5,519 research outputs found
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
Beyond developable: computational design and fabrication with auxetic materials
We present a computational method for interactive 3D design and rationalization of surfaces via auxetic materials, i.e., flat flexible material that can stretch uniformly up to a certain extent. A key motivation for studying such material is that one can approximate doubly-curved surfaces (such as the sphere) using only flat pieces, making it attractive for fabrication. We physically realize surfaces by introducing cuts into approximately inextensible material such as sheet metal, plastic, or leather. The cutting pattern is modeled as a regular triangular linkage that yields hexagonal openings of spatially-varying radius when stretched. In the same way that isometry is fundamental to modeling developable surfaces, we leverage conformal geometry to understand auxetic design. In particular, we compute a global conformal map with bounded scale factor to initialize an otherwise intractable non-linear optimization. We demonstrate that this global approach can handle non-trivial topology and non-local dependencies inherent in auxetic material. Design studies and physical prototypes are used to illustrate a wide range of possible applications
Chaotic Hypothesis, Fluctuation Theorem and singularities
The chaotic hypothesis has several implications which have generated interest
in the literature because of their generality and because a few exact
predictions are among them. However its application to Physics problems
requires attention and can lead to apparent inconsistencies. In particular
there are several cases that have been considered in the literature in which
singularities are built in the models: for instance when among the forces there
are Lennard-Jones potentials (which are infinite in the origin) and the
constraints imposed on the system do not forbid arbitrarily close approach to
the singularity even though the average kinetic energy is bounded. The
situation is well understood in certain special cases in which the system is
subject to Gaussian noise; here the treatment of rather general singular
systems is considered and the predictions of the chaotic hypothesis for such
situations are derived. The main conclusion is that the chaotic hypothesis is
perfectly adequate to describe the singular physical systems we consider, i.e.
deterministic systems with thermostat forces acting according to Gauss'
principle for the constraint of constant total kinetic energy (``isokinetic
Gaussian thermostats''), close and far from equilibrium. Near equilibrium it
even predicts a fluctuation relation which, in deterministic cases with more
general thermostat forces (i.e. not necessarily of Gaussian isokinetic nature),
extends recent relations obtained in situations in which the thermostatting
forces satisfy Gauss' principle. This relation agrees, where expected, with the
fluctuation theorem for perfectly chaotic systems. The results are compared
with some recent works in the literature.Comment: 7 pages, 1 figure; updated to take into account comments received on
the first versio
Heat and Fluctuations from Order to Chaos
The Heat theorem reveals the second law of equilibrium Thermodynamics
(i.e.existence of Entropy) as a manifestation of a general property of
Hamiltonian Mechanics and of the Ergodic Hypothesis, valid for 1 as well as
degrees of freedom systems, {\it i.e.} for simple as well as very
complex systems, and reflecting the Hamiltonian nature of the microscopic
motion. In Nonequilibrium Thermodynamics theorems of comparable generality do
not seem to be available. Yet it is possible to find general, model
independent, properties valid even for simple chaotic systems ({\it i.e.} the
hyperbolic ones), which acquire special interest for large systems: the Chaotic
Hypothesis leads to the Fluctuation Theorem which provides general properties
of certain very large fluctuations and reflects the time-reversal symmetry.
Implications on Fluids and Quantum systems are briefly hinted. The physical
meaning of the Chaotic Hypothesis, of SRB distributions and of the Fluctuation
Theorem is discussed in the context of their interpretation and relevance in
terms of Coarse Grained Partitions of phase space. This review is written
taking some care that each section and appendix is readable either
independently of the rest or with only few cross references.Comment: 1) added comment at the end of Sec. 1 to explain the meaning of the
title (referee request) 2) added comment at the end of Sec. 17 (i.e. appendix
A4) to refer to papers related to the ones already quoted (referee request
Nonlinear waves in Newton's cradle and the discrete p-Schroedinger equation
We study nonlinear waves in Newton's cradle, a classical mechanical system
consisting of a chain of beads attached to linear pendula and interacting
nonlinearly via Hertz's contact forces. We formally derive a spatially discrete
modulation equation, for small amplitude nonlinear waves consisting of slow
modulations of time-periodic linear oscillations. The fully-nonlinear and
unilateral interactions between beads yield a nonstandard modulation equation
that we call the discrete p-Schroedinger (DpS) equation. It consists of a
spatial discretization of a generalized Schroedinger equation with p-Laplacian,
with fractional p>2 depending on the exponent of Hertz's contact force. We show
that the DpS equation admits explicit periodic travelling wave solutions, and
numerically find a plethora of standing wave solutions given by the orbits of a
discrete map, in particular spatially localized breather solutions. Using a
modified Lyapunov-Schmidt technique, we prove the existence of exact periodic
travelling waves in the chain of beads, close to the small amplitude modulated
waves given by the DpS equation. Using numerical simulations, we show that the
DpS equation captures several other important features of the dynamics in the
weakly nonlinear regime, namely modulational instabilities, the existence of
static and travelling breathers, and repulsive or attractive interactions of
these localized structures
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