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Contents: The Possibility of Using Computer to Study the Equation
of Gravitation (Q K Lu); Solving Polynomial Systems by Homotopy
Continuation Methods (T Y Li); Sketch of a New Discipline of
Modeling (E Engeler); The Symmetry Groups of Computer Programs and
Program Equivalence (J R Gabriel); Computations with Rational
Parametric Equations (S C Chou et al.); Computer Versus Paper and
Pencil (M Mignotte); The Finite Basis of an Irreducible Ascending
Set (H Shi); A Note on Wu Wen-Tsun's Non-Degenerate Condition (J Z
Zhang et al.); Mechanical Theorem Proving in Riemann Geometry Using
Wu's Method (S C Chou & X S Gao); and other papers;
There seems to be no doubt that geometry originates from such
practical activ ities as weather observation and terrain survey.
But there are different manners, methods, and ways to raise the
various experiences to the level of theory so that they finally
constitute a science. F. Engels said, "The objective of mathematics
is the study of space forms and quantitative relations of the real
world. " Dur ing the time of the ancient Greeks, there were two
different methods dealing with geometry: one, represented by the
Euclid's "Elements," purely pursued the logical relations among
geometric entities, excluding completely the quantita tive
relations, as to establish the axiom system of geometry. This
method has become a model of deduction methods in mathematics. The
other, represented by the relevant work of Archimedes, focused on
the study of quantitative re lations of geometric objects as well
as their measures such as the ratio of the circumference of a
circle to its diameter and the area of a spherical surface and of a
parabolic sector. Though these approaches vary in style, have their
own features, and reflect different viewpoints in the development
of geometry, both have made great contributions to the development
of mathematics. The development of geometry in China was all along
concerned with quanti tative relations."
This comprehensive monograph provides a self-contained treatment of
the theory of I*-measure, or Sullivan's rational homotopy theory,
from a constructive point of view. It centers on the notion of
calculability which is due to the author himself, as are the
measure-theoretical and constructive points of view in rational
homotopy. The I*-measure is shown to differ from other homology and
homotopy measures in that it is calculable with respect to most of
the important geometric constructions encountered in algebraic
topology. This approach provides a new method of treatment and
leads to various new results. In particular, an axiomatic system of
I*-measure is formulated, quite different in spirit from the usual
Eilenberg-Steenrod axiomatic system for homology, and giving at the
same time an algorithmic method of computation of the I*-measure in
concrete cases. The book will be of interest to researchers in
rational homotopy theory and will provide them with new ideas and
lines of research to develop further.
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