on a general X is irreducible and of dimension 10.
TL;DR: In this paper, the isotopy classification of unions of an M-curve of degree 5 and two lines in ℝP2 satisfying some conditions of maximality and general position is obtained.
Abstract: In this paper the isotopy classification of unions of an M-curve of degree 5 and two lines in ℝP2 satisfying some conditions of maximality and general position is obtained. This classification consists of 20 isotopy types of such unions.
TL;DR: For general algebraic maps of orders higher than or equal to 5, the word-lifting technique for calculating parameters of symbolic sequences meets with the difficulty of lacking explicit expressions of the inverse functions of the maps as discussed by the authors.
TL;DR: This work uses piecewise linear terms to emulate the polynomial nonlinear terms in a typical reaction-diffusion equation, replacing it thus with a set of simple linear inhomogeneous differential equations, yielding an excellent approximation to the exact propagating front.
Abstract: We use piecewise linear terms to emulate the polynomial nonlinear terms in a typical reaction-diffusion equation, replacing it thus with a set of simple linear inhomogeneous differential equations. The resulting analytic solution constitutes an excellent approximation to the exact propagating front, as is explicitly shown in the case of cubic and quintic nonlinearities, yielding also the correct value for the physically selected speed of the observable front. Such a piecewise linear emulation can be used for any nonlinearity, giving therefore a very reliable and accurate method for determining the selected speed of fronts invading unstable states, especially pushed fronts.
TL;DR: In this paper, the mathematical structure of the optical cubic-quintic Schrodinger equation is investigated in a special way by considering a potential depending upon the modulus of the wave-functions involved.
TL;DR: If the quintic polynomial is a solvablePolynomial, then its associated parameter Z in the Brioschi resolvent satisfies Z=g(t) where g( t) is a rational function in ℚ(t), and t is chosen from an appropriate field.
Abstract: The Brioschi resolvent y 5 −10Zy 3 +45Z 2 y−Z 2 is a key component of an algorithm for calculating the roots of a general quintic polynomial. It is obtained from the general quintic polynomial by applying two Tschirnhausen transformations. In this paper it is shown that if the quintic polynomial is a solvable polynomial, then its associated parameter Z in the Brioschi resolvent satisfies Z=g(t) where g(t) is a rational function in ℚ(t) and t is chosen from an appropriate field.
11 Nov 2003
TL;DR: In this paper, a closed-form expression for the effective nonlinear index coefficient was derived for the cubic and quintic index coefficients, respectively, assuming a medium with two-photon nonlinear absorption and both cubic (n 2 ), quintic (n 4 ) nonlinear indices variations.
Abstract: Mach-Zehnder interferometric technique for nonlinearity measurements is briefly described. The optical setup is combined to a charge-coupled device image processing and allows to resolve the spatial profile of the complex nonlinear variation index with only one laser shot in the nonlinear material. Our experimental results in chalcogenide glasses clearly demonstrate the existence of an intensity dependent change of the sign in the nonlinear dephasing. It is shown that the nonlinear index coefficient cannot be correctly described with the usual cubic model. A more convenient theory is developed assuming a medium with two-photon nonlinear absorption and both cubic (n 2 ) and quintic (n 4 ) nonlinear index variations. The resulting closed-form expression for the effective nonlinear index coefficient allows to extract the cubic and quintic index coefficients. Keywords: Mach-Zehnder interferometer, nonlinear refractive index, Fourier transform, nonlinear absorption, quintic materials.
01 Apr 2003