Showing papers on "Equivalence class published in 1973"

On the Classification of Flows on 2-Manifolds†

Abstract: Publisher Summary This chapter presents the classification of flows on 2-manifolds. The set Σ of all structurally stable flows is open and dense in an approximation problem. The chapter presents a classification of all equivalent classes of Σ modulo Σ, thus, completing the solution of the fundamental problem. This is done by establishing a one-to-one correspondence between these equivalence classes and certain distinguished graphs, that is, graphs together with a distinguished set of edges satisfying some conditions. This results in a precise rule for labeling all equivalence classes of Σ modulo ˜ in such a way that each equivalence class appears exactly once in the labeling process. The chapter also discusses the concept of structurally stable flows, the graph of a gradient-like flow, the nonorientable case, without closed orbit, the orientable case with closed orbits, and the nonorientable case with closed orbits.

On the domains of generalized operators

Abstract: The domains of generalized operatorsT: Φ−→Φ+ on rigged Hilbert spaces Φ− ⊂H ⊂ Φ+ are investigated. We introduce an equivalence relation for operators with different domains. Arguments are given for taking Φ+ to be the weak quasi-completion of Φ− and for Φ− to be Mackey quasi-complete. For domains of closed symmetric Hilbert space operators we give a representation for Φ+ and provide certain elements in the equivalence class of the corresponding operator.