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Equivalence relation
About: Equivalence relation is a research topic. Over the lifetime, 5096 publications have been published within this topic receiving 80250 citations.
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01 Jan 1995TL;DR: The Cartan-Kahler existence theorem as discussed by the authors is based on the Cartan's equivalence method, which is a generalization of the Frobenius' theorem of Cartan and Kuhn.
Abstract: 1 Geometric foundations 2 Lie groups 3 Representation theory 4 Jets and contact transformations 5 Differential invariants 6 Symmetries of differential equations 7 Symmetries of variational problems 8 Equivalence of coframes 9 Formulation of equivalence problems 10 Cartan's equivalence method 11 Involution 12 Prolongation of equivalence problems 13 Differential systems 14 Frobenius' theorem 15 The Cartan-Kahler existence theorem
1,078 citations
TL;DR: In this article, the disjoint sum of two connected n-manifolds is obtained by removing a small n-cell from each, and then pasting together the resulting boundaries.
Abstract: DEFINITION. Two closed n-manifolds M, and M2 are h-cobordant1 if the disjoint sum M, + (- M2) is the boundary of some manifold W, where both M1 and (-M2) are deformation retracts of W. It is clear that this is an equivalence relation. The connected sum of two connected n-manifolds is obtained by removing a small n-cell from each, and then pasting together the resulting boundaries. Details will be given in ? 2.
930 citations
TL;DR: The authors prove that, although the state dimension is not preserved, the number of input channels is kept fixed and it is proved that a Lie-Backlund isomorphism can be realized by an endogenous feedback.
Abstract: A new system equivalence relation, using the framework of differential geometry of jets and prolongations of infinite order, is studied. In this setting, two systems are said to be equivalent if any variable of one system may be expressed as a function of the variables of the other system and of a finite number of their time derivatives. This is a Lie-Backlund isomorphism. The authors prove that, although the state dimension is not preserved, the number of input channels is kept fixed. They also prove that a Lie-Backlund isomorphism can be realized by an endogenous feedback. The differentially flat nonlinear systems introduced by the authors (1992) via differential algebraic techniques, are generalized and the new notion of orbitally flat systems is defined. They correspond to systems which are equivalent to a trivial one, with time preservation or not. The endogenous linearizing feedback is explicitly computed in the case of the VTOL aircraft to track given reference trajectories with stability.
742 citations
01 Feb 2013
TL;DR: In this paper, the authors present a survey of Canonical and log canonical singularities and their application in the context of finite equivalence relations, including semi-log-canonical pairs and the Du Bois property.
Abstract: Preface Introduction 1. Preliminaries 2. Canonical and log canonical singularities 3. Examples 4. Adjunction and residues 5. Semi-log-canonical pairs 6. Du Bois property 7. Log centers and depth 8. Survey of further results and applications 9. Finite equivalence relations 10. Appendices References Index.
739 citations