The Experts below are selected from a list of 138 Experts worldwide ranked by ideXlab platform
Haibao Duan - One of the best experts on this subject based on the ideXlab platform.
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Multiplicative rule of Schubert classes
Inventiones mathematicae, 2006Co-Authors: Haibao Duan, Xuezhi ZhaoAbstract:Let G be a compact connected Lie group and let H be the centralizer of a One-Parameter Subgroup in G. Combining the ideas of Bott-Samelson resolutions of Schubert varieties and the enumerative formula on a twisted product of 2 spheres obtained in [Du2], we obtain an explicit formula for multiplying Schubert classes in the flag manifold \(G \diagup H\).
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Multiplicative rule of Schubert classes
Inventiones mathematicae, 2005Co-Authors: Haibao DuanAbstract:Let G be a compact connected Lie group and H, the centralizer of a One-Parameter Subgroup in G. Combining the ideas of Bott-Samelson resulotions of Schubert varieties and the enumerative formula on a twisted products of 2-spheres obatained in {Du2], we obtain a closed formula for multiplying Schubert classes in the flag manifold G/H.
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The Cartan matrix and enumerative calculus
Journal of Symbolic Computation, 2004Co-Authors: Haibao Duan, Xu-an Zhao, Xuezhi ZhaoAbstract:Abstract Let G be a compact connected Lie group and P ⊂ G be the centralizer of a One-Parameter Subgroup in G . We explain a program that reduces integration along a Schubert variety in the flag manifold G / P to the Cartan matrix of G . As applications of the program, we complete the project of explicit computation of the degree and Chern number of an arbitrary Schubert variety started in Zhao and Duan (J. Symbolic Comput. 33 (2002) 507).
Xuezhi Zhao - One of the best experts on this subject based on the ideXlab platform.
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Multiplicative rule of Schubert classes
Inventiones mathematicae, 2006Co-Authors: Haibao Duan, Xuezhi ZhaoAbstract:Let G be a compact connected Lie group and let H be the centralizer of a One-Parameter Subgroup in G. Combining the ideas of Bott-Samelson resolutions of Schubert varieties and the enumerative formula on a twisted product of 2 spheres obtained in [Du2], we obtain an explicit formula for multiplying Schubert classes in the flag manifold \(G \diagup H\).
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The Cartan matrix and enumerative calculus
Journal of Symbolic Computation, 2004Co-Authors: Haibao Duan, Xu-an Zhao, Xuezhi ZhaoAbstract:Abstract Let G be a compact connected Lie group and P ⊂ G be the centralizer of a One-Parameter Subgroup in G . We explain a program that reduces integration along a Schubert variety in the flag manifold G / P to the Cartan matrix of G . As applications of the program, we complete the project of explicit computation of the degree and Chern number of an arbitrary Schubert variety started in Zhao and Duan (J. Symbolic Comput. 33 (2002) 507).
Aidan Sims - One of the best experts on this subject based on the ideXlab platform.
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KMS states on C⁎-algebras associated to higher-rank graphs☆
Journal of Functional Analysis, 2014Co-Authors: Astrid An Huef, Marcelo Laca, Iain Raeburn, Aidan SimsAbstract:Consider a higher-rank graph of rank k. Both the Cuntz–Krieger algebra and the Toeplitz–Cuntz–Krieger algebra of the graph carry natural gauge actions of the torus Tk, and restricting these gauge actions to One-Parameter Subgroups of Tk gives dynamical systems involving actions of the real line. We study the KMS states of these dynamical systems. We find that for large inverse temperatures β, the simplex of KMSβ states on the Toeplitz–Cuntz–Krieger algebra has dimension d One less than the number of vertices in the graph. We also show that there is a preferred dynamics for which there is a critical inverse temperature βc: for β larger than βc, there is a d-dimensional simplex of KMS states; when β=βc and the One-Parameter Subgroup is dense, there is a unique KMS state, and this state factors through the Cuntz–Krieger algebra. As in previous studies for k=1, our main tool is the Perron–Frobenius theory for irreducible non-negative matrices, though here we need a version of the theory for commuting families of matrices.
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KMS states on $C^*$-algebras associated to higher-rank graphs
arXiv: Operator Algebras, 2012Co-Authors: Astrid An Huef, Marcelo Laca, Iain Raeburn, Aidan SimsAbstract:Consider a higher-rank graph of rank k. Both the Cuntz-Krieger algebra and the Toeplitz-Cuntz-Krieger algebra of the graph carry natural gauge actions of the torus T^k, and restricting these gauge actions to One-Parameter Subgroups of T^k gives dynamical systems involving actions of the real line. We study the KMS states of these dynamical systems. We find that for large inverse temperatures \beta, the simplex of KMS_\beta states on the Toeplitz-Cuntz-Krieger algebra has dimension d One less than the number of vertices in the graph. We also show that there is a preferred dynamics for which there is a critical inverse temperature \beta_c: for \beta larger than \beta_c, there is a d-dimensional simplex of KMS states; when \beta=\beta_c and the One-Parameter Subgroup is dense, there is a unique KMS state, and this state factors through the Cuntz-Krieger algebra. As in previous studies for k=1, our main tool is the Perron-Frobenius theory for irreducible nonnegative matrices, though here we need a version of the theory for commuting families of matrices.
