Topic
Orientability
About: Orientability is a research topic. Over the lifetime, 325 publications have been published within this topic receiving 5563 citations. The topic is also known as: orientable manifold & orientation.
Papers published on a yearly basis
Papers
Book•
25 Mar 2002
TL;DR: In this article, the Symplectic Group Sp(2n) and its subsets have been studied in the context of index functions and Morse indices, and the Rabinowitz Conjecture has been shown to be equivalent to the dual action principle.
Abstract: I The Symplectic Group Sp(2n).- 1 Algebraic Aspects.- 1.1 Symplectic matrices.- 1.2 Symplectic spaces.- 1.3 Eigenvalues of symplectic matrices.- 1.4 Normal forms for the eigenvalue 1.- 1.5 Normal forms for the eigenvalue ?1.- 1.6 Normal forms for eigenvalues in U?R.- 1.7 Normal forms for eigenvalues outside U.- 1.8 Basic normal forms.- 1.9 Perturbations basic normal forms.- 2 Topological Aspects.- 2.1 Structures of Sp(2) and its subsets.- 2.2 The global structure of Sp(2n,R).- 2.3 Hyperbolic symplectic matrix set.- 2.4 Structure of regular sets.- 2.5 Structures of singular sets.- 2.6 Transversality of rotation paths.- 2.7 Orientability of M?,(2n) in Sp(2n).- II The Variational Method.- 3 Hamiltonian Systems and Canonical Transformations.- 3.1 Canonical transformations.- 3.2 Generating functions.- 4 The Variational Functional.- 4.1 The Galerkin approximation.- 4.2 The L2-Variational Structure.- 4.3 The saddle point reduction.- 4.4 The dimension theorem on kernels.- 4.5 Certain estimates.- III Index Theory.- 5 Index Functions for Symplectic Paths.- 5.1 Paths in Sp(2).- 5.2 Non-degenerate paths in Sp(2n).- 5.3 Index properties of non-degenerate paths.- 5.4 Perturbations of degenerate paths.- 6 Properties of Index Functions.- 6.1 Index functions and Morse indices.- 6.2 An axiom approach and further properties.- 7 Relations with other Morse Indices.- 7.1 The Galerkin approximation.- 7.2 Second order Hamiltonian systems.- 7.3 Lagrangian systems.- IV Iteration Theory.- 8 Precise Iteration Formulae.- 8.1 Paths in Sp(2).- 8.2 Hyperbolic and elliptic paths.- 8.3 General symplectic paths.- 9 Bott-type Iteration Formulae.- 9.1 Splitting numbers.- 9.2 Bott-type formulae.- 9.3 Abstract precise iteration formulae.- 10 Iteration Inequalities.- 10.1 Estimates via mean index and initial index.- 10.2 Successive estimates.- 10.3 Controlling iteration numbers via indices.- 11 The Common Index Jump Theorem.- 11.1 A common selection theorem.- 11.2 The common index jump theorem.- 12 Index Iteration Theory for Closed Geodesics.- 12.1 Morse index theory.- 12.2 Splitting numbers.- V Applications.- 13 The Rabinowitz Conjecture.- 13.1 Minimax principle preparations.- 13.2 Controlling the minimal period via indices.- 13.3 Asymptotically linear Hamiltonian systems.- 13.4 Superquadratic Hamiltonian systems.- 13.5 Second order systems.- 13.6 Subharmonics.- 13.7 Notes and comments.- 14 Periodic Lagrangian Orbits on Tori.- 14.1 Critical module preparations.- 14.2 The finite energy homology theory.- 14.3 Critical modules and isomorphisms.- 14.4 Global homological injectivity.- 14.5 Global homological vanishing.- 14.6 Notes and comments.- 15 Closed Characteristics on Convex Hypersurfaces.- 15.1 Index theorem for dual action principle.- 15.2 Variational properties.- 15.3 Critical orbits and index jumps.- 15.4 Existence and multiplicity.- 15.5 Stability results.- 15.6 Symmetric hypersurfaces.- 15.7 Notes and comments.
562 citations
TL;DR: The main result is that ordered topological models are (roughly speaking) equivalent with respect to the class of objects which can be modelled (i.e. withrespect to dimension and orientability).
Abstract: In boundary representation, a geometric object is represented by the union of a ‘topological’ model, which describes the topology of the modelled object, and an ‘embedding’ model, which describes the embedding of the object, for instance in three-dimensional Euclidean space. In recent years, numerous topological models have been developed for boundary representation, and there have been important developments with respect to dimension and orientability. In the main, two types of topological models can be distinguished. ‘Incidence graphs’ are graphs or hypergraphs, where the nodes generally represent the cells of the modelled subdivision (vertex, edge, face, etc.), and the edges represent the adjacency and incidence relations between these cells. ‘Ordered’ models use a single type of basic element (more or less explicitly defined), on which ‘element functions’ act; the cells of the modelled subdivision are implicitly defined in this type of model. In this paper some of the most representative ordered topological models are compared using the concepts of the n- dimensional generalized map and the n- dimensional map. The main result is that ordered topological models are (roughly speaking) equivalent with respect to the class of objects which can be modelled (i.e. with respect to dimension and orientability).
335 citations
TL;DR: In this paper it is shown that every coordinatization (representation) of a matroid over an ordered field induces an orientation of the matroid, and that every unimodular matroid has an orientation that is induced by a coordinatisation and is unique in a certain straightforward sense.
262 citations
Book•
1 Jan 1998TL;DR: In this article, Thom Spectra and (Co)homology theories are used for orientability and orientations, including K- and KO-Orientability, and complex (Co)-bordism with singularities.
Abstract: Notation, Conventions and Other Preliminaries.- Spectra and (Co)homology Theories.- Phantoms.- Thom Spectra.- Orientability and Orientations.- K- and KO-Orientability.- Complex (Co)bordism.- (Co)bordism with Singularities.- Complex (Co)bordism with Singularities.
204 citations
TL;DR: In this article, the authors study the differences and the overlaps between the Oseen-Frank theory and the constrained Landau-de Gennes theory, and show that for simply-connected domains and in the natural energy class W1,2 the two theories coincide.
Abstract: Uniaxial nematic liquid crystals are modelled in the Oseen–Frank theory through a unit vector field n. This theory has the apparent drawback that it does not respect the head-to-tail symmetry in which n should be equivalent to −n. This symmetry is preserved in the constrained Landau–de Gennes theory that works with the tensor \({Q=s \left(n\otimes n-\frac{1}{3} Id\right)}\). We study the differences and the overlaps between the two theories. These depend on the regularity class used as well as on the topology of the underlying domain. We show that for simply-connected domains and in the natural energy class W1,2 the two theories coincide, but otherwise there can be differences between the two theories, which we identify. In the case of planar domains with holes and various boundary conditions, for the simplest form of the energy functional, we completely characterise the instances in which the predictions of the constrained Landau–de Gennes theory differ from those of the Oseen–Frank theory.
191 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 13 |
| 2020 | 15 |
| 2019 | 10 |
| 2018 | 13 |
| 2017 | 16 |
| 2016 | 14 |