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Register a new userLocal recovery of the compressional and shear speeds from the hyperbolic DN map
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We study the isotropic elastic wave equation in a bounded domain with boundary. We show that local knowledge of the Dirichlet-to-Neumann map determines uniquely the speed of the p-wave locally if there is a strictly convex foliation with respect to it, and similarly for the s-wave speed.
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We show that given two hyperbolic Dirichlet to Neumann maps associated to two Riemannian metrics of a Riemannian manifold with boundary which coincide near the boundary are close then the lens data of the two metrics is the same. As a consequence, we prove uniqueness of recovery a conformal factor (sound speed) locally under some conditions on the latter.
We study the recovery of piecewise analytic density and stiffness tensor of a three-dimensional domain from the local dynamical Dirichlet-to-Neumann map. We give global uniqueness results if the medium is transversely isotropic with known axis of symmetry or orthorhombic with known symmetry planes on each subdomain. We also obtain uniqueness of a fully anisotropic stiffness tensor, assuming that it is piecewise constant and that the interfaces which separate the subdomains have curved portions. The domain partition need not to be known. Precisely, we show that a domain partition consisting of subanalytic sets is simultaneously uniquely determined.
In this paper we consider the hyperbolic-elliptic Ishimori initial-value problem. We prove that such system is locally well-posed for small data in $H^{s}$ level space, for $s> 3/2$. The new ingredient is that we develop the methods of Ionescu and Kenig cite{IK} and cite{IK2} to approach the problem in a perturbative way.
We study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity $lambdabeta e^{beta u }$, forced by an additive space-time white noise. We prove local and global well-posedness of these equations, depending on the sign of $lambda$ and the size of $beta^2 > 0$, and invariance of the associated Gibbs measures. See the abstract of the paper for a more precise abstract. (Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here.)
The Data Processing Inequality (DPI) says that the Umegaki relative entropy $S(rho||sigma) := {rm Tr}[rho(log rho - log sigma)]$ is non-increasing under the action of completely positive trace preserving (CPTP) maps. Let ${mathcal M}$ be a finite dimensional von Neumann algebra and ${mathcal N}$ a von Neumann subalgebra if it. Let ${mathcal E}_tau$ be the tracial conditional expectation from ${mathcal M}$ onto ${mathcal N}$. For density matrices $rho$ and $sigma$ in ${mathcal N}$, let $rho_{mathcal N} := {mathcal E}_tau rho$ and $sigma_{mathcal N} := {mathcal E}_tau sigma$. Since ${mathcal E}_tau$ is CPTP, the DPI says that $S(rho||sigma) geq S(rho_{mathcal N}||sigma_{mathcal N})$, and the general case is readily deduced from this. A theorem of Petz says that there is equality if and only if $sigma = {mathcal R}_rho(sigma_{mathcal N} )$, where ${mathcal R}_rho$ is the Petz recovery map, which is dual to the Accardi-Cecchini coarse graining operator ${mathcal A}_rho$ from ${mathcal M} $ to ${mathcal N} $. In it simplest form, our bound is $$S(rho||sigma) - S(rho_{mathcal N} ||sigma_{mathcal N} ) geq left(frac{1}{8pi}right)^{4} |Delta_{sigma,rho}|^{-2} | {mathcal R}_{rho_{mathcal N}} -sigma|_1^4 $$ where $Delta_{sigma,rho}$ is the relative modular operator. We also prove related results for various quasi-relative entropies. Explicitly describing the solutions set of the Petz equation $sigma = {mathcal R}_rho(sigma_{mathcal N} )$ amounts to determining the set of fixed points of the Accardi-Cecchini coarse graining map. Building on previous work, we provide a throughly detailed description of the set of solutions of the Petz equation, and obtain all of our results in a simple self, contained manner.
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