On Global Univalence Theorems
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977
T Parthasarathy
On
Global Univalence Theorems
Springer-Verlag Berlin Heidelberg New York 1983
Author
1. Parthasarathy Indian Statistical Institute, Delhi Centre 7, S.J.S. Sansanwal Marg., New Delhi 110016, India
AMS Subject Classifications (1980): 26-02,26810,90-02, 90A 14, 90C30 ISBN 3-540-11988-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-11988-4 Springer-Verlag New York Heidelberg Berlin
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© by Springer-Verlag Berlin Heidelberg 1983 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
PREFACE This volume of lecture notes contains results on global univalent mappings. Some of the material of this volume had been given as seminar talks at the Department of Mathematics, university of Kansas, Lawrence during 1978-79 and at the Indian Statistical Institute, Delhi Centre during 1979-80. Even though the classical local inverse function theorem is well-known, Gale-Nikaido's global univalent results obtained in (1965) are not known to many mathematicians that I have sampled.
Recently some significant contributions have
been made in this area notably by Garcia-Zangwill (1979), Mas-Colell (1979) and Scarf-Hirsch-ChilniskY (1980).
Global univalent results are as important as local
univalent results and as such I thought it is worthwhile to make these results well-known to the mathematical
at large.
Also I believe that there are
very many interesting open problems which are worth solving in this branch of Mathematics.
I have also included a number of applications from different
disciplines like Differential Equations, Mathematical Programming, Statistics etc.
Mathematical
Some of the results have appeared onlY in Journals
and we are bringing them to-gether in one nlace. These notes contain some new results.
For example Proposition 2, Theorem 4
in Chapter II, Theorem 4, Theorem 5 in Chapter III, Theorem 2" in Chapter V, Theorem 8 in Chapter VI, Theorem 2 in Chapter VII, Theorem 9 in Chapter VIII are new results. It is next to impossible to cover all the known results on global univalent mappings for lack of space and time.
For example a notable omission could be the
role played by univalent mappings whose domain is comnlex numbers.
We have also
not done enough justice to the nroblem when a PL-function will be a homeomorphism in view of the growLng importance of such functions.
We have certainly given
references where an interested reader can get more information. I am grateful to Professors : Andreu Mas-Colell, Ruben
Albrecht Dold
and an anonymous referee for their several constructive suggestions on various parts
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