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International audience; This study introduces a new signal analysis method, based on a semiclassical approach. The main idea in this method is to interpret a pulse-shaped signal as a potential of a Schrödinger operator and then to use the discrete spectrum of this operator for the analysis of the signal. We present some numerical examples and the first results obtained with this method on the analysis of arterial blood pressure waveforms.

International audience; In this paper we propose a muscle tissue model -- valid for striated muscles in general, and for the myocardium in particular -- based on a multi-scale physiological description. This model extends and refines an earlier-proposed formulation by allowing to account for all major energy exchanges and balances, from the chemical activity coupled with oxygen supply to the production of actual mechanical work, namely, the biological function of the tissue. We thus perform a thorough analysis of the energy mechanisms prevailing at the various scales, and we proceed to propose a complete discretization strategy -- in time and space -- respecting the same balance laws. This will be crucial in future works to adequately model the many important physiological -- normal and pathological -- phenomena associated with these energy considerations.

International audience; Having in mind applications to the fault-detection/diagnosis of lossless electrical networks, here we consider some inverse scattering problems for Schrödinger operators over star-shaped graphs. We restrict ourselves to the case of minimal experimental setup consisting in measuring, at most, two reflection coefficients when an infinite homogeneous (potential-less) branch is added to the central node. First, by studying the asymptotic behavior of only one reflection coefficient in the high-frequency limit, we prove the identiability of the geometry of this star-shaped graph: the number of edges and their lengths. Next, we study the potential identification problem by inverse scattering, noting that the potentials represent the inhomogeneities due to the soft faults in the network wirings (potentials with bounded H1-norms). The main result states that, under some assumptions on the geometry of the graph, the measurement of two reflection coefficients, associated to two different sets of boundary conditions at the external vertices of the tree, determines uniquely the potentials; it can be seen as a generalization of the theorem of the two boundary spectra on an interval.

In this paper, we study a scalar conservation law that models a highly re-entrant manufacturing system as encountered in semi-conductor production. As a generalization of \cite{CKWang}, the velocity function possesses both the local and nonlocal character. We prove the existence and uniqueness of the weak solution to the Cauchy problem with initial and boundary data in $L^{\infty}$. We also obtain the stability (continuous dependence) of both the solution and the out-flux with respect to the initial and boundary data. Finally, we prove the existence of an optimal control that minimizes, in the $L^p$-sense with $p\in [1,\infty)$, the difference between the actual out-flux and a forecast demand over a fixed time period., Comment: 32 pages, 12 figures

International audience; In vertebrates, the gonadotrope axis is made up of the hypothalamus, belonging to the central nervous system, the pituitary gland and the gonads. These organs communicate with one another within entangled endocrine loops. From a dynamical viewpoint, the gonadotrope axis exhibits remarkable features, such as the pulsatile mode of secretion of the master reproductive hormone GnRH (gonadotropin releasing hormone) and the permanent renewal of the gonadic tissues related to the maturation of the germ cell lines. We will present a multilevel and multiscale modelling approach of the gonadotrope axis focusing on the ovulation process. On the gonadic level, we address the question of the selection of ovarian follicles within the framework of controlled conservation laws. We have to deal with a 2D EDP with integro-differential expressions in the velocity and source terms. On the hypothalamic level, we design a model of GnRH secretion within the framework of weakly coupled nonlinear oscillators. The model outputs reproduce the transition from the pulsatile to the surge pattern and the changes in pulse frequency. To explain the behaviour of the model and meet qualitative and quantitative endocrinological specifications, we explore the whole sequence of bifurcations occurring within the model.; Chez les vertébrés, l'axe gonadotrope est constitué par l'hypothalamus (structure du système nerveux central), l'hypophyse et les gonades, dont les activités sont coordonnées par des boucles de rétrocontrôle endocrines. Le fonctionnement de ce système exhibe des propriétés remarquables sur le plan dynamique, telles que le caractère pulsatile de la sécrétion de la GnRH (gonadotropin releasing hormone) par les neurones hypothalamiques, et le remaniement permanent (au cours de la vie reproductive) des tissus gonadiques, en liaison avec la maturation des cellules germinales. Il s'agit ici de présenter une approche de modélisation et de contrôle multi-échelles et multi-niveaux de l'axe gonadotrope. Au niveau gonadique, nous décrirons un modèle multi-échelles du processus de sélection des follicules ovulatoires, au sein de la population de follicules ovariens en croissance, rentrant dans le cadre des lois de conservation contrôlées. Au niveau hypothalamique, nous présenterons un modèle de sécrétion de la GnRH, dans le cadre du couplage d'oscillateurs non linéaires. Les sorties du modèle reproduisent non seulement la transition entre régime pulsatile et décharge ovulatoire, mais aussi l'accéleration en fréquence avant la décharge, et l'existence d'une pause après la décharge. Nous étudions la séquence complète de bifurcations sous-jacentes au modèle, afin de respecter fidèlement les spécifications qualitatives et quantitatives du cahier des charges endocrine, qui portent sur les rapports en durée, amplitude et fréquence, entre les différents événements sécrétoires

