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Yoshitsugu Takei - One of the best experts on this subject based on the ideXlab platform.
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exact wkb analysis of a schrodinger Equation with a merging triplet of two simple poles and one simple turning point ii its relevance to the mathieu Equation and the Legendre Equation
Advances in Mathematics, 2014Co-Authors: Shingo Kamimoto, Takahiro Kawai, Yoshitsugu TakeiAbstract:Abstract We develop the exact WKB analysis of an M2P1T (merging two simple poles and one simple turning point) Schrodinger Equation. In Part II, using a WKB-theoretic transformation to the algebraic Mathieu Equation constructed in Part I, we calculate the alien derivative of its Borel transformed WKB solutions at each fixed singular point relevant to the simple poles through the analysis of Borel transformed WKB solutions of the Legendre Equations. In the course of the calculation of the alien derivative we make full use of microdifferential operators whose symbols are given by the infinite series that appear in the coefficients of the algebraic Mathieu Equation and the Legendre Equation.
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microlocal analysis of fixed singularities of wkb solutions of a schrodinger Equation with a merging triplet of two simple poles and a simple turning point
2012Co-Authors: Shingo Kamimoto, Takahiro Kawai, Yoshitsugu TakeiAbstract:We first show that the WKB-theoretic canonical form of an M2P1T (merging two poles and one turning point) Schrodinger Equation is given by the algebraic Mathieu Equation. We further show that, in analyzing the structure of WKB solutions of a Mathieu Equation near fixed singular points relevant to simple poles of the Equation, we can focus our attention on the pole part of the Equation so that we may reduce it to the Legendre Equation. The Borel transformation of WKB-theoretic transformations thus obtained gives rise to microdifferential relations, which lead to the microlocal analysis of the Borel transformed WKB solutions of an M2P1T Equation near their fixed singular points. The fully detailed account of the results will be given in Kamimoto et al. (Exact WKB analysis of a Schrodinger Equation with a merging triplet of two simple poles and one simple turning point—its relevance to the Mathieu Equation and the Legendre Equation, 2011).
Shingo Kamimoto - One of the best experts on this subject based on the ideXlab platform.
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exact wkb analysis of a schrodinger Equation with a merging triplet of two simple poles and one simple turning point ii its relevance to the mathieu Equation and the Legendre Equation
Advances in Mathematics, 2014Co-Authors: Shingo Kamimoto, Takahiro Kawai, Yoshitsugu TakeiAbstract:Abstract We develop the exact WKB analysis of an M2P1T (merging two simple poles and one simple turning point) Schrodinger Equation. In Part II, using a WKB-theoretic transformation to the algebraic Mathieu Equation constructed in Part I, we calculate the alien derivative of its Borel transformed WKB solutions at each fixed singular point relevant to the simple poles through the analysis of Borel transformed WKB solutions of the Legendre Equations. In the course of the calculation of the alien derivative we make full use of microdifferential operators whose symbols are given by the infinite series that appear in the coefficients of the algebraic Mathieu Equation and the Legendre Equation.
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microlocal analysis of fixed singularities of wkb solutions of a schrodinger Equation with a merging triplet of two simple poles and a simple turning point
2012Co-Authors: Shingo Kamimoto, Takahiro Kawai, Yoshitsugu TakeiAbstract:We first show that the WKB-theoretic canonical form of an M2P1T (merging two poles and one turning point) Schrodinger Equation is given by the algebraic Mathieu Equation. We further show that, in analyzing the structure of WKB solutions of a Mathieu Equation near fixed singular points relevant to simple poles of the Equation, we can focus our attention on the pole part of the Equation so that we may reduce it to the Legendre Equation. The Borel transformation of WKB-theoretic transformations thus obtained gives rise to microdifferential relations, which lead to the microlocal analysis of the Borel transformed WKB solutions of an M2P1T Equation near their fixed singular points. The fully detailed account of the results will be given in Kamimoto et al. (Exact WKB analysis of a Schrodinger Equation with a merging triplet of two simple poles and one simple turning point—its relevance to the Mathieu Equation and the Legendre Equation, 2011).
Takahiro Kawai - One of the best experts on this subject based on the ideXlab platform.
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exact wkb analysis of a schrodinger Equation with a merging triplet of two simple poles and one simple turning point ii its relevance to the mathieu Equation and the Legendre Equation
Advances in Mathematics, 2014Co-Authors: Shingo Kamimoto, Takahiro Kawai, Yoshitsugu TakeiAbstract:Abstract We develop the exact WKB analysis of an M2P1T (merging two simple poles and one simple turning point) Schrodinger Equation. In Part II, using a WKB-theoretic transformation to the algebraic Mathieu Equation constructed in Part I, we calculate the alien derivative of its Borel transformed WKB solutions at each fixed singular point relevant to the simple poles through the analysis of Borel transformed WKB solutions of the Legendre Equations. In the course of the calculation of the alien derivative we make full use of microdifferential operators whose symbols are given by the infinite series that appear in the coefficients of the algebraic Mathieu Equation and the Legendre Equation.
