The Experts below are selected from a list of 2988 Experts worldwide ranked by ideXlab platform
Joao P Nunes - One of the best experts on this subject based on the ideXlab platform.
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complex time evolution in geometric quantization and generalized coherent state transforms
Journal of Functional Analysis, 2013Co-Authors: William D Kirwin, J M Mourao, Joao P NunesAbstract:Abstract For the coTangent bundle T ⁎ K of a compact Lie group K, we study the complex-time evolution of the Vertical Tangent bundle and the associated geometric quantization Hilbert space L 2 ( K ) under an infinite-dimensional family of Hamiltonian flows. For each such flow, we construct a generalized coherent state transform (CST), which is a unitary isomorphism between L 2 ( K ) and a certain weighted L 2 -space of holomorphic functions. For a particular set of choices, we show that this isomorphism is naturally decomposed as a product of a Heisenberg-type evolution (for complex time −τ) within L 2 ( K ) , followed by a polarization-changing geometric-quantization evolution (for complex time +τ). In this case, our construction yields the usual generalized Segal–Bargmann transform of Hall. We show that the infinite-dimensional family of Hamiltonian flows can also be understood in terms of Thiemannʼs “complexifier” method (which generalizes the construction of adapted complex structures).
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complex time evolution in geometric quantization and generalized coherent state transforms
arXiv: Differential Geometry, 2012Co-Authors: William D Kirwin, J M Mourao, Joao P NunesAbstract:For the coTangent bundle $T^{*}K$ of a compact Lie group $K$, we study the complex-time evolution of the Vertical Tangent bundle and the associated geometric quantization Hilbert space $L^{2}(K)$ under an infinite-dimensional family of Hamiltonian flows. For each such flow, we construct a generalized coherent state transform (CST), which is a unitary isomorphism between $L^{2}(K)$ and a certain weighted $L^{2}$-space of holomorphic functions. For a particular set of choices, we show that this isomorphism is naturally decomposed as a product of a Heisenberg-type evolution (for complex time $-\tau$) within $L^{2}(K)$, followed by a polarization--changing geometric quantization evolution (for complex time $+\tau$). In this case, our construction yields the usual generalized Segal--Bargmann transform of Hall. We show that the infinite-dimensional family of Hamiltonian flows can also be understood in terms of Thiemann's "complexifier" method (which generalizes the construction of adapted complex structures). We will also investigate some properties of the generalized CSTs, and discuss how their existence can be understood in terms of Mackey's generalization of the Stone-von Neumann theorem.
Sağlamer, Ayse Funda - One of the best experts on this subject based on the ideXlab platform.
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? -Recurrent Sasakian Finsler Structures
2018Co-Authors: Çalişkan Nesrin, Sağlamer, Ayse FundaAbstract:In this paper, ? -recurrent Sasakian Finsler structures on horizontal and Vertical Tangent bundlediffusions and their various geometric properties are studied
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phi-Recurrent Sasakian Finsler Structures
INT ELECTRONIC JOURNAL GEOMETRY, 2018Co-Authors: Özbir, Nesrin Çalışkan, Sağlamer, Ayse FundaAbstract:WOS: 000448884100007In this paper, phi-recurrent Sasakian Finsler structures on horizontal and Vertical Tangent bundle diffusions and their various geometric properties are studied
William D Kirwin - One of the best experts on this subject based on the ideXlab platform.
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complex time evolution in geometric quantization and generalized coherent state transforms
Journal of Functional Analysis, 2013Co-Authors: William D Kirwin, J M Mourao, Joao P NunesAbstract:Abstract For the coTangent bundle T ⁎ K of a compact Lie group K, we study the complex-time evolution of the Vertical Tangent bundle and the associated geometric quantization Hilbert space L 2 ( K ) under an infinite-dimensional family of Hamiltonian flows. For each such flow, we construct a generalized coherent state transform (CST), which is a unitary isomorphism between L 2 ( K ) and a certain weighted L 2 -space of holomorphic functions. For a particular set of choices, we show that this isomorphism is naturally decomposed as a product of a Heisenberg-type evolution (for complex time −τ) within L 2 ( K ) , followed by a polarization-changing geometric-quantization evolution (for complex time +τ). In this case, our construction yields the usual generalized Segal–Bargmann transform of Hall. We show that the infinite-dimensional family of Hamiltonian flows can also be understood in terms of Thiemannʼs “complexifier” method (which generalizes the construction of adapted complex structures).
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complex time evolution in geometric quantization and generalized coherent state transforms
arXiv: Differential Geometry, 2012Co-Authors: William D Kirwin, J M Mourao, Joao P NunesAbstract:For the coTangent bundle $T^{*}K$ of a compact Lie group $K$, we study the complex-time evolution of the Vertical Tangent bundle and the associated geometric quantization Hilbert space $L^{2}(K)$ under an infinite-dimensional family of Hamiltonian flows. For each such flow, we construct a generalized coherent state transform (CST), which is a unitary isomorphism between $L^{2}(K)$ and a certain weighted $L^{2}$-space of holomorphic functions. For a particular set of choices, we show that this isomorphism is naturally decomposed as a product of a Heisenberg-type evolution (for complex time $-\tau$) within $L^{2}(K)$, followed by a polarization--changing geometric quantization evolution (for complex time $+\tau$). In this case, our construction yields the usual generalized Segal--Bargmann transform of Hall. We show that the infinite-dimensional family of Hamiltonian flows can also be understood in terms of Thiemann's "complexifier" method (which generalizes the construction of adapted complex structures). We will also investigate some properties of the generalized CSTs, and discuss how their existence can be understood in terms of Mackey's generalization of the Stone-von Neumann theorem.