International audience; A parsimonious representation of signals is a mathematic model parametrized with a small number of parameters. Such models are useful for analysis, interpolation, filtering, feature extraction, and data compression. A new parsimonious model is presented in this paper based on scattering transforms. It is closely related to the eigenvalues and eigenfunctions of the linear Schrödinger equation. The efficiency of this method is illustrated in this paper with examples of both synthetic and real signals.

Actuator fault diagnosis is often studied under strong assumptions on available sensors: typically it is assumed that either the sensors are fault-free or there is a sufficient redundancy among the sensors. The purpose of this paper is to propose a new method for actuator fault diagnosis which is robust to sensor distortion. It does not require sensor redundancy to compensate sensor distortion. The essential assumption is that sensor distortions are strictly monotonous. Despite the nonlinear and unknown nature of distortions, such sensors still provide useful information for fault diagnosis. A numerical example is provided to illustrate the efficiency of the proposed method. Copyright © 2006 IFAC

We show that a planar slow–fast piecewise-linear (PWL) system with three zones admits limit cycles that share a lot of similarity with van der Pol canards, in particular an explosive growth. Using phase-space compactification, we show that these quasi-canard cycles are strongly related to a bifurcation at infinity. Furthermore, we investigate a limiting case in which we show the existence of a continuum of canard homoclinic connections that coexist for a single-parameter value and with amplitude ranging from an order of ε to an order of 1, a phenomenon truly associated with the non-smooth character of this system and which we call super-explosion .

International audience; Continuation techniques have been known to successfully describe bifurcation diagrams appearing in slow-fast systems with more than one slow variable (see, e.g., [M. Desroches, B. Krauskopf, and H. M. Osinga, Nonlinearity, 23 (2010), pp. 739--765]). In this paper we investigate the usefulness of numerical continuation techniques dealing with some solved and some open problems in the study of planar singular perturbations. More precisely, we first verify known theoretical results (thereby showing the reliability of this numerical tool) on the appearance of multiple limit cycles of relaxation-oscillation type and on the existence of multiple critical periods in well-chosen annuli of slow-fast periodic orbits in the plane. We then apply the technique to study the period function in detail.

International audience; The cusp singularity --a point at which two curves of fold points meet-- is a prototypical example in Takens' classification of singularities in constrained equations, which also includes folds, folded saddles, folded nodes, among others. In this article, we study cusp singularities in singularly perturbed systems for sufficiently small values of the perturbation parameter, in the regime in which these systems exhibit fast and slow dynamics. Our main result is an analysis of the cusp point using the method of geometric desingularization, also known as the blow-up method, from the field of geometric singular perturbation theory. Our analysis of the cusp singularity was inspired by the nerve impulse example of Zeeman, and we also apply our main theorem to it. Finally, a brief review of geometric singular perturbation theory for the two elementary singularities from the Takens' classification occurring for the nerve impulse example --folds and folded saddles -- is included to make this article self-contained.

