1,072 research outputs found
All functions are (locally) -harmonic (up to a small error) - and applications
The classical and the fractional Laplacians exhibit a number of similarities,
but also some rather striking, and sometimes surprising, structural
differences.
A quite important example of these differences is that any function
(regardless of its shape) can be locally approximated by functions with locally
vanishing fractional Laplacian, as it was recently proved by Serena Dipierro,
Ovidiu Savin and myself.
This informal note is an exposition of this result and of some of its
consequences
Doubly nonlocal reaction-diffusion equation and the emergence of species
The paper is devoted to a reaction-diffusion equation with doubly nonlocal
nonlinearity arising in various applications in population dynamics. One of the
integral terms corresponds to the nonlocal consumption of resources while
another one describes reproduction with different phenotypes. Linear stability
analysis of the homogeneous in space stationary solution is carried out.
Existence of travelling waves is proved in the case of narrow kernels of the
integrals. Periodic travelling waves are observed in numerical simulations.
Existence of stationary solutions in the form of pulses is shown, and
transition from periodic waves to pulses is studied. In the applications to the
speciation theory, the results of this work signify that new species can emerge
only if they do not have common offsprings. Thus, it is shown how Darwin's
definition of species as groups of morphologically similar individuals is
related to Mayr's definition as groups of individuals that can breed only among
themselves.Comment: 15 pages, 4 figure
Image Restoration Using Joint Statistical Modeling in Space-Transform Domain
This paper presents a novel strategy for high-fidelity image restoration by
characterizing both local smoothness and nonlocal self-similarity of natural
images in a unified statistical manner. The main contributions are three-folds.
First, from the perspective of image statistics, a joint statistical modeling
(JSM) in an adaptive hybrid space-transform domain is established, which offers
a powerful mechanism of combining local smoothness and nonlocal self-similarity
simultaneously to ensure a more reliable and robust estimation. Second, a new
form of minimization functional for solving image inverse problem is formulated
using JSM under regularization-based framework. Finally, in order to make JSM
tractable and robust, a new Split-Bregman based algorithm is developed to
efficiently solve the above severely underdetermined inverse problem associated
with theoretical proof of convergence. Extensive experiments on image
inpainting, image deblurring and mixed Gaussian plus salt-and-pepper noise
removal applications verify the effectiveness of the proposed algorithm.Comment: 14 pages, 18 figures, 7 Tables, to be published in IEEE Transactions
on Circuits System and Video Technology (TCSVT). High resolution pdf version
and Code can be found at: http://idm.pku.edu.cn/staff/zhangjian/IRJSM
Visco-potential free-surface flows and long wave modelling
In a recent study [DutykhDias2007] we presented a novel visco-potential free
surface flows formulation. The governing equations contain local and nonlocal
dissipative terms. From physical point of view, local dissipation terms come
from molecular viscosity but in practical computations, rather eddy viscosity
should be used. On the other hand, nonlocal dissipative term represents a
correction due to the presence of a bottom boundary layer. Using the standard
procedure of Boussinesq equations derivation, we come to nonlocal long wave
equations. In this article we analyse dispersion relation properties of
proposed models. The effect of nonlocal term on solitary and linear progressive
waves attenuation is investigated. Finally, we present some computations with
viscous Boussinesq equations solved by a Fourier type spectral method.Comment: 29 pages, 13 figures. Some figures were updated. Revised version for
European Journal of Mechanics B/Fluids. Other author's papers can be
downloaded from http://www.lama.univ-savoie.fr/~dutyk
Density Functional Simulation of Spontaneous Formation of Vesicle in Block Copolymer Solutions
We carry out numerical simulations of vesicle formation based on the density
functional theory for block copolymer solutions. It is shown by solving the
time evolution equations for concentrations that a polymer vesicle is
spontaneously formed from the homogeneous state. The vesicle formation
mechanism obtained by our simulation agree with the results of other
simulations based on the particle models as well as experiments. By changing
parameters such as the volume fraction of polymers or the Flory-Huggins
interaction parameter between the hydrophobic subchains and solvents, we can
obtain the spherical micelles, cylindrical micelles or bilayer structures, too.
We also show that the morphological transition dynamics of the micellar
structures can be reproduced by controlling the Flory-Huggins interaction
parameter.Comment: 29 pages, 11 figures, to appear in J. Chem. Phy
Nonlinear operators on graphs via stacks
International audienceWe consider a framework for nonlinear operators on functions evaluated on graphs via stacks of level sets. We investigate a family of transformations on functions evaluated on graph which includes adaptive flat and non-flat erosions and dilations in the sense of mathematical morphology. Additionally, the connection to mean motion curvature on graphs is noted. Proposed operators are illustrated in the cases of functions on graphs, textured meshes and graphs of images
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