The Experts below are selected from a list of 321 Experts worldwide ranked by ideXlab platform
Clement De Seguins-pazzis - One of the best experts on this subject based on the ideXlab platform.
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The geometric realization of a simplicial Hausdorff space is Hausdorff
Topology and its Applications, 2013Co-Authors: Clement De Seguins-pazzisAbstract:Abstract We show that the thin geometric realization of a simplicial Hausdorff space is Hausdorff. This proves a long-standing conjecture of Graeme Segal stating that the thin geometric realization of a simplicial k-space is a k-space.
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The geometric realization of a simplicial Hausdorff space is Hausdorff
2010Co-Authors: Clement De Seguins-pazzisAbstract:It is shown that the thin geometric realization of a simplicial Hausdorff space is Hausdorff. This proves a famous claim by Graeme Segal that the thin geometric realisation of a simplicial k-space is a k-space.
Wei Wu - One of the best experts on this subject based on the ideXlab platform.
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Quantized Gromov–Hausdorff distance
Journal of Functional Analysis, 2006Co-Authors: Wei WuAbstract:Abstract A quantized metric space is a matrix order unit space equipped with an operator space version of Rieffel's Lip-norm. We develop for quantized metric spaces an operator space version of quantum Gromov–Hausdorff distance. We show that two quantized metric spaces are completely isometric if and only if their quantized Gromov–Hausdorff distance is zero. We establish a completeness theorem. As applications, we show that a quantized metric space with 1-exact underlying matrix order unit space is a limit of matrix algebras with respect to quantized Gromov–Hausdorff distance, and that matrix algebras converge naturally to the sphere for quantized Gromov–Hausdorff distance.
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Quantized Gromov-Hausdorff distance
arXiv: Operator Algebras, 2005Co-Authors: Wei WuAbstract:A quantized metric space is a matrix order unit space equipped with an operator space version of Rieffel's Lip-norm. We develop for quantized metric spaces an operator space version of quantum Gromov-Hausdorff distance. We show that two quantized metric spaces are completely isometric if and only if their quantized Gromov-Hausdorff distance is zero. We establish a completeness theorem. As applications, we show that a quantized metric space with 1-exact underlying matrix order unit space is a limit of matrix algebras with respect to quantized Gromov-Hausdorff distance, and that matrix algebras converge naturally to the sphere for quantized Gromov-Hausdorff distance.
Yusu Wang - One of the best experts on this subject based on the ideXlab platform.
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computing the gromov Hausdorff distance for metric trees
International Symposium on Algorithms and Computation, 2015Co-Authors: Pankaj K Agarwal, Abhinandan Nath, Anastasios Sidiropoulos, Yusu WangAbstract:The Gromov-Hausdorff distance is a natural way to measure distance between two metric spaces. We give the first proof of hardness and first non-trivial approximation algorithm for computing the Gromov-Hausdorff distance for geodesic metrics in trees. Specifically, we prove it is \(\mathrm {NP}\)-hard to approximate the Gromov-Hausdorff distance better than a factor of 3. We complement this result by providing a polynomial time \(O(\min \{n, \sqrt{rn}\})\)-approximation algorithm where r is the ratio of the longest edge length in both trees to the shortest edge length. For metric trees with unit length edges, this yields an \(O(\sqrt{n})\)-approximation algorithm.
Marton Elekes - One of the best experts on this subject based on the ideXlab platform.
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Hausdorff and packing dimension of fibers and graphs of prevalent continuous maps
Advances in Mathematics, 2016Co-Authors: Richard Balka, Marton Elekes, Udayan B DarjiAbstract:The notions of shyness and prevalence generalize the property of being zero and full Haar measure to arbitrary (not necessarily locally compact) Polish groups. The main goal of the paper is to answer the following question: What can we say about the Hausdorff and packing dimension of the fibers of prevalent continuous maps? Let K be an uncountable compact metric space. We prove that a prevalent f∈C(K,Rd) has many fibers with almost maximal Hausdorff dimension. This generalizes a theorem of Dougherty and yields that a prevalent f∈C(K,Rd) has graph of maximal Hausdorff dimension, generalizing a result of Bayart and Heurteaux. We obtain similar results for the packing dimension. We show that for a prevalent f∈C([0,1]m,Rd) the set of y∈f([0,1]m) for which dimHf−1(y)=m contains a dense open set having full measure with respect to the occupation measure λm∘f−1, where dimH and λm denote the Hausdorff dimension and the m-dimensional Lebesgue measure, respectively. We also prove an analogous result when [0,1]m is replaced by any self-similar set satisfying the open set condition. We cannot replace the occupation measure with Lebesgue measure in the above statement: We show that the functions f∈C[0,1] for which positively many level sets are singletons form a non-shy set in C[0,1]. In order to do so, we generalize a theorem of Antunovic, Burdzy, Peres and Ruscher. As a complementary result we prove that the functions f∈C[0,1] for which dimHf−1(y)=1 for all y∈(minf,maxf) form a non-shy set in C[0,1]. We also prove sharper results in which large Hausdorff dimension is replaced by positive measure with respect to generalized Hausdorff measures, which answers a problem of Fraser and Hyde.
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topological Hausdorff dimension and level sets of generic continuous functions on fractals
Chaos Solitons & Fractals, 2012Co-Authors: Richard Balka, Zoltan Buczolich, Marton ElekesAbstract:Abstract In an earlier paper we introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. For a compact metric space K let dimH K and dimtH K denote its Hausdorff and topological Hausdorff dimension, respectively. We proved that this new dimension describes the Hausdorff dimension of the level sets of the generic continuous function on K, namely sup { dim H f - 1 ( y ) : y ∈ R } = dim tH K - 1 for the generic f ∈ C(K), provided that K is not totally disconnected, otherwise every non-empty level set is a singleton. We also proved that if K is not totally disconnected and sufficiently homogeneous then dimH f−1(y) = dimtH K − 1 for the generic f ∈ C(K) and the generic y ∈ f(K). The most important goal of this paper is to make these theorems more precise. As for the first result, we prove that the supremum is actually attained on the left hand side of the first equation above, and also show that there may only be a unique level set of maximal Hausdorff dimension. As for the second result, we characterize those compact metric spaces for which for the generic f ∈ C(K) and the generic y ∈ f(K) we have dimH f−1(y) = dimtH K − 1. We also generalize a result of B. Kirchheim by showing that if K is self-similar then for the generic f ∈ C(K) for every y ∈ int f ( K ) we have dimH f−1(y) = dimtH K − 1. Finally, we prove that the graph of the generic f ∈ C(K) has the same Hausdorff and topological Hausdorff dimension as K.
Gang Xiao - One of the best experts on this subject based on the ideXlab platform.
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study on an improved Hausdorff distance for multi sensor image matching
Communications in Nonlinear Science and Numerical Simulation, 2012Co-Authors: Jianming Wu, Zhongliang Jing, Zheng Wu, Yan Feng, Gang XiaoAbstract:Abstract A new modifying Hausdorff distance image matching algorithm was proposed in this paper. After the corners of two images was extracted using Harris corner detector, a kind of Hausdorff distance integrating points set coincidence numbers was presented to aim at the traditional Hausdorff distance. The accuracy of matching was improved by this modifying. Hausdorff distance coefficient matrix is calculating by corners neighborhood’s related matching. The initial matching point-pairs are obtained by the rule that the small coefficient is good matching. Finally the wrong matching point-pairs are deleted by the distance-ration invariant, the right matching point-pairs are acquired. Experimental results show that the proposed method can be easily and quickly to process the multiple sensor images.
