Rabinowitz Floer Homology for Tentacular Hamiltonians

2020 ◽  
Author(s):  
Federica Pasquotto ◽  
Robert Vandervorst ◽  
Jagna Wiśniewska
Keyword(s):  
Periodic Orbits ◽  
Moduli Spaces ◽  
Level Sets ◽  
General Framework ◽  
Floer Homology ◽  
Contact Manifolds ◽  
Weinstein Conjecture ◽  
Rabinowitz Floer Homology ◽  
Closed Characteristics ◽  
Definition Of

Abstract This paper extends the definition of Rabinowitz Floer homology to non-compact hypersurfaces. We present a general framework for the construction of Rabinowitz Floer homology in the non-compact setting under suitable compactness assumptions on the periodic orbits and the moduli spaces of Floer trajectories. We introduce a class of hypersurfaces arising as the level sets of specific Hamiltonians: strongly tentacular Hamiltonians for which the compactness conditions are satisfied, cf. [ 21], thus enabling us to define the Rabinowitz Floer homology for this class. Rabinowitz Floer homology in turn serves as a tool to address the Weinstein conjecture and establish existence of closed characteristics for non-compact contact manifolds.

scholarly journals Symplectic cobordisms and the strong Weinstein conjecture

2012 ◽  
Vol 153 (2) ◽  
pp. 261-279 ◽  
Author(s):  
HANSJÖRG GEIGES ◽  
KAI ZEHMISCH
Keyword(s):  
Moduli Spaces ◽  
Quantitative Character ◽  
Contact Manifolds ◽  
Symplectic Capacity ◽  
Contact Type ◽  
Weinstein Conjecture ◽  
Cotangent Bundles ◽  
Higher Dimensional ◽  
Definition Of ◽  
Symplectic Cobordisms

AbstractWe study holomorphic spheres in certain symplectic cobordisms and derive information about periodic Reeb orbits in the concave end of these cobordisms from the non-compactness of the relevant moduli spaces. We use this to confirm the strong Weinstein conjecture (predicting the existence of null-homologous Reeb links) for various higher-dimensional contact manifolds, including contact type hypersurfaces in subcritical Stein manifolds and in some cotangent bundles. The quantitative character of this result leads to the definition of a symplectic capacity.

scholarly journals Translated points on hypertight contact manifolds

2018 ◽  
Vol 10 (02) ◽  
pp. 289-322
Author(s):  
Matthias Meiwes ◽  
Kathrin Naef
Keyword(s):  
Floer Homology ◽  
Contact Manifold ◽  
Contact Form ◽  
Contact Manifolds ◽  
Rabinowitz Floer Homology

A contact manifold admitting a supporting contact form without contractible Reeb orbits is called hypertight. In this paper we construct a Rabinowitz–Floer homology associated to an arbitrary supporting contact form for a hypertight contact manifold [Formula: see text], and use this to prove versions of a conjecture of Sandon [17] on the existence of translated points and to show that positive loops of contactomorphisms give rise to non-contractible Reeb orbits.

scholarly journals Bounds for tentacular Hamiltonians

2018 ◽  
Vol 12 (01) ◽  
pp. 209-265 ◽  
Author(s):  
Federica Pasquotto ◽  
Jagna Wiśniewska
Keyword(s):  
Floer Homology ◽  
Energy Levels ◽  
Symplectic Manifolds ◽  
Rabinowitz Floer Homology ◽  
The Subject ◽  
Definition Of

This paper represents a first step toward the extension of the definition of Rabinowitz Floer homology to non-compact energy hypersurfaces in exact symplectic manifolds. More concretely, we study under which conditions it is possible to establish [Formula: see text]-bounds for the Floer trajectories of a Hamiltonian with non-compact energy levels. Moreover, we introduce a class of Hamiltonians, called tentacular Hamiltonians which satisfy the conditions: how to define Rabinowitz Floer homology for these examples will be the subject of a follow-up paper.

scholarly journals Computing the Rabinowitz Floer homology of tentacular hyperboloids

