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Gerald Beer - One of the best experts on this subject based on the ideXlab platform.
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Uniform Continuity and a new bornology for a metric space
Set-valued and Variational Analysis, 2018Co-Authors: Gerald Beer, Isabel M Garrido, Ana S MeronoAbstract:In the context of functions between metric spaces, Continuity is preserved by Uniform convergence on the bornology of relatively compact subsets while Cauchy Continuity is preserved under Uniform convergence on the bornology of totally bounded subsets. We identify a new bornology for a metric space containing the bornology of Bourbaki bounded sets on which Uniform convergence preserves Uniform Continuity. Further, for real-valued Uniformly continuous functions, the function space is a ring (with respect to pointwise multiplication) if and only if the two bornologies agree. We show that Cauchy Continuity is preserved by Uniform convergence on compact subsets if and only if the domain space is complete, and that Uniform Continuity is preserved under Uniform convergence on totally bounded subsets if and only if the domain space has UC completion. Finally, Uniform Continuity is preserved under Uniform convergence on compact subsets if and only if the domain space is a UC-space. We prove a simple omnibus density result for Lipschitz functions within a larger class of continuous functions equipped with a topology of Uniform convergence on a bornology and apply that to each of our three function classes.
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the alexandroff property and the preservation of strong Uniform Continuity
Applied general topology, 2013Co-Authors: Gerald BeerAbstract:In this paper we extend the theory of strong Uniform Continuity and strong Uniform convergence, developed in the setting of metric spaces in, to the Uniform space setting, where again the notion of shields plays a key role. Further, we display appropriate bornological/variational modifications of classical properties of Alexandroff [1] and of Bartle for nets of continuous functions, that combined with pointwise convergence, yield Continuity of the limit for functions between metric spaces.
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Uniform Continuity Uniform convergence and shields
Set-valued and Variational Analysis, 2010Co-Authors: Gerald Beer, Sandro LeviAbstract:Let \(\mathcal{B}\) be a bornology in a metric space \(\langle X,d \rangle\), that is, a cover of X by nonempty subsets that also forms an ideal. In Beer and Levi (J Math Anal Appl 350:568–589, 2009), the authors introduced the variational notions of strong Uniform Continuity of a function on \(\mathcal{B}\) as an alternative to Uniform Continuity of the restriction of the function to each member of \(\mathcal{B}\), and the topology of strong Uniform convergence on \(\mathcal{B}\) as an alternative to the classical topology of Uniform convergence on \(\mathcal{B}\). Here we continue this study, showing that shields as introduced in Beer, Costantini and Levi (Bornological Convergence and Shields, Mediterranean J. Math, submitted) play a pivotal role. For example, restricted to continuous functions, the topology of strong Uniform convergence on \(\mathcal{B}\) reduces to the classical topology if and only if the natural closure of the bornology is shielded from closed sets. The paper also further develops the theory of shields and their applications.
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strong Uniform Continuity
Journal of Mathematical Analysis and Applications, 2009Co-Authors: Gerald Beer, Sandro LeviAbstract:Abstract Let B be an ideal of subsets of a metric space 〈 X , d 〉 . This paper considers a strengthening of the notion of Uniform Continuity of a function restricted to members of B which reduces to ordinary Continuity when B consists of the finite subsets of X and agrees with Uniform Continuity on members of B when B is either the power set of X or the family of compact subsets of X. The paper also presents new function space topologies that are well suited to this strengthening. As a consequence of the general theory, we display necessary and sufficient conditions for Continuity of the pointwise limit of a net of continuous functions.
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Uniform Continuity on bounded sets and the attouch wets topology
Proceedings of the American Mathematical Society, 1991Co-Authors: Gerald Beer, Anna Di ConcilioAbstract:Let CL(X) be the nonempty closed subsets of a metrizable space X. If d is a compatible metric, the metrizable Attouch-Wets topology Taw (d) on CL(X) is the topology of Uniform convergence of distance functionals associated with elements of CL(X) on bounded subsets of X. The main result of this paper shows that two compatible metrics d and p determine the same Attouch-Wets topologies if and only if they determine the same bounded sets and the same class of functions that are Uniformly continpous on bounded sets.
Anant Sahai - One of the best experts on this subject based on the ideXlab platform.
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On the Uniform Continuity of the rate-distortion function
2008 IEEE International Symposium on Information Theory, 2008Co-Authors: Hari Palaiyanur, Anant SahaiAbstract:It is well known that the rate-distortion function for a finite alphabet IID source with distribution p, denoted R(p,D), is Uniformly continuous in its arguments. We prove an explicit bound on |R(p,D) - R(q,D)| for distributions p, q in terms of the variational distance parp-qpar1. A simple and elementary proof shows that |R(p,D) - R(q,D)| = O(-parp - qpar1 log parp - qpar1), with constants depending on the distortion measure. The Uniform Continuity of the rate-distortion function has the same behavior as the Uniform Continuity of entropy in the order sense. The bounds are used for several applications. First, a simple sampling algorithm is presented to compute the rate-distortion function for an arbitrarily varying source to within a given accuracy. The Uniform Continuity bound is used here to roughly quantify the tradeoff between complexity and accuracy. Second, we comment on the problem of approximating the rate-distortion function for an unknown IID source to within a desired precision.
