3,582 research outputs found
Discrete differential forms and applications to surface tiling
The geometry of manifolds has been extensively studied for centuries — though almost exclusively from a differential point of view. Unfortunately, well-established theoretical geometric foundations do not directly translate to discrete meshes: discretizations of inherently-continuous
notions such as curvatures and geodesics may lose their geometric and/or variational properties.
In this talk, we will introduce the notion of discrete differential forms and show how they provide
differential, yet readily discretizable computational foundations [1]. We will describe how key
geometric properties built into their description can more readily yield robust numerical
computations which are true to the underlying continuous equations: they exactly preserve
invariants of continuous models in the discrete computational realm.
These discrete forms will be put to good
use, first for surface flows and conformal
parameterizations, then for the design of pure quadrilateral tiling of arbitrary 2-manifolds [2]. We
will also briefly mention other applications (fluid animation, vector field design) benefiting greatly
from this principled, discrete approach to geometry and computations
Cohomology of matching rules
Quasiperiodic patterns described by polyhedral "atomic surfaces" and
admitting matching rules are considered. It is shown that the cohomology ring
of the continuous hull of such patterns is isomorphic to that of the complement
of a torus to an arrangement of thickened affine tori of codimension
two. Explicit computation of Betti numbers for several two-dimensional tilings
and for the icosahedral Ammann-Kramer tiling confirms in most cases the results
obtained previously by different methods. The cohomology groups of have a natural structure of a right module over the group ring of
the space symmetry group of the pattern and can be decomposed in a direct sum
of its irreducible representations. An example of such decomposition is shown
for the Ammann-Kramer tiling
Exclusion processes: short range correlations induced by adhesion and contact interactions
We analyze the out-of-equilibrium behavior of exclusion processes where
agents interact with their nearest neighbors, and we study the short-range
correlations which develop because of the exclusion and other contact
interactions. The form of interactions we focus on, including adhesion and
contact-preserving interactions, is especially relevant for migration processes
of living cells. We show the local agent density and nearest-neighbor two-point
correlations resulting from simulations on two dimensional lattices in the
transient regime where agents invade an initially empty space from a source and
in the stationary regime between a source and a sink. We compare the results of
simulations with the corresponding quantities derived from the master equation
of the exclusion processes, and in both cases, we show that, during the
invasion of space by agents, a wave of correlations travels with velocity v(t)
~ t^(-1/2). The relative placement of this wave to the agent density front and
the time dependence of its height may be used to discriminate between different
forms of contact interactions or to quantitatively estimate the intensity of
interactions. We discuss, in the stationary density profile between a full and
an empty reservoir of agents, the presence of a discontinuity close to the
empty reservoir. Then, we develop a method for deriving approximate
hydrodynamic limits of the processes. From the resulting systems of partial
differential equations, we recover the self-similar behavior of the agent
density and correlations during space invasion
Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities, I: The Dirichlet Problem
Consider a planar, bounded, -connected region , and let
\bord\Omega be its boundary. Let be a cellular decomposition of
\Omega\cup\bord\Omega, where each 2-cell is either a triangle or a
quadrilateral. From these data and a conductance function we construct a
canonical pair where is a genus singular flat surface tiled
by rectangles and is an energy preserving mapping from
onto .Comment: 27 pages, 11 figures; v2 - revised definition (now denoted by the
flux-gradient metric (1.9)) in section 1 and minor modifications of proofs;
corrected typo
A nodal domain theorem for integrable billiards in two dimensions
Eigenfunctions of integrable planar billiards are studied - in particular,
the number of nodal domains, , of the eigenfunctions are considered. The
billiards for which the time-independent Schr\"odinger equation (Helmholtz
equation) is separable admit trivial expressions for the number of domains.
Here, we discover that for all separable and non-separable integrable
billiards, satisfies certain difference equations. This has been
possible because the eigenfunctions can be classified in families labelled by
the same value of , given a particular , for a set of quantum
numbers, . Further, we observe that the patterns in a family are similar
and the algebraic representation of the geometrical nodal patterns is found.
Instances of this representation are explained in detail to understand the
beauty of the patterns. This paper therefore presents a mathematical connection
between integrable systems and difference equations.Comment: 13 pages, 5 figure
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