The Experts below are selected from a list of 6702 Experts worldwide ranked by ideXlab platform
Masaaki Yoshida - One of the best experts on this subject based on the ideXlab platform.
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singularities of flat fronts and their caustics and an example arising from the hyperbolic schwarz map of a Hypergeometric Equation
Results in Mathematics, 2009Co-Authors: Takeshi Sasaki, Masaaki YoshidaAbstract:For parallel families of flat fronts and their caustic surfaces in the hyperbolic 3-space, singularities such as cuspidal edges and swallowtails are studied. An example arising from a Hypergeometric differential Equation is closely studied.
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surface singularities appeared in the hyperbolic schwarz map for the Hypergeometric Equation
2009Co-Authors: Takeshi Sasaki, Masaaki YoshidaAbstract:Surface singularities, swallowtail and cuspidal edge, appear in the hyperbolic Schwarz map for the Hypergeometric differential Equation. Such singularities are studied in detail. After an overview of classical staffs, the Hypergeometric Equation and the Schwarz map, the hyperbolic Schwarz map is introduced. We study the singularities of this map, whose target is the hyperbolic 3-space, and visualize its image when the monodromy group is a finite group or a typical Fuchsian group. Several confluences of swallowtails are also observed.
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a family of schottky groups arising from the Hypergeometric Equation
arXiv: Complex Variables, 2004Co-Authors: Takashi Ichikawa, Masaaki YoshidaAbstract:We study a complex 3-dimensional family of classical Schottky groups of genus 2 as monodromy groups of the Hypergeometric Equation. We find non-trivial loops in the deformation space; these correspond to continuous integer-shifts of the parameters of the Equation.
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on schottky groups arising from the Hypergeometric Equation with imaginary exponents
Proceedings of the American Mathematical Society, 2003Co-Authors: Takashi Ichikawa, Masaaki YoshidaAbstract:In an article by Sasaki and Yoshida (2000), we encountered Schottky groups of genus 2 as monodromy groups of the Hypergeometric Equation with purely imaginary exponents. In this paper we study automorphic functions for these Schottky groups, and give a conjectural infinite product formula for the elliptic modular function A.
Takeshi Sasaki - One of the best experts on this subject based on the ideXlab platform.
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singularities of flat fronts and their caustics and an example arising from the hyperbolic schwarz map of a Hypergeometric Equation
Results in Mathematics, 2009Co-Authors: Takeshi Sasaki, Masaaki YoshidaAbstract:For parallel families of flat fronts and their caustic surfaces in the hyperbolic 3-space, singularities such as cuspidal edges and swallowtails are studied. An example arising from a Hypergeometric differential Equation is closely studied.
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surface singularities appeared in the hyperbolic schwarz map for the Hypergeometric Equation
2009Co-Authors: Takeshi Sasaki, Masaaki YoshidaAbstract:Surface singularities, swallowtail and cuspidal edge, appear in the hyperbolic Schwarz map for the Hypergeometric differential Equation. Such singularities are studied in detail. After an overview of classical staffs, the Hypergeometric Equation and the Schwarz map, the hyperbolic Schwarz map is introduced. We study the singularities of this map, whose target is the hyperbolic 3-space, and visualize its image when the monodromy group is a finite group or a typical Fuchsian group. Several confluences of swallowtails are also observed.
Corentin Vallée - One of the best experts on this subject based on the ideXlab platform.
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Multidomain spectral method for the Gauss Hypergeometric function
Numerical Algorithms, 2019Co-Authors: Siegfried Crespo, Marco Fasondini, Christian Klein, Nikola Stoilov, Corentin ValléeAbstract:We present a multidomain spectral approach for Fuchsian ordinary differential Equations in the particular case of the Hypergeometric Equation. Our hybrid approach uses Frobenius’ method and Moebius transformations in the vicinity of each of the singular points of the Hypergeometric Equation, which leads to a natural decomposition of the real axis into domains. In each domain, solutions to the Hypergeometric Equation are constructed via the well-conditioned ultraspherical spectral method. The solutions are matched at the domain boundaries to lead to a solution which is analytic on the whole compactified real line ℝ ∪ ∞ $\mathbb {R}\cup {\infty }$ , except for the singular points and cuts of the Riemann surface on which the solution is defined. The solution is further extended to the whole Riemann sphere by using the same approach for ellipses enclosing the singularities. The Hypergeometric Equation is solved on the ellipses with the boundary data from the real axis. This solution is continued as a harmonic function to the interior of the disk by solving the Laplace Equation in polar coordinates with an optimal complexity Fourier–ultraspherical spectral method. In cases where logarithms appear in the solution, a hybrid approach involving an analytical treatment of the logarithmic terms is applied. We show for several examples that machine precision can be reached for a wide class of parameters, but also discuss almost degenerate cases where this is not possible.
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Multidomain Spectral Method for the Gauss Hypergeometric Function
2018Co-Authors: Siegfried Crespo, Marco Fasondini, Christian Klein, Nikola Stoilov, Corentin ValléeAbstract:We present a multidomain spectral approach for Fuchsian ordinary differential Equations in the particular case of the Hypergeometric Equation. Our hybrid approach uses Frobenius’ method and Moebius transformations in the vicinity of each of the singular points of the Hypergeometric Equation, which leads to a natural decomposition of the real axis into domains. In each domain, solutions to the Hypergeometric Equation are constructed via the well-conditioned ultraspherical spectral method. The solutions are matched at the domain boundaries to lead to a solution which is analytic on the whole compactified real line R∪∞, except for the singular points and cuts of the Riemann surface on which the solution is defined. The solution is further extended to the whole Riemann sphere by using the same approach for ellipses enclosing the singularities. The Hypergeometric Equation is solved on the ellipses with the boundary data from the real axis. This solution is continued as a harmonic function to the interior of the disk by solving the Laplace Equation in polar coordinates with an optimal complexity Fourier–ultraspherical spectral method.
