Richtungsfelder und Fernparallelismus in n-dimensionalen Mannigfaltigkeiten

Stiefel, Eduard

doi:10.1007/bf01199559

Cited by 214 publications

(101 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…If in addition the cost function obeys J (W) = J (QW) for any (m × m) unitary matrix Q, then the solutions to (5) and (6) for the subspace spanned by the rows of W lie within the Grassmann manifold of orthonormal subspaces [21]. If J (W) = J (QW), then the solutions to (6) for W lie within the Stiefel manifold of orthonormal matrices [22]. For a general discussion of the geometric structures of the Grassmann and Stiefel manifolds, see [30].…”

Section: Grassmann and Stiefel Manifoldsmentioning

confidence: 99%

“…Although other adaptive methods could be used, gradient methods represent the simplest to implement and thus form the baseline to which others are often compared. Our algorithms are spatio-temporal extensions of gradient techniques on the Grassmann and Stiefel manifolds [21][22][23][24][25][26][27][28][29][30] and can be applied to any appropriatelydefined cost function. We prove that our algorithms in differential form preserve (2) or (4) over time, and thus they adjust W p in the impulse response space of paraunitary systems.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Gradient Adaptive Paraunitary Filter Banks for Spatio-Temporal Subspace Analysis and Multichannel Blind Deconvolution

Douglas

Амари

Kung

2004

The Journal of VLSI Signal Processing-Systems for Signal, Image

View full text Add to dashboard Cite

Abstract. Paraunitary filter banks are important for several signal processing tasks, including coding, multichannel deconvolution and equalization, adaptive beamforming, and subspace processing. In this paper, we consider the task of adapting the impulse response of a multichannel paraunitary filter bank via gradient ascent or descent on a chosen cost function. Our methods are spatio-temporal generalizations of gradient techniques on the Grassmann and Stiefel manifolds, and we prove that they inherently maintain the paraunitariness of the multichannel adaptive system over time. We then discuss the necessary practical approximations, modifications, and simplifications of the methods for solving two relevant signal processing tasks: (i) spatio-temporal subspace analysis and (ii) multichannel blind deconvolution. Simulations indicate that our methods can provide simple, useful solutions to these important problems.

show abstract

Section: Grassmann and Stiefel Manifoldsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Gradient Adaptive Paraunitary Filter Banks for Spatio-Temporal Subspace Analysis and Multichannel Blind Deconvolution

Douglas

Амари

Kung

2004

The Journal of VLSI Signal Processing-Systems for Signal, Image

View full text Add to dashboard Cite

show abstract

“…(M, g) admits a spin structure (see R.P. Geroch [23,24,25] and E. Stiefel [45]) and we choose one. We denote by S (or S A in the abstract index formalism) the spin bundle over M andS (or S A ′ ) the same bundle with the complex structure replaced by its opposite.…”

Section: Spin Structuresmentioning

confidence: 99%

Creation of fermions by rotating charged black holes

Häfner¹

2009

Mémoires Soc. Math. France

View full text Add to dashboard Cite

:This work is devoted to the mathematical study of the Hawking effect for fermions in the setting of the collapse of a rotating charged star. We show that an observer who is located far away from the star and at rest with respect to the Boyer Lindquist coordinates observes the emergence of a thermal state when his proper time goes to infinity. We first introduce a model of the collapse of the star. We suppose that the space-time outside the star is given by the Kerr-Newman metric. The assumptions on the asymptotic behavior of the surface of the star are inspired by the asymptotic behavior of certain timelike geodesics in the Kerr-Newman metric. The Dirac equation is then written using coordinates and a Newman-Penrose tetrad which are adapted to the collapse. This coordinate system and tetrad are based on the so called simple null geodesics. The quantization of Dirac fields in a globally hyperbolic space-time is described. We formulate and prove a theorem about the Hawking effect in this setting. The proof of the theorem contains a minimal velocity estimate for Dirac fields that is slightly stronger than the usual ones and an existence and uniqueness result for solutions of a characteristic Cauchy problem for Dirac fields in the Kerr-Newman space-time. In an appendix we construct explicitly a Penrose compactification of block I of the Kerr-Newman space-time based on simple null geodesics. Résumé :Ce travail est dédiéà l'étude mathématique de l'effet Hawking pour des fermions dans le cadre de l'effondrement d'uneétoile chargée en rotation. On démontre qu'un observateur localisé loin de l'étoile et au repos par rapport aux coordonnées de BoyerLindquist observe l'émergence d'unétat thermal quand son temps propre tend vers l'infini. On introduit d'abord un modèle de l'effondrement de l'étoile. On suppose que l'espace-tempsà l'extérieur de l'étoile est donné par la métrique de Kerr-Newman. Les hypothèses sur le comportement asymptotique de la surface de l'étoile sont inspirées par le comportement asymptotique de certaines géodésiques de type temps dans la métrique de Kerr-Newman. L'équation de Dirac est alorsécrite en utilisant des coordonnées et une tétrade de Newman-Penrose adaptésà l'effondrement. Ce système de coordonnées et cette tétrade sont basés sur des géodésiques qu'on appelle des géodésiques simples isotropes. La quantification des champs de Dirac dans un espace-temps globalement hyperbolique est décrite. On formule un théorème sur l'effet Hawking dans ce cadre. La preuve du théorème contient une estimation de vitesse minimale pour les champs de Dirac légèrement plus forte que les estimations usuelles ainsi qu'un résultat d'existence et d'unicité pour les solutions d'un problème caractéristique pour les champs de Dirac dans l'espace-temps de Kerr-Newman. Dans un appendice, nous construisons explicitement la compactification de Penrose du bloc I de l'espace-temps de Kerr-Newman qui est basée sur les géodésiques simples isotropes.2000 Mathematics Subject Classification : 35P25, 35Q75, 58J45, 83C47, 8...

show abstract

“…Characteristic cohomology classes, defined in modulo 2 coefficients by Stiefel [26] and Whitney [28] and with integral coefficients by Pontrjagin [24], make up the primary source of first-order invariants of smooth manifolds. When their utility was first recognized, it became an obvious goal to study the ways in which they admitted extensions to other categories, such as the categories of topological or PL manifolds; perhaps a clean description of characteristic classes for simplicial complexes could even give useful computational techniques.…”

Section: Introductionmentioning

confidence: 99%

The homotopy type of the matroid grassmannian

Biss¹

2003

Ann. Math.

View full text Add to dashboard Cite

Richtungsfelder und Fernparallelismus in n-dimensionalen Mannigfaltigkeiten

Cited by 214 publications

References 0 publications

Gradient Adaptive Paraunitary Filter Banks for Spatio-Temporal Subspace Analysis and Multichannel Blind Deconvolution

Gradient Adaptive Paraunitary Filter Banks for Spatio-Temporal Subspace Analysis and Multichannel Blind Deconvolution

Creation of fermions by rotating charged black holes

The homotopy type of the matroid grassmannian

Contact Info

Product

Resources

About