Showing papers on "Quintic function published in 1993"
TL;DR: In this paper, the authors give a mathematical account of a recent string theory calcula- tion which predicts the number of rational curves on the generic quintic three-fold, using the interpretation of Yukawa couplings in terms of variations of Hodge structure, a new q-expansion principle for functions on the moduli space of Calabi-Yau manifolds, and the mirror symmetry.
Abstract: We give a mathematical account of a recent string theory calcula- tion which predicts the number of rational curves on the generic quintic three- fold. Our account involves the interpretation of Yukawa couplings in terms of variations of Hodge structure, a new q-expansion principle for functions on the moduli space of Calabi-Yau manifolds, and the "mirror symmetry" phe- nomenon recently observed by string theorists. DEPARTMENT OF MATHEMATICS, DUKE UNIVERSITY, DURHAM, NORTH CAROLINA 27706 E-mail address: drm@math.duke.edu This content downloaded from 157.55.39.224 on Wed, 14 Dec 2016 04:59:36 UTC All use subject to http://about.jstor.org/terms
203 citations
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TL;DR: In this paper, the authors presented formulas for the number of rational (n)-nodal curves in an n-dimensional linear system on a smooth, projective surface, which yields in particular the numbers of rational curves in the system of hyperplane sections of a generic $K3-$surface imbedded in \p{n} by a complete system of curves of genus $n$ as well as the number {\bf17,601,000} of rational plane quintic curves in a generic quintic threefold.
Abstract: For each $1\leq n\leq6$ we present formulas for the number of $n-$nodal curves in an $n-$dimensional linear system on a smooth, projective surface. This yields in particular the numbers of rational curves in the system of hyperplane sections of a generic $K3-$surface imbedded in \p{n} by a complete system of curves of genus $n$ as well as the number {\bf17,601,000} of rational ({\em singular}) plane quintic curves in a generic quintic threefold.
102 citations
TL;DR: A method of interpolating a set of discrete data points to form a composite quintic spline for application in precision machining, which is closer to being arc-length parameterized than cubic splines.
Abstract: The paper presents a method of interpolating a set of discrete data points to form a composite quintic spline for application in precision machining. Precision machining requires tight machining tolerance, i.e. the preselected data points on the original curve are approximately evenly distributed according to a small total curvature between two consecutive data points. With this restriction, the resultant composite quintic splines, in comparison with cubic splines, are closer to being arc-length parameterized. The quintic splines are also likely to have shapes that are close to those of the original curves with no unwanted high-order oscillations. Curves with these properties should be useful for reverse engineering and online motion-command generation for precision machining.
88 citations
TL;DR: In this paper, the authors presented a quintic with 126 nodes, which is the smallest quintic known to exist in Euclidean geometry, except for the Segre Cubic and the Burkhardt Quartic.
48 citations
TL;DR: This paper reduces the problem of solving index form equations in quartic number fields K to the resolution of a cubic equation F ( u, v ) = i and a corresponding system of quadratic equations Q 1, Q 2 = v, which enables a fast algorithm for calculating "small" solutions of index form equation in any Quartic number field.
34 citations
TL;DR: In this paper, the integrability of the cubic-quintic nonlinear Schrodinger equation is investigated numerically, and it is shown that the quintic term leads to a spatiotemporal complexity of wave fields, and illustrate that this behaviour is associated with the stochastic partition of energy in Fourier modes.
Abstract: The integrability of the cubic-quintic nonlinear Schrodinger equation is investigated numerically. By analytically studying the linear stability of the homogeneous state and numerically solving such a continuum Hamiltonian dynamic system, the authors show that the quintic nonlinear term leads to a spatiotemporal complexity of wave fields, and illustrate that this-behaviour is associated with the stochastic partition of energy in Fourier modes. In addition, they show the presence of the stochastic motion is due to the homoclinic orbit crossings.
9 citations
TL;DR: In this article, the problem of wave-number selection by a ramp is considered within the framework of the cubic and quintic Ginzburg-Landau (GL) equations with a sign-changing overcriticality parameter.
Abstract: The problem of wave-number selection by a ramp, i.e., a region smoothly matching sub- and supercritical domains, is considered within the framework of the cubic and quintic Ginzburg-Landau (GL) equations with a sign-changing overcriticality parameter. A local frequency is also allowed to be a smooth function of the spatial coordinate. For the cubic model, a unique value of the selected wave number is found by means of an asymptotic procedure valid when the imaginary parts of coefficients in the GL equation are small, while the group velocity is arbitrary. Under certain conditions, the wave number may lie outside the stability band, which is expected to give rise to a dynamical chaos. In the quintic model, which describes a system with the inverted bifurcation, the selection scenario is much simpler: A front separating a traveling wave and the trivial state is expected to be pinned at the point of the ramp where its velocity, regarded as a function of the local overcriticality, vanishes. Eventually, the wave-number selection is performed by the pinned front. Experimentally, the selection may be realized in the traveling-wave convection, or in a self-oscillatory chemical system.
7 citations
TL;DR: In this paper, exact solitary wave solutions of the one-dimensional quintic complex Ginzburg-Landau equation are obtained using a method derived from the Painlev\'e test for integrability.
Abstract: Exact solitary wave solutions of the one-dimensional quintic complex Ginzburg-Landau equation are obtained using a method derived from the Painlev\'e test for integrability. These solutions are expressed in terms of hyperbolic functions, and include the pulses and fronts found by van Saarloos and Hohenberg. We also find previously unknown sources and sinks. The emphasis is put on the systematic character of the method which breaks away from approaches involving somewhat ad hoc Ans\"atze.
6 citations
TL;DR: In this paper, the authors used pairs of fourth-order polynomials (quartic blending) to construct waveforms that represent the restricted parameters vs time for an oblate Earth.
Abstract: This paper presents a numerical method to rapidly determine visibility periods of a satellite for a ground station having restrictions in azimuth, elevation, and range. The algorithm uses pairs of fourth-order polynomials (quartic blending) to construct waveforms that represent the restricted parameters vs time for an oblate Earth. These waveforms are produced from either uniform or arbitrarily spaced data points; viewing times are obtained by extracting the real roots of localized quintic polynomials. This algorithm works for all orbital eccentricities and perturbed satellite motion, provided the functions do not become discontinuous. Results from this algorithm are almost identical to those obtained by modeling satellites subject to first-order secular perturbations caused by mass anomalies but generated with an 89% decrease in computation time over a 5-s step brute force method. Advantages of this numerical method include compact storage and ease of calculation, making it attractive for supporting ground-based satellite operations. C(T) e ee F(r) fpUM(t) / LIM(0 /RANGE(O H / J_2 h «o P Pi p Pi Q R(t)
4 citations
TL;DR: In this article, the cyclic cubic, quartic, and quintic number fields of a prime conductor p containing units that bear a special relationship to the classical Gaussian periods were examined.
Abstract: In this paper we examine cyclic cubic, quartic, and quintic number fields of prime conductor p containing units that bear a special relationship to the classical Gaussian periods: qj qj+1 + c is a unit for periods q_ and c E Z .
2 citations