15 research outputs found
Director configuration of planar solitons in nematic liquid crystals
The director configuration of disclination lines in nematic liquid crystals
in the presence of an external magnetic field is evaluated. Our method is a
combination of a polynomial expansion for the director and of further
analytical approximations which are tested against a numerical shooting method.
The results are particularly simple when the elastic constants are equal, but
we discuss the general case of elastic anisotropy. The director field is
continuous everywhere apart from a straight line segment whose length depends
on the value of the magnetic field. This indicates the possibility of an
elongated defect core for disclination lines in nematics due to an external
magnetic field.Comment: 12 pages, Revtex, 8 postscript figure
Topological defects in spinor condensates
We investigate the structure of topological defects in the ground states of
spinor Bose-Einstein condensates with spin F=1 or F=2. The type and number of
defects are determined by calculating the first and second homotopy groups of
the order-parameter space. The order-parameter space is identified with a set
of degenerate ground state spinors. Because the structure of the ground state
depends on whether or not there is an external magnetic field applied to the
system, defects are sensitive to the magnetic field. We study both cases and
find that the defects in zero and non-zero field are different.Comment: 10 pages, 1 figure. Published versio
Subset currents on free groups
We introduce and study the space of \emph{subset currents} on the free group
. A subset current on is a positive -invariant locally finite
Borel measure on the space of all closed subsets of consisting of at least two points. While ordinary geodesic currents
generalize conjugacy classes of nontrivial group elements, a subset current is
a measure-theoretic generalization of the conjugacy class of a nontrivial
finitely generated subgroup in , and, more generally, in a word-hyperbolic
group. The concept of a subset current is related to the notion of an
"invariant random subgroup" with respect to some conjugacy-invariant
probability measure on the space of closed subgroups of a topological group. If
we fix a free basis of , a subset current may also be viewed as an
-invariant measure on a "branching" analog of the geodesic flow space for
, whose elements are infinite subtrees (rather than just geodesic lines)
of the Cayley graph of with respect to .Comment: updated version; to appear in Geometriae Dedicat
Property (T) and rigidity for actions on Banach spaces
We study property (T) and the fixed point property for actions on and
other Banach spaces. We show that property (T) holds when is replaced by
(and even a subspace/quotient of ), and that in fact it is
independent of . We show that the fixed point property for
follows from property (T) when 1
. For simple Lie groups and their lattices, we prove that the fixed point property for holds for any if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive Banach spaces.Comment: Many minor improvement