Publications

2015

• Hypergraph conditions for the solvability of the ergodic equation for zero-sum games
  
  • Akian Marianne
  • Gaubert Stephane
  • Hochart Antoine
  , 2015. The ergodic equation is a basic tool in the study of mean-payoff stochastic games. Its solvability entails that the mean payoff is independent of the initial state. Moreover, optimal stationary strategies are readily obtained from its solution. In this paper, we give a general sufficient condition for the solvability of the ergodic equation, for a game with finite state space but arbitrary action spaces. This condition involves a pair of directed hypergraphs depending only on the ``growth at infinity'' of the Shapley operator of the game. This refines a recent result of the authors which only applied to games with bounded payments, as well as earlier nonlinear fixed point results for order preserving maps, involving graph conditions.
• Quasi-Barabanov Semigroups and Finiteness of the $L_2$-Induced Gain for Switched Linear Control Systems: Case of Full-State Observation
  
  • Chitour Yacine
  • Mason Paolo
  • Sigalotti Mario
  , 2015. Motivated by an open problem posed by J.P. Hespanha we extend the notion of Barabanov norm and extremal trajectory to general classes of switching signals. As a consequence we characterize the finiteness of the L2-induced gain for a large set of switched linear control systems in case of full-state observation in terms of the sign of the generalized spectral radius associated with minimal realizations of the original switched system. (10.1109/cdc.2015.7402985)
  DOI : 10.1109/cdc.2015.7402985
• Tropical methods for the localization of eigenvalues and application to their numerical computation
  
  • Marchesini Andrea
  , 2015. In this thesis we use tropical mathematics to locate and numerically compute eigenvalues of matrices and matrix polynomials. The first part of the work focuses on eigenvalues of matrices, while the second part focuses on matrix polynomials and adds a numerical experimental side along the theoretical one. By “locating” an eigenvalue we mean being able to identify some bounds within which it must lie. This can be useful in situations where one only needs approximate eigenvalues; moreover, they make good starting values for iterative eigenvalue-finding algorithms. Rather than full location, our result for matrices is in the form of majorization bounds to control the absolute value of the eigenvalues. These bounds are to some extent a generalization to matrices of a result proved by Ostrowski for polynomials: he showed (albeit with different terminology) that the product of the k largest absolute values of the roots of a polynomial can be bounded from above and below by the product of its k largest tropical (max-times) roots, up to multiplicative factors which are independent of the coefficients of the polynomial. We prove an analogous result for matrices: the product of the k largest absolute values of eigenvalues is bounded, up to a multiplicative factor, by the product of the k largest tropical eigenvalues. It should be noted that tropical eigenvalues can be computed by using the solution to a parametric optimal assignment problem, in a way that is robust with respect to small perturbations in the data. Another thing worth mentioning is that the multiplicative factor in the bound is of combinatorial nature and it is reminiscent of a work by Friedland, who essentially proved a specialization of our result to the particular case k = 1 (i.e. for the largest eigenvalue only). We can interpret the absolute value as an archimedean valuation; in this light, there is a correspondence between the present result and previous work by Akian, Bapat and Gaubert, who dealt with the same problem for matrices over fields with non- archimedean valuation (specifically Puiseux series, with the leading exponent as valuation) and showed in that case more stringent bounds, with no multiplicative factor, and with generic equality rather than upper and lower bounds. The second part of the thesis revolves around the computation of eigenvalues of matrix polynomials. For linear matrix polynomials, stable algorithms such as the QZ method have been known for a long time. Eigenproblems for matrix polynomials of higher degree are usually reduced to the linear case, using a linearization such as the companion linearization. This however can worsen the condition number and backward error of the computed eigenvalue with respect to perturbations in the coefficients of the original polynomial (even if they remain stable in the coefficients of the linearized). To mitigate this inconvenience it is common to perform a scaling of the matrix polynomial before linearizing. Various scaling methods have been proposed. In our work, we introduce a two-sided diagonal scaling strategy based on the tropical eigenvalues of the matrix polynomial obtained by taking entrywise valuation of the original one (and we will consider both the archimedean and non-archimedean case). We study the effect of this scaling on the conditioning and backward error, with both analytic formulas and numerical examples, showing that it can increase the accuracy of the computed eigenvalues by several orders of magnitude.
• Convergent stochastic Expectation Maximization algorithm with efficient sampling in high dimension. Application to deformable template model estimation
  
