Topic
Parabola
About: Parabola is a research topic. Over the lifetime, 590 publications have been published within this topic receiving 6167 citations.
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TL;DR: In this article, it was shown that for a wave ψ in the form of an Airy function the probability density ψ 2 propagates in free space without distortion and with constant acceleration.
Abstract: We show that for a wave ψ in the form of an Airy function the probability density ‖ψ‖2 propagates in free space without distortion and with constant acceleration. This ’’Airy packet’’ corresponds classically to a family of orbits represented by a parabola in phase space; under the classical motion this parabola translates rigidly, and the fact that no other curve has this property shows that the Airy packet is unique in propagating without change of form. The acceleration of the packet (which does not violate Ehrenfest’s theorem) is related to the curvature of the caustic (envelope) of the family of world lines in spacetime. When a spatially uniform force F (t) acts the Airy packet continues to preserve its integrity. We exhibit the solution of Schrodinger’s equation for general F (t) and discuss the motion for some special forms of F (t).
1,298 citations
TL;DR: In this paper, the probability of the parabolic failure domain is computed exactly by inversion of the characteristic function for the Parabolic quadratic form, and the exact results for the probability content of the failure domain obtained from the full second-order Taylor expansion of a failure function at the design point are presented.
Abstract: In second-order reliability methods the failure surface in the standard normal space is approximated by a parabolic surface at the design point. The corresponding probability is computed by asymptotic formulas and by approximation formulas. In this paper the probability content of the parabolic failure domain is computed exactly by inversion of the characteristic function for the parabolic quadratic form. Also, the exact results for the probability content of the failure domain obtained from the full second-order Taylor expansion of the failure function at the design point is presented. The approximating parabola does not depend on the formulation of the failure function as long as this preserves the original failure surface. This invariance characteristic is in general not shared by the approximation obtained using the full second-order Taylor expansion of the failure function at the design point. The exact results for the probability content of the approximating quadratic domains significantly extend the class of problems that can be treated by approximate methods.
288 citations
TL;DR: This work analyzes two recently proposed contours, namely a parabola and a hyperbola, for the numerical inversion of the Laplace transform and determines estimates for the optimal parameters that dene these contours.
Abstract: Some of the most eectiv e methods for the numerical inversion of the Laplace transform are based on the approximation of the Bromwich contour integral. The accuracy of these methods often hinges on a good choice of contour, and several such contours have been proposed in the literature. Here we analyze two recently proposed contours, namely a parabola and a hyperbola. Using a representative model problem, we determine estimates for the optimal parameters that dene these contours. An application to a fractional diusion equation is presented.
237 citations
TL;DR: The piecewise parabolic method (PPM) as discussed by the authors is a finite volume scheme that represents physical variables as averages over a grid zone rather than single values at discrete points, and each zone boundary is treated as a discontinuity.
Abstract: The Piecewise Parabolic Method (PPM), a numerical technique developed in astrophysics for modeling fluid flows with strong shocks and discontinuities is adapted for treating sharp gradients in small-scale meteorological flows. PPM differs substantially from conventional gridpoint techniques in three ways. First, PPM is a finite volume scheme, and thus represents physical variables as averages over a grid zone rather than single values at discrete points. Second, a unique, monotonic parabola is fit to the zone average of each dependent variable using information from neighboring zone averages. As shown in a series of one- and two-dimensional linear advection experiments, the use of parabolas provides for extremely accurate advection, particularly of sharp gradients. Furthermore, the monotonicity constraint renders PPM's solutions free from Gibbs oscillations. PPM's third attribute is that each zone boundary is treated as a discontinuity. Using the method of characteristic the nonlinear flux of qua...
192 citations
TL;DR: In this paper, a method is proposed to allow a more accurateevaluation of thermogravimetric data to identify diffusion or partial diffusion control of scalingkinetics, based on the fitting ofmass-gain data to a parabola over a short time interval.
Abstract: A method is proposed to allow a more accurateevaluation of thermogravimetric data to identifydiffusion or partial diffusion control of scalingkinetics. This method is based on the fitting ofmass-gain data to a parabola over a short time interval.The translation of the time interval over the entiretest time period provides an actual instantaneousparabolic rate constant independently of any transient stage or simultaneous reaction steps. Theusefulness and limitations of this procedure areillustrated from oxidation tests performed on severalmetallic materials (pure nickel, single-crystalsuperalloys, and NbTi-Al alloy).
183 citations