3,812 research outputs found
Biharmonic pattern selection
A new model to describe fractal growth is discussed which includes effects
due to long-range coupling between displacements . The model is based on the
biharmonic equation in two-dimensional isotropic defect-free
media as follows from the Kuramoto-Sivashinsky equation for pattern formation
-or, alternatively, from the theory of elasticity. As a difference with
Laplacian and Poisson growth models, in the new model the Laplacian of is
neither zero nor proportional to . Its discretization allows to reproduce a
transition from dense to multibranched growth at a point in which the growth
velocity exhibits a minimum similarly to what occurs within Poisson growth in
planar geometry. Furthermore, in circular geometry the transition point is
estimated for the simplest case from the relation
such that the trajectories become stable at the growing surfaces in a
continuous limit. Hence, within the biharmonic growth model, this transition
depends only on the system size and occurs approximately at a distance far from a central seed particle. The influence of biharmonic patterns on
the growth probability for each lattice site is also analysed.Comment: To appear in Phys. Rev. E. Copies upon request to
[email protected]
A note on the Almansi property
The first goal of this note is to study the Almansi property on an
m-dimensional model in the sense of Greene and Wu and, more generally, in a
Riemannian geometric setting. In particular, we shall prove that the only model
on which the Almansi property is verified is the Euclidean space R^m. In the
second part of the paper we shall study Almansi's property and biharmonicity
for functions which depend on the distance from a given submanifold. Finally,
in the last section we provide an extension to the semi-Euclidean case R^{p,q}
which includes the proof of the classical Almansi property in R^m as a special
instance.Comment: Dedicated to Prof. Renzo Caddeo, to appear in Mediterranean Journal
of Mathematic
Optical Diffraction from Fractals with a Structural Transition
A macroscopic characterization of fractals showing up a structural transition
from dense to multibranched growth is made using optical diffraction theory.
Such fractals are generated via the numerical solution of the 2D Poisson and
Biharmonic equations and are compared to more 'regular'irreversible clusters
such as diffusion limited and Laplacian aggregates. The optical diffraction
method enables to identify a decrease of the fractal dimension above the
structural point.Comment: LaTex file; 6 figures upon request to [email protected]**.
IC/94/7
Some classifications of biharmonic hypersurfaces with constant scalar curvature
We give some classifications of biharmonic hypersurfaces with constant scalar
curvature. These include biharmonic Einstein hypersurfaces in space forms,
compact biharmonic hypersurfaces with constant scalar curvature in a sphere,
and some complete biharmonic hypersurfaces of constant scalar curvature in
space forms and in a non-positively curved Einstein space. Our results provide
additional cases (Theorem 2.3 and Proposition 2.8) that supports the conjecture
that a biharmonic submanifold in a sphere has constant mean curvature, and two
more cases that support Chen's conjecture on biharmonic hypersurfaces
(Corollaries 2.2,2.7).Comment: 11 page
Efficient, sparse representation of manifold distance matrices for classical scaling
Geodesic distance matrices can reveal shape properties that are largely
invariant to non-rigid deformations, and thus are often used to analyze and
represent 3-D shapes. However, these matrices grow quadratically with the
number of points. Thus for large point sets it is common to use a low-rank
approximation to the distance matrix, which fits in memory and can be
efficiently analyzed using methods such as multidimensional scaling (MDS). In
this paper we present a novel sparse method for efficiently representing
geodesic distance matrices using biharmonic interpolation. This method exploits
knowledge of the data manifold to learn a sparse interpolation operator that
approximates distances using a subset of points. We show that our method is 2x
faster and uses 20x less memory than current leading methods for solving MDS on
large point sets, with similar quality. This enables analyses of large point
sets that were previously infeasible.Comment: Conference CVPR 201
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