  • Kuhn Estelle
  • Allassonnière Stéphanie
  • Durrleman Stanley
  , 2015, pp.np. Estimation in the deformable template model is a big challenge in image analysis. The issue is to estimate an atlas of a population. This atlas contains a template and the corresponding geometrical variability of the observed shapes. The goal is to propose an accurate estimation algorithm with low computational cost and with theoretical guaranties of relevance. This becomes very demanding when dealing with high dimensional data, which is particularly the case of medical images. The use of an optimized Monte Carlo Markov Chain method for a stochastic Expectation Maximization algorithm, is proposed to estimate the model parameters by maximizing the likelihood. A new Anisotropic Metropolis Adjusted Langevin Algorithm is used as transition in the MCMC method. First it is proven that this new sampler leads to a geometrically uniformly ergodic Markov chain. Furthermore, it is proven also that under mild conditions, the estimated parameters converge almost surely and are asymptotically Gaussian distributed. The methodology developed is then tested on handwritten digits and some 2D and 3D medical images for the deformable model estimation. More widely, the proposed algorithm can be used for a large range of models in many fields of applications such as pharmacology or genetic. The technical proofs are detailed in an appendix.
• The Landau-Lifshitz-Gilbert equation driven by Gaussian noise
  
  • Hocquet Antoine
  , 2015. This thesis is devoted to the influence of a noise term in the stochastic Landau-Lifshitz-Gilbert Equation (SLLG). It is a nonlinear stochastic partial differential equation with a non-convex constraint on the modulus of the solutions. First, we study in chapter 1 the question of local solvability. Using classical properties of stochastic integration with Banach space-valued processes, we propose a mild formulation, and give the existence and uniqueness of a local solution in any dimension, for a Gaussian noise, regular in space. Secondly, we focus on the specific study of SLLG in a two-dimensional space domain. Chapter 2 deals with the existence of a strong solution, in the probabilistic sense. Using the energy formula, we give a method to obtain uniquely a global solution in time. Chapter 3 gives uniqueness of weak solutions, provided that the energy satisfies a super-martingale property. This is the stochastic counterpart of a known deterministic result giving the uniqueness of weak solutions, knowing that the energy decreases. Chapter 4 gives the existence, in the so-called ``overdamped case'', of solutions that blow-up in finite time. We prove that, unlike the deterministic case, a singularity may appear with positive probability, regardless of the initial data chosen. Then we return to the case of general dimension of space, providing in chapter 5 a new time semi-discrete scheme for SLLG. This chapter is based on an article in collaboration with F. Alouges and A. De Bouard. We show the convergence in law of a projection-type scheme for SLLG, which has the advantage of respecting exactly the local constraint on the magnitude. This scheme treats the case of a rather general noise term regularized in space but infinite-dimensional. In Chapter 6, we show how to implement it with a finite element dicretization in space, and we give a practical method for approaching a regular noise in this framework. We also evidence numerical blow-up of the solutions, despite the presence of a gyromagnetic term, and of a more general noise than that of Chapter 4.
• Learning spatio-temporal trajectories from manifold-valued longitudinal data
  
