11,387 research outputs found
Rigidity of frameworks on expanding spheres
A rigidity theory is developed for bar-joint frameworks in
whose vertices are constrained to lie on concentric -spheres with
independently variable radii. In particular, combinatorial characterisations
are established for the rigidity of generic frameworks for with an
arbitrary number of independently variable radii, and for with at most
two variable radii. This includes a characterisation of the rigidity or
flexibility of uniformly expanding spherical frameworks in .
Due to the equivalence of the generic rigidity between Euclidean space and
spherical space, these results interpolate between rigidity in 1D and 2D and to
some extent between rigidity in 2D and 3D. Symmetry-adapted counts for the
detection of symmetry-induced continuous flexibility in frameworks on spheres
with variable radii are also provided.Comment: 22 pages, 2 figures, updated reference
A characterisation of generically rigid frameworks on surfaces of revolution
A foundational theorem of Laman provides a counting characterisation of the
finite simple graphs whose generic bar-joint frameworks in two dimensions are
infinitesimally rigid. Recently a Laman-type characterisation was obtained for
frameworks in three dimensions whose vertices are constrained to concentric
spheres or to concentric cylinders. Noting that the plane and the sphere have 3
independent locally tangential infinitesimal motions while the cylinder has 2,
we obtain here a Laman-Henneberg theorem for frameworks on algebraic surfaces
with a 1-dimensional space of tangential motions. Such surfaces include the
torus, helicoids and surfaces of revolution. The relevant class of graphs are
the (2,1)-tight graphs, in contrast to (2,3)-tightness for the plane/sphere and
(2,2)-tightness for the cylinder. The proof uses a new characterisation of
simple (2,1)-tight graphs and an inductive construction requiring generic
rigidity preservation for 5 graph moves, including the two Henneberg moves, an
edge joining move and various vertex surgery moves.Comment: 23 pages, 5 figures. Minor revisions - most importantly, the new
version has a different titl
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
Building block libraries and structural considerations in the self-assembly of polyoxometalate and polyoxothiometalate systems
Inorganic metal-oxide clusters form a class of compounds that are unique in their topological and electronic versatility and are becoming increasingly more important in a variety of applications. Namely, Polyoxometalates (POMs) have shown an unmatched range of physical properties and the ability to form structures that can bridge several length scales. The formation of these molecular clusters is often ambiguous and is governed by self-assembly processes that limit our ability to rationally design such molecules. However, recent years have shown that by considering new building block principles the design and discovery of novel complex clusters is aiding our understanding of this process. Now with current progress in thiometalate chemistry, specifically polyoxothiometalates (POTM), the field of inorganic molecular clusters has further diversified allowing for the targeted development of molecules with specific functionality. This chapter discusses the main differences between POM and POTM systems and how this affects synthetic methodologies and reactivities. We will illustrate how careful structural considerations can lead to the generation of novel building blocks and further deepen our understanding of complex systems
Frameworks with coordinated edge motions
We develop a rigidity theory for bar-joint frameworks in Euclidean -space
in which specified classes of edges are allowed to change length in a
coordinated fashion that requires differences of lengths to be preserved within
each class. Rigidity for these coordinated frameworks is a generic property,
and we characterize the rigid graphs in terms of redundant rigidity in the
standard -dimensional rigidity matroid. We also interpret our main results
in terms of matroid unions.Comment: 16 pages, 4 figure
Remarks on the combinatorial intersection cohomology of fans
We review the theory of combinatorial intersection cohomology of fans
developed by Barthel-Brasselet-Fieseler-Kaup, Bressler-Lunts, and Karu. This
theory gives a substitute for the intersection cohomology of toric varieties
which has all the expected formal properties but makes sense even for
non-rational fans, which do not define a toric variety. As a result, a number
of interesting results on the toric and polynomials have been extended
from rational polytopes to general polytopes. We present explicit complexes
computing the combinatorial IH in degrees one and two; the degree two complex
gives the rigidity complex previously used by Kalai to study . We present
several new results which follow from these methods, as well as previously
unpublished proofs of Kalai that implies and
.Comment: 34 pages. Typos fixed; final version, to appear in Pure and Applied
Math Quarterl
Hypoconstrained Jammed Packings of Nonspherical Hard Particles: Ellipses and Ellipsoids
Continuing on recent computational and experimental work on jammed packings
of hard ellipsoids [Donev et al., Science, vol. 303, 990-993] we consider
jamming in packings of smooth strictly convex nonspherical hard particles. We
explain why the isocounting conjecture, which states that for large disordered
jammed packings the average contact number per particle is twice the number of
degrees of freedom per particle (\bar{Z}=2d_{f}), does not apply to
nonspherical particles. We develop first- and second-order conditions for
jamming, and demonstrate that packings of nonspherical particles can be jammed
even though they are hypoconstrained (\bar{Z}<2d_{f}). We apply an algorithm
using these conditions to computer-generated hypoconstrained ellipsoid and
ellipse packings and demonstrate that our algorithm does produce jammed
packings, even close to the sphere point. We also consider packings that are
nearly jammed and draw connections to packings of deformable (but stiff)
particles. Finally, we consider the jamming conditions for nearly spherical
particles and explain quantitatively the behavior we observe in the vicinity of
the sphere point.Comment: 33 pages, third revisio
A characterization of generically rigid frameworks on surfaces of revolution
A foundational theorem of Laman provides a counting characterization of the finite simple graphs whose generic bar-joint frameworks in two dimensions are infinitesimally rigid. Recently a Laman-type characterization was obtained for frameworks in three dimensions whose vertices are constrained to concentric spheres or to concentric cylinders. Noting that the plane and the sphere have 3 independent locally tangential infinitesimal motions while the cylinder has 2, we obtain here a Laman-type theorem for frameworks on algebraic surfaces with a 1-dimensional space of tangential motions. Such surfaces include the torus, helicoids, and surfaces of revolution. The relevant class of graphs are the (2,1)-tight graphs, in contrast to (2,3)-tightness for the plane/sphere and (2,2)-tightness for the cylinder. The proof uses a new characterization of simple (2,1)-tight graphs and an inductive construction requiring generic rigidity preservation for 5 graph moves, including the two Henneberg moves, an edge joining move, and various vertex surgery moves. Read More: http://epubs.siam.org/doi/abs/10.1137/13091319
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