Topic
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.
Papers published on a yearly basis
Papers
More filters
••
TL;DR: In this article, the determinant theory for finite factors was introduced and the trace of a generalized nilpotent operator is shown to lie in the convex hull of its spectrum.
Abstract: Murray and J. v. Neumann classified factors (i.e., central rings of operators) by means of a "relative dimension function." They studied, extensively, those factors for which this dimension function has a finite range ("finite factors") and showed that these factors (and these factors alone) admit a trace function' with the standard algebraic properties. In an attempt to establish, what seems to us to be a further important algebraic property of the trace, viz., the trace of a generalized nilpotent operator is zero, and, more generally, the trace of an operator lies in the convex hull of its spectrum, we were led to the introduction of a determinant theory for finite factors. This paper will be concerned, principally, with the development of this theory. We might note that it is a simple algebraic matter to prove that the trace T(N) of a proper nilpotent N is zero. In fact, if N' = 0 and E is the projection on the closure of the range of N, then EN = N so that (NE)'-' = N'-'E = 0. Then T(N) = T(EN) = T(NE) = 0, by induction on n. That the normalized trace lies in the convex hull of the spectrum of a finite-dimensional matrix follows at once by bringing the matrix to super-diagonal form, whereby the normalized trace appears as the "center of gravity" of the spectrum. This fact together with the theory developed in R.O. IV, Chapter IV, yields the same result for operators in an approximately finite factor. Furthermore, it is an immediate consequence of the spectral theorem that the trace of a normal operator in an arbitrary finite factor lies in the convex hull of its spectrum. None of these easily proved facts enabled us to conclude the result for arbitrary non-normal operators in nonapproximately finite factors. However, the general result was established as a byproduct of the determinant theory. In ?2 we define the determinant on regular operators in a factor of type II1,, and establish the properties of this determinant. The proof that the trace lies in the convex hull of the spectrum is given in ?3 as an application of the results of ?2. The uniqueness of the determinant is established in ?4 by means of an algebraic characterization. The final section, ?5, begins with a discussion of the normalization which has taken place in the definition of the determinant. A
229 citations
••
TL;DR: A linguistic multi-criteria group decision-making method is developed that converts linguistic variables into clouds and then aggregated using cloud aggregation operators, and is compared to existing methods to confirm its feasibility and rationality.
229 citations
••
TL;DR: Three experiments are presented that demonstrate that solvers use at least two sources of information to make operator selections in the building sticks task (BST): information from their past history of using the operators and information from the current context of the problem.
229 citations
••
TL;DR: In this paper, the authors characterize the reproducing kernel Hilbert spaces whose elements are p-integrable functions in terms of the boundedness of the integral operator whose kernel is the Reproducing Kernel.
Abstract: We characterize the reproducing kernel Hilbert spaces whose elements are p-integrable functions in terms of the boundedness of the integral operator whose kernel is the reproducing kernel. Moreover, for p = 2, we show that the spectral decomposition of this integral operator gives a complete description of the reproducing kernel, extending the Mercer theorem.
229 citations
•
19 Jun 1978
TL;DR: In this article, a visual aid system for assisting an operator while he maneuvers a vehicle comprises a closed circuit video system having distance measuring apparatus, which enables the operator to see behind the vehicle and, through the use of the distance indicating apparatus, the operator can tell how far the rear of the vehicle is from objects seen via the video system.
Abstract: A visual aid system for assisting an operator while he maneuvers a vehicle comprises a closed circuit video system having distance measuring apparatus. The system enables the operator to see behind the vehicle and, through the use of the distance indicating apparatus, the operator can tell how far the rear of the vehicle is from objects seen via the video system.
227 citations