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microlocal analysis of fixed singularities of wkb solutions of a schrodinger Equation with a merging triplet of two simple poles and a simple turning point
2012Co-Authors: Shingo Kamimoto, Takahiro Kawai, Yoshitsugu TakeiAbstract:We first show that the WKB-theoretic canonical form of an M2P1T (merging two poles and one turning point) Schrodinger Equation is given by the algebraic Mathieu Equation. We further show that, in analyzing the structure of WKB solutions of a Mathieu Equation near fixed singular points relevant to simple poles of the Equation, we can focus our attention on the pole part of the Equation so that we may reduce it to the Legendre Equation. The Borel transformation of WKB-theoretic transformations thus obtained gives rise to microdifferential relations, which lead to the microlocal analysis of the Borel transformed WKB solutions of an M2P1T Equation near their fixed singular points. The fully detailed account of the results will be given in Kamimoto et al. (Exact WKB analysis of a Schrodinger Equation with a merging triplet of two simple poles and one simple turning point—its relevance to the Mathieu Equation and the Legendre Equation, 2011).
Frederic Barbaresco - One of the best experts on this subject based on the ideXlab platform.
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geometric theory of heat from souriau lie groups thermodynamics and koszul hessian geometry applications in information geometry for exponential families
2016Co-Authors: Frederic BarbarescoAbstract:We introduce the Symplectic Structure of Information Geometry based on Souriau’s Lie Group Thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its moment space, defining physical observables like energy, heat, and moment as pure geometrical objects. Using Geometric (Planck) Temperature of Souriau model and Symplectic cocycle notion, the Fisher metric is identified as a Souriau Geometric Heat Capacity. Souriau model is based on affine representation of Lie Group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. In the framework of Lie Group Thermodynamics, an Euler-Poincare Equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant Poincare-Cartan-Souriau integral. The Souriau-Fisher metric is linked to KKS (Kostant-Kirillov-Souriau) 2-form that associates a canonical homogeneous symplectic manifold to the co-adjoint orbits. We apply this model in the framework of Information Geometry for the action of an affine Group for exponentiel families, and provide some illustrations of use cases for multivariate Gaussian densities. Information Geometry is presented in the context of seminal work of Frechet and his Clairaut-Legendre Equation. Souriau model of Statistical Physics is validated as compatible with Balian gauge model of thermodynamics. We recall the precursor work of Casalis on affine group invariance for natural exponential families.
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geometric theory of heat from souriau lie groups thermodynamics and koszul hessian geometry applications in information geometry for exponential families
Entropy, 2016Co-Authors: Frederic BarbarescoAbstract:We introduce the symplectic structure of information geometry based on Souriau’s Lie group thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its moment space, defining physical observables like energy, heat, and moment as pure geometrical objects. Using geometric Planck temperature of Souriau model and symplectic cocycle notion, the Fisher metric is identified as a Souriau geometric heat capacity. The Souriau model is based on affine representation of Lie group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. In the framework of Lie group thermodynamics, an Euler-Poincare Equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant Poincare-Cartan-Souriau integral. The Souriau-Fisher metric is linked to KKS (Kostant–Kirillov–Souriau) 2-form that associates a canonical homogeneous symplectic manifold to the co-adjoint orbits. We apply this model in the framework of information geometry for the action of an affine group for exponential families, and provide some illustrations of use cases for multivariate gaussian densities. Information geometry is presented in the context of the seminal work of Frechet and his Clairaut-Legendre Equation. The Souriau model of statistical physics is validated as compatible with the Balian gauge model of thermodynamics. We recall the precursor work of Casalis on affine group invariance for natural exponential families.
Zonglin Jiang - One of the best experts on this subject based on the ideXlab platform.
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Approximate analytic solution of heat conduction in hollow semi-spheres flying at hypersonic speed
International Communications in Heat and Mass Transfer, 2013Co-Authors: Guilai Han, Zonglin JiangAbstract:Heat transfer in blunt noses of hypersonic vehicles with coolant inside can be approximately considered as heat conduction in hollow semi-sphere with aerodynamic heating on the outer boundary and enhanced cooling on the inner boundary. Theoretical investigations of temperature field in hollow semi-spheres were carried out by solving the two-dimensional axsymmetric conduction Equation, which could be transformed into Legendre Equation when the separation of variables is applied. However, for such a semi-sphere flying at hypersonic speed, the distribution of heat transfer rates as an outer boundary condition is so complex that the integration in the Legendre solution is nearly impossible to be completed. In this paper, a 4th order Legendre polynomial, derived by the method of undetermined coefficients, was adopted to approach the local similarity solution of hypersonic aerodynamic heating and simplify the integration process, by which an approximate solution could be set up for the temperature field. The approximate solution is also validated by comparing the analytical results with data from numerical simulations, in which the conduction Equation is solved with the improved Richardson scheme. Both analytical and numerical results are compared to each other and match quite well.