17,440 research outputs found
Generic rigidity of reflection frameworks
We give a combinatorial characterization of generic minimally rigid
reflection frameworks. The main new idea is to study a pair of direction
networks on the same graph such that one admits faithful realizations and the
other has only collapsed realizations. In terms of infinitesimal rigidity,
realizations of the former produce a framework and the latter certifies that
this framework is infinitesimally rigid.Comment: 14 pages, 2 figure
Symmetry as a sufficient condition for a finite flex
We show that if the joints of a bar and joint framework are
positioned as `generically' as possible subject to given symmetry constraints
and possesses a `fully-symmetric' infinitesimal flex (i.e., the
velocity vectors of the infinitesimal flex remain unaltered under all symmetry
operations of ), then also possesses a finite flex which
preserves the symmetry of throughout the path. This and other related
results are obtained by symmetrizing techniques described by L. Asimov and B.
Roth in their paper `The Rigidity Of Graphs' from 1978 and by using the fact
that the rigidity matrix of a symmetric framework can be transformed into a
block-diagonalized form by means of group representation theory. The finite
flexes that can be detected with these symmetry-based methods can in general
not be found with the analogous non-symmetric methods.Comment: 26 pages, 10 figure
The orbit rigidity matrix of a symmetric framework
A number of recent papers have studied when symmetry causes frameworks on a
graph to become infinitesimally flexible, or stressed, and when it has no
impact. A number of other recent papers have studied special classes of
frameworks on generically rigid graphs which are finite mechanisms. Here we
introduce a new tool, the orbit matrix, which connects these two areas and
provides a matrix representation for fully symmetric infinitesimal flexes, and
fully symmetric stresses of symmetric frameworks. The orbit matrix is a true
analog of the standard rigidity matrix for general frameworks, and its analysis
gives important insights into questions about the flexibility and rigidity of
classes of symmetric frameworks, in all dimensions.
With this narrower focus on fully symmetric infinitesimal motions, comes the
power to predict symmetry-preserving finite mechanisms - giving a simplified
analysis which covers a wide range of the known mechanisms, and generalizes the
classes of known mechanisms. This initial exploration of the properties of the
orbit matrix also opens up a number of new questions and possible extensions of
the previous results, including transfer of symmetry based results from
Euclidean space to spherical, hyperbolic, and some other metrics with shared
symmetry groups and underlying projective geometry.Comment: 41 pages, 12 figure
Natural realizations of sparsity matroids
A hypergraph G with n vertices and m hyperedges with d endpoints each is
(k,l)-sparse if for all sub-hypergraphs G' on n' vertices and m' edges, m'\le
kn'-l. For integers k and l satisfying 0\le l\le dk-1, this is known to be a
linearly representable matroidal family.
Motivated by problems in rigidity theory, we give a new linear representation
theorem for the (k,l)-sparse hypergraphs that is natural; i.e., the
representing matrix captures the vertex-edge incidence structure of the
underlying hypergraph G.Comment: Corrected some typos from the previous version; to appear in Ars
Mathematica Contemporane
Euclidean distance geometry and applications
Euclidean distance geometry is the study of Euclidean geometry based on the
concept of distance. This is useful in several applications where the input
data consists of an incomplete set of distances, and the output is a set of
points in Euclidean space that realizes the given distances. We survey some of
the theory of Euclidean distance geometry and some of the most important
applications: molecular conformation, localization of sensor networks and
statics.Comment: 64 pages, 21 figure
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