Topic
Transversality
About: Transversality is a research topic. Over the lifetime, 1175 publications have been published within this topic receiving 25736 citations.
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Book•
1 Jan 1973
TL;DR: In this article, the Whitney C? topology is used to classify singularities on 2-manifolds. But the Thom-Boardman invariants are not included in this classification.
Abstract: I: Preliminaries on Manifolds.- 1. Manifolds.- 2. Differentiable Mappings and Submanifolds.- 3. Tangent Spaces.- 4. Partitions of Unity.- 5. Vector Bundles.- 6. Integration of Vector Fields.- II: Transversality.- 1. Sard's Theorem.- 2. Jet Bundles.- 3. The Whitney C? Topology.- 4. Transversality.- 5. The Whitney Embedding Theorem.- 6. Morse Theory.- 7. The Tubular Neighborhood Theorem.- III: Stable Mappings.- 1. Stable and Infinitesimally Stable Mappings.- 2. Examples.- 3. Immersions with Normal Crossings.- 4. Submersions with Folds.- IV: The Malgrange Preparation Theorem.- 1. The Weierstrass Preparation Theorem.- 2. The Malgrange Preparation Theorem.- 3. The Generalized Malgrange Preparation Theorem.- V: Various Equivalent Notions of Stability.- 1. Another Formulation of Infinitesimal Stability.- 2. Stability Under Deformations.- 3. A Characterization of Trivial Deformations.- 4. Infinitesimal Stability => Stability.- 5. Local Transverse Stability.- 6. Transverse Stability.- 7. Summary.- VI: Classification of Singularities, Part I: The Thom-Boardman Invariants.- 1. The Sr Classification.- 2. The Whitney Theory for Generic Mappings between 2-Manifolds.- 3. The Intrinsic Derivative.- 4. The Sr,s Singularities.- 5. The Thom-Boardman Stratification.- 6. Stable Maps Are Not Dense.- VII: Classification of Singularities, Part II: The Local Ring of a Singularity.- 1. Introduction.- 2. Finite Mappings.- 3. Contact Classes and Morin Singularities.- 4. Canonical Forms for Morin Singularities.- 5. Umbilics.- 6. Stable Mappings in Low Dimensions.- A. Lie Groups.- Symbol Index.
1,942 citations
TL;DR: In this paper, the authors introduce a gravitational catastrophe machine and a set of generic properties of curves, such as transversality, normal values, regular values and smooth manifolds.
Abstract: 1. Introductory example: a gravitational catastrophe machine 2. Curves, and functions on them 3. More about functions 4. Regular values and smooth manifolds 5. Envelopes 6. Unfoldings 7. Unfoldings: applications 8. Transversality 9. Generic properties of curves 10. More on unfoldings 11. Singular points, several variables and generic surfaces Appendix: Null sets and Sard's theorem.
569 citations
TL;DR: In this article, the transversality question for perturbed nonlinear Cauchy-Riemann equations on the cylinder was resolved by a continuation theorem derived from a generalization of the Carleman similarity principle.
Abstract: Our goal in this paper is to settle some transversality question for the perturbed nonlinear Cauchy-Riemann equations on the cylinder These results play a central role in the denition of symplectic Floer homology and hence in the proof of the Arnold conjecture There is currently no other reference to these transversality results in the open literature Our approach does not require Aronszajn’s theorem Instead we derive the unique continuation theorem from a generalization of the Carleman similarity principle
411 citations
1 Jan 1993
TL;DR: The Legendre condition as discussed by the authors is a necessary condition for the solution of the Euler equation (16.1), and the Transversality condition is sufficient for the transversality.
Abstract: The Euler equation. A necessary condition for the solution of (16.1). An alternative form of the Euler equation. The Legendre condition. A necessary condition for the solution of (16.1). Sufficient conditions for the solution of (16.1). Transversality condition. Adding condition (16.5) gives sufficient conditions.
391 citations
TL;DR: In this article, the problem of optimal response with nonsmooth (generally speaking, nonfunctional) constraints imposed on the state variables is considered, and the convergence of the approximating problems to the initial problem with constraints is proved, and for general assumptions the necessary conditions of optimality resembling the Pontriagin maximum principle are derived for the generalized solutions of the original problem.
389 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 21 |
| 2024 | 21 |
| 2023 | 60 |
| 2022 | 115 |
| 2021 | 43 |
| 2020 | 59 |