2,858 research outputs found
Weil-Petersson Teichm\"uller space
The paper presents some recent results on the Weil-Petersson geometry theory
of the universal Teichm\"uller space, a topic which is important in
Teichm\"uller theory and has wide applications to various areas such as
mathematical physics, differential equation and computer vision.
\noindent (1) It is shown that a sense-preserving homeomorphism on the
unit circle belongs to the Weil-Petersson class, namely, can be extended to
a quasiconformal mapping to the unit disk whose Beltrami coefficient is
squarely integrable in the Poincar\'e metric if and only if is absolutely
continuous such that belongs to the Sobolev class .
This solves an open problem posed by Takhtajan-Teo [TT2] in 2006 and
investigated later by Figalli [Fi], Gay-Balmaz-Marsden-Ratiu ([GMR], [GR]) and
others.
\noindent The intrinsic characterization (1) of the Weil-Petersson class has
the following applications which are also explored in this paper:
\noindent (2) It is proved that there exists a quasisymmetric homeomorphism
of the Weil-Petersson class which belongs neither to the Sobolev class
nor to the Lipschitz class , which was conjectured
very recently by Gay-Balmaz-Ratiu [GR] when studying the classical
Euler-Poincar\'e equation in the new setting that the involved sense-preserving
homeomorphisms on the unit circle belong to the Weil-Petersson class.
\noindent (3) It is proved that the flows of the vector fields
on the unit circle are contained in the Weil-Petersson class, which was also
conjectured by Gay-Balmaz-Ratiu [GR] during their above mentioned research.
\noindent (4) A new metric is introduced on the Weil-Petersson Teichm\"uller
space and is shown to be topologically equivalent to the Weil-Petersson metric.Comment: 33 page
Dimensional study of the dynamical arrest in a random Lorentz gas
The random Lorentz gas is a minimal model for transport in heterogeneous
media. Upon increasing the obstacle density, it exhibits a growing subdiffusive
transport regime and then a dynamical arrest. Here, we study the dimensional
dependence of the dynamical arrest, which can be mapped onto the void
percolation transition for Poisson-distributed point obstacles. We numerically
determine the arrest in dimensions d=2-6. Comparing the results with standard
mode-coupling theory reveals that the dynamical theory prediction grows
increasingly worse with . In an effort to clarify the origin of this
discrepancy, we relate the dynamical arrest in the RLG to the dynamic glass
transition of the infinite-range Mari-Kurchan model glass former. Through a
mixed static and dynamical analysis, we then extract an improved dimensional
scaling form as well as a geometrical upper bound for the arrest. The results
suggest that understanding the asymptotic behavior of the random Lorentz gas
may be key to surmounting fundamental difficulties with the mode-coupling
theory of glasses.Comment: 9 pages, 6 figure
Nucleation in binary polymer blends: Effects of foreign mesoscopic spherical particles
We study nucleation in binary polymer blends in the presence of mesoscopic spherical particles using self-consistent field theory, considering both heterogeneous and homogeneous nucleation mechanisms. Heterogeneous nucleation is found to be highly sensitive to surface selectivity and particle size, with rather subtle dependence on the particle size. Particles that preferentially adsorb the nucleating species generally favor heterogeneous nucleation. For sufficiently strong adsorption, barrierless nucleation is possible. By comparing the free energy barrier for homogeneous and heterogeneous nucleation, we construct a kinetic phase diagram
Model of random packings of different size balls
We develop a model to describe the properties of random assemblies of
polydisperse hard spheres. We show that the key features to describe the system
are (i) the dependence between the free volume of a sphere and the various
coordination numbers between the species, and (ii) the dependence of the
coordination numbers with the concentration of species; quantities that are
calculated analytically. The model predicts the density of random close packing
and random loose packing of polydisperse systems for a given distribution of
ball size and describes packings for any interparticle friction coefficient.
The formalism allows to determine the optimal packing over different
distributions and may help to treat packing problems of non-spherical particles
which are notoriously difficult to solve.Comment: 6 pages, 6 figure
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