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.

This book presents the classical theorems about simply connected smooth 4-manifolds: intersection forms and homotopy type, oriented and spin bordism, the index theorem, Wall's diffeomorphisms and h-cobordism, and Rohlin's theorem. Most of the proofs are new or are returbishings of post proofs; all are geometric and make us of handlebody theory. There is a new proof of Rohlin's theorem using spin structures. There is an introduction to Casson handles and Freedman's work including a chapter of unpublished proofs on exotic R4's. The reader needs an understanding of smooth manifolds and characteristic classes in low dimensions. The book should be useful to beginning researchers in 4-manifolds.

This book aims to give a user friendly tutorial of an interdisciplinary research topic (fronts or interfaces in random media) to senior undergraduates and beginning grad uate students with basic knowledge of partial differential equations (PDE) and prob ability. The approach taken is semiformal, using elementary methods to introduce ideas and motivate results as much as possible, then outlining how to pursue rigor ous theorems, with details to be found in the references section. Since the topic concerns both differential equations and probability, and proba bility is traditionally a quite technical subject with a heavy measure theoretic com ponent, the book strives to develop a simplistic approach so that students can grasp the essentials of fronts and random media and their applications in a self contained tutorial. The book introduces three fundamental PDEs (the Burgers equation, Hamilton– Jacobi equations, and reaction–diffusion equations), analysis of their formulas and front solutions, and related stochastic processes. It builds up tools gradually, so that students are brought to the frontiers of research at a steady pace. A moderate number of exercises are provided to consolidate the concepts and ideas. The main methods are representation formulas of solutions, Laplace meth ods, homogenization, ergodic theory, central limit theorems, large deviation princi ples, variational principles, maximum principles, and Harnack inequalities, among others. These methods are normally covered in separate books on either differential equations or probability. It is my hope that this tutorial will help to illustrate how to combine these tools in solving concrete problems.

An interdisciplinary introduction to the structural and scattering properties of complex photonic media, focusing on deterministic aperiodic structures and their conceptual roots in geometry and number theory. An essential tool for students at the graduate or advanced undergraduate level.

This IMA Volume in Mathematics and its Applications RANDOM MEDIA represents the proceedings of a workshop which was an integral part of the 1984-85 IMA program on STOCHASTIC DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS We are grateful to the Scientific Committee: Daniel Stroock (Chairman) \~ende 11 Fl emi ng Theodore Harris Pierre-Louis Lions Steven Orey George Papanicolaou for planning and implementing an exciting and stimulating year-long program. We especi ally thank George Papani col aOIJ for organi zi ng a workshop which produced fruitful interactions between mathematicians and scientists from both academia and industry. George R. Sell Hans I~ei nherger PREFACE During September 1985 a workshop on random media was held at the Institute for Mathematics and its Applications at the University of Minnesota. This was part of the program for the year on Probability and Stochastic Processes at IMA. The main objective of the workshop was to bring together researchers who work in a broad area including applications and mathematical methodology. The papers in this volume give an idea of what went on and they also represent a cross section of problems and methods that are currently of interest.

Electrical Engineering Wave Propagation and Scattering in Random Media A volume in the IEEE/OUP Series on Electromagnetic Wave Theory Donald G. Dudley, Series Editor This IEEE Classic Reissue presents a unified introduction to the fundamental theories and applications of wave propagation and scattering in random media. Now for the first time, the two volumes of Wave Propagation and Scattering in Random Media previously published by Academic Press in 1978 are combined into one comprehensive volume. This book presents a clear picture of how waves interact with the atmosphere, terrain, ocean, turbulence, aerosols, rain, snow, biological tissues, composite material, and other media. The theories presented will enable you to solve a variety of problems relating to clutter, interference, imaging, object detection, and communication theory for various media. This book is expressly designed for engineers and scientists who have an interest in optical, microwave, or acoustic wave propagation and scattering. Topics covered include: Wave characteristics in aerosols and hydrometeors Optical and acoustic scattering in sea water Scattering from biological materials Pulse scattering and beam wave propagation in such media Optical diffusion in tissues and blood Transport and radiative transfer theory Kubelka—Munk flux theory and plane-parallel problem Multiple scattering theory Wave fluctuations in turbulence Strong fluctuation theory Rough surface scattering Remote sensing and inversion techniques Imaging through various media About the IEEE/OUP Series on Electromagnetic Wave Theory Formerly the IEEE Press Series on Electromagnetic Waves, this joint series between IEEE Press and Oxford University Press offers outstanding coverage of the field with new titles as well as reprintings and revisions of recognized classics that maintain long-term archival significance in electromagnetic waves and applications. Designed specifically for graduate students, practicing engineers, and researchers, this series provides affordable volumes that explore electromagnetic waves and applications beyond the undergraduate level. See page il of the front matter for a listing of books in this series.