  • Schiratti Jean-Baptiste
  • Allassonniere Stéphanie
  • Colliot Olivier
  • Durrleman Stanley
  , 2015. We propose a Bayesian mixed-effects model to learn typical scenarios of changes from longitudinal manifold-valued data, namely repeated measurements of the same objects or individuals at several points in time. The model allows to estimate a group-average trajectory in the space of measurements. Random variations of this trajectory result from spatiotemporal transformations, which allow changes in the direction of the trajectory and in the pace at which trajectories are followed. The use of the tools of Riemannian geometry allows to derive a generic algorithm for any kind of data with smooth constraints, which lie therefore on a Riemannian manifold. Stochastic approximations of the Expectation-Maximization algorithm is used to estimate the model parameters in this highly non-linear setting. The method is used to estimate a data-driven model of the progressive impairments of cognitive functions during the onset of Alzheimer’s disease. Experimental results show that the model correctly put into correspondence the age at which each in- dividual was diagnosed with the disease, thus validating the fact that it effectively estimated a normative scenario of disease progression. Random effects provide unique insights into the variations in the ordering and timing of the succession of cognitive impairments across different individuals.
• Learning spatiotemporal trajectories from manifold-valued longitudinal data
  
  • Schiratti Jean-Baptiste
  • Allassonniere Stéphanie
  • Colliot Olivier
  • Durrleman Stanley
  , 2015 (28). We propose a Bayesian mixed-effects model to learn typical scenarios of changes from longitudinal manifold-valued data, namely repeated measurements of the same objects or individuals at several points in time. The model allows to estimate a group-average trajectory in the space of measurements. Random variations of this trajectory result from spatiotemporal transformations, which allow changes in the direction of the trajectory and in the pace at which trajectories are followed. The use of the tools of Riemannian geometry allows to derive a generic algorithm for any kind of data with smooth constraints, which lie therefore on a Riemannian manifold. Stochastic approximations of the Expectation-Maximization algorithm is used to estimate the model parameters in this highly non-linear setting. The method is used to estimate a data-driven model of the progressive impairments of cognitive functions during the onset of Alzheimer's disease. Experimental results show that the model correctly put into correspondence the age at which each individual was diagnosed with the disease, thus validating the fact that it effectively estimated a normative scenario of disease progression. Random effects provide unique insights into the variations in the ordering and timing of the succession of cognitive impairments across different individuals.
• Geometric and asymptotic properties associated with linear switched systems
  
  • Chitour Yacine
  • Gaye Moussa
  • Mason Paolo
  Journal of Differential Equations, Elsevier, 2015, 259 (11), pp.5582-5616. Consider a continuous-time linear switched system on $\mathbb{R}^n$ associated with a compact convex set of matrices. When it is irreducible and its largest Lyapunov exponent is zero there always exists a Barabanov norm associated with the system. This paper deals with two types of issues: $(a)$ properties of Barabanov norms such as uniqueness up to homogeneity and strict convexity; $(b)$ asymptotic behaviour of the extremal solutions of the linear switched system.Regarding Issue $(a)$, we provide partial answers and propose four related open problems. As for Issue $(b)$, we establish, when $n=3$, a Poincar\'e-Bendixson theorem under a regularity assumption on the set of matrices. We then revisit a noteworthy result of N.E. Barabanov describing the asymptotic behaviour of linear switched system on $\mathbb{R}^3$ associated with a pair of Hurwitz matrices $\{A,A+bc^T\}$. After pointing out a fatal gap in Barabanov's proof we partially recover his result by alternative arguments. (10.1016/j.jde.2015.07.001)
  DOI : 10.1016/j.jde.2015.07.001
• Discrete limit and monotonicity properties of the Floquet eigenvalue in an age structured cell division cycle model
  
  • Gaubert Stéphane
  • Lepoutre Thomas
  Journal of Mathematical Biology, Springer, 2015, 71 (6). We consider a cell population described by an age-structured partial differential equation with time periodic coefficients. We assume that division only occurs after a minimal age (majority) and within certain time intervals. We study the asymptotic behavior of the dominant Floquet eigenvalue, or Perron-Frobenius eigenvalue, representing the growth rate, as a function of the majority age, when the division rate tends to infinity (divisions become instantaneous). We show that the dominant Floquet eigenvalue converges to a staircase function with an infinite number of steps, determined by a discrete dynamical system. As an intermediate result, we give a structural condition which guarantees that the dominant Floquet eigenvalue is a nondecreasing function of the division rate. We also give a counter example showing that the latter monotonicity property does not hold in general. (10.1007/s00285-015-0874-3)
  DOI : 10.1007/s00285-015-0874-3
• Formulas for phase recovering from phaseless scattering data at fixed frequency
  
