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Upper bounds on the topological Betti numbers of Vietoris-Rips complexes are established, and examples of such complexes with high Betti numbers are given.
L. Vietoris, Über den höheren Zusammenhang kompaktor Räume und eine Klasse von zusammenhangstreuen Abbildungen, Math. Ann
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M. Gromov, Hyperbolic groups, in Essays in Group Theory, MSRI Publ. 8, Springer-Verlag, 1987
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A. Björner, Topological Methods, Handbook of Combinatorics
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G. Carlsson, V. De Silva, Topological estimation using witness complexes, in SPBG Symposium on Point-Based Graphics
2004
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G. Carlsson, A. Collins, L. Guibas, A. Zomorodian, Persistence barcodes for shapes, Intl. J. Shape Modeling
2005
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E. Carlsson, G. Carlsson, V. De Silva, An algebraic topological method for feature identification, Intl. J. Computational Geometry and Applications
2006
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V. De Silva, R. Ghrist, Coordinate-free coverage in sensor networks with controlled boundaries via homology, Intl. J. Robotics Research
2006
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T. Tao, V. Vu Additive Combinatorics
2006
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E. Chambers, V. De Silva, J. Erickson, R. Ghrist, Rips complexes of planar point sets, to appear in Discrete and Computational Geometry, math arXiv: 0712.0395v1
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M. Elkin, An Improved Construction of Progression-Free Sets, math arXiv: 0801.4310v1
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V. De Silva, R. Ghrist, Coverage in sensor networks via persistent homology, Alg. & Geom. Top
2007
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A. Muhammad, A. Jadbabaie, Dynamic Coverage Verification in mobile sensor networks via switched higher order Laplacians, Robotics: Science & Systems
2007
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G. Carlsson, V. De Silva, T. Ishkhanov, A. Zomorodian, On the Local Behavior of Spaces of Natural Images, International Journal of Computer Vision
2008
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F. Chazal, S. Oudot, Towards persistence-based reconstruction in Euclidean spaces, in proc. 2008 ACM Symposium of Computational Geometry
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