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This book gives the reader a thorough knowledge of the basic topological ideas necessary for studying differential manifolds. These topics include immersions and imbeddings, approach techniques, and the Morse classification of surfaces and their cobordism. The author keeps the mathematical prerequisites to a minimum; this and the emphasis on the geometric and intuitive aspects of the subject make the book an excellent and useful introduction for the student. There are numerous excercises on many different levels ranging from practical applications of the theorems to significant further development of the theory and including some open research problems.
"A very valuable book. In little over 200 pages, it presents a
well-organized and surprisingly comprehensive treatment of most of
the basic material in differential topology, as far as is
accessible without the methods of algebraic topology....There is an
abundance of exercises, which supply many beautiful examples and
much interesting additional information, and help the reader to
become thoroughly familiar with the material of the main text."
-MATHEMATICAL REVIEWS
Hirsch, Devaney, and Smale s classic "Differential Equations,
Dynamical Systems, and an Introduction to Chaos" has been used by
professors as the primary text for undergraduate and graduate level
courses covering differential equations. It provides a theoretical
approach to dynamical systems and chaos written for a diverse
student population among the fields of mathematics, science, and
engineering. Prominent experts provide everything students need to
know about dynamical systems as students seek to develop sufficient
mathematical skills to analyze the types of differential equations
that arise in their area of study. The authors provide rigorous
exercises and examples clearly and easily by slowly introducing
linear systems of differential equations. Calculus is required as
specialized advanced topics not usually found in elementary
differential equations courses are included, such as exploring the
world of discrete dynamical systems and describing chaotic
systems.
Classic text bythree of the world s most prominent mathematicians
Continues the tradition of expository excellenceContains updated
material and expanded applications for use in applied studies"
An extraordinary mathematical conference was held 5-9 August 1990
at the University of California at Berkeley: From Topology to
Computation: Unity and Diversity in the Mathematical Sciences An
International Research Conference in Honor of Stephen Smale's 60th
Birthday The topics of the conference were some of the fields in
which Smale has worked: * Differential Topology * Mathematical
Economics * Dynamical Systems * Theory of Computation * Nonlinear
Functional Analysis * Physical and Biological Applications This
book comprises the proceedings of that conference. The goal of the
conference was to gather in a single meeting mathemati cians
working in the many fields to which Smale has made lasting con
tributions. The theme "Unity and Diversity" is enlarged upon in the
section entitled "Research Themes and Conference Schedule." The
organizers hoped that illuminating connections between seemingly
separate mathematical sub jects would emerge from the conference.
Since such connections are not easily made in formal mathematical
papers, the conference included discussions after each of the
historical reviews of Smale's work in different fields. In
addition, there was a final panel discussion at the end of the
conference.
The intention of the authors is to examine the relationship between
piecewise linear structure and differential structure: a
relationship, they assert, that can be understood as a homotopy
obstruction theory, and, hence, can be studied by using the
traditional techniques of algebraic topology. Thus the book attacks
the problem of existence and classification (up to isotopy) of
differential structures compatible with a given combinatorial
structure on a manifold. The problem is completely "solved" in the
sense that it is reduced to standard problems of algebraic
topology. The first part of the book is purely geometrical; it
proves that every smoothing of the product of a manifold M and an
interval is derived from an essentially unique smoothing of M. In
the second part this result is used to translate the classification
of smoothings into the problem of putting a linear structure on the
tangent microbundle of M. This in turn is converted to the homotopy
problem of classifying maps from M into a certain space PL/O. The
set of equivalence classes of smoothings on M is given a natural
abelian group structure.
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