Operator (computer programming)

About: Operator (computer programming) is a research topic. Over the lifetime, 40896 publications have been published within this topic receiving 671452 citations. The topic is also known as: operator symbol & operator name.

Conformally invariant powers of the Laplacian — A complete nonexistence theorem

Abstract: Conformally invariant operators and the equations they determine play a central role in the study of manifolds with pseudo-Riemannian, Riemannian, conformai and related structures. This observation dates back to at least the very early part of the last century when it was shown that the equations of massless particles on curved space-time exhibit conformai invariance. In this setting a key operator is the con formally invariant wave operator which has leading term a pseudo-Laplacian. The Riemannian signature variant of this operator is a fundamental tool in the Yam abe problem on compact manifolds. Here one seeks to find a metric, from a given conformai class, that has constant scalar curvature. Recently it has become clear that higher order analogues of these operators, viz., conformally invariant operators on weighted functions (i.e., conformai densities) with leading term a power of the Laplacian, have a central role in generating and solving other curvature prescription problems as well as other problems in geometric spectral theory and mathematical physics [2, 5, 15]. In the flat setting, the existence of such operators dates back to [16], where it is shown that, on 4-dimensional Minkowski space, for k G N = {1,2,...}, the kth power of the flat wave operator Ak, acting on densities of the appropriate weight, is invariant under the action of the conformai group. More generally, if ?[w] denotes the space of conformai densities of weight uiGl, then on a flat conformai manifold of dimension n > 3 (and any signature) there exists, for each k E N, a unique conformally invariant operator

On Permutation Representations for Scheduling Problems

Abstract: In this paper we concentrate on job shop scheduling as a representative of constrained combinatorial problems. We introduce a new permutation representation for this problem. Three crossover operators, different in tending to preserve the relative order, the absolute order, and the position in the permutation, are defined. By experiment we observe the strongest phenotypical correlation between parents and offspring when respecting the absolute order. It is shown that a genetic algorithm using an operator which preserves the absolute order also obtains a superior solution quality.

Second order periodic differential operators. Threshold properties and homogenization

Abstract: The vector periodic differential operators (DO’s) A admitting a factorization A = X ∗X , where X is a first order homogeneous DO, are considered in L2(R). Many operators of mathematical physics have this form. The effects that depend only on a rough behavior of the spectral expansion of A in a small neighborhood of zero are called threshold effects at the point λ = 0. An example of a threshold effect is the behavior of a DO in the small period limit (the homogenization effect). Another example is related to the negative discrete spectrum of the operator A−αV , α > 0, where V (x) ≥ 0 and V (x) → 0 as |x| → ∞. “Effective characteristics”, such as the homogenized medium, effective mass, effective Hamiltonian, etc., arise in these problems. The general approach to these problems proposed in this paper is based on the spectral perturbation theory for operator-valued functions admitting analytic factorization. Most of the arguments are carried out in abstract terms. As to applications, the main attention is paid to homogenization of DO’s.

An overview of periodic elliptic operators

Abstract: The article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas. Close attention is paid to studying analytic properties of Bloch and Fermi varieties, which influence significantly most properties of such operators. The approaches described are applicable not only to the standard model example of Schrodinger operator with periodic electric potential −∆ + V (x), but to a wide variety of elliptic periodic equations and systems, equations on graphs, ∂-operator, and other operators on abelian coverings of compact bases. Many important applications are mentioned. However, due to the size restrictions, they are not dealt with in details.