  • Novikov Roman
  Bulletin des Sciences Mathématiques, Elsevier, 2015, 139 (8), pp.923–936. We consider quantum and acoustic wave propagation at fixed frequency for compactly supported scatterers in dimension d ≥ 2. In these framework we give explicit formulas for phase recovering from appropriate phaseless scattering data. As a corollary, we give global uniqueness results for quantum and acoustic inverse scattering at fixed frequency without phase information. (10.1016/j.bulsci.2015.04.005)
  DOI : 10.1016/j.bulsci.2015.04.005
• Convex Relaxations for Permutation Problems
  
  • Fogel F.
  • Jenatton R.
  • Bach Francis
  • d'Aspremont A.
  SIAM Journal on Matrix Analysis and Applications, Society for Industrial and Applied Mathematics, 2015, 36 (4). (10.1137/130947362)
  DOI : 10.1137/130947362
• Image processing in the semidiscrete group of rototranslations
  
  • Prandi Dario
  • Boscain Ugo
  • Gauthier Jean-Paul
  , 2015. It is well-known, since [12], that cells in the primary visual cortex V1 do much more than merely signaling position in the visual field: most cortical cells signal the local orientation of a contrast edge or bar – they are tuned to a particular local orientation. This orientation tuning has been given a mathematical interpretation in a sub-Riemannian model by Petitot, Citti, and Sarti [14,6]. According to this model, the primary visual cortex V1 lifts grey-scale images, given as functions f : R 2 → [0, 1], to functions Lf defined on the projectivized tangent bundle of the plane P T R 2 = R 2 × P 1. Recently, in [1], the authors presented a promising semidiscrete variant of this model where the Euclidean group of roto-translations SE(2), which is the double covering of P T R 2 , is replaced by SE(2, N), the group of translations and discrete rotations. In particular , in [15], an implementation of this model allowed for state-of-the-art image inpaintings. In this work, we review the inpainting results and introduce an application of the semidiscrete model to image recognition. We remark that both these applications deeply exploit the Moore structure of SE(2, N) that guarantees that its unitary representations behaves similarly to those of a compact group. This allows for nice properties of the Fourier transform on SE(2, N) exploiting which one obtains numerical advantages. 1 The semi-discrete model The starting point of our work is the sub-Riemannian model of the primary visual cortex V1 [14,6], and our recent contributions [3,1,2,4]. This model has also been deeply studied in [8,11]. In the sub-Riemannian model, V1 is modeled as the projective tangent bundle P T R 2 ∼ = R 2 × P 1 , whose double covering is the roto-translation group SE(2) = R 2 S 1 , endowed with a left-invariant sub-Riemannian structure that mimics the connections between neurons. In particular , grayscale visual stimuli f : R 2 → [0, 1] feeds V1 neurons N = (x, θ) ∈ P T R 2 with an extracellular voltage Lf (ξ) that is widely accepted to be given by Lf (ξ) = f, Ψ ξ. The functions {Ψ ξ } ξ∈P T R 2 are the receptive fields. A good fit is Ψ (x,θ) = π(x, θ)Ψ where Ψ is the Gabor filter (a sinusoidal multiplied by a Gaussian function) and π(x, θ)Ψ (y) := Ψ (R −θ (x − y)).
• Finding scattering data for a time-harmonic wave equation with first order perturbation from the Dirichlet-to-Neumann map
  
