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Published in:
Mathematical Preliminaries Read chapter Read first chapter

2018 | OriginalPaper | Chapter

10. Universal and Dimensional Rigidities

Author : Abdo Y. Alfakih

Published in: Euclidean Distance Matrices and Their Applications in Rigidity Theory

Publisher: Springer International Publishing

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Abstract

In this chapter, we study the universal rigidity problem of bar frameworks and the related problem of dimensional rigidity. The main tools in tackling these two problems are the Cayley configuration spectrahedron \(\mathcal{F}\), defined in (8.​10), and Ω, the stress matrix, defined in (8.​13). The more general problem of universally linked pair of nonadjacent nodes is also studied and the results are interpreted in terms of the Strong Arnold Property and the notion of nondegeneracy in semidefinite programming.

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previous chapter Local and Infinitesimal Rigidities
Footnotes
1
Affine motions are often defined without the nonsingularity assumption on A. However, this assumption is more convenient for our purposes.
 
Literature
6.
9.
A.Y. Alfakih, On the universal rigidity of generic bar frameworks. Contrib. Discret. Math. 5, 7–17 (2010)MathSciNetMATH
11.
A.Y. Alfakih, On stress matrices of chordal bar frameworks in general positions, 2012. arXiv/1205.3990
13.
A.Y. Alfakih, Universal rigidity of bar frameworks in general position: a Euclidean distance matrix approach, in Distance Geometry: Theory, Methods, and Applications, ed. by A. Mucherino, C. Lavor, L. Liberti, N. Maculan (Springer, Berlin, 2013), pp. 3–22CrossRef
14.
A.Y. Alfakih, On Farkas lemma and dimensional rigidity of bar frameworks. Linear Algebra Appl. 486, 504–522 (2015)MathSciNetCrossRef
16.
A.Y. Alfakih, Universal rigidity of bar frameworks via the geometry of spectrahedra. J. Glob. Optim. 67, 909–924 (2017)MathSciNetCrossRef
17.
A.Y. Alfakih, V.-H. Nyugen, On affine motions and universal rigidity of tensegrity frameworks. Linear Algebra Appl. 439, 3134–3147 (2013)MathSciNetCrossRef
20.
A.Y. Alfakih, Y. Ye, On affine motions and bar frameworks in general positions. Linear Algebra Appl. 438, 31–36 (2013)MathSciNetCrossRef
22.
A.Y. Alfakih, N. Taheri, Y. Ye, On stress matrices of (d + 1)-lateration frameworks in general position. Math. Program. 137, 1–17 (2013)MathSciNetCrossRef
26.
F. Alizadeh, J.A. Haeberly, M.L. Overton, Complementarity and nondegeneracy in semidefinite programming. Math. Program. Ser. B 77, 111–128 (1997)MathSciNetMATH
39.
56.
Y. Colin De Verdière, Sur un nouvel invariant des graphes et un critère de planarité. J. Combin. Theory Ser B 50, 11–21 (1990)MathSciNetCrossRef
57.
60.
R. Connelly, Tensegrity structures: why are they stable? in Rigidity Theory and Applications, ed. by M.F. Thorpe, P.M. Duxbury (Kluwer Academic/Plenum Publishers, New York, 1999), pp. 47–54
61.
63.
64.
R. Connelly, S. Gortler, Universal rigidity of complete bipartite graphs. Discret. Comput. Geom. 57, 281–304 (2017)MathSciNetCrossRef
90.
S.J. Gortler, D.P. Thurston, Characterizing the universal rigidity of generic frameworks. Discret. Comput. Geom. 51, 1017–1036 (2014)MathSciNetCrossRef
118.
T. Jordán, V.-H. Nguyen, On universal rigid frameworks on the line. Contrib. Discret. Math. 10, 10–21 (2015)MATH
132.
M. Laurent, A. Varvitsiotis, Positive semidefinite matrix completion, universal rigidity and the strong Arnold property. Linear Algebra Appl. 452, 292–317 (2014)MathSciNetCrossRef
138.
L. Lovász, Geometric representations of graphs. Unpublished lecture notes, 2016
177.
A.M.-C. So, Semidefinite Programming Approach to the Graph Realization Problem: Theory, Applications and Extensions. PhD thesis, Stanford University, 2007
179.
A.M.-C. So, Y. Ye, Theory of semidefinite programming for sensor network localization. Math. Prog. Ser. B 109, 367–384 (2007)MathSciNetCrossRef
201.
Z. Zhu, A.M.-C. So, Y. Ye, Universal rigidity: towards accurate and efficient localization of wireless networks, in Proceedings IEEE INFOCOM, 2010
Metadata
Title
Universal and Dimensional Rigidities
Author
Abdo Y. Alfakih
Copyright Year
2018
Book
Euclidean Distance Matrices and Their Applications in Rigidity Theory

Print ISBN: 978-3-319-97845-1

Electronic ISBN: 978-3-319-97846-8

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-97846-8_10

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