1,720,976 research outputs found
Quantum limits of sub-Laplacians via joint spectral calculus
Full text linkWe establish two results concerning the Quantum Limits (QLs) of some
sub-Laplacians. First, under a commutativity assumption on the vector fields
involved in the definition of the sub- Laplacian, we prove that it is possible
to split any QL into several pieces which can be studied separately, and which
come from well-characterized parts of the associated sequence of
eigenfunctions. Secondly, building upon this result, we study in detail the QLs
of a particular family of sub-Laplacians defined on products of compact
quotients of Heisenberg groups. We express the QLs through a disintegration of
measure result which follows from a natural spectral decomposition of the
sub-Laplacian in which harmonic oscillators appear. Both results are based on
the construction of an adequate elliptic operator commuting with the
sub-Laplacian, and on the associated joint spectral calculus. They illustrate
the fact that, because of the possible high degeneracies in the spectrum, the
spectral theory of sub-Laplacians is very rich
Equations sous-elliptiques : contrôle, singularités et théorie spectrale
No full textIn this thesis at the boundary between analysis and geometry, we study some subelliptic partial differential equations (PDEs) with modern tools coming from sub-Riemannian geometry and microlocal analysis. We first study the controllability and observability of some subelliptic PDEs: we show that in directions requiring more brackets to be generated, the propagation of energy (and hence the observability) takes more time. Our results apply with full generality to linear subelliptic wave equations, but also to some Schrödinger-type and damped wave equations. Then, we study the propagation of singularities in subelliptic wave equations: we show that singularities propagate only along null-bicharacteristics and abnormal extremal lifts of singular curves. This result makes a bridge with classical notions in sub-Riemannian geometry. We illustrate it in the Martinet case: we construct initial data whose singularities propagate along any singular curve at any speed between 0 and 1. Finally, we study the eigenfunctions of some families of subelliptic Laplacians, in the high-frequency limit: we show that their limits, called quantum limits, can be decomposed in an infinite number of pieces, corresponding to an infinite number of dynamics on the underlying manifold.Dans cette thèse à la frontière entre analyse et géométrie, nous étudions des équations aux dérivées partielles (EDPs) sous-elliptiques en utilisant des outils récents de géométrie sous-Riemannienne et d'analyse microlocale. Nous étudions tout d'abord la contrôlabilité et l'observabilité d'EDPs sous-elliptiques, en montrant que plus une direction demande de crochets de Lie pour être engendrée, plus la propagation de l'énergie (et donc l'observabilité) se fait lentement dans cette direction. Nos résultats s'appliquent de façon générale aux équations d'ondes sous-elliptiques linéaires, mais aussi à des équations de type Schrödinger et à des équations d'ondes amorties. Ensuite, nous étudions la propagation des singularités dans les équations d'ondes sous-elliptiques : nous montrons que les singularités ne se propagent que le long des bicaractéristiques nulles et le long des relèvements anormaux extrémaux de courbes singulières. Ce résultat fait donc le lien avec des notions classiques de géométrie sous-Riemannienne. Nous l'illustrons dans le cas Martinet, en construisant des données initiales dont les singularités se propagent le long des courbes singulières à n'importe quelle vitesse entre 0 et 1. Enfin, nous étudions les fonctions propres de certaines familles de Laplaciens sous-elliptiques, dans la limite des hautes fréquences : nous montrons que leurs limites, appelées limites quantiques, peuvent être décomposées en une infinité de morceaux, correspondant à une infinité de dynamiques classiques sur la variété sous-jacente
Unstable optimal transport maps
No full textThe stability of optimal transport maps with respect to perturbations of the marginals is a question of interest for several reasons, ranging from the justification of the linearized optimal transport framework to numerical analysis and statistics. Under various assumptions on the source measure, it is known that optimal transport maps are stable with respect to variations of the target measure. In this note, we focus on the mechanisms that can, on the contrary, lead to instability. We identify two of them, which we illustrate through examples of absolutely continuous source measures in for which optimal transport maps are less stable, or even very unstable. We first show that instability may arise from the unboundedness of the density: we exhibit a source density on the unit ball of which blows up superpolynomially at two points of the boundary and for which optimal transport maps are highly unstable. Then we prove that even for uniform densities on bounded open sets, optimal transport maps can be rather unstable close enough to configurations where uniqueness of optimal plans is lost
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Subelliptic equations : control, singularities and spectral theory
