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Theory and Applications of Distance Geometry
01 Nov 1970-
About: The article was published on 1970-11-01 and is currently open access. It has received 994 citations till now.
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01 Feb 1993TL;DR: Inequalities for mixed volumes 7. Selected applications Appendix as discussed by the authors ] is a survey of mixed volumes with bounding boxes and quermass integrals, as well as a discussion of their applications.
Abstract: 1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.
3,954 citations
TL;DR: An overview of the measurement techniques in sensor network localization and the one-hop localization algorithms based on these measurements are provided and a detailed investigation on multi-hop connectivity-based and distance-based localization algorithms are presented.
1,870 citations
Cites methods from "Theory and Applications of Distance..."
...An interesting development in the area is the use of the Cayley–Menger determinant [60,61] to reduce the impact of distance measurement errors on the location estimate [62,63]....
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IBM1
TL;DR: In this paper, it was shown that multiquadric surface interpolation is always solvable, thereby settling a conjecture of R Franke, which is a conjecture that was later proved in the present paper.
Abstract: Among other things, we prove that multiquadric surface interpolation is always solvable, thereby settling a conjecture of R Franke
1,476 citations
TL;DR: Efficient algorithms for embedding graphs low-dimensionally with a small distortion, and a new deterministic polynomial-time algorithm that finds a (nearly tight) cut meeting this bound.
Abstract: In this paper we explore some implications of viewing graphs asgeometric objects. This approach offers a new perspective on a number of graph-theoretic and algorithmic problems. There are several ways to model graphs geometrically and our main concern here is with geometric representations that respect themetric of the (possibly weighted) graph. Given a graphG we map its vertices to a normed space in an attempt to (i) keep down the dimension of the host space, and (ii) guarantee a smalldistortion, i.e., make sure that distances between vertices inG closely match the distances between their geometric images. In this paper we develop efficient algorithms for embedding graphs low-dimensionally with a small distortion. Further algorithmic applications include:
Given faithful low-dimensional representations of statistical data, it is possible to obtain meaningful and efficientclustering. This is one of the most basic tasks in pattern-recognition. For the (mostly heuristic) methods used in the practice of pattern-recognition, see [20], especially chapter 6. Our studies of multicommodity flows also imply that every embedding of (the metric of) ann-vertex, constant-degree expander into a Euclidean space (of any dimension) has distortion Ω(logn). This result is tight, and closes a gap left open by Bourgain [12].
1,133 citations
Cites background from "Theory and Applications of Distance..."
...Remark 3.7. The case c -- 1, i.e., the characterization of metric spaces that isometrically embed in Euclidean space is classical (see [ 11 ])....
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01 Feb 1969
TL;DR: In this article, it was shown that for a metrically convex space, the conclusion of Banach's theorem still holds, and that one need only assume that ip(t) 0, together with a semicontinuity condition on \[/.
Abstract: where \p is some function defined on the closure of the range of p. In [3], Rakotch proved that if \p(t) =ct(t)t, where a is decreasing and a(t) <1 for i>0, then a mapping satisfying (3) has a unique fixed point x0. It is an easy exercise to show that if \p(t) =a(t)t, where a is increasing, and a(/)0, together with a semicontinuity condition on \[/. For a metrically convex space, even this latter condition may be dropped. A number of examples are given to show that the results do in fact improve upon those mentioned above. We wish to thank the referee for suggesting the improved version of Theorem 1 which is presented in this paper. We begin with some preliminary results on metrically convex spaces. Definition 1 [l, p. 41]. A metric space X is said to be metrically convex if for each x, yEX, there is a z^x, y for which p(x, y) =p(x, z) +p(z, y)-
980 citations