Local and Global Properties of the World

The essence of the method of physics is inseparably connected with the problem of interplay between local and global properties of the universe. In the present paper we discuss this interplay as it is present in three major departments of contemporary physics: general relativity, quantum mechanics and some attempts at quantizing gravity (especially geometrodynamics and its recent successors in the form of various pregeometry conceptions). It turns out that all big interpretative issues involved in this problem point towards the necessity of changing from the standard space-time geometry to some radically new, most probably non-local, generalization. We argue that the recent noncommutative geometry offers attractive possibilities, and gives us a conceptual insight into its algebraic foundations. Noncommutative spaces are, in general, non-local, and their applications to physics, known at present, seem very promising. One would expect that beneath the Planck threshold there reigns a ``noncommutative pregeometry'', and only when crossing this threshold the usual space-time geometry emerges.Comment: 43 pages, latex, no figures, changes: authors and abstract added to the body of pape

Scale-factor duality in string Bianchi cosmologies

We apply the scale factor duality transformations introduced in the context of the effective string theory to the anisotropic Bianchi-type models. We find dual models for all the Bianchi-types [except for types $VIII$ and $IX$] and construct for each of them its explicit form starting from the exact original solution of the field equations. It is emphasized that the dual Bianchi class $B$ models require the loss of the initial homogeneity symmetry of the dilatonic scalar field.Comment: 18 pages, no figure

O(d,d)-invariance in inhomogeneous string cosmologies with perfect fluid

In the first part of the present paper, we show that O(d,d)-invariance usually known in a homogeneous cosmological background written in terms of proper time can be extended to backgrounds depending on one or several coordinates (which may be any space-like or time-like coordinate(s)). In all cases, the presence of a perfect fluid is taken into account and the equivalent duality transformation in Einstein frame is explicitly given. In the second part, we present several concrete applications to some four-dimensional metrics, including inhomogeneous ones, which illustrate the different duality transformations discussed in the first part. Note that most of the dual solutions given here do not seem to be known in the literature.Comment: 25 pages, no figures, Latex. Accepted for publication in General Relativity and Gravitatio

A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity

The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smooth regions as well as reproducing faithful contours and textures. The most recent ones, proposed in the past decade, share an hybrid heritage highlighting the multiscale and oriented nature of edges and patterns in images. This paper presents a panorama of the aforementioned literature on decompositions in multiscale, multi-orientation bases or dictionaries. They typically exhibit redundancy to improve sparsity in the transformed domain and sometimes its invariance with respect to simple geometric deformations (translation, rotation). Oriented multiscale dictionaries extend traditional wavelet processing and may offer rotation invariance. Highly redundant dictionaries require specific algorithms to simplify the search for an efficient (sparse) representation. We also discuss the extension of multiscale geometric decompositions to non-Euclidean domains such as the sphere or arbitrary meshed surfaces. The etymology of panorama suggests an overview, based on a choice of partially overlapping "pictures". We hope that this paper will contribute to the appreciation and apprehension of a stream of current research directions in image understanding.Comment: 65 pages, 33 figures, 303 reference

Vibrational stability of stars in thermal imbalance : A solution in terms of asymptotic expansions

Nous avons contruit la solution des équations générales décrivant les oscillations d'étoiles présentant un défaut de balance énergétique (thermal imbalance) en termes d'un développement asymptotique jusqu'au premier ordre en le paramètre Π/ts, où Π est la période d'oscillation adiabatique du mode fondamental et ts, une échelle de temps «lente» de l'ordre du temps de Kelvin-Helmholtz. La solution ainsi obtenue nous permet de définir, sans ambiguïté, un coefficient de stabilité vibrationnelle pour un modèle stellaire arbitraire et de dériver une formule générale reliant les résultats d'analyses de stabilité en termes de δr et de δr/r. L'application au cas simple de la contraction homologue nous permet de retrouver, d'une manière beaucoup plus directe, des résultats obtenus par Cox, Hansen et Davey (1973).Demaret Jacques. Vibrational stability of stars in thermal imbalance : A solution in terms of asymptotic expansions. In: Bulletin de la Classe des sciences, tome 60, 1974. pp. 183-190

