403 research outputs found
Crystal frameworks, symmetry and affinely periodic flexes
Symmetry equations are obtained for the rigidity matrices associated with
various forms of infinitesimal flexibility for an idealised bond-node crystal
framework \C in \bR^d. These equations are used to derive symmetry-adapted
Maxwell-Calladine counting formulae for periodic self-stresses and affinely
periodic infinitesimal mechanisms. The symmetry equations also lead to general
Fowler-Guest formulae connecting the character lists of subrepresentations of
the crystallographic space and point groups which are associated with bonds,
nodes, stresses, flexes and rigid motions. A new derivation is also given for
the Borcea-Streinu rigidity matrix and the correspondence between its nullspace
and the space of affinely periodic infinitesimal flexes.Comment: This preprint has some new diagrams and clarifications. A final
version will appear in the New York Journal of Mathematic
Generic rigidity with forced symmetry and sparse colored graphs
We review some recent results in the generic rigidity theory of planar
frameworks with forced symmetry, giving a uniform treatment to the topic. We
also give new combinatorial characterizations of minimally rigid periodic
frameworks with fixed-area fundamental domain and fixed-angle fundamental
domain.Comment: 21 pages, 2 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
Periodic Body-And-Bar Frameworks
Periodic body-and-bar frameworks are abstractions of crystalline structures made of rigid bodies connected by fixed-length bars and subject to the action of a lattice of translations. We give a Maxwell–Laman characterization for minimally rigid periodic body-and-bar frameworks in terms of their quotient graphs. As a consequence we obtain efficient polynomial time algorithms for their recognition based on matroid partition and pebble games
Liftings and stresses for planar periodic frameworks
We formulate and prove a periodic analog of Maxwell's theorem relating
stressed planar frameworks and their liftings to polyhedral surfaces with
spherical topology. We use our lifting theorem to prove deformation and
rigidity-theoretic properties for planar periodic pseudo-triangulations,
generalizing features known for their finite counterparts. These properties are
then applied to questions originating in mathematical crystallography and
materials science, concerning planar periodic auxetic structures and ultrarigid
periodic frameworks.Comment: An extended abstract of this paper has appeared in Proc. 30th annual
Symposium on Computational Geometry (SOCG'14), Kyoto, Japan, June 201
The rigidity of infinite graphs
A rigidity theory is developed for the Euclidean and non-Euclidean placements
of countably infinite simple graphs in R^d with respect to the classical l^p
norms, for d>1 and 1<p<\infty. Generalisations are obtained for the Laman and
Henneberg combinatorial characterisations of generic infinitesimal rigidity for
finite graphs in the Euclidean plane. Also Tay's multi-graph characterisation
of the rigidity of generic finite body-bar frameworks in d-dimensional
Euclidean space is generalised to the non-Euclidean l^p norms and to countably
infinite graphs. For all dimensions and norms it is shown that a generically
rigid countable simple graph is the direct limit of an inclusion tower of
finite graphs for which the inclusions satisfy a relative rigidity property.
For d>2 a countable graph which is rigid for generic placements in R^d may fail
the stronger property of sequential rigidity, while for d=2 the equivalence
with sequential rigidity is obtained from the generalised Laman
characterisations. Applications are given to the flexibility of non-Euclidean
convex polyhedra and to the infinitesimal and continuous rigidity of compact
infinitely-faceted simplicial polytopes.Comment: 51 page
Polynomials for Crystal Frameworks and the Rigid Unit Mode Spectrum
To each discrete translationally periodic bar-joint framework \C in \bR^d
we associate a matrix-valued function \Phi_\C(z) defined on the d-torus. The
rigid unit mode spectrum \Omega(\C) of \C is defined in terms of the
multi-phases of phase-periodic infinitesimal flexes and is shown to correspond
to the singular points of the function z \to \rank \Phi_\C(z) and also to the
set of wave vectors of harmonic excitations which have vanishing energy in the
long wavelength limit. To a crystal framework in Maxwell counting equilibrium,
which corresponds to \Phi_\C(z) being square, the determinant of \Phi_\C(z)
gives rise to a unique multi-variable polynomial p_\C(z_1,\dots,z_d). For
ideal zeolites the algebraic variety of zeros of p_\C(z) on the d-torus
coincides with the RUM spectrum. The matrix function is related to other
aspects of idealised framework rigidity and flexibility and in particular leads
to an explicit formula for the number of supercell-periodic floppy modes. In
the case of certain zeolite frameworks in dimensions 2 and 3 direct proofs are
given to show the maximal floppy mode property (order ). In particular this
is the case for the cubic symmetry sodalite framework and some other idealised
zeolites.Comment: Final version with new examples and figures, and with clearer
streamlined proof
Isotopy classes for 3-periodic net embeddings
Entangled embedded periodic nets and crystal frameworks are defined, along with their {dimension type}, {homogeneity type}, {adjacency depth} and {periodic isotopy type}. We obtain periodic isotopy classifications for various families of embedded nets with small quotient graphs. We enumerate the 25 periodic isotopy classes of depth 1 embedded nets with a single vertex quotient graph. Additionally, we classify embeddings of n-fold copies of {pcu} with all connected components in a parallel orientation and n vertices in a repeat unit, and determine their maximal symmetry periodic isotopes. We also introduce the methodology of linear graph knots on the flat 3-torus [0, 1)^3. These graph knots, with linear edges, are spatial embeddings of the labelled quotient graphs of an embedded net which are associated with its periodicity bases
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