McGraw-Hill, New York.

presented very clearly and intuitively, with some physical motivation, though with less rigor than some may prefer. The treatment of singularly perturbed two-point boundary value problems limits itself to linear second-order equations, but turning points are allowed, both at an endpoint and at an interior point. By using a WKB approach, Cheng is able to solve some examples with boundary layer resonance and two higher-order turning point problems that were presumably solved incorrectly by Bender and Orszag. The final chapter treats nonlinear oscillations. It generalizes the Poincaré– Linstedt procedure as a renormalization of angular frequency and two-timing as a renormalized two-scale method, and ends by illustrating the renormalization group method. The subtleties addressed by Professor Cheng will be of interest to experts and more curious students who are up to handling the D (difficult) problems of Bender and Orszag. As suggested by von Neumann's famous remark, " Mathematical ideas originate in empirics, although the genealogy is sometimes long and obscure, " it is not always easy to identify a specific moment at which a definition-theorem-proof mathematical topic crystalizes out of more applied questions. But this gem of a monograph, written with a quintessentially French style and clarity, provides a self-contained account of a recently emerged topic that will be seen as a cornerstone for future theoretical developments. The basic idea of stochastic fragmentation is to start with a unit mass particle. By time t this has split into particles of masses (xi), where in the conservative case we require i xi = 1 and more generally we require i xi ≤ 1. Different particles evolve independently, a mass-x particle splitting at some stochastic rate λx into particles whose relative masses (xj/x, j ≥ 1) follow some probability distribution µx(·). (So the model neglects detailed three-dimensional geometry; the shape of a particle is assumed not to affect its propensity to split, and different particles do not interact.) Especially tractable is the self-similar case, where µx = µ1 and λx = x α for some scaling exponent α. Such processes are closely related to classical topics in theoretical and applied probability—the log-masses form a continuous-time branching random walk, and the mass of the particle containing a tagged atom forms a continuous-time Markov process on state space (0, 1]. Chapter 1 gives a clear description (assuming a first-year graduate knowledge of measure-theoretic probability at the level of [4], say) of the general setup, …