Catchment region point symmetry

In order to prove this result, we shall show that the catchment regions C(X,i) and C (X,, i ) overlap, and we shall apply the catchment region point symmetry theorem to C (X, i ) and to a point K from the overlapping region. 

The first of these theorems, the catchment region point symmetry theorem states the following ... 

Within each catchment region the highest point symmetry occurs at the critical point [33]. 

Interrelations of point symmetry groups of nuclear arrangements, the catchment regions of chemical species, and the fundamental groups of reaction mechanisms... 

Here we shall be concerned with a subfamily of special critical levels those where the point symmetry g(K(X,i)) of the nuclear conflguiation of the critical point K(A,i) is unique within a small open neighborhood N (X,i)) of the critical point within its catchment region C(X,i) ... 

The proof of this statement follows from the fact that if the point symmetry g(K) of a nuclear configuration K e C(X,i)), different from the critical point, K K(X,i), is the same as the point symmetry g(K(X,i)) at the critical point K(X,i) of the same catchment region C(X,i), then this symmetry g(K(X,i)) must be present along the entire steepest descent path from the point K to the critical point g(K(X,i)). Since the steepest descent path must enter the open neighborhood N(K(A,i)) in order to reach the critical point K(X,i), the open neighborhood N(K(X,i)) must also contain a continuum of points with the same symmetry g(K(X,i)). Consequently, if N(K(X,i)) does not have points K K(X,i) with symmetry g(K(X,i)), then the catchment region C(X,i) caruiot have such points either, and the point symmetry g(K( i,i)) is unique within the entire catchment region C(X4). 

Of course, if the critical point of the catchment region is an element of the level set, then within the homotopy classes of paths passing through the catchment region the maximal local symmetry is g(K(X,i)). Nevertheless, within the family of homotopic paths, the paths with this maximal locd symmetry form a subset of measure zero. 

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