This paper proposes a generalization of the conjugate gradient (CG) method used to solve the equation $Ax=b$ for a symmetric positive definite matrix $A$ of large size $n$. The generalization consists of permitting the scalar control parameters (= stepsizes in gradient and conjugate gradient directions) to be replaced by matrices, so that multiple descent and conjugate directions are updated simultaneously. Implementation involves the use of multiple agents or threads and is referred to as cooperative CG (cCG), in which the cooperation between agents resides in the fact that the calculation of each entry of the control parameter matrix now involves information that comes from the other agents. For a sufficiently large dimension $n$, the use of an optimal number of cores gives the result that the multithread implementation has worst case complexity $O(n^{2+1/3})$ in exact arithmetic. Numerical experiments, that illustrate the interest of theoretical results, are carried out on a multicore computer., Comment: Expanded version of manuscript submitted to the IEEE-CDC 2012 (Conference on Decision and Control)

In this work we study mixed mode oscillations in a model of secretion of GnRH (Gonadotropin Releasing Hormone). The model is a phantom burster consisting of two feedforward coupled FitzHugh-Nagumo systems, with three time scales. The forcing system (Regulator) evolves on the slowest scale and acts by moving the slow nullcline of the forced system (Secretor). There are three modes of dynamics: pulsatility (transient relaxation oscillation), surge (quasi steady state) and small oscillations related to the passage of the slow nullcline through a fold point of the fast nullcline. We derive a variety of reductions, taking advantage of the mentioned features of the system. We obtain two results; one on the local dynamics near the fold in the parameter regime corresponding to the presence of small oscillations and the other on the global dynamics, more specifically on the existence of an attracting limit cycle. Our local result is a rigorous characterization of small canards and sectors of rotation in the case of folded node with an additional time scale, a feature allowing for a clear geometric argument. The global result gives the existence of an attracting unique limit cycle, which, in some parameter regimes, remains attracting and unique even during passages through a canard explosion., 38 pages, 16 figures

We present a numerical strategy to compute one-parameter families of isolas of equilibrium solutions in ODEs. Isolas are solution branches closed in parameter space. Numerical continuation is required to compute one single isola since it contains at least one unstable segment. We show how to use pseudo-arclength predictor–corrector schemes in order to follow an entire isola in parameter space, as an individual object, by posing a suitable algebraic problem. We continue isolas of equilibria in a two-dimensional dynamical system, the so-called continuous stirred tank reactor model, and also in a three-dimensional model related to plasma physics. We then construct a toy model and follow a family of isolas past a fold and illustrate how to initiate the computation close to a formation center, using approximate ellipses in a model inspired by the van der Pol equation. We also show how to introduce node adaptivity in the discretization of the isola, so as to concentrate on nodes in region with higher curvature. We conclude by commenting on the extension of the proposed numerical strategy to the case of isolas of periodic orbits.

International audience; In this paper we study the well-posedness of different models of population of leaky integrate- and- re neurons with a population density approach. The synaptic interaction between neurons is modeled by a potential jump at the reception of a spike. We study populations that are self excitatory or self inhibitory. We distinguish the cases where this interaction is instantaneous from the one where there is a repartition of conduction delays. In the case of a bounded density of delays both excitatory and inhibitory population models are shown to be well-posed. But without conduction delay the solution of the model of self excitatory neurons may blow up. We analyze the di erent behaviours of the model with jumps compared to its di usion approximation.

