1,304 research outputs found
Algebraic methods for parameterized codes and invariants of vanishing ideals over finite fields
Let K be a finite field with q elements and let X be a subset of a projective
space P^{s-1}, over the field K, which is parameterized by Laurent monomials.
Let I(X) be the vanishing ideal of X. Some of the main contributions of this
paper are in determining the structure of I(X) and some of their invariants. It
is shown that I(X) is a lattice ideal. We introduce the notion of a
parameterized code arising from X and present algebraic methods to compute and
study its dimension, length and minimum distance. For a parameterized code
arising from a connected graph we are able to compute its length and to make
our results more precise. If the graph is non-bipartite, we show an upper bound
for the minimum distance. We also study the underlying geometric structure of
X.Comment: Finite Fields Appl., to appea
Cohen-Macaulay graphs and face vectors of flag complexes
We introduce a construction on a flag complex that, by means of modifying the
associated graph, generates a new flag complex whose -factor is the face
vector of the original complex. This construction yields a vertex-decomposable,
hence Cohen-Macaulay, complex. From this we get a (non-numerical)
characterisation of the face vectors of flag complexes and deduce also that the
face vector of a flag complex is the -vector of some vertex-decomposable
flag complex. We conjecture that the converse of the latter is true and prove
this, by means of an explicit construction, for -vectors of Cohen-Macaulay
flag complexes arising from bipartite graphs. We also give several new
characterisations of bipartite graphs with Cohen-Macaulay or Buchsbaum
independence complexes.Comment: 14 pages, 3 figures; major updat
Casimir-like tunneling-induced electronic forces
We study the quantum forces that act between two nearby conductors due to
electronic tunneling. We derive an expression for these forces by calculating
the flux of momentum arising from the overlap of evanescent electronic fields.
Our result is written in terms of the electronic reflection amplitudes of the
conductors and it has the same structure as Lifshitz's formula for the
electromagnetically mediated Casimir forces. We evaluate the tunneling force
between two semiinfinite conductors and between two thin films separated by an
insulating gap. We discuss some applications of our results.Comment: 8 pages, 3 figs, submitted to Proc. of QFEXT'05, to be published in
J. Phys.
Axially symmetric membranes with polar tethers
Axially symmetric equilibrium configurations of the conformally invariant
Willmore energy are shown to satisfy an equation that is two orders lower in
derivatives of the embedding functions than the equilibrium shape equation, not
one as would be expected on the basis of axial symmetry. Modulo a translation
along the axis, this equation involves a single free parameter c.If c\ne 0, a
geometry with spherical topology will possess curvature singularities at its
poles. The physical origin of the singularity is identified by examining the
Noether charge associated with the translational invariance of the energy; it
is consistent with an external axial force acting at the poles. A one-parameter
family of exact solutions displaying a discocyte to stomatocyte transition is
described.Comment: 13 pages, extended and revised version of Non-local sine-Gordon
equation for the shape of axi-symmetric membrane
High-multipolar effects on the Casimir force: the non-retarded limit
We calculate exactly the Casimir force or dispersive force, in the
non-retarded limit, between a spherical nanoparticle and a substrate beyond the
London's or dipolar approximation. We find that the force is a non-monotonic
function of the distance between the sphere and the substrate, such that, it is
enhanced by several orders of magnitude as the sphere approaches the substrate.
Our results do not agree with previous predictions like the Proximity theorem
approach.Comment: 7 pages including 2 figures. Submitted to Europjysics Letter
Generally covariant state-dependent diffusion
Statistical invariance of Wiener increments under SO(n) rotations provides a
notion of gauge transformation of state-dependent Brownian motion. We show that
the stochastic dynamics of non gauge-invariant systems is not unambiguously
defined. They typically do not relax to equilibrium steady states even in the
absence of extenal forces. Assuming both coordinate covariance and gauge
invariance, we derive a second-order Langevin equation with state-dependent
diffusion matrix and vanishing environmental forces. It differs from previous
proposals but nevertheless entails the Einstein relation, a Maxwellian
conditional steady state for the velocities, and the equipartition theorem. The
over-damping limit leads to a stochastic differential equation in state space
that cannot be interpreted as a pure differential (Ito, Stratonovich or else).
At odds with the latter interpretations, the corresponding Fokker-Planck
equation admits an equilibrium steady state; a detailed comparison with other
theories of state-dependent diffusion is carried out. We propose this as a
theory of diffusion in a heat bath with varying temperature. Besides
equilibrium, a crucial experimental signature is the non-uniform steady spatial
distribution.Comment: 24 page
Matrix geometries and Matrix Models
We study a two parameter single trace 3-matrix model with SO(3) global
symmetry. The model has two phases, a fuzzy sphere phase and a matrix phase.
Configurations in the matrix phase are consistent with fluctuations around a
background of commuting matrices whose eigenvalues are confined to the interior
of a ball of radius R=2.0. We study the co-existence curve of the model and
find evidence that it has two distinct portions one with a discontinuous
internal energy yet critical fluctuations of the specific heat but only on the
low temperature side of the transition and the other portion has a continuous
internal energy with a discontinuous specific heat of finite jump. We study in
detail the eigenvalue distributions of different observables.Comment: 20 page
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