TAILIEUCHUNG - Research report: "The w - the germ of function defined on the manifold r-reticular"

Tham khảo luận văn - đề án 'research report: "the w - the germ of function defined on the manifold r-reticular"', luận văn - báo cáo phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | -DETERMINANCY OF FUNCTION GERMS ON r-RETICULAR MANIFOLDS NGO DINH QUOC a Abstract. In this paper we generalize the w-determinancy of a function germ f e s n under the group Diffr n on r-reticular manifolds and give a necessary and sufficient condition that a function germ f e e n is an w-determinancy under the group Diffr n . The main result is Theorem . 1. Introduction In many years the following question When is a smooth function f determined in a neighbourhood of a point 0 by one of its Taylor polynomials at 0 in the sense that every another function having the same Taylor polynomial coincides with f around 0 up to a smooth change of coordinates has attracted considerable attention. In general an any function does not have this property at an arbitrary point for instance a flat function cannot be determined by its Taylor polynomial in the above sense. But some noticeable results have been obtained in 2 5 and 8 . In 1968 John Mather 5 gave two conditions one sufficient and the other necessary for a function to be determined by its k-th Taylor polynomial. In 1977 H. H. Vui N. H. Due N. T. Cuong N. T. Dai N. s. Minh 2 gave necessary and sufficient conditions that a function f e e n is -determined under the group Diff n . In 1980 H. H. Vui gave some results for map-germs to be u- determined by its Taylor polynomial under the subgroup K X1 and K X1 of the group K and K which consisting of all coordinate transformations preserving 0 and x1 0 . The aim of this paper is to give a necessary and sufficient condition for a function f e e n to be -determined under the group Diffr n and to generalize the u-determinancy of function-germs on a manifold with boundary which is introduced by H. H. Vui in 1980. The paper contains two sections. The first section deals with the basic notations. In the section 2 we prove a necessary and sufficient condition that a function f e e n is u-determined under the group Diffr n . Our main result is Theorem . 2. Preliminaries We denote

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