We study Laplacian transport by the Dirichlet-to-Neumann formalism in isotropic media (γ = I). Our main results concern the solution of the localisation inverse problem of absorbing domains and its relative Dirichlet-to-Neumann operator . In this paper, we define explicitly operator , and we show that Green-Ostrogradski theorem is adopted to this type of problem in three dimensional case.

Abstract:
by decomposing the laplacian on the heisenberg group into a family of parametrized partial differential operators lt ,t ∈ r \ {0}, and using parametrized fourier-wigner transforms, we give formulas and estimates for the strongly continuous one-parameter semigroup generated by lt, and the inverse of lt . using these formulas and estimates, we obtain sobolev estimates for the one-parameter semigroup and the inverse of the laplacian.

Abstract:
We reconstruct a Riemannian manifold and a Hermitian vector bundle with compatible connection from the hyperbolic Dirichlet-to-Neumann operator associated with the wave equation of the connection Laplacian. The boundary data is local and the reconstruction is up to the natural gauge transformations of the problem. As a corollary we derive an elliptic analogue of the main result which solves a Calderon problem for connections on a cylinder.

Abstract:
By decomposing the Laplacian on the Heisenberg group into a family of parametrized partial differential operators Lt ,t ∈ R {0}, and using parametrized Fourier-Wigner transforms, we give formulas and estimates for the strongly continuous one-parameter semigroup generated by Lt, and the inverse of Lt . Using these formulas and estimates, we obtain Sobolev estimates for the one-parameter semigroup and the inverse of the Laplacian. Mediante descomposición del Laplaceano sobre el grupo de Heisenberg en una familia de operadores diferenciales parciales parametrizados Lt, t ∈ R {0}, y usando transformada de Fourier-Wigner parametrizada, damos fórmulas y estimativas para la continuidad fuerte del semigrupo generado por Lt, y la inversa de Lt. Usando esas fórmulas y estimativas obtenemos estimativas de Sobolev para el semigrupo a un parámetro y la inversa del Laplaceano.

Abstract:
In this paper we develop a time reversal method for the radiative transport equation to solve two problems: an inverse problem for the recovery of an initial condition from boundary measurements, and the exact boundary controllability of the transport field with finite steering time. Absorbing and scattering effects, modeled by coefficients with low regularity, are incorporated in the formulation of these problems. This time reversal approach leads to a convergent iterative procedure to reconstruct the initial condition provided that the scattering coefficient is sufficiently small in the $L^{\infty}$ norm. Then, using duality arguments, we show that the solvability of the inverse problem leads to exact controllability of the transport field with minimum-norm control obtained constructively. The solution approach to both of these problems may have medical applications in areas such as optical imaging and optimization of radiation delivery for cancer therapy.

Abstract:
In this paper we present an iterative method, inspired by the inverse iteration with shift technique of finite linear algebra, designed to find the eigenvalues and eigenfunctions of the Laplacian with homogeneous Dirichlet boundary condition for arbitrary bounded domains $\Omega\subset R^{N}$. This method, which has a direct functional analysis approach, does not approximate the eigenvalues of the Laplacian as those of a finite linear operator. It is based on the uniform convergence away from nodal surfaces and can produce a simple and fast algorithm for computing the eigenvalues with minimal computational requirements, instead of using the ubiquitous Rayleigh quotient of finite linear algebra. Also, an alternative expression for the Rayleigh quotient in the associated infinite dimensional Sobolev space which avoids the integration of gradients is introduced and shown to be more efficient. The method can also be used in order to produce the spectral decomposition of any given function $u\in L^{2}(\Omega)$.

Abstract:
We prove uniqueness results for a Calderon type inverse problem for the Hodge Laplacian acting on graded forms on certain manifolds in three dimensions. In particular, we show that partial measurements of the relative-to-absolute or absolute-to-relative boundary value maps uniquely determine a zeroth order potential. The method is based on Carleman estimates for the Hodge Laplacian with relative or absolute boundary conditions, and on the construction of complex geometric optics solutions which reduce the Calderon type problem to a tensor tomography problem for 2-tensors. The arguments in this paper allow to establish partial data results for elliptic systems that generalize the scalar results due to Kenig-Sjostrand-Uhlmann.

Abstract:
We examine the impact of a complex absorbing potential on electron transport, both in the continuum and on a lattice. This requires the use of non-Hermitian Hamiltonians; the required formalism is briefly outlined. The lattice formulation allows us to study the interesting problem of an electron interacting with a stationary spin, and the subsequent time evolution of the electron and spin properties as the electron is absorbed after the initial interaction. Remarkably, the properties of the localized spin are affected 'at-a-distance' by the interaction of the (now entangled) electron with a complex potential.

Abstract:
We study an inverse problem for nonlinear elliptic equations modelled after the p-Laplacian. It is proved that the boundary values of a conductivity coefficient are uniquely determined from boundary measurements given by a nonlinear Dirichlet-to-Neumann map. The result is constructive and local, and gives a method for determining the coefficient at a boundary point from measurements in a small neighborhood. The proofs work with the nonlinear equation directly instead of being based on linearization. In the complex valued case we employ complex geometrical optics type solutions based on p-harmonic exponentials, while for the real case we use p-harmonic functions first introduced by Wolff.

Abstract:
We study a tight binding model including both on site disorder and coupling of the electrons to randomly oriented magnetic moments. The transport properties are calculated via the Kubo-Greenwood scheme, using the exact eigenstates of the disordered system and large system size extrapolation of the low frequency optical conductivity. We first benchmark our method in the model with only structural disorder and then use it to map out the transport regimes and metal- insulator transitions in problems involving (i) scattering from random magnetic moments, and (ii) the combined effect of structural disorder and magnetic scattering. We completely map out the dependence of the d.c conductivity on electron density (n) the structural disorder (\Delta) and the magnetic coupling (J'), and locate the insulator-metal phase boundary in the space of n-\Delta-J'. These results serve as a reference for understanding transport in systems ranging from magnetic semiconductors to double exchange `colossal magnetoresistance' systems. A brief version of this study appears in our earlier paper Europhys. Lett. vol 65, 75 (2004).