Fast-slow systems are studied usually by "geometrical dissection" [Borisyuk & Rinzel, 2005]. The fast dynamics exhibit attractors which may bifurcate under the influence of the slow dynamics which is seen as a parameter of the fast dynamics. A generic solution comes close to a connected component of the stable invariant sets of the fast dynamics. As the slow dynamics evolves, this attractor may lose its stability and the solution eventually reaches quickly another connected component of attractors of the fast dynamics and the process may repeat. This scenario explains quite well relaxation oscillations and more complicated oscillations like bursting. More recently, in relation both with theory of dynamical systems [Dumortier & Roussarie, 1996] and with applications to physiology [Desroches et al., 2008; Toporikova et al., 2008], a new interest has emerged in canard cycles. These orbits share the property that they remain for a while close to an unstable invariant set (either singular set or periodic orbits of the fast dynamics). Although canards were first discovered when the transition points were folds, in this article, we focus on the case where one or several transition points (or "jumps") are instead transcritical. We present several new surprising effects like the "amplification of canards" or the "exceptionally fast recovery" on both (1 + 1)-systems and (2 + 1)-systems associated with tritrophic food chain dynamics. Finally, we also mention their possible relevance to the notion of resilience which has been coined out in ecology [Holling, 1973; Ludwig et al., 1997; Martin, 2004].

International audience; This article proposes an algebraic method to fault diagnosis for uncertain linear systems. The main advantage of this new approach is to realize fault diagnosis only from knowledge of input and output measurements without identifying explicitly model parameters. Using tools and results of algebraic identification and pseudospectra analysis, the issues of robustness of the pro- posed approach compared to the model order and noise measurement are examined. Numerical examples are provided and discussed to illustrate the efficiency of the proposed fault diagnosis method.

In this paper, we consider the 1-D wave equation where the spatial domain is a bounded interval. Assuming the initial conditions to be known, we are here interested in identifying an unknown source term, while we take the Neumann derivative of the solution on one of the boundaries as the measurement output. Applying a back-and-forth iterative scheme and constructing well-chosen observers (that will be applied in an offline manner), we retrieve the source term from the measurement output in the minimal observation time.

This paper presents the design and implementation of a numerical method to simulate a multiscale model describing the selection process in ovarian follicles. The PDE model consists in a quasi-linear hyperbolic system of large size, namely Nf × Nf, ruling the time evolution of the cell density functions of Nf follicles (in practice Nf is of the order of a few to twenty). These equations are weakly coupled through the sum of the first order moments of the density functions. The time-dependent equations make use of two structuring variables, age and maturity, which play the roles of space variables. The problem is naturally set over a compact domain of R2. The formulation of the time-dependent controlled transport coefficients accounts for available biological knowledge on follicular cell kinetics. We introduce a dedicated numerical scheme that is amenable to parallelization, by taking advantage of the weak coupling. Numerical illustrations assess th e relevance of the proposed method both in term of accuracy and HPC achievements. Ce document présente la conception et l’implémentation d’une méthode numérique servant à simuler un modèle multiéchelle décrivant le processus de sélection des follicules ovariens. Le modèle EDP consiste en un système hyperbolique quasi linéaire de grande taille, typiquement Nf × Nf, gouvernant l’évolution des fonctions de densité cellulaire pour Nf follicules (en pratique Nf est de l’ordre de quelques-uns à une vingtaine). Ces équations d’évolution utilisent deux variables structurantes, l’âge et la maturité, qui jouent le rôle de variables d’espace. Le problème est naturellement posé sur un domaine compact de R2. La formulation du transport à coefficients variables au cours du temps en fonction du contrôle est issue des connaissances disponibles sur la cinétique cellulaire au sein des follicules ovariens. Nous présentons un schéma numérique dédié au problème parallélisable en tirant parti du faible couplage. Des simulations numériques permettent d’évaluer la pertinence de la méthode proposée tant en terme de précision que de calcul haute performance.

International audience; We prove that the fidelity between the quantum state governed by a continuous time stochastic master equation driven by a Wiener process and its associated quantum-filter state is a sub-martingale. This result is a generalization to non-pure quantum states where fidelity does not coincide in general with a simple Frobenius inner product. This result implies the stability of such filtering process but does not necessarily ensure the asymptotic convergence of such quantum-filters.

We establish large deviation properties valid for almost every sample path of a class of stationary mixing processes ( X 1 , … , X n , … ) . These properties are inherited from those of S n = ∑ i = 1 n X i and describe how the local fluctuations of almost every realization of S n deviate from the almost sure behavior. These results apply to the fluctuations of Brownian motion, Birkhoff averages on hyperbolic dynamics, as well as branching random walks. Also, they lead to new insights into the “randomness” of the digits of expansions in integer bases of Pi. We formulate a new conjecture, supported by numerical experiments, implying the normality of Pi.

