Complexity Analysis via Approach Spaces

Colebunders, Eva; Wachter, Sarah De; Schellekens, Michel

doi:10.1007/s10485-013-9302-2

Cited by 6 publications

(7 citation statements)

References 10 publications

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“…In particular, we obtain existing fixed-point theorems due to D. Scott (Gierz et al 2003) and K. Martin (Martin 2000b;Martin 2000a) for both monotone and non-monotone maps as special cases. Concrete applications of these fixedpoint theorems to complexity analysis are currently being worked on in collaboration with M. Schellekens and will appear in a forthcoming paper (Colebunders et al 2012). The investigation of the relation between our present context and the theory of measurements as developed by K. Martin (Martin 2000a;Martin 2000b) is also the subject of ongoing research.…”

Section: Introductionmentioning

confidence: 89%

Fixed points of contractive maps on dcpo's

Colebunders

Wachter

Löwen

2013

Math. Struct. Comp. Sci.

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In this paper we study approach structures on dcpo's. A dcpo (X, 6) will be endowed with several other structures: the Scott topology; an approach structure generated by a collection of weightable quasi metrics on X; and a collection W of weights corresponding to the quasi metrics. Understanding the interaction between these structures on X will eventually lead to some fixed-point theorems for the morphisms in the category of approach spaces, which are called contractions. Existing fixed-point theorems on both monotone and non-monotone maps are obtained as special cases.

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Section: Introductionmentioning

confidence: 89%

Fixed points of contractive maps on dcpo's

Colebunders

Wachter

Löwen

2013

Math. Struct. Comp. Sci.

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“…From now on, for simplicity in terminology we drop the words "extended" and "pseudo", so in this respect our terminology differs from what is commonly used. It is however conform with the terminology in [2], [7] and [6], [19], [11]. We denote by qMet the category of all quasi-metric spaces with non-expansive maps as morphisms and by Met the full subcategory of all metric spaces.…”

Section: Preliminaries On Approach Spacesmentioning

confidence: 99%

“…Such isometric settings get more and more attention like for instance in the study of approximation by Lipschitz functions in [12], of cofinal completeness and UC-property in [1], in investigations on hyperconvexity in [15] and on the non-symmetric analogue of the Urysohn metric space in [16] and [17]. For other applications the larger context of approach spaces with contractions is even more suitable as was recently shown in the context of probability m easures [2], [3] and [4], or complexity analysis [6] and [7].…”

Section: Introductionmentioning

confidence: 99%

Normality in terms of distances and contractions

Colebunders

Sioen

Haute

2018

Journal of Mathematical Analysis and Applications

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The main purpose of this paper is to explore normality in terms of distances between points and sets. We prove some important consequences on realvalued contractions, i.e. functions not enlarging the distance, showing that as in the classical context of closures and continuous maps, normality in terms of distances based on an appropriate numerical notion of γ-separation of sets, has far reaching consequences on real valued contractive maps, where the real line is endowed with the Euclidean metric. We show that normality is equivalent to (1) separation of γ-separated sets by some Urysohn contractive map, (2) to Katětov-Tong's interpolation, stating that for bounded positive realvalued functions, between an upper and a larger lower regular function, there exists a contractive interpolating map and (3) to Tietze's extension theorem stating that certain contractions defined on a subspace can be contractively extended to the whole space.The appropriate setting for these investigations is the category of approach spaces, but the results have (quasi)-metric counterparts in terms of non-expansive maps. Moreover when restricted to topological spaces, classical normality and its equivalence to separation by a Urysohn continuous map, to Katětov-Tong's interpolation for semicontinuous maps and to Tietze's extension theorem for continuous maps are recovered.

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“…The structural flexibility of approach theory entails the existence of canonical approach spaces in branches of mathematical analysis such as functional analysis ( [LS00], [SV03], [LV04], [SV04], [SV06], [SV07]), hyperspace theory ( [LS96], [LS98],[LS00']), domain theory ( [CDL11], [CDL14], [CDS14]), and probability theory and statistics ( [BLV11], [BLV13], [BLV16]). A careful study of these approach spaces has resulted in new insights and applications in these branches.…”

Section: Introductionmentioning

confidence: 99%

“…The structural flexibility of approach theory entails the existence of canonical approach spaces in branches of mathematical analysis such as functional analysis ([LS00], [SV03], [LV04], [SV04], [SV06], [SV07]), hyperspace theory ( [LS96], [LS98],[LS00']), domain theory ( [CDL11], [CDL14], [CDS14]), and Key words and phrases. approach theory, Ascoli's Theorem, contractive approach structure, Dini's Theorem, (relative) Hausdorff measure of non-compactness, measure of non-uniform integrability, Prokhorov's Theorem, Wasserstein metric.…”

Section: Introductionmentioning

confidence: 99%

An application of approach theory to the relative Hausdorff measure of non-compactness for the Wasserstein metric

Berckmoes¹,

Hellemans²,

Sioen³

et al. 2017

Journal of Mathematical Analysis and Applications

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Abstract. After shortly reviewing the fundamentals of approach theory as introduced by R. Lowen in 1989, we show that this theory is intimately related with the well-known Wasserstein metric on the space of probability measures with a finite first moment on a complete and separable metric space. More precisely, we introduce a canonical approach structure, called the contractive approach structure, and prove that it is metrized by the Wasserstein metric. The key ingredients of the proof of this result are Dini's Theorem, Ascoli's Theorem, and the fact that the class of real-valued contractions on a metric space has some nice stability properties. We then combine the obtained result with Prokhorov's Theorem to establish inequalities between the relative Hausdorff measure of non-compactness for the Wasserstein metric and a canonical measure of non-uniform integrability.

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Complexity Analysis via Approach Spaces

Cited by 6 publications

References 10 publications

Fixed points of contractive maps on dcpo's

Fixed points of contractive maps on dcpo's

Normality in terms of distances and contractions

An application of approach theory to the relative Hausdorff measure of non-compactness for the Wasserstein metric

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