228 research outputs found
Smectic Phases with Cubic Symmetry: The Splay Analog of the Blue Phase
We report on a construction for smectic blue phases, which have quasi-long
range smectic translational order as well as long range cubic or hexagonal
order. Our proposed structures fill space with a combination of minimal surface
patches and cylindrical tubes. We find that for the right range of material
parameters, the favorable saddle-splay energy of these structures can stabilize
them against uniform layered structures.Comment: 4 pages, 4 eps figures, RevTe
Smectic blue phases: layered systems with high intrinsic curvature
We report on a construction for smectic blue phases, which have quasi-long
range smectic translational order as well as three dimensional crystalline
order. Our proposed structures fill space by adding layers on top of a minimal
surface, introducing either curvature or edge defects as necessary. We find
that for the right range of material parameters, the favorable saddle-splay
energy of these structures can stabilize them against uniform layered
structures. We also consider the nature of curvature frustration between mean
curvature and saddle-splay.Comment: 15 pages, 11 figure
Buckling Instabilities of a Confined Colloid Crystal Layer
A model predicting the structure of repulsive, spherically symmetric,
monodisperse particles confined between two walls is presented. We study the
buckling transition of a single flat layer as the double layer state develops.
Experimental realizations of this model are suspensions of stabilized colloidal
particles squeezed between glass plates. By expanding the thermodynamic
potential about a flat state of confined colloidal particles, we derive
a free energy as a functional of in-plane and out-of-plane displacements. The
wavevectors of these first buckling instabilities correspond to three different
ordered structures. Landau theory predicts that the symmetry of these phases
allows for second order phase transitions. This possibility exists even in the
presence of gravity or plate asymmetry. These transitions lead to critical
behavior and phases with the symmetry of the three-state and four-state Potts
models, the X-Y model with 6-fold anisotropy, and the Heisenberg model with
cubic interactions. Experimental detection of these structures is discussed.Comment: 24 pages, 8 figures on request. EF508
The dynamics of thin vibrated granular layers
We describe a series of experiments and computer simulations on vibrated
granular media in a geometry chosen to eliminate gravitationally induced
settling. The system consists of a collection of identical spherical particles
on a horizontal plate vibrating vertically, with or without a confining lid.
Previously reported results are reviewed, including the observation of
homogeneous, disordered liquid-like states, an instability to a `collapse' of
motionless spheres on a perfect hexagonal lattice, and a fluctuating,
hexagonally ordered state. In the presence of a confining lid we see a variety
of solid phases at high densities and relatively high vibration amplitudes,
several of which are reported for the first time in this article. The phase
behavior of the system is closely related to that observed in confined
hard-sphere colloidal suspensions in equilibrium, but with modifications due to
the effects of the forcing and dissipation. We also review measurements of
velocity distributions, which range from Maxwellian to strongly non-Maxwellian
depending on the experimental parameter values. We describe measurements of
spatial velocity correlations that show a clear dependence on the mechanism of
energy injection. We also report new measurements of the velocity
autocorrelation function in the granular layer and show that increased
inelasticity leads to enhanced particle self-diffusion.Comment: 11 pages, 7 figure
Monge's transport problem in the Heisenberg group
We prove the existence of solutions to Monge transport problem between two
compactly supported Borel probability measures in the Heisenberg group equipped
with its Carnot-Caratheodory distance assuming that the initial measure is
absolutely continuous with respect to the Haar measure of the group
An insight into polarization states of solid-state organic lasers
The polarization states of lasers are crucial issues both for practical
applications and fundamental research. In general, they depend in a combined
manner on the properties of the gain material and on the structure of the
electromagnetic modes. In this paper, we address this issue in the case of
solid-state organic lasers, a technology which enables to vary independently
gain and mode properties. Different kinds of resonators are investigated:
in-plane micro-resonators with Fabry-Perot, square, pentagon, stadium, disk,
and kite shapes, and external vertical resonators. The degree of polarization P
is measured in each case. It is shown that although TE modes prevail generally
(P>0), kite-shaped micro-laser generates negative values for P, i.e. a flip of
the dominant polarization which becomes mostly TM polarized. We at last
investigated two degrees of freedom that are available to tailor the
polarization of organic lasers, in addition to the pump polarization and the
resonator geometry: upon using resonant energy transfer (RET) or upon pumping
the laser dye to an higher excited state. We then demonstrate that
significantly lower P factors can be obtained.Comment: 12 pages, 12 figure
Order and Frustration in Chiral Liquid Crystals
This paper reviews the complex ordered structures induced by chirality in
liquid crystals. In general, chirality favors a twist in the orientation of
liquid-crystal molecules. In some cases, as in the cholesteric phase, this
favored twist can be achieved without any defects. More often, the favored
twist competes with applied electric or magnetic fields or with geometric
constraints, leading to frustration. In response to this frustration, the
system develops ordered structures with periodic arrays of defects. The
simplest example of such a structure is the lattice of domains and domain walls
in a cholesteric phase under a magnetic field. More complex examples include
defect structures formed in two-dimensional films of chiral liquid crystals.
The same considerations of chirality and defects apply to three-dimensional
structures, such as the twist-grain-boundary and moire phases.Comment: 39 pages, RevTeX, 14 included eps figure
Isometric group actions on Banach spaces and representations vanishing at infinity
Our main result is that the simple Lie group acts properly
isometrically on if . To prove this, we introduce property
({\BP}_0^V), for be a Banach space: a locally compact group has
property ({\BP}_0^V) if every affine isometric action of on , such
that the linear part is a -representation of , either has a fixed point
or is metrically proper. We prove that solvable groups, connected Lie groups,
and linear algebraic groups over a local field of characteristic zero, have
property ({\BP}_0^V). As a consequence for unitary representations, we
characterize those groups in the latter classes for which the first cohomology
with respect to the left regular representation on is non-zero; and we
characterize uniform lattices in those groups for which the first -Betti
number is non-zero.Comment: 28 page
First and second variation formulae for the sub-Riemannian area in three-dimensional pseudo-hermitian manifolds
We calculate the first and the second variation formula for the
sub-Riemannian area in three dimensional pseudo-hermitian manifolds. We
consider general variations that can move the singular set of a C^2 surface and
non-singular variation for C_H^2 surfaces. These formulas enable us to
construct a stability operator for non-singular C^2 surfaces and another one
for C2 (eventually singular) surfaces. Then we can obtain a necessary condition
for the stability of a non-singular surface in a pseudo-hermitian 3-manifold in
term of the pseudo-hermitian torsion and the Webster scalar curvature. Finally
we classify complete stable surfaces in the roto-traslation group RT .Comment: 36 pages. Misprints corrected. Statement of Proposition 9.8 slightly
changed and Remark 9.9 adde
Hodge Theory on Metric Spaces
Hodge theory is a beautiful synthesis of geometry, topology, and analysis,
which has been developed in the setting of Riemannian manifolds. On the other
hand, spaces of images, which are important in the mathematical foundations of
vision and pattern recognition, do not fit this framework. This motivates us to
develop a version of Hodge theory on metric spaces with a probability measure.
We believe that this constitutes a step towards understanding the geometry of
vision.
The appendix by Anthony Baker provides a separable, compact metric space with
infinite dimensional \alpha-scale homology.Comment: appendix by Anthony W. Baker, 48 pages, AMS-LaTeX. v2: final version,
to appear in Foundations of Computational Mathematics. Minor changes and
addition
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