International audience; This article proposes an algebraic method to fault diagnosis for uncertain linear systems. The main advantage of this new approach is to realize fault diagnosis only from knowledge of input and output measurements without identi- fying explicitly model parameters. Using tools and results of algebraic identification and pseudospectra analysis, the issues of robustness of the proposed approach compared to the model order and noise measurement are examined. Numerical examples are provided and discussed to illustrate the efficiency of the proposed fault diagnosis method.

International audience; We propose an estimation algorithm for large deviations spectra of measures and functions. The algorithm converges for natural examples of multifractals.

International audience; This paper introduces a new paradigm, called cooperative computation, for the solution of systems of linear equations with symmetric coefficient matrices. The simplest version of the algorithm consists of two agents, each one computing the solution of the whole system, using an iterative method. Infrequent unidirectional communication occurs from one agent to the other, either periodically, or probabilistically, thus characterizing the computation as parallel and asynchronous. Every time one agent communicates its current approximation of the solution to the other, the receiving agent carries out a least squares computation to replace its current value by an affine combination of the current approximations, and the algorithm continues until a stopping criterion is met. Deterministic and probabilistic variants of this algorithm are introduced and shown to be efficient, specifically in relation to the popular Barzilai-Borwein algorithm, particularly for ill-conditioned matrices.

International audience; In today's Internet, a large part of the traffic is carried using the TCP transport protocol. Characterization of the variations of TCP traffic is thus a major challenge, both for resource provisioning and Quality of Service purposes. However, most existing models are limited to the prediction of the (almost-sure) mean TCP throughput and are unable to characterize deviations from this value. In this paper, we propose a method to describe the deviations of a long TCP flow's throughput from its almost-sure mean value. This method relies on an ergodic large-deviations result, which was recently proved to hold on almost every single realization for a large class of stochastic processes. Applying this result to a Markov chain modeling the congestion window's evolution of a long-lived TCP flow, we show that it is practically possible to quantify and to statistically bound the throughput's variations at different scales of interest for applications. Our Markov-chain model can take into account various network conditions and we demonstrate the accuracy of our method's prediction in different situations using simulations, experiments and real-world Internet traffic. In particular, in the classical case of Bernoulli losses, we demonstrate: i) the consistency of our method with the widely-used square-root formula predicting the almost-sure mean throughput, and ii) its ability to additionally predict finer properties reflecting the traffic variability at different scales.

International audience; Positive T-martingales were developed as a general framework that extends the positive measure-valued martingales and are meant to model intermittent turbulence. We extend their scope by allowing the martingale to take complex values. We focus on martingales constructed on the interval T = [0, 1] and replace random measures by random functions. We specify a large class of such martingales for which we provide a general sufficient condition for almost sure uniform convergence to a nontrivial limit. Such a limit yields new examples of naturally generated multifractal processes that may be of use in multifractal signals modeling.

