Practical needs in geographic information systems (GIS) have led to the investigation of formal and sound methods of describing spatial relations. Arter an introduction to the basic ideas and notions of topology, a novel theory of topological spatial relations between sets is developed in which the relations are defined in tenns of the intersections of the boundaries and interiors of two sets. By consideringemply and non-emptyas the values of'the intersections, a total of sixteen topological spatial relations is described, each of which can be realized in R 2 • This set is reduced to nine relations if the sets are restricted to spatial regions, a fairly broad class of subsets of a connected topological space with an application to GIS. It is shown that these relations correspond to some of the standard set theoretical and topological spatial relations between sets such as equality, disjointness and containment in the interior. .
The connection matrix theory for Morse decompositions is introduced. The connection matrices are matrices of maps between the homology indices of the sets in the Morse decomposition. The connection matrices cover, in a natural way, the homology index braid of the Morse decomposition and provide information about the structure of the Morse decomposition. The existence of connection matrices of Morse decompositions is established, and examples illustrating applications of the connection matrix are provided.
ABSTRACT. On a Morse decomposition of an invariant set in a flow there are partial orderings defined by the flow. These are called admissible orderings of the Morse decomposition. The index filtrations for a total ordering of a Morse decomposition are generalized in this paper with the definition and proof of existence of index filtrations for adInissible partial orderings of a Morse decomposition.It is shown that associated to an index filtration there is a collection of chain complexes and chain maps called the chain complex braid of the index filtration. The homology index braid of the corresponding admissible ordering of the Morse decomposition is obtained by passing to homology in the chain complex braid.Introduction. In the classical Morse theory a gradient flow of a function defined on a manifold is examined. The function is assumed to have finitely many critical points. The statement of Morse theory then relates the dimensions of the unstable invariant manifolds of these critical points to algebraic invariants of the whole manifold.In Conley [1] and Conley and Zehnder [2] these ideas are extended to a setting where the manifold is replaced with a compact invariant set S in a locally compact local flow in a Hausdorff space with a flow. The critical points are replaced with a collection M of mutually disjoint compact invariant subsets of S. The gradient structure is replaced with a total order that is defined on M and respected by the flow on the complement, in S, of the union of the sets in M.The collection M is called a Morse decomposition of S. The total order on Mis called an admissible (total) ordering of the Morse decomposition. Associated to an admissible ordering of a Morse decomposition there is a distinguished collection of compact invariant subsets of S. This collection, which includes the Morse decomposition, is called the collection of Morse sets of the admissible ordering. Using an index filtration for an admissible ordering of a Morse decomposition Conley and Zehnder [2] exhibit algebraic relationships between the Conley indices of the associated Morse sets.
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