Abstract kinetic equations with positive collision operators

We consider "forward-backward" parabolic equations in the abstract form $Jd \psi / d x + L \psi = 0$, $0< x < \tau \leq \infty$, where $J$ and $L$ are operators in a Hilbert space $H$ such that $J=J^*=J^{-1}$, $L=L^* \geq 0$, and $\ker L = 0$. The following theorem is proved: if the operator $B=JL$ is similar to a self-adjoint operator, then associated half-range boundary problems have unique solutions. We apply this theorem to corresponding nonhomogeneous equations, to the time-independent Fokker-Plank equation $\mu \frac {\partial \psi}{\partial x} (x,\mu) = b(\mu) \frac {\partial^2 \psi}{\partial \mu^2} (x, \mu)$, $0<x<\tau$, $\mu \in \R$, as well as to other parabolic equations of the "forward-backward" type. The abstract kinetic equation $T d \psi/dx = - A \psi (x) + f(x)$, where $T=T^*$ is injective and $A$ satisfies a certain positivity assumption, is considered also.Comment: 20 pages, LaTeX2e, version 2, references have been added, changes in the introductio

Reprint of Testing scattering matrices: a compendium of recipes

Scattering matrices describe the transformation of the Stokes parameters of a beam of radiation upon scattering of that beam. The problems of testing scattering matrices for scattering by one particle and for single scattering by an assembly of particles are addressed. The treatment concerns arbitrary particles, orientations and scattering geometries. A synopsis of tests that appear to be the most useful ones from a practical point of view is presented. Special attention is given to matrices with uncertainties due, e.g., to experimental errors. In particular, it is shown how a matrix E(mod) can be constructed which is closest (in the sense of the Frobenius norm) to a given real 4 x 4 matrix E such that E(mod) is a proper scattering matrix of one particle or of an assembly of particles, respectively, Criteria for the rejection of E are also discussed. To illustrate the theoretical treatment a practical example is treated. Finally, it is shown that all results given for scattering matrices of one particle are applicable for all pure Mueller matrices, while all results for scattering matrices of assemblies of particles hold for sums of pure Mueller matrices