Bijective mapping

The principles of statistical mechanics can be applied to a dynamical systeni provided that it obeys Louiville s Theorem (that is, it preserves volumes in phase space) and that its energy remains constant. The first requirement is easy since all reversible rules 4>r define bijective mappings of the phase space volume... 

Since a is assumed to be an automorphism, a is a morphism. Thus, as z G yr, za e yara. From z G yr we also obtain ypDzq = apqr. Thus, as a is assumed to be a bijective map, yapa fl za(qa) = apqr. It follows that apqr = apaqara. 

Now we have shown that v and v are inverse to each other. In particular, v is a bijective map from U T to V T, and v is a bijective map from V T to U T. Thus, the above given isomorphism and its analogue yield that,... 

For each element x in yT, we define xx to be the uniquely defined element in yxf, where t stands for the uniquely defined element in T with x yt. Then X is a bijective map from yT to yxT satisfying yx = yx an(i = -W We claim that x is faithful. 

Note that h Gx. Note also that h is a bijective map from xT to xT. Since T is assumed to be not thin, 1 yt h. Thus, we just have to show that h is faithful. 

Thus, we have shown that y is a bijective map from X to xT x xU. 

That 4> is a bijective map from TUxTu to (TxT)q(Uxu) follows immediately from the definition of . 

Theorem 11.2.2 Assume that L does not contain thin elements. Then K i—> (.K) is a bijective map from the power set of L to the set of all closed subsets of L). 

Corollary 11.4.8 Each faithful map from yV to X extends faithfully to a bijective map from yV U zV to yxY U zxV1. ... 

Recall that L is called a Coxeter set if L is constrained and satisfies the exchange condition. Recall also that L is called spherical if S- (L) is not empty. Recall, finally, that a closed subset T of S is called faithfully embedded in S if, for any two elements y in X and z in yT, each faithful map x from y, z to X extends to a bijective map from yT to yyT. 

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