The selected lectures cover the topology and geometry of real singularities, singularities of mappings and the vanishing homology of images and discriminants, notes on non-isolated singularities, algorithms for primary decomposition
in Singular, singularities of maps and characteristic classes, and centers and limit cycles in polynomial systems of ordinary differential equations.
The results obtained on the failed sample included weight loss steps associated with the dehydrohalogenation of the polymer, the primary decomposition
of the polymer, and the combustion of carbon black and char.
However, B is an integrally closed Noetherian domain, and so we can replace factorization of functions by primary decomposition of principal ideals.
[intersection] [N.sub.r] a primary decomposition of N in M with associated prime ideals [Mathematical Expression Omitted].
f) Let I [subset] N [(omega)] be an ideal with compact zero-set, and let I = [I.sub.1][subset]...[subset][I.sub.r] be a primary decomposition with associated prime ideals [p.sub.i] = [Mathematical Expressions Omitted], 1 [less than or equal to] i [less than or equal to] r.
Section 4 is devoted to separation; in fact, we prove more, namely, that primary decompositions of modules are preserved by the base change [Mathematical Expression Omitted].
Let us remark that the latter proof hides primary decompositions. In fact, we have actually shown that primary decompositions of principal ideals are preserved by the base change [Mathematical Expression Omitted].