The familiar cascade measures are sequences of random positive measures obtained on $[0,1]$ via $b$-adic independent cascades. To generalize them, this paper allows the random weights invoked in the cascades to take real or complex values. This yields sequences of random functions whose possible strong or weak limits are natural candidates for modeling multifractal phenomena. Their asymptotic behavior is investigated, yielding a sufficient condition for almost sure uniform convergence to nontrivial statistically self-similar limits. Is the limit function a monofractal function in multifractal time? General sufficient conditions are given under which such is the case, as well as examples for which no natural time change can be used. In most cases when the sufficient condition for convergence does not hold, we show that either the limit is 0 or the sequence diverges almost surely. In the later case, a functional central limit theorem holds, under some conditions. It provides a natural normalization making the sequence converge in law to a standard Brownian motion in multifractal time., Comment: Published in at http://dx.doi.org/10.1214/09-AAP665 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Folded saddle-nodes occur generically in one parameter families of singularly perturbed systems with two slow variables. We show that these folded singularities are the organizing centers for two main delay phenomena in singular perturbation problems: canards and delayed Hopf bifurcations. We combine techniques from geometric singular perturbation theory—the blow-up technique—and from delayed Hopf bifurcation theory—complex time path analysis—to analyze the flow near such folded saddle-nodes. In particular, we show the existence of canards as intersections of stable and unstable slow manifolds. To derive these canard results, we extend the singularly perturbed vector field into the complex domain and study it along elliptic paths. This enables us to extend the invariant slow manifolds beyond points where normal hyperbolicity is lost. Furthermore, we define a way-in/way-out function describing the maximal delay expected for generic solutions passing through a folded saddle-node singularity. Branch points associated with the change from a complex to a real eigenvalue structure in the variational equation along the critical (slow) manifold make our analysis significantly different from the classical delayed Hopf bifurcation analysis where these eigenvalues are complex only.

International audience; Iterative methods for linear systems with a symmetric positive definite coefficient matrix are designed from a control-theoretic viewpoint. In particular, it is shown that a control-theoretic approach loosely based on m-step dead beat control of the error or residual system, with a suitable definition of error norm can be utilized to design new iterative methods that are competitive with the popular Barzilai-Borwein method, that is well known to be an efficient method with low computational cost. Numerical experiments are reported on to confirm the claimed results.

We achieve the multifractal analysis of a class of complex valued statistically self-similar continuous functions. For we use multifractal formalisms associated with pointwise oscillation exponents of all orders. Our study exhibits new phenomena in multifractal analysis of continuous functions. In particular, we find examples of statistically self-similar such functions obeying the multifractal formalism and for which the support of the singularity spectrum is the whole interval $[0,\infty]$., 37 pages, 8 figures

A Lyapunov-based approach for the trajectory generation of an $N$-dimensional Schr{\"o}dinger equation in whole $\RR^N$ is proposed. For the case of a quantum particle in an $N$-dimensional decaying potential the convergence is precisely analyzed. The free system admitting a mixed spectrum, the dispersion through the absolutely continuous part is the main obstacle to ensure such a stabilization result. Whenever, the system is completely initialized in the discrete part of the spectrum, a Lyapunov strategy encoding both the distance with respect to the target state and the penalization of the passage through the continuous part of the spectrum, ensures the approximate stabilization., Comment: 34 pages

Gibbs measures μ on cookie-cutter sets are the archetype of multifractal measures on Cantor sets. We compute the singularity spectrum of the inverse measure of μ. Such a measure is discrete (it is constituted only by Dirac masses), it satisfies a multifractal formalism and its Lq-spectrum possesses one point of non-differentiability. The results rely on heterogeneous ubiquity theorems.

We consider an ensemble of quantum systems whose average evolution is described by a density matrix, solution of a Lindblad-Kossakowski differential equation. We focus on the special case where the decoherence is only due to a highly unstable excited state and where the spontaneously emitted photons are measured by a photo-detector. We propose a systematic method to eliminate the fast and asymptotically stable dynamics associated to the excited state in order to obtain another differential equation for the slow part. We show that this slow differential equation is still of Lindblad-Kossakowski type, that the decoherence terms and the measured output depend explicitly on the amplitudes of quasi-resonant applied field, i.e., the control. Beside a rigorous proof of the slow/fast (adiabatic) reduction based on singular perturbation theory, we also provide a physical interpretation of the result in the context of coherence population trapping via dark states and decoherence-free subspaces. Numerical simulations illustrate the accuracy of the proposed approximation for a 5-level systems., Comment: 6 pages, 2 figures

International audience; Mandelbrot multiplicative cascades provide a construction of a dynamical system on a set of probability measures de ned by inequalities on moments. To be more speci c, beyond the rst iteration, the trajectories take values in the set of xed points of smoothing transformations (i.e., some generalized stable laws). Studying this system leads to a central limit theorem and to its functional version. The limit Gaussian process can also be obtained as limit of an 'additive cascade' of independent normal variables.

