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Local, Dimensional and Universal Rigidities: A Unified Gram Matrix Approach

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Rigidity and Symmetry

Part of the book series: Fields Institute Communications ((FIC,volume 70))

Abstract

This chapter is a unified treatment, based on projected Gram matrices (PGMs), of the problems of local, dimensional, and universal rigidities of bar frameworks. This PGM-based approach makes these problems amenable to semidefinite programming methodology; and naturally gives rise to results expressed in terms of Gale matrices. We survey known results emphasizing numerical examples and proofs which highlight the salient aspects of this approach.

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Notes

  1. 1.

    The rank function is lower semi-continuous on the set of matrices of order n − 1.

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Acknowledgements

Research supported by the Natural Sciences and Engineering Research Council of Canada.The author would like to thank an anonymous referee for his/her remarks which improved the presentation of this chapter.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Windsor, Windsor, ON, N9B 3P4, Canada

    A. Y. Alfakih

Authors
  1. A. Y. Alfakih
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Correspondence to A. Y. Alfakih .

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Editors and Affiliations

  1. Department of Mathematics, Cornell University, Ithaca, New York, USA

    Robert Connelly

  2. Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada

    Asia Ivić Weiss

  3. Department of Mathematics & Statistics, York University, Toronto, Ontario, Canada

    Walter Whiteley

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Alfakih, A.Y. (2014). Local, Dimensional and Universal Rigidities: A Unified Gram Matrix Approach. In: Connelly, R., Ivić Weiss, A., Whiteley, W. (eds) Rigidity and Symmetry. Fields Institute Communications, vol 70. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0781-6_3

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