Lectures on Twisted Rabinowitz-Floer Homology

Rabinowitz-Floer homology is the Morse-Bott homology in the sense of Floer associated with the Rabinowitz action functional introduced by Kai Cieliebak and Urs Frauenfelder in 2009. In this manuscript, we consider a generalisation of this theory to a Rabinowitz-Floer homology of a Liouville automorphism. As an application, we show the existence of noncontractible periodic Reeb orbits on quotients of symmetric star-shaped hypersurfaces. In particular, this theory applies to lens spaces. Moreover, we prove a forcing theorem, which guarantees the existence of a contractible twisted closed characteristic on a displaceable twisted stable hypersurface in a symplectically aspherical geometrically bounded symplectic manifold if there exists a contractible twisted closed characteristic belonging to a Morse-Bott component, with energy difference smaller or equal to the displacement energy of the displaceable hypersurface.Comment: 95 pages, 14 figures. arXiv admin note: text overlap with arXiv:2105.1393

First steps in twisted Rabinowitz-Floer homology

Rabinowitz-Floer homology is the Morse-Bott homology in the sense of Floer associated with the Rabinowitz action functional introduced by Kai Cieliebak and Urs Frauenfelder in 2009. In our work, we consider a generalisation of this theory to a Rabinowitz-Floer homology of a Liouville automorphism. As an application, we show the existence of noncontractible periodic Reeb orbits on quotients of symmetric star-shaped hypersurfaces. In particular, our theory applies to lens spaces. Moreover, we show a forcing theorem, which guarantees the existence of a contractible twisted closed characteristic on a displaceable twisted stable hypersurface in a symplectically aspherical geometrically bounded symplectic manifold if there exists a contractible twisted closed characteristic belonging to a Morse-Bott component, with energy difference smaller or equal to the displacement energy of the displaceable hypersurface

Rabinowitz Floer homology and coisotropic intersections

학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2014. 2. Urs Frauenfelder.Urs Frauenfelder와 Kai Cieliebak은 Paul Rabinowitz가 자율적 해밀턴 시스템에서 주기궤도들 찾기 위해 제안한 라그랑즈 승수 함수를 사용하여 Rabinowitz Floer homology 이론을 개발하였다. 이 논문에서는 우리는 임의의 여차원을 가지는 여등방성 부분다양체 위의 역학구조를 분석하는데 적합한 여러개의 Lagrange 상수들을 가지는 일반화된 Rabinowitz 함수를 연구할 것이다. 우리는 일반화된 Rabinowitz 함수를 사용하여 여등방성 궤적 교차점, 여등방성 부분 다양체의 전치가능성, 그리고 여등방성 부분다양체의 Rabinowitz Floer homology 등에 관해 연구할 것이다. 우리는 또한 Rabinowitz Floer homology의 Künneth 공식을 유도하여 무한개의 여등방 궤적 교차점을 가지는 여등방성 부분다양체들을 찾을 것이다. 이 연구는 여러 개의 운동 상수 (보존량) 를 가지는 운동 시스템을 연구하는데 중요한 역할을 할 것이다.Rabinowitz Floer homology theory was developed by Kai Cieliebak and Urs Frauenfelder using a Lagrange multiplier action functional, which was introduced by Paul Rabinowitz in order to detect periodic orbits of autonomous Hamiltonian systems. In this thesis, we study a generalized Rabinowitz action functional with several Lagrange multipliers, which is well suited for exploring dynamics on coisotropic submanifolds of arbitrary codimensions. Using this, we investigate among others, the existence problem of leafwise coisotropic intersection points, displaceability of coisotropic submanifolds, and Rabinowitz Floer homology for coisotropic submanifolds. We also derive a Künneth formula for the Rabinowitz Floer homology of product coisotropic submanifolds, and this enables us to find a class of coisotropic submanifolds which have infinitely many leafwise coisotropic intersection points. This study will serve as a crucial tool for exploring autonomous dynamical systems with several integrals.Abstract i 1 Preliminaries on symplectic geometry 1 1.1 Hamiltonian dieomorphisms 2 1.2 Coisotropic submanifolds 3 1.3 Examples of contact coisotropic submanifolds 9 2 Statement of the results 14 2.1 Assumptions on manifolds 15 2.2 Main theorem 17 2.3 Leafwise coisotropic intersections 18 2.4 Leafwise displacement energy 22 2.5 Rabinowitz Floer homology 23 2.6 Künneth formula 25 2.7 List of related results 27 3 The Rabinowitz action functional with several Lagrange multipliers 28 3.1 The Rabinowitz action functional for coisotropic submanifolds 28 3.2 The perturbed Rabinowitz action functional 30 3.2.1 Compactness 34 3.3 Proof of Theorem A 42 4 The existence of a periodic orbit and the leafwise displacement energy 49 4.1 Proof of Theorem D 50 5 Rabinowitz Floer homology 53 5.1 Boundary Operator 54 5.2 Continuation Homomorphism 58 5.3 Proof of Theorem E 60 5.4 Filtered Rabinowitz Floer Homology 61 5.5 Proof of Theorem B 62 5.6 Proof of Theorem C 65 6 Künneth formula in Rabinowitz Floer homology 69 6.1 Rabinowitz action functional for product manifolds 69 6.1.1 Compactness 71 6.2 Proof of Theorem F 74 6.3 Proof of Theorem G 78 7 Infinitely many leafwise intersection points 83 7.1 Proofs of Corollary F and Corollary G 83 Abstract (in Korean) 95 Acknowledgement (in Korean) 97Docto

