51,373 research outputs found
Self-averaging of Wigner transforms in random media
We establish the self-averaging properties of the Wigner transform of a
mixture of states in the regime when the correlation length of the random
medium is much longer than the wave length but much shorter than the
propagation distance. The main ingredients in the proof are the error estimates
for the semiclassical approximation of the Wigner transform by the solution of
the Liouville equations, and the limit theorem for two-particle motion along
the characteristics of the Liouville equations. The results are applied to a
mathematical model of the time-reversal experiments for the acoustic waves, and
self-averaging properties of the re-transmitted wave are proved
Beyond Relativism? Re-engaging Wittgenstein
Relativism is the view that there are as many
worlds as there are ways of thinking and expressing the
worlds that are expressed. That is to say, things are
related to the ways in which we express them. Thus
philosophers assert that the way we express our thoughts
in language even affects the way we perceive the world.
Relativism is a reaction against the view that there is one
and only one way of describing the world. Therefore,
relativists argue that the different conceptual abilities and
habits are liable to result in different ways of seeing the
world
Existence of radial solution for a quasilinear equation with singular nonlinearity
We prove that the equation \begin{eqnarray*} -\Delta_p u =\lambda\Big(
\frac{1} {u^\delta} + u^q + f(u)\Big)\;\text{ in } \, B_R(0) u =0 \,\text{ on}
\; \partial B_R(0), \quad u>0 \text{ in } \, B_R(0) \end{eqnarray*} admits a
weak radially symmetric solution for sufficiently small,
and . We achieve this by combining a blow-up
argument and a Liouville type theorem to obtain a priori estimates for the
regularized problem. Using a variant of a theorem due to Rabinowitz we derive
the solution for the regularized problem and then pass to the limit.Comment: 16 page
Cauchy problem for Ultrasound Modulated EIT
Ultrasound modulation of electrical or optical properties of materials offers
the possibility to devise hybrid imaging techniques that combine the high
electrical or optical contrast observed in many settings of interest with the
high resolution of ultrasound. Mathematically, these modalities require that we
reconstruct a diffusion coefficient for , a bounded domain
in \Rm^n, from knowledge of for , where
is the solution to the elliptic equation in
with on .
This inverse problem may be recast as a nonlinear equation, which formally
takes the form of a 0-Laplacian. Whereas Laplacians with are
well-studied variational elliptic non-linear equations, is a limiting
case with a convex but not strictly convex functional, and the case
admits a variational formulation with a functional that is not convex. In this
paper, we augment the equation for the 0-Laplacian with full Cauchy data at the
domain's boundary, which results in a, formally overdetermined, nonlinear
hyperbolic equation.
The paper presents existence, uniqueness, and stability results for the
Cauchy problem of the 0-Laplacian. In general, the diffusion coefficient
can be stably reconstructed only on a subset of described as
the domain of influence of the space-like part of the boundary for
an appropriate Lorentzian metric. Global reconstructions for specific
geometries or based on the construction of appropriate complex geometric optics
solutions are also analyzed.Comment: 26 pages, 6 figure
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