Stabilité vibrationnelle d'étoiles supermassives stabilisées dynamiquement par une rotation uniforme ou différentielle

We calculate the linear e-folding time of pulsation for vibrationally unstable main-sequence supermassive stars (105-3.109M, where M0 is the solar mass), dynamically stabilized by a uniform or differential rotation, using some simplifying assumptions, none of which seems critical. For the most massive stars, stabilized by a differential rotation (108-3.109 M0) which, following Hoyle and Fowler, could represent quasi-stellar objects, the e-folding time of pulsation is of the order of their life time (about 2.106 years) in the main-sequence stage, for a large interval of values of the central angular velocity of rotation. So, the phenomenon of vibrational instability cannot affect significantly the hydrogen burning stage of these stars. Vibrational instability can, however, excite the adiabatic pulsations of some supermassive stars and so explain the optical periodicities observed for certain «quasars », like 3 C 345 and 3 C 273. If it is sufficiently violent, it can give birth to a stellar wind, which can be the source of the clouds of gas moving outward at very high velocities, which seem to be necessary to explain the multiple absorption red-shifts of some «quasars ». Finally, supermassive stars dynamically stabilized by a differential rotation, but vibrationally unstable, could play an important rôle in the formation of the helium observed throughout the universe.Demaret Jacques. Stabilité vibrationnelle d'étoiles supermassives stabilisées dynamiquement par une rotation uniforme ou différentielle. In: Bulletin de la Classe des sciences, tome 58, 1972. pp. 68-85

General relativistic stellar stability

We give here a derivation of the general relativistic non adiabatic radial pulsation equation for spherically symmetric non rotating stars. The influence of general relativistic corrections on the non adiabatic terms is treated up to the first order in the relativistic parameter. This implies that the unperturbed model must be considered in a state of "quasi-equilibrium" in the sense that its physical characteristics are slowly varying functions of time, due to the conversion of proper mass into energy, caused by the thermonuclear reactions. Eckart's formalism is used to take into account the presence of a radiative or convective flux but the influence of the radiative viscosity terms is neglected in the derivation of the structure and pulsation equations. Relativistic expressions are given for the vibrational stability coefficient and the secular root, the last one being used to recover and clarify the general relativistic secular instability of supermassive stars above a certain critical mass, found by Appenzeller and Kippenhahn.Demaret Jacques. General relativistic stellar stability. In: Bulletin de la Classe des sciences, tome 62, 1976. pp. 377-407

Bouncing Quantum Cosmologies

Spatially homogeneous cosmological models of Bianchi types I, V and IX (as well as their isotropic counterparts : the Einstein-de Sitter, open and closed Friedmann-Robertson-Walker (FRW) models, respectively), filled with a perfect fluid with an equation of state of the type p = (γ - 1 )ρ (γ = constant comprised between 1 and 2) are quantized in the framework of an ADM-type canonical quantization scheme introduced by Demaret and Moncrief. This method which generalizes a method due to Lund and originally restricted to space-times filled with a pressureless fluid (p = 0) is based on Schutz's Hamiltonian theory of a relativistic perfect fluid, extending to general relativity Seliger and Whitham's velocity-potential version of classical hydrodynamics. It leads to a Schrödinger-like equation for the wave function of the universe, which admits a true physical interpretation in terms of probability as in orthodox quantum mechanics. Contrary to Misner's early results based on a different quantization scheme, our method leads to the conclusion that, apart from a set of measure zero of models comprising FRW models as well as Bianchi models filled with a fluid with a stiff equation of state (p = ρ), all Bianchi models are non-singular, in the sense that the quantum wave function becomes zero at the classical singularity. Such a non-singular model can be interpreted as a contracting universe bouncing into an expanding universe, due to quantum gravitational fluctuations. A similar conclusion of non-singularity of a quantum Bianchi I universe has recently been obtained by Narlikar by using a different method of quantization, i.e. the method of path integration.Demaret Jacques. Bouncing Quantum Cosmologies. In: Bulletin de la Classe des sciences, tome 66, 1980. pp. 473-486