8,250 research outputs found
Tangent unit-vector fields: nonabelian homotopy invariants and the Dirichlet energy
Let O be a closed geodesic polygon in S 2 . Maps from O into S 2 are said to satisfy tangent boundary conditions if the edges of O are mapped into the geodesics which contain them. Taking O to be an octant of S 2 , we compute the infimum Dirichlet energy, E(H), for continuous maps satisfying tangent boundary conditions of arbitrary homotopy type H. The expression for E(H) involves a topological invariant – the spelling length – associated with the (nonabelian) fundamental group of the n-times punctured two-sphere, π1 (S 2 − {s1 , . . . , sn }, ∗). The lower bound for E(H) is obtained from combinatorial group theory arguments, while the upper bound is obtained by constructing explicit representatives which, on all but an arbitrarily small subset of O, are alternatively locally conformal or anticonformal. For conformal and anticonformal classes (classes containing wholly conformal and anticonformal representatives respectively), the expression for E(H) reduces to a previous result involving the degrees of a set of regular values s1 , . . . , sn in the target S 2 space. These degrees may be viewed as invariants associated with the abelianization of π1 (S 2 − {s1 , . . . , sn }, ∗). For nonconformal classes, however, E(H) may be strictly greater than the abelian bound. This stems from the fact that, for nonconformal maps, the number of preimages of certain regular values may necessarily be strictly greater than the absolute value of their degrees.\ud
\ud
This work is motivated by the theoretical modelling of nematic liquid crystals in confined polyhedral geometries. The results imply new lower and upper bounds for the Dirichlet energy (one-constant Oseen-Frank energy) of reflection-symmetric tangent unit-vector fields in a rectangular prism
Lower bound for energies of harmonic tangent unit-vector fields on convex polyhedra
We derive a lower bound for energies of harmonic maps of convex polyhedra in
to the unit sphere with tangent boundary conditions on the
faces. We also establish that maps, satisfying tangent boundary
conditions, are dense with respect to the Sobolev norm, in the space of
continuous tangent maps of finite energy.Comment: Acknowledgment added, typos removed, minor correction
Maximum Distance Between the Leader and the Laggard for Three Brownian Walkers
We consider three independent Brownian walkers moving on a line. The process
terminates when the left-most walker (the `Leader') meets either of the other
two walkers. For arbitrary values of the diffusion constants D_1 (the Leader),
D_2 and D_3 of the three walkers, we compute the probability distribution
P(m|y_2,y_3) of the maximum distance m between the Leader and the current
right-most particle (the `Laggard') during the process, where y_2 and y_3 are
the initial distances between the leader and the other two walkers. The result
has, for large m, the form P(m|y_2,y_3) \sim A(y_2,y_3) m^{-\delta}, where
\delta = (2\pi-\theta)/(\pi-\theta) and \theta =
cos^{-1}(D_1/\sqrt{(D_1+D_2)(D_1+D_3)}. The amplitude A(y_2,y_3) is also
determined exactly
Spin-glass-like state in GdCu: role of phase separation and magnetic frustration
We report investigations on the ground state magnetic properties of
intermetallic compound GdCu through dc magnetization measurements. GdCu
undergoes first order martensitic type structural transition over a wide
temperature window of coexisting phases. The high temperature cubic and the low
temperature orthorhombic phases have different magnetic character and they show
antiferromagnetic and helimagnetic orderings below 145 K and 45 K respectively.
We observe clear signature of a glassy magnetic phase below the helimagnetic
ordering temperature, which is marked by thermomagnetic irreversibility, aging
and memory effects. The glassy magnetic phase in GdCu is found to be rather
intriguing with its origin lies in the interfacial frustration due to distinct
magnetic character of the coexisting phases.Comment: Physical Review B 83, 134427 (2011
Universal Asymptotic Statistics of Maximal Relative Height in One-dimensional Solid-on-solid Models
We study the probability density function of the maximum relative
height in a wide class of one-dimensional solid-on-solid models of finite
size . For all these lattice models, in the large limit, a central limit
argument shows that, for periodic boundary conditions, takes a
universal scaling form , with the width of the fluctuating interface and the Airy
distribution function. For one instance of these models, corresponding to the
extremely anisotropic Ising model in two dimensions, this result is obtained by
an exact computation using transfer matrix technique, valid for any .
These arguments and exact analytical calculations are supported by numerical
simulations, which show in addition that the subleading scaling function is
also universal, up to a non universal amplitude, and simply given by the
derivative of the Airy distribution function .Comment: 13 pages, 4 figure
- …