13,395 research outputs found
Melting and Rippling Phenomenan in Two Dimensional Crystals with localized bonding
We calculate Root Mean Square (RMS) deviations from equilibrium for atoms in
a two dimensional crystal with local (e.g. covalent) bonding between close
neighbors. Large scale Monte Carlo calculations are in good agreement with
analytical results obtained in the harmonic approximation. When motion is
restricted to the plane, we find a slow (logarithmic) increase in fluctuations
of the atoms about their equilibrium positions as the crystals are made larger
and larger. We take into account fluctuations perpendicular to the lattice
plane, manifest as undulating ripples, by examining dual layer systems with
coupling between the layers to impart local rigidly (i.e. as in sheets of
graphene made stiff by their finite thickness). Surprisingly, we find a rapid
divergence with increasing system size in the vertical mean square deviations,
independent of the strength of the interplanar coupling. We consider an
attractive coupling to a flat substrate, finding that even a weak attraction
significantly limits the amplitude and average wavelength of the ripples. We
verify our results are generic by examining a variety of distinct geometries,
obtaining the same phenomena in each case.Comment: 17 pages, 28 figure
Symmetry adapted Assur decompositions
Assur graphs are a tool originally developed by mechanical engineers to
decompose mechanisms for simpler analysis and synthesis. Recent work has
connected these graphs to strongly directed graphs, and decompositions of the
pinned rigidity matrix. Many mechanisms have initial configurations which are
symmetric, and other recent work has exploited the orbit matrix as a symmetry
adapted form of the rigidity matrix. This paper explores how the decomposition
and analysis of symmetric frameworks and their symmetric motions can be
supported by the new symmetry adapted tools.Comment: 40 pages, 22 figure
On the maximal number of real embeddings of minimally rigid graphs in , and
Rigidity theory studies the properties of graphs that can have rigid
embeddings in a euclidean space or on a sphere and which in
addition satisfy certain edge length constraints. One of the major open
problems in this field is to determine lower and upper bounds on the number of
realizations with respect to a given number of vertices. This problem is
closely related to the classification of rigid graphs according to their
maximal number of real embeddings.
In this paper, we are interested in finding edge lengths that can maximize
the number of real embeddings of minimally rigid graphs in the plane, space,
and on the sphere. We use algebraic formulations to provide upper bounds. To
find values of the parameters that lead to graphs with a large number of real
realizations, possibly attaining the (algebraic) upper bounds, we use some
standard heuristics and we also develop a new method inspired by coupler
curves. We apply this new method to obtain embeddings in . One of
its main novelties is that it allows us to sample efficiently from a larger
number of parameters by selecting only a subset of them at each iteration.
Our results include a full classification of the 7-vertex graphs according to
their maximal numbers of real embeddings in the cases of the embeddings in
and , while in the case of we achieve this
classification for all 6-vertex graphs. Additionally, by increasing the number
of embeddings of selected graphs, we improve the previously known asymptotic
lower bound on the maximum number of realizations. The methods and the results
concerning the spatial embeddings are part of the proceedings of ISSAC 2018
(Bartzos et al, 2018)
Folding of the triangular lattice in a discrete three-dimensional space: Density-matrix-renormalization-group study
Folding of the triangular lattice in a discrete three-dimensional space is
investigated numerically. Such ``discrete folding'' has come under through
theoretical investigation, since Bowick and co-worker introduced it as a
simplified model for the crumpling of the phantom polymerized membranes. So
far, it has been analyzed with the hexagon approximation of the cluster
variation method (CVM). However, the possible systematic error of the
approximation was not fully estimated; in fact, it has been known that the
transfer-matrix calculation is limited in the tractable strip widths L \le 6.
Aiming to surmount this limitation, we utilized the density-matrix
renormalization group. Thereby, we succeeded in treating strip widths up to
L=29 which admit reliable extrapolations to the thermodynamic limit. Our data
indicate an onset of a discontinuous crumpling transition with the latent heat
substantially larger than the CVM estimate. It is even larger than the latent
heat of the planar (two dimensional) folding, as first noticed by the preceding
CVM study. That is, contrary to our naive expectation, the discontinuous
character of the transition is even promoted by the enlargement of the
embedding-space dimensions. We also calculated the folding entropy, which
appears to lie within the best analytical bound obtained previously via
combinatorics arguments
Stress management in composite biopolymer networks
Living tissues show an extraordinary adaptiveness to strain, which is crucial
for their proper biological functioning. The physical origin of this mechanical
behaviour has been widely investigated using reconstituted networks of collagen
fibres, the principal load-bearing component of tissues. However, collagen
fibres in tissues are embedded in a soft hydrated polysaccharide matrix which
generates substantial internal stresses whose effect on tissue mechanics is
unknown. Here, by combining mechanical measurements and computer simulations,
we show that networks composed of collagen fibres and a hyaluronan matrix
exhibit synergistic mechanics characterized by an enhanced stiffness and
delayed strain-stiffening. We demonstrate that the polysaccharide matrix has a
dual effect on the composite response involving both internal stress and
elastic reinforcement. Our findings elucidate how tissues can tune their
strain-sensitivity over a wide range and provide a novel design principle for
synthetic materials with programmable mechanical properties
Properties of Bott manifolds and cohomological rigidity
The cohomological rigidity problem for toric manifolds asks whether the
cohomology ring of a toric manifold determines the topological type of the
manifold. In this paper, we consider the problem with the class of one-twist
Bott manifolds to get an affirmative answer to the problem. We also generalize
the result to quasitoric manifolds. In doing so, we show that the twist number
of a Bott manifold is well-defined and is equal to the cohomological complexity
of the cohomology ring of the manifold. We also show that any cohomology Bott
manifold is homeomorphic to a Bott manifold. All these results are also
generalized to the case with -coefficients, where is the localized ring at 2.Comment: 22 page
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