For a system consisting of a random medium with rough boundaries, the governing (Bethe-Salpeter) equation for boundary-value transport problems can be written in a form such that the medium and the boundaries are treatedon an equal footing. This enables several expressions for the solution to be obtained by interchanging the roles of the medium and the boundaries, thus allowing the most convenient one to be selected according to the specific situation and the information required. This book presents a unified theory based on the Bethe-Salpeter equation with particular attention being paid to: boundary-value problems of transport, layer problems, a fixed scatterer imbedded in a bounded random medium, construction of an optical scattering matrix for a complete system, and optical wave propagation in a turbulent medium. The last topic is treated in terms of first moment equations combined with the cluster expansion and, second, the two-scale method based on the Lagrange variational principle.

The past decade has witnessed breakthroughs in the understanding of the wave localization phenomena and its implications for wave multiple scattering in inhomogeneous media. This book brings together review articles written by noted researchers in this field in a tutorial manner so as to give the readers a coherent picture of its status. It would be valuable both as an up-to-date reference for active researchers as well as a readable source for students looking to gain an understanding of the latest results.

The instant New York Times bestseller End the struggle, speak up for what you need, and experience the freedom of being truly yourself. Healthy boundaries. We all know we should have them--in order to achieve work/life balance, cope with toxic people, and enjoy rewarding relationships with partners, friends, and family. But what do "healthy boundaries" really mean--and how can we successfully express our needs, say "no," and be assertive without offending others? Licensed counselor, sought-after relationship expert, and one of the most influential therapists on Instagram Nedra Glover Tawwab demystifies this complex topic for today's world. In a relatable and inclusive tone, Set Boundaries, Find Peace presents simple-yet-powerful ways to establish healthy boundaries in all aspects of life. Rooted in the latest research and best practices used in cognitive behavioral therapy (CBT), these techniques help us identify and express our needs clearly and without apology--and unravel a root problem behind codependency, power struggles, anxiety, depression, burnout, and more.

Science and engineering have been great sources of problems and inspiration for generations of mathematicians. This is probably true now more than ever as numerous challenges in science and technology are met by mathematicians. One of these challenges is understanding propagation of waves of different nature in systems of complex structure. This book contains the proceedings of the research conference, ``Waves in Periodic and Random Media''. Papers are devoted to a number of related themes, including spectral theory of periodic differential operators, Anderson localization and spectral theory of random operators, photonic crystals, waveguide theory, mesoscopic systems, and designer random surfaces. Contributions are written by prominent experts and are of interest to researchers and graduate students in mathematical physics.

Based on his many years of professional experience at leading companies in communications technology, the author describes an analytical solution for wave propagation over the sea surface in an atmospheric boundary layer. His approach allows the detailed analysis of combined effects of diffraction, refraction and scattering in random media. While specific applications covered are targeted at radio wave propagation over the sea surface, a similar approach is applicable to many problems in underwater acoustics, seismology, solid matter physics and astrophysics.

W S HALL School of Computing and Mathematics, University of Teesside, Middlesbrough, TS1 3BA UK G OLIVETO Division of Structural Engineering, Department of Civil and Environmental Engineering, University of Catania, Viale A. Doria 6, 95125 Catania, Italy Soil-Structure Interaction is a challenging multidisciplinary subject which covers several areas of Civil Engineering. Virtually every construction is connected to the ground and the interaction between the artefact and the foundation medium may affect considerably both the superstructure and the foundation soil. The Soil-Structure Interaction problem has become an important feature of Structural Engineering with the advent of massive constructions on soft soils such as nuclear power plants, concrete and earth dams. Buildings, bridges, tunnels and underground structures may also require particular attention to be given to the problems of Soil-Structure Interaction. Dynamic Soil-Structure Interaction is prominent in Earthquake Engineering problems. The complexity of the problem, due also to its multidisciplinary nature and to the fact of having to consider bounded and unbounded media of different mechanical characteristics, requires a numerical treatment for any application of engineering significance. The Boundary Element Method appears to be well suited to solve problems of Soil- Structure Interaction through its ability to discretize only the boundaries of complex and often unbounded geometries. Non-linear problems which often arise in Soil-Structure Interaction may also be treated advantageously by a judicious mix of Boundary and Finite Element discretizations.

This book describes state-of-the-art advances and applications of the unified transform and its relation to the boundary element method. The authors present the solution of boundary value problems from several different perspectives, in particular the type of problems modeled by partial differential equations (PDEs). They discuss recent applications of the unified transform to the analysis and numerical modeling of boundary value problems for linear and integrable nonlinear PDEs and the closely related boundary element method, a well-established numerical approach for solving linear elliptic PDEs. The text is divided into three parts. Part I contains new theoretical results on linear and nonlinear evolutionary and elliptic problems. New explicit solution representations for several classes of boundary value problems are constructed and rigorously analyzed. Part II is a detailed overview of variational formulations for elliptic problems. It places the unified transform approach in a classic context alongside the boundary element method and stresses its novelty. Part III presents recent numerical applications based on the boundary element method and on the unified transform.