Hori K. - One of the best experts on this subject based on the ideXlab platform.
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Solitary magnetostrophic Rossby waves in spherical shells
2020Co-Authors: Hori K., Tobias S. M., Jones C. A.Abstract:Finite-amplitude hydromagnetic Rossby waves in the magnetostrophic regime are studied. We consider the slow mode, which travels in the opposite direction to the hydrodynamic or fast mode, in the presence of a toroidal magnetic field and zonal flow by means of quasi-geostrophic models for thick spherical shells. The weakly-nonlinear, long waves are derived asymptotically using a reductive perturbation method. The problem at the first order is found to obey a second-order ODE, leading to a Hypergeometric Equation for a Malkus field and a confluent Heun Equation for an electrical-wire field, and is nonsingular when the wave speed approaches the mean flow. Investigating its neutral, nonsingular eigensolutions for different basic states, we find the evolution is described by the Korteweg-de Vries Equation. This implies that the nonlinear slow wave forms solitons and solitary waves. These may take the form of a coherent eddy, such as a single anticyclone. We speculate on the relation of the anti-cyclone to the asymmetric gyre seen in Earth's fluid core, and in state-of-the-art dynamo DNS.Comment: 12 pages, 4 figure
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Solitary magnetostrophic Rossby waves in spherical shells
'Cambridge University Press (CUP)', 2020Co-Authors: Hori K., Sm Tobias, Ca JonesAbstract:Finite-amplitude hydromagnetic Rossby waves in the magnetostrophic regime are studied. We consider the slow mode, which travels in the opposite direction to the hydrodynamic or fast mode, in the presence of a toroidal magnetic field and zonal flow by means of quasi-geostrophic models for thick spherical shells. The weakly nonlinear long waves are derived asymptotically using a reductive perturbation method. The problem at the first order is found to obey a second-order ordinary differential Equation, leading to a Hypergeometric Equation for a Malkus field and a confluent Heun Equation for an electrical wire field, and is non-singular when the wave speed approaches the mean flow. Investigating its neutral non-singular eigensolutions for different basic states, we find the evolution is described by the Korteweg–de Vries Equation. This implies that the nonlinear slow wave forms solitons and solitary waves. These may take the form of a coherent eddy, such as a single anticyclone. We speculate on the relation of the anticyclone to the asymmetric gyre seen in the Earth's fluid core, and in state-of-the-art dynamo direct numerical simulations
Siegfried Crespo - One of the best experts on this subject based on the ideXlab platform.
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Multidomain spectral method for the Gauss Hypergeometric function
Numerical Algorithms, 2019Co-Authors: Siegfried Crespo, Marco Fasondini, Christian Klein, Nikola Stoilov, Corentin ValléeAbstract:We present a multidomain spectral approach for Fuchsian ordinary differential Equations in the particular case of the Hypergeometric Equation. Our hybrid approach uses Frobenius’ method and Moebius transformations in the vicinity of each of the singular points of the Hypergeometric Equation, which leads to a natural decomposition of the real axis into domains. In each domain, solutions to the Hypergeometric Equation are constructed via the well-conditioned ultraspherical spectral method. The solutions are matched at the domain boundaries to lead to a solution which is analytic on the whole compactified real line ℝ ∪ ∞ $\mathbb {R}\cup {\infty }$ , except for the singular points and cuts of the Riemann surface on which the solution is defined. The solution is further extended to the whole Riemann sphere by using the same approach for ellipses enclosing the singularities. The Hypergeometric Equation is solved on the ellipses with the boundary data from the real axis. This solution is continued as a harmonic function to the interior of the disk by solving the Laplace Equation in polar coordinates with an optimal complexity Fourier–ultraspherical spectral method. In cases where logarithms appear in the solution, a hybrid approach involving an analytical treatment of the logarithmic terms is applied. We show for several examples that machine precision can be reached for a wide class of parameters, but also discuss almost degenerate cases where this is not possible.
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Multidomain Spectral Method for the Gauss Hypergeometric Function
2018Co-Authors: Siegfried Crespo, Marco Fasondini, Christian Klein, Nikola Stoilov, Corentin ValléeAbstract:We present a multidomain spectral approach for Fuchsian ordinary differential Equations in the particular case of the Hypergeometric Equation. Our hybrid approach uses Frobenius’ method and Moebius transformations in the vicinity of each of the singular points of the Hypergeometric Equation, which leads to a natural decomposition of the real axis into domains. In each domain, solutions to the Hypergeometric Equation are constructed via the well-conditioned ultraspherical spectral method. The solutions are matched at the domain boundaries to lead to a solution which is analytic on the whole compactified real line R∪∞, except for the singular points and cuts of the Riemann surface on which the solution is defined. The solution is further extended to the whole Riemann sphere by using the same approach for ellipses enclosing the singularities. The Hypergeometric Equation is solved on the ellipses with the boundary data from the real axis. This solution is continued as a harmonic function to the interior of the disk by solving the Laplace Equation in polar coordinates with an optimal complexity Fourier–ultraspherical spectral method.