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Partial differential equations and group theory. New perspectives for applications. (English) Zbl 0808.35002

Mathematics and its Applications (Dordrecht). 293. Dordrecht: Kluwer Academic Publishers. ix, 473 p. Dfl. 340.00; US$ 204.00; £136.00/hbk (1994).
The author of this book has provided a very well presented study of differential algebraic geometry and its applications to mathematical physics and engineering sciences. The emphasis is given mainly to characteristic examples and applications.
A few things deserve special mention for the content of this book. The author shows how the methods of differential elimination presented here will help to determine compatibility conditions on input and/or output for a better understanding of the foundations of control theory. He also unifies fairly well differential geometry and differential algebra in a new well balanced subject with the title differential algebraic geometry.
The main topics treated in this book include a self-contained introduction to the study of linear and nonlinear systems of PDE at an introductory graduate level with emphasis to concrete examples whose origin belongs to engineering physics as well as a brief study of the Galois theory for systems of PDE which is called “differential Galois theory”.
The author has provided a logically balanced and authoritative account of the different branches and problems of pure and applied mathematics which enter in this treatise, and it will prove an essential source for scientists, graduate students, and researchers.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
12H05 Differential algebra
93B27 Geometric methods
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