Astrid An Huef - One of the best experts on this subject based on the ideXlab platform.
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KMS states on C⁎-algebras associated to higher-rank graphs☆
Journal of Functional Analysis, 2014Co-Authors: Astrid An Huef, Marcelo Laca, Iain Raeburn, Aidan SimsAbstract:Consider a higher-rank graph of rank k. Both the Cuntz–Krieger algebra and the Toeplitz–Cuntz–Krieger algebra of the graph carry natural gauge actions of the torus Tk, and restricting these gauge actions to One-Parameter Subgroups of Tk gives dynamical systems involving actions of the real line. We study the KMS states of these dynamical systems. We find that for large inverse temperatures β, the simplex of KMSβ states on the Toeplitz–Cuntz–Krieger algebra has dimension d One less than the number of vertices in the graph. We also show that there is a preferred dynamics for which there is a critical inverse temperature βc: for β larger than βc, there is a d-dimensional simplex of KMS states; when β=βc and the One-Parameter Subgroup is dense, there is a unique KMS state, and this state factors through the Cuntz–Krieger algebra. As in previous studies for k=1, our main tool is the Perron–Frobenius theory for irreducible non-negative matrices, though here we need a version of the theory for commuting families of matrices.
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KMS states on $C^*$-algebras associated to higher-rank graphs
arXiv: Operator Algebras, 2012Co-Authors: Astrid An Huef, Marcelo Laca, Iain Raeburn, Aidan SimsAbstract:Consider a higher-rank graph of rank k. Both the Cuntz-Krieger algebra and the Toeplitz-Cuntz-Krieger algebra of the graph carry natural gauge actions of the torus T^k, and restricting these gauge actions to One-Parameter Subgroups of T^k gives dynamical systems involving actions of the real line. We study the KMS states of these dynamical systems. We find that for large inverse temperatures \beta, the simplex of KMS_\beta states on the Toeplitz-Cuntz-Krieger algebra has dimension d One less than the number of vertices in the graph. We also show that there is a preferred dynamics for which there is a critical inverse temperature \beta_c: for \beta larger than \beta_c, there is a d-dimensional simplex of KMS states; when \beta=\beta_c and the One-Parameter Subgroup is dense, there is a unique KMS state, and this state factors through the Cuntz-Krieger algebra. As in previous studies for k=1, our main tool is the Perron-Frobenius theory for irreducible nonnegative matrices, though here we need a version of the theory for commuting families of matrices.
Xavier Pennec - One of the best experts on this subject based on the ideXlab platform.
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Geodesics, Parallel Transport & One-Parameter Subgroups for Diffeomorphic Image Registration
International Journal of Computer Vision, 2013Co-Authors: Marco Lorenzi, Xavier PennecAbstract:Computational anatomy aims at developing models to understand the anatomical variability of organs and tissues. A widely used and validated instrument for comparing the anatomy in medical images is non-linear diffeomorphic registration which is based on a rich mathematical background. For instance, the “large deformation diffeomorphic metric mapping” (LDDMM) framework defines a Riemannian setting by providing a right invariant metric on the tangent spaces, and solves the registration problem by computing geodesics parametrized by time-varying velocity fields. A simpler alternative based on stationary velocity fields (SVF) has been proposed, using the One-Parameter Subgroups from Lie groups theory. In spite of its better computational efficiency, the geometrical setting of the SVF is more vague, especially regarding the relationship between One-Parameter Subgroups and geodesics. In this work, we detail the properties of finite dimensional Lie groups that highlight the geometric foundations of One-Parameter Subgroups. We show that One can define a proper underlying geometric structure (an affine manifold) based on the canonical Cartan connections, for which One-Parameter Subgroups and their translations are geodesics. This geometric structure is perfectly compatible with all the group operations (left, right composition and inversion), contrarily to left- (or right-) invariant Riemannian metrics. Moreover, we derive closed-form expressions for the parallel transport. Then, we investigate the generalization of such properties to infinite dimensional Lie groups. We suggest that some of the theoretical objections might actually be ruled out by the practical implementation of both the LDDMM and the SVF frameworks for image registration. This leads us to a more practical study comparing the Parameterization (initial velocity field) of metric and Cartan geodesics in the specific optimization context of longitudinal and inter-subject image registration.Our experimental results suggests that stationarity is a good approximation for longitudinal deformations, while metric geodesics notably differ from stationary Ones for inter-subject registration, which involves much larger and non-physical deformations. Then, we turn to the practical comparison of five parallel transport techniques along One-Parameter Subgroups. Our results point out the fundamental role played by the numerical implementation, which may hide the theoretical differences between the different schemes. Interestingly, even if the parallel transport generally depends on the path used, an experiment comparing the Cartan parallel transport along the One-Parameter Subgroup and the LDDMM (metric) geodesics from inter-subject registration suggests that our parallel transport methods are not so sensitive to the path.
