Showing papers on "Auxiliary function published in 1987"

Polynomials with Low Height and Prescribed Vanishing

Abstract: In a recent paper [2] we obtained an improved formulation of Siegel’s classical result([9],Bd. I,p. 213, Hilfssatz) on small solutions of systems of linear equations. Our purpose here is to illustrate the use of this new version of Siegel’s lemma in the problem of constructing a simple type of auxiliary polynomial. More precisely, let k be an algebraic number field, O k its ring of integers, α1,α2,…,αJ distinct, nonzero algebraic numbers (which are not necesarily in k), and m1,m2,…,mJ positive integers. We will be interested in determining nontrivial polynomials P(X) in 0 K [X] which have degree less than N, vanish at each αj with multiplicity at least mj and have low height. In particular, the height of such plynomials will be bounded from above by a simple function of the degrees and heights of the algebraic numbers αj and the remaining data in the problem: m1,m2,…mJ, N and the field constants associated with k.

Absorption and emission line profile coefficients of multilevel atoms—IV. Velocity-averaged generalized redistribution functions for three-photon processes

Abstract: The atomic generalized redistribution functions for three-photon processes, derived in the previous paper of this series, are formulated in terms of linear superpositions of newly introduced auxiliary functions qI−qVI, thus extending the traditional formalism of redistribution functions for two-photon processes. The corresponding velocity-averaged laboratory functions QI−QVI of these auxiliary functions are derived in both their angle-dependent and angle-averaged forms. Since the expressions found for QI−QVI are quite complicated, the so-called disentangled approximation is employed that uses the representative values of the generalized redistribution function at an orthogonal triad of photon directions rather than the angle-averaged function itself. This approximation yields relatively simple expressions which can be used in radiative transfer calculations.

A sew method for global optixal control

Abstract: A variation-free method for global optimal control is proposed. The method is based on the appropriately constructed measures and leads to numerical algorithms yielding (in the limit) the exact global minimum value and the subset of all global minimizers within a given set of functions. It does not employ variational principles nor variation-based techniques. 1. INTRODUCTIOS Traditional methods of optimal control are based on some kind of variational optimality condition which is applied either to find right away the locally optimal solution or to arrange for an approximation process. Yost cf currently used conditions are variation-based , thus, requiring certain smoothness and convexity assumptions, complete information about the state of the system and, sometimes, the existence of (certain continuously differentiable auxiliary function, or other properties. The method