472 research outputs found
Geometry of lipid vesicle adhesion
The adhesion of a lipid membrane vesicle to a fixed substrate is examined
from a geometrical point of view. This vesicle is described by the Helfrich
hamiltonian quadratic in mean curvature; it interacts by contact with the
substrate, with an interaction energy proportional to the area of contact. We
identify the constraints on the geometry at the boundary of the shared surface.
The result is interpreted in terms of the balance of the force normal to this
boundary. No assumptions are made either on the symmetry of the vesicle or on
that of the substrate. The strong bonding limit as well as the effect of
curvature asymmetry on the boundary are discussed.Comment: 7 pages, some major changes in sections III and IV, version published
in Physical Review
The Finite Field Kakeya Problem
A Besicovitch set in AG(n,q) is a set of points containing a line in every
direction. The Kakeya problem is to determine the minimal size of such a set.
We solve the Kakeya problem in the plane, and substantially improve the known
bounds for n greater than 4.Comment: 13 page
Molecular Dynamics Study of the Nematic-Isotropic Interface
We present large-scale molecular dynamics simulations of a nematic-isotropic
interface in a system of repulsive ellipsoidal molecules, focusing in
particular on the capillary wave fluctuations of the interfacial position. The
interface anchors the nematic phase in a planar way, i.e., the director aligns
parallel to the interface. Capillary waves in the direction parallel and
perpendicular to the director are considered separately. We find that the
spectrum is anisotropic, the amplitudes of capillary waves being larger in the
direction perpendicular to the director. In the long wavelength limit, however,
the spectrum becomes isotropic and compares well with the predictions of a
simple capillary wave theory.Comment: to appear in Phys. Rev.
A Hilton-Milner theorem for vector spaces
We show for k = 2 that if q = 3 and n = 2k + 1, or q = 2 and n = 2k + 2, then any intersecting family F of k-subspaces of an n-dimensional vector space over GF(q) with nF¿F F = 0 has size at most (formula). This bound is sharp as is shown by Hilton-Milner type families. As an application of this result, we determine the chromatic number of the corresponding q-Kneser graphs
The Effect of Thermal Fluctuations on Schulman Area Elasticity
We study the elastic properties of a two-dimensional fluctuating surface
whose area density is allowed to deviate from its optimal (Schulman) value. The
behavior of such a surface is determined by an interplay between the
area-dependent elastic energy, the curvature elasticity, and the entropy. We
identify three different elastic regimes depending on the ratio
between the projected (frame) and the saturated areas. We show that thermal
fluctuations modify the elastic energy of stretched surfaces (),
and dominate the elastic energy of compressed surfaces (). When
the elastic energy is not much affected by the fluctuations; the
frame area at which the surface tension vanishes becomes smaller than and
the area elasticity modulus increases.Comment: 12 pages, to appear in Euro. Phys. J.
Good Random Matrices over Finite Fields
The random matrix uniformly distributed over the set of all m-by-n matrices
over a finite field plays an important role in many branches of information
theory. In this paper a generalization of this random matrix, called k-good
random matrices, is studied. It is shown that a k-good random m-by-n matrix
with a distribution of minimum support size is uniformly distributed over a
maximum-rank-distance (MRD) code of minimum rank distance min{m,n}-k+1, and
vice versa. Further examples of k-good random matrices are derived from
homogeneous weights on matrix modules. Several applications of k-good random
matrices are given, establishing links with some well-known combinatorial
problems. Finally, the related combinatorial concept of a k-dense set of m-by-n
matrices is studied, identifying such sets as blocking sets with respect to
(m-k)-dimensional flats in a certain m-by-n matrix geometry and determining
their minimum size in special cases.Comment: 25 pages, publishe
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