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.

A boundedness criterion for generalized Caldéron-Zygmund operators

Abstract: In 1965, A. P. Calderon showed the L2-boundedness of the so-called first Calderon commutator. This is one of the first examples of a non-convolution operator associated to a kernel which satisfies certain size and smoothness properties comparable to those of the kernel of the Hilbert transform. These properties, together with the L2-boundedness, imply the LP-boundedness for all p's in ]1, + oo[. Many operators in analysis, such as certain classes of pseudo-differential operators and the Cauchy integral operator on a curve, are associated with kernels having these properties. For these operators, one of the major questions is if they are bounded on L2. We are going to give necessary and sufficient conditions for such an operator to be bounded on L2. They are essentially that the images of the function 1 under the actions of the operator and its adjoint both lie in BMO. In the case of the aforementioned first commutator this can be checked by an integration by parts. In the first section we present some basic notions and state the theorem, which is proved in Sections 2 and 3. In Section 4 we show how to recover some classical results. In Sections 5 and 6 we construct a functional calculus for small perturbations of A, and in Section 7 we show a connection between the theory of Calderon-Zygmund operators and Kato's conjecture. It is a pleasure to express our thanks to R. R. Coifman and Y. Meyer for suggesting many elegant simplifications in our proofs and most of the applications. We also wish to thank Stephen Semmes for several pertinent remarks.

The uncertain OWA operator

Abstract: The ordered weighted averaging (OWA) operator was introduced by Yager1 to provide a method for aggregating several inputs that lie between the max and min operators. In this article, we investigate the uncertain OWA operator in which the associated weighting parameters cannot be specified, but value ranges can be obtained and each input argument is given in the form of an interval of numerical values. The problem of ranking a set of interval numbers and obtaining the weights associated with the uncertain OWA operator is studied. © 2002 Wiley Periodicals, Inc.

Methods of algebraic geometry in the theory of non-linear equations

Abstract: CONTENTSIntroduction § 1. The Akhiezer function and the Zakharov-Shabat equations § 2. Commutative rings of differential operators § 3. The two-dimensional Schrodinger operator and the algebras associated with it § 4. The problem of multi-dimensional -algebraic operators Appendix 1. The Hamiltonian formalism in equations of Lax and Novikov types Appendix 2. Elliptic and rational solutions of the K-dV equations and systems of many particles Concluding Remarks References

Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. i. mapping theorems and ordering of functions of noncommuting operators.

Abstract: A new calculus for functions of noncommuting operators is developed, based on the notion of mapping of functions of operators onto $c$-number functions. The class of linear mappings, each member of which is characterized by an entire analytic function of two complex variables, is studied in detail. Closed-form solutions for such mappings and for the inverse mappings are obtained and various properties of these mappings are studied. It is shown that the most commonly occurring rules of association between operators and $c$-numbers (the Weyl, the normal, the antinormal, the standard, and the antistandard rules) belong to this class and are, in fact, the simplest ones in a clearly defined sense. It is shown further that the problem of expressing an operator in an ordered form according to some prescribed rule is equivalent to an appropriate mapping of the operator on a $c$-number space. The theory provides a systematic technique for the solution of numerous quantum-mechanical problems that were treated in the past by ad hoc methods, and it furnishes a new approach to many others. This is illustrated by a number of examples relating to mappings and ordering of operators.