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Book review

Author(s) Neuman, Frantisek
Title Global properties of linear ordinary differential equations
Publisher Kluwer Academic Publishers
Year of publication 1992
Reviewed by G. Micula

This monograph presents an authoritative, unified description of the methods and results concerning the global properties of linear differential equations of order n (n >= 2) in the real domain. It does not, however, seek to be comprehensive. Rather, it contains a selection of results which richly illustrate the unified approach presented. By using recent methods and results of algebra, topology, differential geometry, functional analysis, theory of functional equations and linear differential equations of the second order, and by introducing several original methods, global solutions of problems previously studied only locally are provided. The structure of global transformations is described algebraically, and a new geometrical approach is introduced which leads to global canonical forms suitable for Cartan's moving frame-of-references method.

The theory discussed also provides effective tools for solving some open problems, especially relating to the distribution of zeros of solutions. In addition, the theory of functional equations plays an important role in investigating the asymptotic behaviour of solutions. The book also contains applications to other fields of mathematics, especially to differential geometry and functional equations. Some related results and further possible areas of research are mentioned at the end of the book.

This monograph is written for mathematicians working with differential equations but, due to the application of modern aspects of other fields of mathematics, also for those working in algebra, topology, differential geometry, functional-differential equations and functional equations. This book can also be of use for specialists in computer sciences, physics, chemistry, engineering, biology, astronomy and others whose work involves the use of linear differential equations.