David Hilbert | EPFL Graph Search

David Hilbert, né en 1862 à Königsberg et mort en 1943 à Göttingen, est un mathématicien allemand. Il est souvent considéré comme un des plus grands mathématiciens du . Il a créé ou développé un large éventail d'idées fondamentales, que ce soit la théorie des invariants, l'axiomatisation de la géométrie ou les fondements de l'analyse fonctionnelle (avec les espaces de Hilbert). L'un des exemples les mieux connus de sa position de chef de file est sa présentation, en 1900, de ses fameux problèmes qui ont durablement influencé les recherches mathématiques du . Hilbert et ses étudiants ont fourni une portion significative de l'infrastructure mathématique nécessaire à l'éclosion de la mécanique quantique et de la relativité générale. Il a adopté et défendu avec vigueur les idées de Georg Cantor en théorie des ensembles et sur les nombres transfinis. Il est aussi connu comme l'un des fondateurs de la théorie de la démonstration, de la logique mathématique et a clairement distingué les mat

Harmonic symmetrization of convex sets and of Finsler structures, with applications to Hilbert geometry

Marc Troyanov

David Hilbert discovered in 1895 an important metric that is canonically associated to any convex domain

\Omega

in the Euclidean (or projective) space. This metric is known to be Finslerian, and the usual proof assumes a certain degree of smoothness of the boundary of

\Omega

and refers to a theorem by Busemann and Mayer that produces the norm of a tangent vector from the distance function. In this paper, we develop a new approach for the study of the Hilbert metric where no differentiability is assumed. The approach exhibits the Hilbert metric on a domain as a symmetrization of a natural weak metric, known as the Funk metric. The Funk metric is described as a tautological weak Finsler metric, in which the unit ball at each tangent space is naturally identified with the domain

\Omega

itself. The Hilbert metric is then identified with the reversible tautological weak Finsler structure on

\Omega

, and the unit ball at each point is described as the harmonic symmetrization of the unit ball of the Funk metric. Properties of the Hilbert metric then follow from general properties of harmonic symmetrizations of weak Finsler structures.

2009

Efficient geometric integrators for nonadiabatic quantum dynamics. II. The diabatic representation

Seonghoon Choi, Julien Roulet, Jiri Vanicek

Exact nonadiabatic quantum evolution preserves many geometric properties of the molecular Hilbert space. In the first paper of this series ["Paper I," S. Choi and J. Vaníček, J. Chem. Phys. 150, 204112 (2019)], we presented numerical integrators of arbitrary-order of accuracy that preserve these geometric properties exactly even in the adiabatic representation, in which the molecular Hamiltonian is not separable into kinetic and potential terms. Here, we focus on the separable Hamiltonian in diabatic representation, where the split-operator algorithm provides a popular alternative because it is explicit and easy to implement, while preserving most geometric invariants. Whereas the standard version has only second-order accuracy, we implemented, in an automated fashion, its recursive symmetric compositions, using the same schemes as in Paper I, and obtained integrators of arbitrary even order that still preserve the geometric properties exactly. Because the automatically generated splitting coefficients are redundant, we reduce the computational cost by pruning these coefficients and lower memory requirements by identifying unique coefficients. The order of convergence and preservation of geometric properties are justified analytically and confirmed numerically on a one-dimensional two-surface model of NaI and a three-dimensional three-surface model of pyrazine. As for efficiency, we find that to reach a convergence error of 10^−10, a 600-fold speedup in the case of NaI and a 900-fold speedup in the case of pyrazine are obtained with the higher-order compositions instead of the second-order split-operator algorithm. The pyrazine results suggest that the efficiency gain survives in higher dimensions.

2019

Weak metrics on Euclidean domains

Marc Troyanov

A weak metric on a set is a function that satisfies the axioms of a metric except the symmetry and the separation axioms. The aim of this paper is to present some interesting weak metrics and to study some of their properties. In particular, we introduce a weak metric, called the Apollonian weak metric, on any subset of a Euclidean space which is either bounded or whose boundary is unbounded. We relate this weak metric to some familiar metrics such as the Poincaré metric, the Klein-Hilbert metric, the Funk metric and the part metric which all play important roles in classical and in recent work on geometric function theory.

2007

Harmonic symmetrization of convex sets and of Finsler structures, with applications to Hilbert geometry

Marc Troyanov

David Hilbert discovered in 1895 an important metric that is canonically associated to any convex domain

\Omega

in the Euclidean (or projective) space. This metric is known to be Finslerian, and the usual proof assumes a certain degree of smoothness of the boundary of

\Omega

\Omega

itself. The Hilbert metric is then identified with the reversible tautological weak Finsler structure on

\Omega

2009