Voglaire Yannick - One of the best experts on this subject based on the ideXlab platform.
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Atiyah classes and dg-Lie algebroids for matched pairs
2017Co-Authors: Batakidis Panagiotis, Voglaire YannickAbstract:For every Lie pair $(L,A)$ of algebroids we construct a dg-manifold structure on the $\ZZ$-graded manifold $\M=L[1]\oplus L/A$ such that the inclusion $\iota: A[1] \to \M$ and the projection $p:\M\to L[1]$ are morphisms of dg-manifolds. The Vertical Tangent bundle $T^p\M$ then inherits a structure of dg-Lie algebroid over $\M$. When the Lie pair comes from a matched pair of Lie algebroids, we show that the inclusion $\iota$ induces a quasi-isomorphism that sends the Atiyah class of this dg-Lie algebroid to the Atiyah class of the Lie pair. We also show how (Atiyah classes of) Lie pairs and dg-Lie algebroids give rise to (Atiyah classes of) dDG-algebras
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Atiyah classes and dg-Lie algebroids for matched pairs
'Elsevier BV', 2017Co-Authors: Batakidis Panagiotis, Voglaire YannickAbstract:For every Lie pair $(L,A)$ of algebroids we construct a dg-manifold structure on the $\mathbb{Z}$-graded manifold $\mathcal M=L[1]\oplus L/A$ such that the inclusion $\iota: A[1] \to \mathcal M$ and the projection $p:\mathcal M\to L[1]$ are morphisms of dg-manifolds. The Vertical Tangent bundle $T^p\mathcal M$ then inherits a structure of dg-Lie algebroid over $\mathcal M$. When the Lie pair comes from a matched pair of Lie algebroids, we show that the inclusion $\iota$ induces a quasi-isomorphism that sends the Atiyah class of this dg-Lie algebroid to the Atiyah class of the Lie pair. We also show how (Atiyah classes of) Lie pairs and dg-Lie algebroids give rise to (Atiyah classes of) dDG-algebras.Comment: 22 pages, to appear in J. Geom. Phy
Zhao Xinyu - One of the best experts on this subject based on the ideXlab platform.
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Spatially quasi-periodic water waves of infinite depth
2021Co-Authors: Wilkening Jon, Zhao XinyuAbstract:We formulate the two-dimensional gravity-capillary water wave equations in a spatially quasi-periodic setting and present a numerical study of solutions of the initial value problem. We propose a Fourier pseudo-spectral discretization of the equations of motion in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on a torus. We adopt a conformal mapping formulation and employ a quasi-periodic version of the Hilbert transform to determine the normal velocity of the free surface. Two methods of time-stepping the initial value problem are proposed, an explicit Runge-Kutta (ERK) method and an exponential time-differencing (ETD) scheme. The ETD approach makes use of the small-scale decomposition to eliminate stiffness due to surface tension. We perform a convergence study to compare the accuracy and efficiency of the methods on a traveling wave test problem. We also present an example of a periodic wave profile containing Vertical Tangent lines that is set in motion with a quasi-periodic velocity potential. As time evolves, each wave peak evolves differently, and only some of them overturn. Beyond water waves, we argue that spatial quasi-periodicity is a natural setting to study the dynamics of linear and nonlinear waves, offering a third option to the usual modeling assumption that solutions either evolve on a periodic domain or decay at infinity.Comment: 32 pages, 4 figures; expanded introduction, added two appendice
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Spatially quasi-periodic water waves of infinite depth
2020Co-Authors: Jo Wilkening, Zhao XinyuAbstract:We formulate the two-dimensional gravity-capillary water wave equations in a spatially quasi-periodic setting and present a numerical study of solutions of the initial value problem. We adopt a conformal mapping formulation and employ a quasi-periodic version of the Hilbert transform to determine the normal velocity of the free surface. Four methods of time-stepping the initial value problem are proposed, two explicit Runge-Kutta (ERK) methods and two exponential time-differencing (ETD) schemes. The ETD approach makes use of the small-scale decomposition to eliminate stiffness due to surface tension. We perform a convergence study to compare the accuracy and efficiency of the methods on a traveling wave test problem. We also present an example of a periodic wave profile containing Vertical Tangent lines that is set in motion with a quasi-periodic velocity potential. As time evolves, some of the waves overturn while others flatten out. Beyond water waves, we argue that spatial quasi-periodicity is a natural setting to study the dynamics of linear and nonlinear waves, offering a third option to the usual modeling assumption that solutions either evolve on a periodic domain or decay at infinity.Comment: 26 pages, 4 figure