International audience; We consider a nonrelativistic charged particle in a one-dimensional infinite square potential well. This quantum system is subjected to a control, which is a uniform (in space) timedepending electric field. It is represented by a complex probability amplitude solution of a Schr¨odinger equation on a one-dimensional bounded domain, with Dirichlet boundary conditions. We prove the almost global practical stabilization of the eigenstates by explicit feedback laws.

Statistically self-similar measures on $[0,1]$ are limit of multiplicative cascades of random weights distributed on the $b$-adic subintervals of $[0,1]$. These weights are i.i.d, positive, and of expectation $1/b$. We extend these cascades naturally by allowing the random weights to take negative values. This yields martingales taking values in the space of continuous functions on $[0,1]$. Specifically, we consider for each $H\in (0,1)$ the martingale $(B_{n})_{n\geq1}$ obtained when the weights take the values $-b^{-H}$ and $b^{-H}$, in order to get $B_n$ converging almost surely uniformly to a statistically self-similar function $B$ whose H\"{o}lder regularity and fractal properties are comparable with that of the fractional Brownian motion of exponent $H$. This indeed holds when $H\in(1/2,1)$. Also the construction introduces a new kind of law, one that it is stable under random weighted averaging and satisfies the same functional equation as the standard symmetric stable law of index $1/H$. When $H\in(0,1/2]$, to the contrary, $B_n$ diverges almost surely. However, a natural normalization factor $ a_n$ makes the normalized correlated random walk $ B_n / a_n$ converge in law, as $n$ tends to $\infty$, to the restriction to $[0,1]$ of the standard Brownian motion. Limit theorems are also associated with the case $H>1/2$., Comment: 17 pages, 4 figures. To appear in the Annales de l'Institut Henri Poincare (B)

International audience; We consider a pure two-state quantum system illuminated by two lasers. A photo-detector captures the fluorescence of the system. We build an invariant observer which yields a (local) estimation of the wave function (or density matrix) and the two key parameters (laser de-tuning and the atom- laser coupling strength) of the hamiltonian. The design exploits the symmetries of the system and can be interpreted geometrically. The convergence proof is based on averaging arguments. Simulation with noise illustrates the robustness of the obtained estimation algorithm.

The study of the scattering data for a star-shape network of LC-transmission lines is transformed into the scattering analysis of a Schr\"odinger operator on the same graph. The boundary conditions coming from the Kirchhoff rules ensure the existence of a unique self-adjoint extension of the mentioned Schr\"odinger operator. While the graph consists of a number of infinite branches and a number finite ones, all joining at a central node, we provide a construction of the scattering solutions. Under non-degenerate circumstances (different wave travelling times for finite branches), we show that the study of the reflection coefficient in the high-frequency regime must provide us with the number of the infinite branches as well as the the wave travelling times for finite ones., Comment: 6 pages

We are interested in two properties of real numbers: the first one is the property of being well-approximated by some dense family of real numbers {xn}n≥1, such as rational numbers and more generally algebraic numbers, and the second one is the property of having given digit frequencies in some b-adic expansion.We combine these two ways of classifying the real numbers, in order to provide a finer classification. We exhibit sets S of points x which are approximated at a given rate by some of the {xn}n, those xn being selected according to their digit frequencies. We compute the Hausdorff dimension of any countable intersection of such sets S, and prove that these sets enjoy the so-called large intersection property.

International audience; We describe an approach that we propose to model the electromechanical behavior of the heart, and to use the model in a data assimilation procedure in order to perform an identification of the parameters and state. The modeling of the heart tissue is based on an electrically-activated contraction law formulated via multiscale considerations and is consistent with various physiological and thermomechanical key requirements. The global heart system also incorporates a simplified lumped modeling of the blood compartments. We report on numerical simulations and on validations of our model in reference and pathological conditions. Furthermore, the data assimilation procedure is intended to give access to quantities of interest for diagnosis purposes, and we present some promising results in this direction.