2021 ◽  
Vol 17 (0) ◽  
pp. 353
Author(s):  
Alexander Fauck ◽  
Will J. Merry ◽  
Jagna Wiśniewska
Keyword(s):  
Floer Homology ◽  
Homology Groups ◽  
Weinstein Conjecture ◽  
Singular Homology ◽  
Rabinowitz Floer Homology ◽  
Compact Case ◽  
Chain Map ◽  
A Chain

<p style='text-indent:20px;'>We compute the Rabinowitz Floer homology for a class of non-compact hyperboloids <inline-formula><tex-math id="M1">\begin{document}$ \Sigma\simeq S^{n+k-1}\times\mathbb{R}^{n-k} $\end{document}</tex-math></inline-formula>. Using an embedding of a compact sphere <inline-formula><tex-math id="M2">\begin{document}$ \Sigma_0\simeq S^{2k-1} $\end{document}</tex-math></inline-formula> into the hypersurface <inline-formula><tex-math id="M3">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula>, we construct a chain map from the Floer complex of <inline-formula><tex-math id="M4">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula> to the Floer complex of <inline-formula><tex-math id="M5">\begin{document}$ \Sigma_0 $\end{document}</tex-math></inline-formula>. In contrast to the compact case, the Rabinowitz Floer homology groups of <inline-formula><tex-math id="M6">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula> are both non-zero and not equal to its singular homology. As a consequence, we deduce that the Weinstein Conjecture holds for any strongly tentacular deformation of such a hyperboloid.</p>

Diagnosis Rule Discovery Based on Causality in the Context of Fault Diagnosis in Rotating Machinery

2014 ◽  
Vol 532 ◽  
pp. 113-117
Author(s):  
Zhou Jin ◽  
Ru Jing Wang ◽  
Jie Zhang
Keyword(s):  
Fault Diagnosis ◽  
General Framework ◽  
Complex Structure ◽  
Rotating Machinery ◽  
Massive Data ◽  
Formal Definition ◽  
Rule Discovery ◽  
Rule Based ◽  
Simple Implementation ◽  
Definition Of

The rotating machineries in a factory usually have the characteristics of complex structure and highly automated logic, which generated a large amounts of monitoring data. It is an infeasible task for uses to deal with the massive data and locate fault timely. In this paper, we explore the causality between symptom and fault in the context of fault diagnosis in rotating machinery. We introduce data mining into fault diagnosis and provide a formal definition of causal diagnosis rule based on statistic test. A general framework for diagnosis rule discovery based on causality is provided and a simple implementation is explored with the purpose of providing some enlightenment to the application of causality discovery in fault diagnosis of rotating machinery.

scholarly journals Parallel Robot Translational Performance Evaluation through Direction-Selective Index (DSI)

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
G. Boschetti ◽  
R. Rosa ◽  
A. Trevisani
Keyword(s):  
Performance Evaluation ◽  
Performance Index ◽  
General Framework ◽  
Dynamic Performance ◽  
Parallel Manipulators ◽  
Parallel Robot ◽  
Performance Indexes ◽  
Definition Of ◽  
Performance Variations

Performance indexes usually provide global evaluations of robot performances mixing their translational and/or rotational capabilities. This paper proposes a definition of performance index, called direction-selective index (DSI), which has been specifically developed for parallel manipulators and can provide uncoupled evaluations of robot translational capabilities along relevant directions. The DSI formulation is first presented within a general framework, highlighting its relationship with traditional manipulability definitions, and then applied to a family of parallel manipulators (4-RUU) of industrial interest. The investigation is both numerical and experimental and allows highlighting the two chief advantages of the proposed DSIs over more conventional manipulability indexes: not only are DSIs more accurate in predicting the workspace regions where manipulators can best perform translational movements along specific directions, but also they allow foreseeing satisfactorily the dynamic performance variations within the workspace, though being purely kinematic indexes. The experiments have been carried out on an instrumented 4-RUU commercial robot.

scholarly journals ESTIMATES AND COMPUTATIONS IN RABINOWITZ–FLOER HOMOLOGY

2009 ◽  
Vol 01 (04) ◽  
pp. 307-405 ◽  
Author(s):  
ALBERTO ABBONDANDOLO ◽  
MATTHIAS SCHWARZ
Keyword(s):  
Maximum Principle ◽  
Energy Level ◽  
Exact Sequence ◽  
Floer Homology ◽  
Energy Functional ◽  
Convex Compact ◽  
Closed Geodesics ◽  
Action Functional ◽  
Uniform Estimates ◽  
Rabinowitz Floer Homology

