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Book review

Author(s) Constantin, Gheorghe
Istratescu, Ioana
Title Elements of Probabilistic Analysis with Applications
Publisher Kluwer Academic Publishers
Year of publication 1989
Reviewed by Mihai Postolache

This is a very specialized monograph which presents a coherent exposition of some important analytic methods and results for the exploration and understanding of fixed point theory for operators on probabilistic structures as well as for random operators on metric or linear spaces. Also, some important applications to random equations solving (for example the random wave propagation equation), optimization, stochastic approximation etc. are given.

The book includes seven chapters, an introduction, a comprehensive bibliography, a table of author index and subject index, a table of contents aufficiently detailed. The first of these chapters deals with probabilistic structures and related topics. The second, gives the basic elements of the measurable multivalued mappings and randomness. In the third chapter, fixed point theory in probabilistic structures is presented. Chapter 4, deals with some random fixed point theorems and in Chapter 5 are investigated the applications of the fixed point theory for random operator equations. In Chapter 6 some analysis problems and methods for random equations are presented. In the last chapter some applications of random operators are treated. A few examples illustrate the main concepts. Printed in Romania, this book is very well produced.

Written for more advanced students in mathematics, graduate students, research scientists as well as for applied mathematicians, this book provides a detailed investigation of the properties of the fixed point for stochastic models. This topic is fundamental in biology, statistical physics, the theory of random operator equations and so on. Parts of this book will be useful for biologists, applied physicists etc. An introductory course to probability theory, functional analysis and topology is assumed.

The presentation of the book is very rigorous, strict proofs of the stated results are given. But the notation is simple and the concepts are clearly presented. The book is aufficiently self contained to enable independent research. Purchasing this book is a good investment for individuals who have seriously background in mathematics as well as for libraries alike.