Classification theory

As we mentioned at the beginning, Xl" inherits structures from X. First of all, it is a scheme. It is projective if X is projective. These follows from Grothendieck s construction of Hilbert schemes. A nontrivial example is a result by Beauville [6] Xl" has a holomorphic symplectic form when X has one. When X is projective, X has a holomorphic symplectic form only when X is a X3 surface or an abelian surface by the classification theory. We also have interesting noncompact examples X = or X = r S where E is a Riemann surface. These examples are particularly nice because of the existence of a C -action, which naturally induces an action on Xl" . (See Chapter 7.)... 

Satake, I. Classification Theory of Semi-Simple Algebraic Groups (New York Marcel Dekker, 1971). 

A formal classification theory for such terms, which are neither very exact nor very inexact, was introduced by Zadeh, who, among others, saw the need for generalizing the theory of sets to admit the possibility of unsharp boundaries between classes of imprecisely defined objects. Whereas the characteristic function can only have a value of 0 or 1 (in harmony with the Aristotelian logic of the excluded middle), membership in a fuzzy set is a matter of degree. This degree (or grade) of membership (ix) may have any value in the continuum of the real interval [0,1] in the special case where the value of /x is restricted to 0 or 1, the degree of membership becomes the characteristic function. 

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A third cause of weakness is that, in all materials, surface irregularities provide the initiation points for crack growth. This follows the classifical theories of Griffith for metals and brittle materials, extended to tough materials by Irwin. These showed that there is a critical length of a crack beyond which it grows rapidly and leads to catastrophic failure. 

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