No full textDans cette thèse à la frontière entre analyse et géométrie, nous étudions des équations aux dérivées partielles (EDPs) sous-elliptiques en utilisant des outils récents de géométrie sous-Riemannienne et d'analyse microlocale. Nous étudions tout d'abord la contrôlabilité et l'observabilité d'EDPs sous-elliptiques, en montrant que plus une direction demande de crochets de Lie pour être engendrée, plus la propagation de l'énergie (et donc l'observabilité) se fait lentement dans cette direction. Nos résultats s'appliquent de façon générale aux équations d'ondes sous-elliptiques linéaires, mais aussi à des équations de type Schrödinger et à des équations d'ondes amorties. Ensuite, nous étudions la propagation des singularités dans les équations d'ondes sous-elliptiques : nous montrons que les singularités ne se propagent que le long des bicaractéristiques nulles et le long des relèvements anormaux extrémaux de courbes singulières. Ce résultat fait donc le lien avec des notions classiques de géométrie sous-Riemannienne. Nous l'illustrons dans le cas Martinet, en construisant des données initiales dont les singularités se propagent le long des courbes singulières à n'importe quelle vitesse entre 0 et 1. Enfin, nous étudions les fonctions propres de certaines familles de Laplaciens sous-elliptiques, dans la limite des hautes fréquences : nous montrons que leurs limites, appelées limites quantiques, peuvent être décomposées en une infinité de morceaux, correspondant à une infinité de dynamiques classiques sur la variété sous-jacente.In this thesis at the boundary between analysis and geometry, we study some subelliptic partial differential equations (PDEs) with modern tools coming from sub-Riemannian geometry and microlocal analysis. We first study the controllability and observability of some subelliptic PDEs: we show that in directions requiring more brackets to be generated, the propagation of energy (and hence the observability) takes more time. Our results apply with full generality to linear subelliptic wave equations, but also to some Schrödinger-type and damped wave equations. Then, we study the propagation of singularities in subelliptic wave equations: we show that singularities propagate only along null-bicharacteristics and abnormal extremal lifts of singular curves. This result makes a bridge with classical notions in sub-Riemannian geometry. We illustrate it in the Martinet case: we construct initial data whose singularities propagate along any singular curve at any speed between 0 and 1. Finally, we study the eigenfunctions of some families of subelliptic Laplacians, in the high-frequency limit: we show that their limits, called quantum limits, can be decomposed in an infinite number of pieces, corresponding to an infinite number of dynamics on the underlying manifold
Exact observability properties of subelliptic wave and Schrödinger equations
No full textIn this survey paper, we report on recent works concerning exact observability (and, by duality, exact controllability) properties of subelliptic wave and Schrödinger-type equations. These results illustrate the slowdown of propagation in directions transverse to the horizontal distribution. The proofs combine sub-Riemannian geometry, semi-classical analysis, spectral theory and non-commutative harmonic analysis
Propagation of singularities for subelliptic wave equations
Full text linkH{\"o}rmander's propagation of singularities theorem does not fully describe
the propagation of singularities in subelliptic wave equations, due to the
existence of doubly characteristic points. In the present work, building upon a
visionary conference paper by R. Melrose \cite{Mel86}, we prove that
singularities of subelliptic wave equations only propagate along
null-bicharacteristics and abnormal extremals, which are well-known curves in
optimal control theory. As a consequence, we characterize the singular support
of subelliptic wave kernels outside the diagonal. These results show that
abnormal extremals play an important role in the classical-quantum
correspondence between sub-Riemannian geometry and sub-Laplacians
Catching all geodesics of a manifold with moving balls and application to controllability of the wave equation
No full textInternational audienceWe address the problem of catching all speed geodesics of a Riemannian manifold with a moving ball: given a compact Riemannian manifold and small parameters \e>0 and , is it possible to find and an absolutely continuous map satisfying and such that any geodesic of traveled at speed meets the open ball B_g(x(t),\e)\subset M within time ? Our main motivation comes from the control of the wave equation: our results show that the controllability of the wave equation can sometimes be improved by allowing the domain of control to move adequately, even very slowly. We first prove that, in any Riemannian manifold satisfying a geodesic recurrence condition (GRC), our problem has a positive answer for any and , and we give examples of Riemannian manifolds for which (GRC) is satisfied. Then, we build an explicit example of a domain (with flat metric) containing convex obstacles, not satisfying (GRC), for which our problem has a negative answer if \e and are small enough, i.e., no sufficiently small ball moving sufficiently slowly can catch all geodesics of
Infinite-time observability of the wave equation with time-varying observation domains under a geodesic recurrence condition
Full text linkOur goal is to relate the observation (or control) of the wave equation on observation domains which evolve in time with some dynamical properties of the geodesic flow. In comparison to the case of static domains of observation, we show that the observability of the wave equation in any dimension of space can be improved by allowing the domain of observation to move. We first prove that, for any domain Ω satisfying a geodesic recurrence condition (GRC), it is possible to observe the wave equation in infinite time on a ball of radius ε moving in Ω at finite speed v, where ε > 0 and v > 0 can be taken arbitrarily small, whereas the wave equation in Ω may not be observable on any static ball of radius ε. We comment on the recurrence condition: we give examples of Riemannian manifolds (Ω, g) for which (GRC) is satisfied, and, using a construction inspired by the Birkhoff-Smale homoclinic theorem, we show that there exist Riemannian manifolds (Ω, g) for which (GRC) is not satisfied. Then we prove that on the 2-dimensional torus and on Zoll manifolds, it is possible to observe the wave equation in finite time with moving balls. Finally, we establish a result of spectral observability (or of concentration of eigenfunctions) on time-dependent domains
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