By American College of Surgeons
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Oligomorphic permutation groups For a natural number n, a permutation group is n-transitive if it acts transitively on the set of n-tuples of distinct elements of Ω, and n-set-transitive if it acts transitively on the set of n-element subsets of Ω. We say a permutation group is highly transitive if it is n-transitive for all n, and highly set-transitive if it is n-set-transitive for all n. Oligomorphic permutation groups generalise these classes. Thus, we let Fn (G) denote the number of orbits of G on the set of n-tuples of distinct elements, and fn (G) the number of orbits on n-element subsets.
Thus, the substitution rule for the wreath product can be regarded as the prototype of a wide generalisation of the exponential principle in combinatorics (cf. [16, 33]). If G is the automorphism group of the Fra¨ıss´e limit of a Fra¨ıss´e class C, then GWrS is similarly associated with the class of disjoint unions of C-structures; the exponential principle applies to this situation. In general, if H is associated with a Fra¨ıss´e class D, then GWrH is associated with the class of disjoint unions of C-structures with a D-structure on the set of parts.
There is a filtration of coherent subsheaves 0 = V0 ⊂ V1 ⊂ · · · ⊂ Vn−1 ⊂ Vn = E such that each successive quotient Vi /Vi−1 , i ∈ [1 , n], is polystable, and µ(Vi /Vi−1 ) = µ(V ). More precisely, Vi /Vi−1 is the maximal polystable subsheaf of E/Vi−1 , which is also called the socle of E/Vi−1 . 26) 34 I. Biswas and A. J. Parameswaran be the maximal Zariski open dense subset such that each Vi /Vi−1 , i ∈ [1 , n], is locally free. It can be shown that E possesses a monodromy group as well as a monodromy bundle over the open subset UE .
Yearbook : 1980 by American College of Surgeons
by Daniel
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