3,244 research outputs found
Generic Rigidity Matroids with Dilworth Truncations
We prove that the linear matroid that defines generic rigidity of
-dimensional body-rod-bar frameworks (i.e., structures consisting of
disjoint bodies and rods mutually linked by bars) can be obtained from the
union of graphic matroids by applying variants of Dilworth
truncation times, where denotes the number of rods. This leads to
an alternative proof of Tay's combinatorial characterizations of generic
rigidity of rod-bar frameworks and that of identified body-hinge frameworks
Linking Rigid Bodies Symmetrically
The mathematical theory of rigidity of body-bar and body-hinge frameworks
provides a useful tool for analyzing the rigidity and flexibility of many
articulated structures appearing in engineering, robotics and biochemistry. In
this paper we develop a symmetric extension of this theory which permits a
rigidity analysis of body-bar and body-hinge structures with point group
symmetries. The infinitesimal rigidity of body-bar frameworks can naturally be
formulated in the language of the exterior (or Grassmann) algebra. Using this
algebraic formulation, we derive symmetry-adapted rigidity matrices to analyze
the infinitesimal rigidity of body-bar frameworks with Abelian point group
symmetries in an arbitrary dimension. In particular, from the patterns of these
new matrices, we derive combinatorial characterizations of infinitesimally
rigid body-bar frameworks which are generic with respect to a point group of
the form .
Our characterizations are given in terms of packings of bases of signed-graphic
matroids on quotient graphs. Finally, we also extend our methods and results to
body-hinge frameworks with Abelian point group symmetries in an arbitrary
dimension. As special cases of these results, we obtain combinatorial
characterizations of infinitesimally rigid body-hinge frameworks with
or symmetry - the most common symmetry groups
found in proteins.Comment: arXiv:1308.6380 version 1 was split into two papers. The version 2 of
arXiv:1308.6380 consists of Sections 1 - 6 of the version 1. This paper is
based on the second part of the version 1 (Sections 7 and 8
Nucleation-free rigidity
When all non-edge distances of a graph realized in as a {\em
bar-and-joint framework} are generically {\em implied} by the bar (edge)
lengths, the graph is said to be {\em rigid} in . For ,
characterizing rigid graphs, determining implied non-edges and {\em dependent}
edge sets remains an elusive, long-standing open problem.
One obstacle is to determine when implied non-edges can exist without
non-trivial rigid induced subgraphs, i.e., {\em nucleations}, and how to deal
with them.
In this paper, we give general inductive construction schemes and proof
techniques to generate {\em nucleation-free graphs} (i.e., graphs without any
nucleation) with implied non-edges. As a consequence, we obtain (a) dependent
graphs in that have no nucleation; and (b) nucleation-free {\em
rigidity circuits}, i.e., minimally dependent edge sets in . It
additionally follows that true rigidity is strictly stronger than a tractable
approximation to rigidity given by Sitharam and Zhou
\cite{sitharam:zhou:tractableADG:2004}, based on an inductive combinatorial
characterization.
As an independently interesting byproduct, we obtain a new inductive
construction for independent graphs in . Currently, very few such inductive
constructions are known, in contrast to
One brick at a time: a survey of inductive constructions in rigidity theory
We present a survey of results concerning the use of inductive constructions
to study the rigidity of frameworks. By inductive constructions we mean simple
graph moves which can be shown to preserve the rigidity of the corresponding
framework. We describe a number of cases in which characterisations of rigidity
were proved by inductive constructions. That is, by identifying recursive
operations that preserved rigidity and proving that these operations were
sufficient to generate all such frameworks. We also outline the use of
inductive constructions in some recent areas of particularly active interest,
namely symmetric and periodic frameworks, frameworks on surfaces, and body-bar
frameworks. We summarize the key outstanding open problems related to
inductions.Comment: 24 pages, 12 figures, final versio
Maxwell-Laman counts for bar-joint frameworks in normed spaces
The rigidity matrix is a fundamental tool for studying the infinitesimal
rigidity properties of Euclidean bar-joint frameworks. In this paper we
generalize this tool and introduce a rigidity matrix for bar-joint frameworks
in arbitrary finite dimensional real normed vector spaces. Using this new
matrix, we derive necessary Maxwell-Laman-type counting conditions for a
well-positioned bar-joint framework in a real normed vector space to be
infinitesimally rigid. Moreover, we derive symmetry-extended counting
conditions for a bar-joint framework with a non-trivial symmetry group to be
isostatic (i.e., minimally infinitesimally rigid). These conditions imply very
simply stated restrictions on the number of those structural components that
are fixed by the various symmetry operations of the framework. Finally, we
offer some observations and conjectures regarding combinatorial
characterisations of 2-dimensional symmetric, isostatic bar-joint frameworks
where the unit ball is a quadrilateral.Comment: 17 page
Hyperbanana Graphs
A bar-and-joint framework is a finite set of points together with specified
distances between selected pairs. In rigidity theory we seek to understand when
the remaining pairwise distances are also fixed. If there exists a pair of
points which move relative to one another while maintaining the given distance
constraints, the framework is flexible; otherwise, it is rigid.
Counting conditions due to Maxwell give a necessary combinatorial criterion
for generic minimal bar-and-joint rigidity in all dimensions. Laman showed that
these conditions are also sufficient for frameworks in R^2. However, the
flexible "double banana" shows that Maxwell's conditions are not sufficient to
guarantee rigidity in R^3. We present a generalization of the double banana to
a family of hyperbananas. In dimensions 3 and higher, these are
(infinitesimally) flexible, providing counterexamples to the natural
generalization of Laman's theorem
Rigidity of Frameworks Supported on Surfaces
A theorem of Laman gives a combinatorial characterisation of the graphs that
admit a realisation as a minimally rigid generic bar-joint framework in
\bR^2. A more general theory is developed for frameworks in \bR^3 whose
vertices are constrained to move on a two-dimensional smooth submanifold \M.
Furthermore, when \M is a union of concentric spheres, or a union of parallel
planes or a union of concentric cylinders, necessary and sufficient
combinatorial conditions are obtained for the minimal rigidity of generic
frameworks.Comment: Final version, 28 pages, with new figure
Rigidity and flexibility of biological networks
The network approach became a widely used tool to understand the behaviour of
complex systems in the last decade. We start from a short description of
structural rigidity theory. A detailed account on the combinatorial rigidity
analysis of protein structures, as well as local flexibility measures of
proteins and their applications in explaining allostery and thermostability is
given. We also briefly discuss the network aspects of cytoskeletal tensegrity.
Finally, we show the importance of the balance between functional flexibility
and rigidity in protein-protein interaction, metabolic, gene regulatory and
neuronal networks. Our summary raises the possibility that the concepts of
flexibility and rigidity can be generalized to all networks.Comment: 21 pages, 4 figures, 1 tabl
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