The Rabinowitz–Floer homology of a Liouville domain W is the Floer homology of the Rabinowitz free period Hamiltonian action functional associated to a Hamiltonian whose zero energy level is the boundary of W. This invariant has been introduced by K. Cieliebak and U. Frauenfelder and has already found several applications in symplectic topology and in Hamiltonian dynamics. Together with A. Oancea, the same authors have recently computed the Rabinowitz–Floer homology of the cotangent disk bundle D* M of a closed Riemannian manifold M, by means of an exact sequence relating the Rabinowitz–Floer homology of D* M with its symplectic homology and cohomology. The first aim of this paper is to present a chain level construction of this exact sequence. In fact, we show that this sequence is the long homology sequence induced by a short exact sequence of chain complexes, which involves the Morse chain complex and the Morse differential complex of the energy functional for closed geodesics on M. These chain maps are defined by considering spaces of solutions of the Rabinowitz–Floer equation on half-cylinders, with suitable boundary conditions which couple them with the negative gradient flow of the geodesic energy functional. The second aim is to generalize this construction to the case of a fiberwise uniformly convex compact subset W of T* M whose interior part contains a Lagrangian graph. Equivalently, W is the energy sublevel associated to an arbitrary Tonelli Lagrangian L on TM and to any energy level which is larger than the strict Mañé critical value of L. In this case, the energy functional for closed geodesics is replaced by the free period Lagrangian action functional associated to a suitable calibration of L. An important issue in our analysis is to extend the uniform estimates for the solutions of the Rabinowitz–Floer equation — both on cylinders and on half-cylinders — to Hamiltonians which have quadratic growth in the momenta. These uniform estimates are obtained by the Aleksandrov integral version of the maximum principle. In the case of half-cylinders, they are obtained by an Aleksandrov-type maximum principle with Neumann conditions on part of the boundary.

scholarly journals A General Framework for Implicit and Explicit Debiasing of Distributional Word Vector Spaces

2020 ◽  
Vol 34 (05) ◽  
pp. 8131-8138
Author(s):  
Anne Lauscher ◽  
Goran Glavaš ◽  
Simone Paolo Ponzetto ◽  
Ivan Vulić
Keyword(s):  
General Framework ◽  
Semantic Information ◽  
Evaluation Framework ◽  
Vector Spaces ◽  
Racial Biases ◽  
Definition Of ◽  
Cross Lingual ◽  
Experimental Findings ◽  
Implicit And Explicit ◽  
Embedding Methods

Distributional word vectors have recently been shown to encode many of the human biases, most notably gender and racial biases, and models for attenuating such biases have consequently been proposed. However, existing models and studies (1) operate on under-specified and mutually differing bias definitions, (2) are tailored for a particular bias (e.g., gender bias) and (3) have been evaluated inconsistently and non-rigorously. In this work, we introduce a general framework for debiasing word embeddings. We operationalize the definition of a bias by discerning two types of bias specification: explicit and implicit. We then propose three debiasing models that operate on explicit or implicit bias specifications and that can be composed towards more robust debiasing. Finally, we devise a full-fledged evaluation framework in which we couple existing bias metrics with newly proposed ones. Experimental findings across three embedding methods suggest that the proposed debiasing models are robust and widely applicable: they often completely remove the bias both implicitly and explicitly without degradation of semantic information encoded in any of the input distributional spaces. Moreover, we successfully transfer debiasing models, by means of cross-lingual embedding spaces, and remove or attenuate biases in distributional word vector spaces of languages that lack readily available bias specifications.

scholarly journals Orderability, contact non-squeezing, and Rabinowitz Floer homology

2018 ◽  
Vol 16 (6) ◽  
pp. 1481-1547
Author(s):  
Peter Albers ◽  
Will J. Merry
Keyword(s):  
Floer Homology ◽  
Rabinowitz Floer Homology
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