Showing papers on "Quintic function published in 1987"

Comparing the jerk with other global models of the geomagnetic field from 1960 to 1978

Abstract: About 3300 satellite values of geomagnetic intensity and about 700 observatory values of annual mean magnetic vector components from 1960 to 1978 were fitted by three global models of the geomagnetic field B. Each model includes a spatially constant external field whose time dependence is a constant plus another constant times the Dst index, and each model accepts a time-independent station correction at each observatory. The time dependence of the internal Gauss coefficients is either cubic, quintic, or biquadratic (two independent quadratics, one before and one after January 1, 1970); and g1(0) also has an induced term proportional to the Dst index. The rms residual of the data fit is the same for the cubic and biquadratic models and insignificantly smaller for the quintic model. The quintic and biquadratic models have 1164 adjustable parameters, and the cubic has 1038. At a high level of significance the parameters of the best fitting biquadratic rule out a physical model for the magnetic impulse of 1969 in which the level surfaces of electrical conductivity in the lower mantle are approximately spherical, and the radial magnetic field at the core-mantle boundary goes from one quadratic time dependence to another in a year or less.

On principal ideal testing in totally complex quartic fields and the determination of certain cyclotomic constants

Abstract: Let Y? be any totally complex quartic field. Two algorithms are described for determining whether or not any given ideal in Y is principal. One of these algorithms is very efficient in practice, but its complexity is difficult to analyze; the other algorithm is computationally more elaborate but, in this case, a complexity analysis can be provided. These ideas are applied to the problem of determining the cyclotomic numbers of order 5 for a prime p 1 (mod 5). Given any quadratic (or quintic) nonresidue of p, it is shown that these cyclotomic numbers can be efficiently computed in 0((log p)3) binary operations.

Geometry relevant to the binary quintic

Abstract: The aim in this paper is to indicate one way of interpreting covariants of a binary quintic F geometrically, interpretations having been found recently [7] for its quadratic covariant 2 C 2 , called Γ in [7], and its invariant I 4 . The symbol d C n will denote a covariant of order n in the binary variables x, y and degree d in the coefficients of F, I d being used in preference to d I 0 for invariants . The sum d + n is 4 for both 2 C 2 and I 4 , and no other covariant affords as small a sum; so it is natural to have begun by interpreting these two and to use them as auxiliaries in interpreting others.

Quantitative analysis of the reconstruction performance of interpolants

Abstract: The analysis presented provides a quantitative measure of the reconstruction or interpolation performance of linear, shift-invariant interpolants. The performance criterion is the mean square error of the difference between the sampled and reconstructed functions. The analysis is applicable to reconstruction algorithms used in image processing and to many types of splines used in numerical analysis and computer graphics. When formulated in the frequency domain, the mean square error clearly separates the contribution of the interpolation method from the contribution of the sampled data. The equations provide a rational basis for selecting an optimal interpolant; that is, one which minimizes the mean square error. The analysis has been applied to a selection of frequently used data splines and reconstruction algorithms: parametric cubic and quintic Hermite splines, exponential and nu splines (including the special case of the cubic spline), parametric cubic convolution, Keys' fourth-order cubic, and a cubic with a discontinuous first derivative. The emphasis in this paper is on the image-dependent case in which no a priori knowledge of the frequency spectrum of the sampled function is assumed.