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ISIT - On the Uniform Continuity of the rate-distortion function
2008 IEEE International Symposium on Information Theory, 2008Co-Authors: Hari Palaiyanur, Anant SahaiAbstract:It is well known that the rate-distortion function for a finite alphabet IID source with distribution p, denoted R(p,D), is Uniformly continuous in its arguments. We prove an explicit bound on |R(p,D) - R(q,D)| for distributions p, q in terms of the variational distance parp-qpar1. A simple and elementary proof shows that |R(p,D) - R(q,D)| = O(-parp - qpar1 log parp - qpar1), with constants depending on the distortion measure. The Uniform Continuity of the rate-distortion function has the same behavior as the Uniform Continuity of entropy in the order sense. The bounds are used for several applications. First, a simple sampling algorithm is presented to compute the rate-distortion function for an arbitrarily varying source to within a given accuracy. The Uniform Continuity bound is used here to roughly quantify the tradeoff between complexity and accuracy. Second, we comment on the problem of approximating the rate-distortion function for an unknown IID source to within a desired precision.
Sandro Levi - One of the best experts on this subject based on the ideXlab platform.
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Uniform Continuity Uniform convergence and shields
Set-valued and Variational Analysis, 2010Co-Authors: Gerald Beer, Sandro LeviAbstract:Let \(\mathcal{B}\) be a bornology in a metric space \(\langle X,d \rangle\), that is, a cover of X by nonempty subsets that also forms an ideal. In Beer and Levi (J Math Anal Appl 350:568–589, 2009), the authors introduced the variational notions of strong Uniform Continuity of a function on \(\mathcal{B}\) as an alternative to Uniform Continuity of the restriction of the function to each member of \(\mathcal{B}\), and the topology of strong Uniform convergence on \(\mathcal{B}\) as an alternative to the classical topology of Uniform convergence on \(\mathcal{B}\). Here we continue this study, showing that shields as introduced in Beer, Costantini and Levi (Bornological Convergence and Shields, Mediterranean J. Math, submitted) play a pivotal role. For example, restricted to continuous functions, the topology of strong Uniform convergence on \(\mathcal{B}\) reduces to the classical topology if and only if the natural closure of the bornology is shielded from closed sets. The paper also further develops the theory of shields and their applications.
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strong Uniform Continuity
Journal of Mathematical Analysis and Applications, 2009Co-Authors: Gerald Beer, Sandro LeviAbstract:Abstract Let B be an ideal of subsets of a metric space 〈 X , d 〉 . This paper considers a strengthening of the notion of Uniform Continuity of a function restricted to members of B which reduces to ordinary Continuity when B consists of the finite subsets of X and agrees with Uniform Continuity on members of B when B is either the power set of X or the family of compact subsets of X. The paper also presents new function space topologies that are well suited to this strengthening. As a consequence of the general theory, we display necessary and sufficient conditions for Continuity of the pointwise limit of a net of continuous functions.
Mark M Wilde - One of the best experts on this subject based on the ideXlab platform.
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optimal Uniform Continuity bound for conditional entropy of classical quantum states
Quantum Information Processing, 2020Co-Authors: Mark M WildeAbstract:In this short note, I show how a recent result of Alhejji and Smith (A tight Uniform Continuity bound for equivocation, 2019. arXiv:1909.00787v1) regarding an optimal Uniform Continuity bound for classical conditional entropy leads to an optimal Uniform Continuity bound for quantum conditional entropy of classical–quantum states. The bound is optimal in the sense that there always exists a pair of classical–quantum states saturating the bound, and so, no further improvements are possible. An immediate application is a Uniform Continuity bound for the entanglement of formation that improves upon the one previously given by Winter (Commun Math Phys 347(1):291–313, 2016. arXiv:1507.07775). Two intriguing open questions are raised regarding other possible Uniform Continuity bounds for conditional entropy: one about quantum–classical states and another about fully quantum bipartite states.
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Optimal Uniform Continuity bound for conditional entropy of classical–quantum states
Quantum Information Processing, 2020Co-Authors: Mark M WildeAbstract:In this short note, I show how a recent result of Alhejji and Smith (A tight Uniform Continuity bound for equivocation, 2019. arXiv:1909.00787v1) regarding an optimal Uniform Continuity bound for classical conditional entropy leads to an optimal Uniform Continuity bound for quantum conditional entropy of classical–quantum states. The bound is optimal in the sense that there always exists a pair of classical–quantum states saturating the bound, and so, no further improvements are possible. An immediate application is a Uniform Continuity bound for the entanglement of formation that improves upon the one previously given by Winter (Commun Math Phys 347(1):291–313, 2016. arXiv:1507.07775). Two intriguing open questions are raised regarding other possible Uniform Continuity bounds for conditional entropy: one about quantum–classical states and another about fully quantum bipartite states.
