646 research outputs found
Optimal Estimates for the Electric Field in Two-Dimensions
The purpose of this paper is to set out optimal gradient estimates for
solutions to the isotropic conductivity problem in the presence of adjacent
conductivity inclusions as the distance between the inclusions goes to zero and
their conductivities degenerate. This difficult question arises in the study of
composite media. Frequently in composites, the inclusions are very closely
spaced and may even touch. It is quite important from a practical point of view
to know whether the electric field (the gradient of the potential) can be
arbitrarily large as the inclusions get closer to each other or to the boundary
of the background medium.
In this paper, we establish both upper and lower bounds on the electric field
in the case where two circular conductivity inclusions are very close but not
touching. We also obtain such bounds when a circular inclusion is very close to
the boundary of a circular domain which contains the inclusion. The novelty of
these estimates, which improve and make complete our earlier results published
in Math. Ann., is that they give an optimal information about the blow-up of
the electric field as the conductivities of the inclusions degenerate.Comment: 26 page
Robust Cardiac Motion Estimation using Ultrafast Ultrasound Data: A Low-Rank-Topology-Preserving Approach
Cardiac motion estimation is an important diagnostic tool to detect heart
diseases and it has been explored with modalities such as MRI and conventional
ultrasound (US) sequences. US cardiac motion estimation still presents
challenges because of the complex motion patterns and the presence of noise. In
this work, we propose a novel approach to estimate the cardiac motion using
ultrafast ultrasound data. -- Our solution is based on a variational
formulation characterized by the L2-regularized class. The displacement is
represented by a lattice of b-splines and we ensure robustness by applying a
maximum likelihood type estimator. While this is an important part of our
solution, the main highlight of this paper is to combine a low-rank data
representation with topology preservation. Low-rank data representation
(achieved by finding the k-dominant singular values of a Casorati Matrix
arranged from the data sequence) speeds up the global solution and achieves
noise reduction. On the other hand, topology preservation (achieved by
monitoring the Jacobian determinant) allows to radically rule out distortions
while carefully controlling the size of allowed expansions and contractions.
Our variational approach is carried out on a realistic dataset as well as on a
simulated one. We demonstrate how our proposed variational solution deals with
complex deformations through careful numerical experiments. While maintaining
the accuracy of the solution, the low-rank preprocessing is shown to speed up
the convergence of the variational problem. Beyond cardiac motion estimation,
our approach is promising for the analysis of other organs that experience
motion.Comment: 15 pages, 10 figures, Physics in Medicine and Biology, 201
Mathematical analysis of plasmonic nanoparticles: the scalar case
Localized surface plasmons are charge density oscillations confined to metallic nanoparticles. Excitation of localized surface plasmons by an electromagnetic field at an incident wavelength where resonance occurs results in a strong light scattering and an enhancement of the local electromagnetic fields. This paper is devoted to the mathematical modeling of plasmonic nanoparticles. Its aim is threefold: (i) to mathematically define the notion of plasmonic resonance and to analyze the shift and broadening of the plasmon resonance with changes in size and shape of the nanoparticles; (ii) to study the scattering and absorption enhancements by plasmon resonant nanoparticles and express them in terms of the polarization tensor of the nanoparticle. Optimal bounds on the enhancement factors are also derived; (iii) to show, by analyzing the imaginary part of the Green function, that one can achieve super-resolution and super-focusing using plasmonic nanoparticles. For simplicity, the Helmholtz equation is used to model electromagnetic wave propagation
On the spectrum of a waveguide with periodic cracks
The spectral problem on a periodic domain with cracks is studied. An
asymptotic form of dispersion relations is calculated under assumption that the
opening of the cracks is small
The essential spectrum of the NeumannâPoincarĂ© operator on a domain with corners
Exploiting the homogeneous structure of a wedge in the complex plane, we compute the spectrum of the anti-linear Ahlfors-Beurling transform acting on the associated Bergman space. Consequently, the similarity equivalence between the Ahlfors--Beurling transform and the Neumann-Poincare operator provides the spectrum of the latter integral operator on a wedge. A localization technique and conformal mapping lead to the first complete description of the essential spectrum of the Neumann-Poincare operator on a planar domain with corners, with respect to the energy norm of the associated harmonic field
Enhancement of near-cloaking. Part II: the Helmholtz equation
The aim of this paper is to extend the method of improving cloaking
structures in the conductivity to scattering problems. We construct very
effective near-cloaking structures for the scattering problem at a fixed
frequency. These new structures are, before using the transformation optics,
layered structures and are designed so that their first scattering coefficients
vanish. Inside the cloaking region, any target has near-zero scattering cross
section for a band of frequencies. We analytically show that our new
construction significantly enhances the cloaking effect for the Helmholtz
equation.Comment: 16pages, 12 fugure
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