2,535 research outputs found
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
Universality theorems for configuration spaces of planar linkages
We prove realizability theorems for vector-valued polynomial mappings,
real-algebraic sets and compact smooth manifolds by moduli spaces of planar
linkages. We also establish a relation between universality theorems for moduli
spaces of mechanical linkages and projective arrangements.Comment: 45 pages, 15 figures. See also
http://www.math.utah.edu/~kapovich/eprints.htm
Homogenization of the nonlinear bending theory for plates
We carry out the spatially periodic homogenization of nonlinear bending
theory for plates. The derivation is rigorous in the sense of
Gamma-convergence. In contrast to what one naturally would expect, our result
shows that the limiting functional is not simply a quadratic functional of the
second fundamental form of the deformed plate as it is the case in nonlinear
plate theory. It turns out that the limiting functional discriminates between
whether the deformed plate is locally shaped like a "cylinder" or not. For the
derivation we investigate the oscillatory behavior of sequences of second
fundamental forms associated with isometric immersions, using two-scale
convergence. This is a non-trivial task, since one has to treat two-scale
convergence in connection with a nonlinear differential constraint.Comment: 36 pages, 4 figures. Major revisions of Sections 2,3 and 4. In
Section 2: Correction of definition of conical and cylindrical part
(Definition 1). In Section 3: Modifications in the proof of Proposition 2 due
to changes in Definition 1. Several new lemmas and other modifications. In
Section 4: Modification of proof of lower bound. Proof of upper bound
completely revised. Several lemmas adde
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
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
Instability of dielectrics and conductors in electrostatic fields
International audienceThis article proves most of the assertion in §116 of Maxwell's treatise on electromagnetism. The results go under the name Earnshaw's Theorem and assert the absence of stable equilibrium configurations of conductors and dielectrics in an external electrostatic field
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