17,046 research outputs found
Asymptotic regime for impropriety tests of complex random vectors
Impropriety testing for complex-valued vector has been considered lately due
to potential applications ranging from digital communications to complex media
imaging. This paper provides new results for such tests in the asymptotic
regime, i.e. when the vector dimension and sample size grow commensurately to
infinity. The studied tests are based on invariant statistics named impropriety
coefficients. Limiting distributions for these statistics are derived, together
with those of the Generalized Likelihood Ratio Test (GLRT) and Roy's test, in
the Gaussian case. This characterization in the asymptotic regime allows also
to identify a phase transition in Roy's test with potential application in
detection of complex-valued low-rank subspace corrupted by proper noise in
large datasets. Simulations illustrate the accuracy of the proposed asymptotic
approximations.Comment: 11 pages, 8 figures, submitted to IEEE TS
Minkowski Tensors of Anisotropic Spatial Structure
This article describes the theoretical foundation of and explicit algorithms
for a novel approach to morphology and anisotropy analysis of complex spatial
structure using tensor-valued Minkowski functionals, the so-called Minkowski
tensors. Minkowski tensors are generalisations of the well-known scalar
Minkowski functionals and are explicitly sensitive to anisotropic aspects of
morphology, relevant for example for elastic moduli or permeability of
microstructured materials. Here we derive explicit linear-time algorithms to
compute these tensorial measures for three-dimensional shapes. These apply to
representations of any object that can be represented by a triangulation of its
bounding surface; their application is illustrated for the polyhedral Voronoi
cellular complexes of jammed sphere configurations, and for triangulations of a
biopolymer fibre network obtained by confocal microscopy. The article further
bridges the substantial notational and conceptual gap between the different but
equivalent approaches to scalar or tensorial Minkowski functionals in
mathematics and in physics, hence making the mathematical measure theoretic
method more readily accessible for future application in the physical sciences
Quantitative Anderson localization of Schr\"odinger eigenstates under disorder potentials
This paper concerns spectral properties of linear Schr\"odinger operators
under oscillatory high-amplitude potentials on bounded domains. Depending on
the degree of disorder, we prove the existence of spectral gaps amongst the
lowermost eigenvalues and the emergence of exponentially localized states. We
quantify the rate of decay in terms of geometric parameters that characterize
the potential. The proofs are based on the convergence theory of iterative
solvers for eigenvalue problems and their optimal local preconditioning by
domain decomposition.Comment: accepted for publication in M3A
Reactive-infiltration instabilities in rocks. Fracture dissolution
A reactive fluid dissolving the surface of a uniform fracture will trigger an
instability in the dissolution front, leading to spontaneous formation of
pronounced well-spaced channels in the surrounding rock matrix. Although the
underlying mechanism is similar to the wormhole instability in porous rocks
there are significant differences in the physics, due to the absence of a
steadily propagating reaction front. In previous work we have described the
geophysical implications of this instability in regard to the formation of long
conduits in soluble rocks. Here we describe a more general linear stability
analysis, including axial diffusion, transport limited dissolution, non-linear
kinetics, and a finite length system.Comment: to be published in J. Fluid. Mec
A lower bound for the principal eigenvalue of the Stokes operator in a random domain
This article is dedicated to localization of the principal eigenvalue (PE) of
the Stokes operator acting on solenoidal vector fields that vanish outside a
large random domain modeling the pore space in a cubic block of porous material
with disordered micro-structure. Its main result is an asymptotically
deterministic lower bound for the PE of the sum of a low compressibility
approximation to the Stokes operator and a small scaled random potential term,
which is applied to produce a similar bound for the Stokes PE. The arguments
are based on the method proposed by F. Merkl and M. V. W\"{u}trich for
localization of the PE of the Schr\"{o}dinger operator in a similar setting.
Some additional work is needed to circumvent the complications arising from the
restriction to divergence-free vector fields of the class of test functions in
the variational characterization of the Stokes PE.Comment: Published in at http://dx.doi.org/10.1214/07-AIHP136 the Annales de
l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques
(http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics
(http://www.imstat.org
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