Hypothèse du continu

En théorie des ensembles, l'hypothèse du continu (HC), due à Georg Cantor, affirme qu'il n'existe aucun ensemble dont le cardinal est strictement compris entre le cardinal de l'ensemble des entiers naturels et celui de l'ensemble des nombres réels. En d'autres termes : tout ensemble strictement plus grand, au sens de la cardinalité, que l'ensemble des entiers naturels doit contenir une « copie » de l'ensemble des nombres réels. Historique Cantor avait démontré (et publié en 1874) que le cardinal de l'ensemble des nombres réels était strictement plus grand que celui des nombres entiers ; il formula plus tard l'hypothèse du continu, qui résultait d'une analyse des sous-ensembles de la droite réelle, et de sa hiérarchisation des cardinaux infinis, mais il tenta en vain de la démontrer. Cette démonstration constituait le premier de la célèbre liste des 23 problèmes de Hilbert, que celui-ci avait établie pour le congrès international des mathématiciens de 1900 à Paris, afin de g

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L'espace contre l'architecture

Grégory Azar

This research work analyses Theo van Doesburg's Counter-constructions presented in Paris in 1923 at the "L'Effort Moderne" exhibition. These Counter-constructions stand as icons of the Modern Movement due to their role as precursors of a new "boundless" spatiality, a role first intuitively perceived by Le Corbusier or Mies van der Rohe and then theorized by Sigfried Giedion. We have attempted in this research a theoretical study of the Counter-constructions' position within the architectural field. That is to say bringing together their aspect of "spatial manifest" and the Dadaist persona of Theo van Doesburg; this alleged contradiction between the progressive and negative dimension of the Counter-constructions being the subject of this research. It is based on the following problematic: understanding what is at stake in the disjointing of the polychrome plane (as a textile surface) and the suppression of the architectural boundary under the guise of spatial continuity ("the breaking of the enclosure" mentioned by Theo van Doesburg and his interest in the notion of mathematical continuum through the theories of the fourth dimension: Henri Poincaré's Analysis situs). That amounts to studying the modalities connecting the notions of dressing and of continuum to a destructive drive, "throwing out" the inside of the house. The methodology turns towards psychoanalysis through the concepts of Verneinung [negation] (Freud) and Moi-peau [Skin-ego] (Anzieu).

EPFL2005

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Geometric, variational discretization of continuum theories

Mathieu Desbrun

This study derives geometric, variational discretization of continuum theories arising in fluid dynamics, magnetohydrodynamics (MHD), and the dynamics of complex fluids. A central role in these discretizations is played by the geometric formulation of fluid dynamics, which views solutions to the governing equations for perfect fluid flow as geodesics on the group of volume-preserving diffeomorphisms of the fluid domain. Inspired by this framework, we construct a finite-dimensional approximation to the diffeomorphism group and its Lie algebra, thereby permitting a variational temporal discretization of geodesics on the spatially discretized diffeomorphism group. The extension to MHD and complex fluid flow is then made through an appeal to the theory of Euler-Poincare systems with advection, which provides a generalization of the variational formulation of ideal fluid flow to fluids with one or more advected parameters. Upon deriving a family of structured integrators for these systems, we test their performance via a numerical implementation of the update schemes on a cartesian grid. Among the hallmarks of these new numerical methods are exact preservation of momenta arising from symmetries, automatic satisfaction of solenoidal constraints on vector fields, good long-term energy behavior, robustness with respect to the spatial and temporal resolution of the discretization, and applicability to irregular meshes. (C) 2011 Elsevier B.V. All rights reserved.

2011

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Lattice formulation of axion inflation. Application to preheating

José Roberto Canivete Cuissa, Daniel Garcia Figueroa

We present a lattice formulation of an interaction phi/Lambda F (F) over tilde between an axion and some U(1) gauge sector with the following properties: it reproduces the continuum theory up to O(dx(mu)(2)) corrections, it preserves exact gauge invariance and shift symmetry on the lattice, and it is suitable for self-consistent expansion of the Universe. The lattice equations of motion can no longer be solved by explicit methods, but we propose an implicit method to overcome this difficulty, which preserves the relevant system constraints down to arbitrary (tunable) precision. As a first application we study, in a comoving grid in (3 +1) dimensions, the last efolds of axion-inflation with quadratic potential and the preheating stage following afterwards. We fully account for the inhomogeneity and non-linearity of the system, including the gauge field contribution to the expansion rate of the Universe and its backreaction into the axion dynamics. We characterize in detail, as a function of the coupling, the energy transfer from the axion to the gauge field. Two coupling regimes are identified, sub- and super-critical, depending on whether the final energy fraction stored in the gauge field is below or above similar to 50% of the total energy. The Universe is very efficiently reheated for super-critical couplings, rapidly entering in a radiation dominated stage. Our results on preheating confirm previously published results.

IOP PUBLISHING LTD2019

Chargement

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La théorie des ensembles est une branche des mathématiques, créée par le mathématicien allemand Georg Cantor à la fin du . La théorie des ensembles se donne comme primitives les notions d'ensemble e

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