10,656 research outputs found
Integral transformation solution of free-space cylindrical vector beams and prediction of modified-Bessel-Gaussian vector beams
A unified description of the free-space cylindrical vector beams is
presented, which is an integral transformation solution to the vector Helmholtz
equation and the transversality condition. The amplitude 2-form of the angular
spectrum involved in this solution can be arbitrarily chosen. When one of the
two elements is zero, we arrive at either transverse-electric or
transverse-magnetic beam mode. In the paraxial condition, this solution not
only includes the known Bessel-Gaussian vector beam and the axisymmetric
Laguerre-Gaussian vector beam that were obtained by solving the paraxial wave
equations, but also predicts two new kinds of vector beam, called the
modified-Bessel-Gaussian vector beam.Comment: 8 pages and 1 figur
Multiplying a Gaussian Matrix by a Gaussian Vector
We provide a new and simple characterization of the multivariate generalized
Laplace distribution. In particular, this result implies that the product of a
Gaussian matrix with independent and identically distributed columns by an
independent isotropic Gaussian vector follows a symmetric multivariate
generalized Laplace distribution
Signed zeros of Gaussian vector fields-density, correlation functions and curvature
We calculate correlation functions of the (signed) density of zeros of
Gaussian distributed vector fields. We are able to express correlation
functions of arbitrary order through the curvature tensor of a certain abstract
Riemann-Cartan or Riemannian manifold. As an application, we discuss one- and
two-point functions. The zeros of a two-dimensional Gaussian vector field model
the distribution of topological defects in the high-temperature phase of
two-dimensional systems with orientational degrees of freedom, such as
superfluid films, thin superconductors and liquid crystals.Comment: 14 pages, 1 figure, uses iopart.cls, improved presentation, to appear
in J. Phys.
Monte Carlo Quasi-Heatbath by approximate inversion
When sampling the distribution P(phi) ~ exp(-|A phi|^2), a global heatbath
normally proceeds by solving the linear system A phi = eta, where eta is a
normal Gaussian vector, exactly. This paper shows how to preserve the
distribution P(phi) while solving the linear system with arbitrarily low
accuracy. Generalizations are presented.Comment: 10 pages, 1 figure; typos corrected, reference added; version to
appear in Phys. Rev.
Intrinsic Gaussian Vector Fields on Manifolds
Various applications ranging from robotics to climate science require
modeling signals on non-Euclidean domains, such as the sphere. Gaussian process
models on manifolds have recently been proposed for such tasks, in particular
when uncertainty quantification is needed. In the manifold setting,
vector-valued signals can behave very differently from scalar-valued ones, with
much of the progress so far focused on modeling the latter. The former,
however, are crucial for many applications, such as modeling wind speeds or
force fields of unknown dynamical systems. In this paper, we propose novel
Gaussian process models for vector-valued signals on manifolds that are
intrinsically defined and account for the geometry of the space in
consideration. We provide computational primitives needed to deploy the
resulting Hodge-Mat\'ern Gaussian vector fields on the two-dimensional sphere
and the hypertori. Further, we highlight two generalization directions:
discrete two-dimensional meshes and "ideal" manifolds like hyperspheres, Lie
groups, and homogeneous spaces. Finally, we show that our Gaussian vector
fields constitute considerably more refined inductive biases than the extrinsic
fields proposed before
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