Topic
Bounded function
About: Bounded function is a research topic. Over the lifetime, 77295 publications have been published within this topic receiving 1321552 citations.
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TL;DR: In this paper, it was shown that for every vector-valued function u Lp(X, p; Rd) there is a unique polar factorization u = V$.s, where $ is a convex function defined on R and s is a measure-preserving mapping from (x, p) into (Q, I. I), provided that u is nondegenerate.
Abstract: Given a probability space (X, p) and a bounded domain R in Rd equipped with the Lebesgue measure 1 . I (normalized so that 10 I = I ), it is shown (under additional technical assumptions on X and Q) that for every vector-valued function u E Lp(X, p; Rd) there is a unique “polar factorization” u = V$.s, where $ is a convex function defined on R and s is a measure-preserving mapping from (X, p) into (Q, I . I), provided that u is nondegenerate, in the sense that p(u-’(E)) = 0 for each Lebesgue negligible subset E of Rd. Through this result, the concepts of polar factorization of real matrices, Helmholtz decomposition of vector fields, and nondecreasing rearrangements of real-valued functions are unified. The Monge-Amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate “Monge-Kantorovich” problem.
1,780 citations
TL;DR: This Second Edition of Global Optimization Using Interval Analysis expands and improves various aspects of its forerunner and features significant new discussions, such as those on the use of consistency methods to enhance algorithm performance.
Abstract: Employing a closed set-theoretic foundation for interval computations, Global Optimization Using Interval Analysis simplifies algorithm construction and increases generality of interval arithmetic. This Second Edition contains an up-to-date discussion of interval methods for solving systems of nonlinear equations and global optimization problems. It expands and improves various aspects of its forerunner and features significant new discussions, such as those on the use of consistency methods to enhance algorithm performance. Provided algorithms are guaranteed to find and bound all solutions to these problems despite bounded errors in data, in approximations, and from use of rounded arithmetic.
1,705 citations
TL;DR: In this article, the authors show that most of the characterizations that were reported thus far in the literature are special cases of the following general result: a standard multilayer feedforward network with a locally bounded piecewise continuous activation function can approximate any continuous function to any degree of accuracy if and only if the network's activation function is not a polynomial.
1,581 citations
TL;DR: This paper proposes an information theoretic criterion for comparing two partitions, or clusterings, of the same data set, called variation of information (VI), and presents it from an axiomatic point of view, showing that it is the only ''sensible'' criterion for compare partitions that is both aligned to the lattice and convexely additive.
1,527 citations
TL;DR: It is shown that in the absence of unmodeled process dynamics, the proposed supervisor can successfully perform its function even if process disturbances are present, provided they are bounded and constant.
Abstract: This paper describes a simple "high-level" controller called a "supervisor" which is capable of switching into feedback with a SISO process, a sequence of linear positioning or set-point controllers from a family of candidate controllers so as to cause the output of the process to approach and track a constant reference input. The process is assumed to be modeled by a SISO linear system whose transfer function is in the union of a number of subclasses, each subclass being small enough so that one of the candidate controllers would solve the positioning problem if the transfer function of the process were to be one of the subclasses' members. Each subclass contains a "nominal process model transfer function" about which the subclass is centred. It is shown that in the absence of unmodeled process dynamics, the proposed supervisor can successfully perform its function even if process disturbances are present, provided they are bounded and constant.
1,415 citations