Overview

This volume presents an unified overview of the methods and results concerning the global properties of linear differential equations of order n (n>2). It does not, however, seek to be comnprehensive. Rather, it contains a selection of results which richly illustrate the unified approach presented. By making use of recent methods and results from many different areas of mathematics and by introducing several original methods, global solutions of problems previously studied only locally are given. 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-reference 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 studying the asymptotic behaviour of solutions. Applications to differential geometry and functional equations are also described. The volume is largely self-contained.

Full Product Details

Author:   Frantisek Neuman
Publisher:   Kluwer Academic Publishers
Imprint:   Kluwer Academic Publishers
Edition:   1992 ed.
Volume:   52
Dimensions:   Width: 16.00cm , Height: 2.00cm , Length: 24.00cm
Weight:   0.820kg
ISBN:  

9780792312697

ISBN 10:   0792312694
Pages:   320
Publication Date:   30 September 1992
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   Out of stock  
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Table of Contents

1. Introduction with historical remark.- 2. Notation, definitions and some basic facts.- 2.1 Generalities.- 2.2 Maps.- 2.3 Topology.- 2.4 Algebraic structures.- 2.5 Vector spaces.- 2.6 Linear differential equations.- 2.7 Functional equations.- 3. Global transformations.- 3.1 Definition of the global transformations.- 3.2 Smoothness of global transformations.- 3.3 Algebraic approach to global transformations.- 3.4 Fundamental problems.- 4. Analytic, algebraic and geometrical aspects of global transformation.- 4.1 Some useful formulas.- 4.2 Global transformations of special classes of linear differential equations.- 4.3 Covariant constructions of linear differential equations.- 4.4 Geometrical approach to global transformations.- 5. Criterion of global equivalence.- 5.1 Bor?vka’s criterion of global equivalence of the second order equations.- 5.2 Criterion of global equivalence of the third and higher order equations.- 6. Stationary groups.- 6.1 Notation and preliminary results.- 6.2 Preparatory results.- 6.3 Subgroups of stationary groups with increasing elements.- 6.4 Stationary groups with decreasing elements.- 6.5 Complete list of stationary groups and characterization of the corresponding equations.- 7. Canonical forms.- 7.1 Notion of canonical forms.- 7.2 The Laguerre-Forsyth and Halphen forms.- 7.3 Cartan’s moving-frame-of-reference method.- 7.4 Hereditary property.- 7.5 Global canonical forms: geometrical approach.- 7.6 Global canonical forms: analytic approach.- 7.7 List of canonical forms of the second and third order equations.- 8. Invariants.- 8.1 Notion of invariant and covariant.- 8.2 Covariants.- 8.3 Local invariants and covariants.- 8.4 Global invariants.- 8.5 Smoothness of coefficients as an invariant.- 9. Equations with solutions of prescribedproperties.- 9.1 Coordinate approach.- 9.2 Asymptotic properties of solutions of the second order equations.- 9.3 Periodic solutions of the second order equations.- 9.4 Geometrical approach.- 10. Zeros of solutions.- 10.1 Notation and definitions.- 10.2 Representation of zeros.- 10.3 Second order equations.- 10.4 Third order equations.- 10.5 Iterative nth order equations.- 10.6 Periodic solutions of nth order equations.- 11. Related results and some applications.- 11.1 Asymptotic properties and zeros of solutions of second order equations.- 11.2 Integral inequalities.- 11.3 Affine geometry of plane curves.- 11.4 Isoperimetric theorems.- 11.5 Related results and comments, possible trends of further research.- 12. Appendix: Two functional equations.- 12.1 Abel functional equation.- 12.2 Euler functional equation for homogeneous functions.- Literature cited in the book and/or for supplementary reading.- Index of names.

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