Ever since Christian Hülsmeyer showed in 1904 that one could use radio waves to detect metallic objects at a distance (the range of the first apparatus was 3000 meters), the race has been on to tease out ever more information from scattered waves. Within months of his first detection demonstration, Hülsmeyer devised a way to determine the distance to the object. At that rate of improvement one might have extrapolated to unimaginable twenty-first-century capabilities. Unfortunately, what has proved to be unimaginable is the difficulty of doing much more than the original device had already accomplished. It would seem that some forms of bionic vision have gone the way of rocket backpacks -that is, until recently.The newest book by Andreas Kirsch with coauthor Natalia Grinberg, The Factorization Method for Inverse Problems, collects over a decade of work by Kirsch and collaborators on a simple method for shape identification in inverse scattering. This book belongs to the next generation of monographs on inverse scattering following the now standard works of Colton and Kress [2] (Inverse Acoustic and Electromagnetic Scattering Theory (1998)) and Isakov [7] (Inverse Problems for Partial Differential Equations).Kirsch's factorization method arose from experimentation with noniterative inverse scattering methods that avoid the computational expense of calculating the solution to the forward problem at each iteration. Noniterative methods attack head-on the inverse problem of determining the scatterer from measured scattered fields by attempting, in principle, simply to invert the scattering operator. In most situations of interest, however, the scattering operator is nonlinear and the inverse problem is ill-posed. Early ideas focused on operator splitting techniques that decompose the scattering operator into a well-posed nonlinear part and an ill-posed linear part, each of which can be inverted stably. Another class of noniterative methods use indicator functions to detect the inconsistency or unsolvability of an easily computed auxiliary problem parameterized by points in space. The shape and location of the object is then determined by those points where the auxiliary problem is solvable. This latter generation of techniques, of which the factorization method is one, is both stable and computationally fast. What separates the factorization method from most of the other noniterative techniques is that it is mathematically complete: the computable criterion for determining the shape and location of the scatterers is both sufficient and necessary while most other techniques rely on only sufficient criteria.The book consists of seven chapters treating the application of the factorization method for, respectively: simplest cases (namely where the far field operator is normal), refinements for more complicated settings (namely, where the far field operator is not normal), so-called mixed boundary value scattering problems, scattering from inhomogeneous media, Maxwell's equations, impedance tomography, and finally a short ...

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A simple modification to Einstein's theory of gravity in terms of a non-Riemannian connection is examined. A new tensor-variational approach yields field equations that possess a covariance similar to the gauge covariance of electromagnetism. These equations are shown to possess solutions analogous to those found in the Einstein-Maxwell system. In particular one finds gravi-electric and gravi-magnetic charges contributing to a spherically symmetric static Reissner-Nordström metric. Such Weyl 'charges' provide a source for the non-Riemannian torsion and metric gradient fields instead of the electromagnetic field. The theory suggests that matter may be endowed with gravitational charges that couple to gravity in a manner analogous to electromagnetic couplings in an electromagnetic field. The nature of gravitational coupling to spinor matter in this theory is also investigated and a solution exhibiting a plane-symmetric gravitational metric wave coupled via non-Riemannian waves to a propagating spinor field is presented.

In this paper we study the inverse scattering problem to determine the shape of a scatterer from either far field data for plane wave incidence or near field data for point sources as incident fields. As the simplest case of an absorbing medium we consider an impedance boundary condition with complex valued impedance λ on the boundary of the obstacle. We extend a new approach which characterizes the domain by those points z ∈ R 3 for which a certain function attains zero as its minimal value. This function is given as the cost functional of an optimization problem and depends explicitly on the data and the points z. An valuable feature of this approach is that it does not assume any a priori knowledge on the number of components of the obstacle or even the type of boundary condition. Some examples show the usefulness of this approach also from the numerical point of view.

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