Control Theory: On the Way to New Application Fields

Control theory is an interdisciplinary field that is located at the crossroads of pure and applied mathematics with systems engineering and the sciences. Recently, deep interactions are emerging with new application areas, such as systems biology, quantum control and information technology. In order to address the new challenges posed by the new application disciplines, a special focus of this workshop has been on the interaction between control theory and mathematical systems biology. To complement these more biology oriented focus, a series of lectures in this workshop was devoted to the control of networks of systems, fundamentals of nonlinear control systems, model reduction and identification, algorithmic aspects in control, as well as open problems in control

An Investigation of Cylindrical Liner Z-pinches as Drivers for Converging Strong Shock Experiments

A cylindrical liner z-pinch configuration has been used to drive converging radia- tive shock waves into different gases. Experiments were carried out on the MAGPIE (1.4 MA, 250 ns rise-time) pulsed-power device at Imperial College London [1]. On application of the current pulse, a series of cylindrical shocks moving at typical velocities of 20 km s-1 are consecutively launched from the inside liner wall into an initially static gas- ll. The drive current skin depth calculated prior to resis- tive heating was slightly less than the liner wall thickness and no bulk motion of the liner occurred. Axial laser probing images show the shock fronts to be smooth and azimuthally symmetric, with instabilities developing downstream of each shock. Evidence for a radiative precursor ahead of the first shock was seen in laser inter- ferometry imaging and time-gated spatially resolved optical spectroscopy. In addition to investigating the shock waves themselves, the timing of the shocks was used together with their trajectories to gain insight into launching mechanisms. This provided information on the response of the liner to the current pulse, which is useful for the benchmarking of magneto-hydrodynamics (MHD) codes. A new load voltage diagnostic provided evidence for two phase transitions occurring within the liner wall. The voltage probe was also fielded on various other z-pinch loads for measurements of energy deposition and inductance. The response of magnetically thick liners was found to differ significantly from the case where the liner wall was thin with respect to the initial skin depth of the current. In the later case the evolution of the liner is dominated by the ablation of plasma much like during the ablation phase of a wire array z-pinch

Research reports: 1985 NASA/ASEE Summer Faculty Fellowship Program

A compilation of 40 technical reports on research conducted by participants in the 1985 NASA/ASEE Summer Faculty Fellowship Program at Marshall Space Flight Center (MSFC) is given. Weibull density functions, reliability analysis, directional solidification, space stations, jet stream, fracture mechanics, composite materials, orbital maneuvering vehicles, stellar winds and gamma ray bursts are among the topics discussed

Annual Research Report, 2009-2010

Annual report of collaborative research projects of Old Dominion University faculty and students in partnership with business, industry and governmenthttps://digitalcommons.odu.edu/or_researchreports/1001/thumbnail.jp