2,598 research outputs found
Approximation learning methods of Harmonic Mappings in relation to Hardy Spaces
A new Hardy space Hardy space approach of Dirichlet type problem based on
Tikhonov regularization and Reproducing Hilbert kernel space is discussed in
this paper, which turns out to be a typical extremal problem located on the
upper upper-high complex plane. If considering this in the Hardy space, the
optimization operator of this problem will be highly simplified and an
efficient algorithm is possible. This is mainly realized by the help of
reproducing properties of the functions in the Hardy space of upper-high
complex plane, and the detail algorithm is proposed. Moreover, harmonic
mappings, which is a significant geometric transformation, are commonly used in
many applications such as image processing, since it describes the energy
minimization mappings between individual manifolds. Particularly, when we focus
on the planer mappings between two Euclid planer regions, the harmonic mappings
are exist and unique, which is guaranteed solidly by the existence of harmonic
function. This property is attractive and simulation results are shown in this
paper to ensure the capability of applications such as planer shape distortion
and surface registration.Comment: 2016 3rd International Conference on Informative and Cybernetics for
Computational Social Systems (ICCSS
The Theory of Quasiconformal Mappings in Higher Dimensions, I
We present a survey of the many and various elements of the modern
higher-dimensional theory of quasiconformal mappings and their wide and varied
application. It is unified (and limited) by the theme of the author's
interests. Thus we will discuss the basic theory as it developed in the 1960s
in the early work of F.W. Gehring and Yu G. Reshetnyak and subsequently explore
the connections with geometric function theory, nonlinear partial differential
equations, differential and geometric topology and dynamics as they ensued over
the following decades. We give few proofs as we try to outline the major
results of the area and current research themes. We do not strive to present
these results in maximal generality, as to achieve this considerable technical
knowledge would be necessary of the reader. We have tried to give a feel of
where the area is, what are the central ideas and problems and where are the
major current interactions with researchers in other areas. We have also added
a bit of history here and there. We have not been able to cover the many recent
advances generalising the theory to mappings of finite distortion and to
degenerate elliptic Beltrami systems which connects the theory closely with the
calculus of variations and nonlinear elasticity, nonlinear Hodge theory and
related areas, although the reader may see shadows of this aspect in parts
Deformations of Annuli on Riemann surfaces with Smallest Mean Distortion
Let and be two circular annuli and let be a radial metric
defined in the annulus . Consider the class of
harmonic mappings between and . It is proved recently by
Iwaniec, Kovalev and Onninen that, if (i.e. if is Euclidean
metric) then is not empty if and only if there holds the
Nitsche condition (and thus is proved the J. C. C. Nitsche conjecture). In this
paper we formulate an condition (which we call Nitsche conjecture) with
corresponds to and define Nitsche harmonic maps. We
determine the extremal mappings with smallest mean distortion for mappings of
annuli w.r. to the metric . As a corollary, we find that Nitsche
harmonic maps are Dirichlet minimizers among all homeomorphisms .
However, outside the -Nitsche condition of the modulus of the annuli,
within the class of homeomorphisms, no such energy minimizers exist. % However,
%outside the Nitsche range of the modulus of the annuli, %within the
class of homeomorphisms, no such energy minimizers exist. This extends some
recent results of Astala, Iwaniec and Martin (ARMA, 2010) where it is
considered the case and .Comment: Some misprints are corrected in this version (see Lemma~5.1
Existence of energy-minimal diffeomorphisms between doubly connected domains
The paper establishes the existence of homeomorphisms between two planar
domains that minimize the Dirichlet energy. Specifically, among all
homeomorphisms f : R -> R* between bounded doubly connected domains such that
Mod (R) < Mod (R*) there exists, unique up to conformal authomorphisms of R, an
energy-minimal diffeomorphism. No boundary conditions are imposed on f.
Although any energy-minimal diffeomorphism is harmonic, our results underline
the major difference between the existence of harmonic diffeomorphisms and the
existence of the energy-minimal diffeomorphisms. The existence of globally
invertible energy-minimal mappings is of primary pursuit in the mathematical
models of nonlinear elasticity and is also of interest in computer graphics.Comment: 34 pages, no figure
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