where [x.sub.1] and [x.sub.s] are called

infimum and supremum, respectively.

where the

infimum is taken over all of the decompositions of f.

The selections that make up the main body of the text are devoted to the chaotic behavior of group actions, halving dynamical systems, dynamical systems of non-algebraic origin, ShearerAEs inequality and

infimum rule for Shannon entropy and topological entropy, loops of transitive interval maps, and a great many other related subject areas.

For every ([omega], t, y, z) the

infimum is achieved in this relation by a pair ([beta], [gamma]) which depends on ([omega], t).

A well-founded tree is a non-empty partially ordered set T = (V S) such that for every two elements t1 t2 their

infimum exists and the set {t ' cV : t ' less than t} is a well- ordered chain for every t cV .

Now, the

infimum of [[sigma].sub.i] is derived by using the following optimal methods.

Thus, the desired integral is smaller than or equal to the

infimum of all such integrals [integral] s(x) dx.

The minimum bound is referred to as the

infimum and the maximum bound is referred to as the supremum.

t is called an L-fuzzy

infimum of x, y if the following conditions are true:

where the

infimum is taken over all reduced and irreducible curves passing through P (see [3] and [7, Chapt.

Further, let [p.sub.l] and [p.sub.h] indicate the

infimum and supremum of the support of F with z [greater than or equal to] [p.sub.h] [greater than or equal to] [p.sub.l] [greater than or equal to] 0 for any F.

The fuzzy number valued Lebesgue outer measure for the fuzzy subset [mu] is defined as m*([mu]) = (0, [??]) where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the

infimum is taken over all countable collection ([I.sub.n]) of open intervals covering R.