Douglas S. Bridges - One of the best experts on this subject based on the ideXlab platform.
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the anti specker property a heine borel property and Uniform Continuity
Archive for Mathematical Logic, 2008Co-Authors: Josef Berger, Douglas S. BridgesAbstract:Working within Bishop’s constructive framework, we examine the connection between a weak version of the Heine–Borel property, a property antithetical to that in Specker’s theorem in recursive analysis, and the Uniform Continuity theorem for integer-valued functions. The paper is a contribution to the ongoing programme of constructive reverse mathematics.
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The anti-Specker property, a Heine–Borel property, and Uniform Continuity
Archive for Mathematical Logic, 2007Co-Authors: Josef Berger, Douglas S. BridgesAbstract:Working within Bishop’s constructive framework, we examine the connection between a weak version of the Heine–Borel property, a property antithetical to that in Specker’s theorem in recursive analysis, and the Uniform Continuity theorem for integer-valued functions. The paper is a contribution to the ongoing programme of constructive reverse mathematics.
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the pseudocompactness of 0 1 is equivalent to the Uniform Continuity theorem
Journal of Symbolic Logic, 2007Co-Authors: Douglas S. Bridges, Hannes DienerAbstract:We prove constructively that, in order to derive the Uniform Continuity theorem for pointwise continuous mappings from a compact metric space into a metric space, it is necessary and sufficient to prove any of a number of equivalent conditions, such as that every pointwise continuous mapping of [0, 1] into R is bounded. The proofs are analytic, making no use of, for example, fan-theoretic ideas. In any variety of constructive mathematics, the status of the (classically valid) Uniform Continuity theorem, UCT Every pointwise continuous mapping of a compact (that is, com plete, totally bounded) metric space into a metric space is Uniformly continuous, reveals a great deal about the kind of constructivity that characterises the variety. For example, UCT holds in intuitionistic mathematics as a result of Brouwer's fan theorem, whereas in the recursive constructive mathematics of the Markov school there is an example of a bounded, pointwise continuous function from [0,1] to [0,1] that fails to be Uniformly continuous [9, Chapters 3 and 5]. It is therefore a problem of some interest in constructive reverse mathematics to discover logical or analytic statements that are equivalent, constructively, to UCT. Berger [4, 3, 5, 6] and Loeb [11] have made important contributions in connection with this problem. The work of the former deals primarily with the connection between UCT and versions of Brouwer's fan theorem. Loeb, on the other hand, works within a strict formal system (whereas we work more informally), and her continuous functions are equipped with stronger information than ours. In this paper we work in Bishop-style constructive mathematics (BISH: mathe matics with intuitionistic logic and some appropriate set-theoretic foundation such as Aczel's CZF [1]), in which it is well known that a Uniformly continuous function from a compact metric space into R is bounded (and actually has a supremum and an infimum). We shall prove a strong converse: if every pointwise continuous mapping of [0,1] into R is bounded, then UCT holds. We also show that UCT is equivalent to other statements, including 'every pointwise continuous mapping of [0,1] into R is Lebesgue integrable'. We begin with some technical material that will enable us to connect pointwise continuous, real-valued functions on [0,1] with such functions on Cantor space 2N. Received November 16, 2006. ? 2007. Association for Symbolic Logic 0022-4812/07/7204-0020/$1.60
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Sequential, pointwise, and Uniform Continuity: A constructive note
Mathematical Logic Quarterly, 2006Co-Authors: Douglas S. BridgesAbstract:The main result of this paper is a weak constructive version of the Uniform Continuity theorem for pointwise continuous, real-valued functions on a convex subset of a normed linear space. Recursive examples are given to show that the hypotheses of this theorem are necessary. The remainder of the paper discusses conditions which ensure that a sequentially continuous function is continuous. MSC: 03F60, 26E40, 46S30.
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strong versus Uniform Continuity a constructive round
Quaestiones Mathematicae, 2003Co-Authors: Peter Schuster, Douglas S. BridgesAbstract:The notion of apartness has recently shown promise as a means of lifting constructive topology from the restrictive context of metric spaces to more general settings. Extending the point-subset apartness axiomatised beforehand, we characterize the constructive meaning of ‘two subsets of a given set lie apart from each other'. We propose axioms for such apartness relations and verify them for the apartness relation associated with an abstract Uniform space. Moreover, we relate Uniform Continuity to strong Continuity, the natural concept for mappings between sets endowed with an apartness structure, which says that if the images of two subsets lie apart from each other, then so do the original subsets. Proofs are carried out with intuitionistic logic, and most of them without the principle of countable choice. Mathematics Subject Classification (2000): 54E05, 54E15, 03F60 Quaestiones Mathematicae 25 (2002), 171-190
