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You searched for subject:(C convexity). Showing records 1 – 2 of 2 total matches.

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University of Pennsylvania

1. Nayak, Soumyashant. On the Diagonals of Projections in Matrix Algebras Over Von Neumann Algebras.

Degree: 2016, University of Pennsylvania

The main focus of this dissertation is on exploring methods to characterize the diagonals of projections in matrix algebras over von Neumann algebras. This may be viewed as a non-commutative version of the generalized Pythagorean theorem and its converse (Carpenter's Theorem) studied by R. Kadison. A combinatorial lemma, which characterizes the permutation polytope of a vector in ℝn in terms of majorization, plays an important role in a proof of the Schur-Horn theorem. The Pythagorean theorem and its converse follow from this as a special case. In the quest for finding a non-commutative version of the lemma alluded to above, the notion of C*-convexity looks promising as the correct generalization for convexity. We make generalizations and improvements of some results known about C*-convex sets. We prove the Douglas lemma for von Neumann algebras and use it to prove some new results on the one-sided ideals of von Neumann algebras. As a useful technical tool, a non-commutative version of the Gram-Schmidt process is proved for finite von Neumann algebras. A complete characterization of the diagonals of projections in full matrix algebras over an abelian C*-algebra is provided in chapter 5. In chapter 6, we study the problem in the case of M2(Mn(C)), the full algebra of 2 x 2 matrices over Mn(C). The example gives us hints regarding the possibility of extracting an underlying notion of convexity for C*-polytopes, which are not necessarily convex.

Subjects/Keywords: C*-convexity; C*-segment; diagonals of projections; Douglas lemma; Schur-Horn theorem; Mathematics

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Nayak, S. (2016). On the Diagonals of Projections in Matrix Algebras Over Von Neumann Algebras. (Thesis). University of Pennsylvania. Retrieved from https://repository.upenn.edu/edissertations/1912

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Nayak, Soumyashant. “On the Diagonals of Projections in Matrix Algebras Over Von Neumann Algebras.” 2016. Thesis, University of Pennsylvania. Accessed March 08, 2021. https://repository.upenn.edu/edissertations/1912.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Nayak, Soumyashant. “On the Diagonals of Projections in Matrix Algebras Over Von Neumann Algebras.” 2016. Web. 08 Mar 2021.

Vancouver:

Nayak S. On the Diagonals of Projections in Matrix Algebras Over Von Neumann Algebras. [Internet] [Thesis]. University of Pennsylvania; 2016. [cited 2021 Mar 08]. Available from: https://repository.upenn.edu/edissertations/1912.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Nayak S. On the Diagonals of Projections in Matrix Algebras Over Von Neumann Algebras. [Thesis]. University of Pennsylvania; 2016. Available from: https://repository.upenn.edu/edissertations/1912

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

2. Gallouët, Thomas. Transport optimal : régularité et applications : Optimal Transport : Regularity and applications.

Degree: Docteur es, Mathématiques, 2012, Lyon, École normale supérieure

Cette thèse comporte deux parties distinctes, toutes les deux liées à la théorie du transport optimal. Dans la première partie, nous considérons une variété riemannienne, deux mesures à densité régulière et un coût de transport, typiquement la distance géodésique quadratique et nous nous intéressons à la régularité de l’application de transport optimal. Le critère décisif à cette régularité s’avère être le signe du tenseur de Ma-Trudinger-Wang (MTW). Nous présentons tout d’abord une synthèse des travaux réalisés sur ce tenseur. Nous nous intéressons ensuite au lien entre la géométrie des lieux d’injectivité et le tenseur MTW. Nous montrons que dans de nombreux cas, la positivité du tenseur MTW implique la convexité des lieux d’injectivité. La deuxième partie de cette thèse est liée aux équations aux dérivées partielles. Certaines peuvent être considérées comme des flots gradients dans l’espace de Wasserstein W2. C’est le cas de l’équation de Keller-Segel en dimension 2. Pour cette équation nous nous intéressons au problème de quantification de la masse lors de l’explosion des solutions ; cette explosion apparaît lorsque la masse initiale est supérieure à un seuil critique Mc. Nous cherchons alors à montrer qu’elle consiste en la formation d’un Dirac de masse Mc. Nous considérons ici un modèle particulaire en dimension 1 ayant le même comportement que l’équation de Keller-Segel. Pour ce modèle nous exhibons des bassins d’attractions à l’intérieur desquels l’explosion se produit avec seulement le nombre critique de particules. Finalement nous nous intéressons au profil d’explosion : à l’aide d’un changement d’échelle parabolique nous montrons que la structure de l’explosion correspond aux points critiques d’une certaine fonctionnelle.

This thesis consists in two distinct parts both related to the optimal transport theory.The first part deals with the regularity of the optimal transport map. The key tool is the Ma-Trundinger-Wang tensor and especially its positivity. We first give a review of the known results about the MTW tensor. We then explore the geometrical consequences of the MTW tensor on the injectivity domain. We prove that in many cases the positivity of MTW implies the convexity of the injectivity domain. The second part is devoted to the behaviour of a Keller-Segel solution in the super critical case. In particular we are interested in the mass quantization problem: we wish to quantify the mass aggregated when the blow-up occurs. In order to study the behaviour of the solution we consider a particle approximation of a Keller-Segel type equation in dimension 1. We define this approximation using the gradient flow interpretation of the Keller-Segel equation and the particular structure of the Wasserstein space in dimension 1. We show two kinds of results; we first prove a stability theorem for the blow-up mechanism: we exhibit basins of attraction in which the solution blows up with only the critical number of particles. We then prove a rigidity theorem for the blow-up mechanism: thanks to a parabolic…

Advisors/Committee Members: Villani, Cédric (thesis director).

Subjects/Keywords: Transport optimal; Régularité; Ma-Trundinger-Wang; MTW; Coût; Variété riemannienne; Convexité; Domaine d'injectivité; Lipschitz; C-convexité; Keller-Segel; Quantification de la masse; Particules; 1D; Explosion; Wasserstein; Flot gradient; Espace métrique; Masse critique; Optimal transport; Regularity; Ma-Trundinger-Wang; MTW; Cost; Riemannian manifold; Convexity; Injectivity domain; Lipschitz continuous; C-convexity; Keller-Segel; Mass quantization; Particles; 1D; Blow-up; Wasserstein; Gradient flow; Metric space; Critical mass

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Gallouët, T. (2012). Transport optimal : régularité et applications : Optimal Transport : Regularity and applications. (Doctoral Dissertation). Lyon, École normale supérieure. Retrieved from http://www.theses.fr/2012ENSL0797

Chicago Manual of Style (16th Edition):

Gallouët, Thomas. “Transport optimal : régularité et applications : Optimal Transport : Regularity and applications.” 2012. Doctoral Dissertation, Lyon, École normale supérieure. Accessed March 08, 2021. http://www.theses.fr/2012ENSL0797.

MLA Handbook (7th Edition):

Gallouët, Thomas. “Transport optimal : régularité et applications : Optimal Transport : Regularity and applications.” 2012. Web. 08 Mar 2021.

Vancouver:

Gallouët T. Transport optimal : régularité et applications : Optimal Transport : Regularity and applications. [Internet] [Doctoral dissertation]. Lyon, École normale supérieure; 2012. [cited 2021 Mar 08]. Available from: http://www.theses.fr/2012ENSL0797.

Council of Science Editors:

Gallouët T. Transport optimal : régularité et applications : Optimal Transport : Regularity and applications. [Doctoral Dissertation]. Lyon, École normale supérieure; 2012. Available from: http://www.theses.fr/2012ENSL0797

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