Interpolation theory

In the next two sections, we discuss the basic theories underlying the essential differences between explicit and implicit methods. To help with this, we need to recall a few elementary steps such as the Newton interpolation formula. Details on the various interpolation theories can be found elsewhere (Burden and Faires 1981). 

The notion of kernel or smoothing function has its root in interpolation theory. In effect, the introduction of a localized kernel function works as a smoothing interpolation filter that generates a locally continuous field. In other words, kernels... 

Habermann, C. Kindermann, F. 2007. Multidimensional Spline Interpolation Theory and Applications. Computational Economics 30(2) 153-169. 

As in 41 1-16, this second disputation is followed by a discourse in which the Servant of Yahweh is first presented to the public (42 1-4), then addressed directly (42 5-9). Duhm s identification of 42 1-4 as the first of his four ebed-Jakwe Dichtungen, composed by a disciple of Second Isaiah in late Persian period Judah, has had the effect, even on those who do not accept his interpolation theory, of isolating this brief passage from its literary context. Read in the context proposed above, 42 1-7 or 42 1-9 would corresponds to 41 8-16 as an address to Israel, Servant of Yahweh, following on the presentation... 

This can be done either by perturbation theory or by solving a secular determinant, arising from (30), using the k = ko solutions as basis functions for other values of k. The former procedure is called k p perturbation theory. An excellent review has been given by Kane. The other scheme is called k p interpolation theory and has received close attention in a recent article by Van Dyke. ... 

Chang CL, Lo SL, Yu SL (2005) Applying fuzzy theory and genetic algorithm to interpolate precipitation. J Hydrol 314 92-104... 

Coming." At this time eschatological discourse was co-opted into the service of the esoteric sciences, most especially into alchemical theory in the twelfth century Latin translation of the Turba philosophorum (Arabic original ca. 900 AD). Christian interpolations were added to this text referring to the death and resurrection of the chemicals in apocalyptic terms." The process of distillation in Christian alchemy symbolised death and resurrection, as well as the union of Macrocosm and Microcosm. To the alchemists the death and resurrection of the stone in the manner of a human being was the clearest indication that alchemy was a divine, not a human science. 

Alternatively one can make use of No Barrier Theory (NBT), which allows calculation of the free energy of activation for such reactions with no need for an empirical intrinsic barrier. This approach treats a real chemical reaction as a result of several simple processes for each of which the energy would be a quadratic function of a suitable reaction coordinate. This allows interpolation of the reaction hypersurface a search for the lowest saddle point gives the free energy of activation. This method has been applied to enolate formation, ketene hydration, carbonyl hydration, decarboxylation, and the addition of water to carbocations. ... 

The classical result for the image potential is -q/4x, independent of the metal, but various theories of the metal which assume an infinite potential barrier for the metal electrons give potentials which are reduced in size near the metal boundary, so that the interaction energy is actually finite24 at x = 0. An interpolation formula which reproduces this behavior is... 

The friction factor per base pair y for rotation of DNA around its symmetry axis was determined from FPA studies of restriction fragments containing N+ 1 =43 and 69 bp.(109) Both fragments are sufficiently short that a substantial amplitude of C (t), and also F (t), resides in their Uniform Mode Zones. Particular values of certain parameters were assumed, namely, the rise per base pair h = 3.4 A, the hydrodynamic radius b = 12 A for transverse motion in Eqs. (4.43)-(4.47) (which are quite insensitive to b), and D, = 1.8 x 106 s-1 for 43 bp and D = 4.8 x 105 s for 69 bp. The latter values were extrapolated or interpolated from the data of Elias and Eden using an inverse cubic relation between DL and L. They are close to the values calculated using the theory of Tirado and Garcia de la Torre.(129)... 

---