  • Agaltsov Alexey
  Journal of Inverse and Ill-posed Problems, De Gruyter, 2015, 23 (6), pp.627-645. We present formulas and equations for finding scattering data from the Dirichlet-to-Neumann map for a time-harmonic wave equation with first order perturbation with compactly supported coefficients. We assume that the coefficients are matrix-valued in general. To our knowledge, these results are new even for the general scalar case. (10.1515/jiip-2015-0014)
  DOI : 10.1515/jiip-2015-0014
• Optimal grasping points identification for a rotational four-fingered aerogripper
  
  • Vazquez Jesus
  • Giacomini Matteo
  • Escareno Juan Antonio
  • Rubio Elsa
  • Sossa Humberto
  , 2015, pp.272-277. In the present paper, we deal with the grasping aspect regarding reactive object retrieving during fast aerial pick-and-place maneuevers. The paper addresses the problem of the optimal grabbing of an object by means of a 5 degrees of freedom (DoF) gripper. Also, is introduced the optimization framework for the identification of the optimal contact points between the end-effector and the object. In particular, we define an objective functional to evaluate the optimality of the contact points and we propose a preliminary strategy to identify the optimal contact points. Simulations and experimental are presented to support the actual proposal. (10.1109/RED-UAS.2015.7441017)
  DOI : 10.1109/RED-UAS.2015.7441017
• Lyapunov exponents for random continuous-time switched systems and stabilizability
  
  • Colonius Fritz
  • Mazanti Guilherme
  , 2015. For linear systems in continuous time with random switching, the Lyapunov exponents are characterized using the Multiplicative Ergodic Theorem for an associated system in discrete time. An application to control systems shows that here a controllability condition implies that arbitrary exponential decay rates for almost sure stabilization can be obtained.
• Convex and Spectral Relaxations for Phase Retrieval, Seriation and Ranking
  
  • Fogel Fajwel
  , 2015. Optimization is often the computational bottleneck in disciplines such as statistics, biology, physics, finance or economics. Many optimization problems can be directly cast in the well- studied convex optimization framework. For non-convex problems, it is often possible to derive convex or spectral relaxations, i.e., derive approximations schemes using spectral or convex optimization tools. Convex and spectral relaxations usually provide guarantees on the quality of the retrieved solutions, which often transcribes in better performance and robustness in practical applications, compared to naive greedy schemes. In this thesis, we focus on the problems of phase retrieval, seriation and ranking from pairwise comparisons. For each of these combinatorial problems we formulate convex and spectral relaxations that are robust, flexible and scalable.
• Virulence evolution at the front line of spreading epidemics
  
  • Griette Quentin
  • Raoul Gaël
  • Gandon Sylvain
  Evolution - International Journal of Organic Evolution, Wiley, 2015, 69 (11), pp.2810-2819. Understanding and predicting the spatial spread of emerging pathogens is a major challenge for the public health management of infectious diseases. Theoretical epidemiology shows that the speed of an epidemic is governed by the life-history characteristics of the pathogen and its ability to disperse. Rapid evolution of these traits during the invasion may thus affect the speed of epidemics. Here we study the influence of virulence evolution on the spatial spread of an epidemic. At the edge of the invasion front, we show that more virulent and transmissible genotypes are expected to win the competition with other pathogens. Behind the front line, however, more prudent exploitation strategies outcompete virulent pathogens. Crucially, even when the presence of the virulent mutant is limited to the edge of the front, the invasion speed can be dramatically altered by pathogen evolution. We support our analysis with individual-based simulations and we discuss the additional effects of demographic stochasticity taking place at the front line on virulence evolution. We confirm that an increase of virulence can occur at the front, but only if the carrying capacity of the invading pathogen is large enough. These results are discussed in the light of recent empirical studies examining virulence evolution at the edge of spreading epidemics. (10.1111/evo.12781)
  DOI : 10.1111/evo.12781
• Stochastic dynamics of adaptive trait and neutral marker driven by eco-evolutionary feedbacks
  
  • Billiard Sylvain
  • Ferriere Regis
  • Méléard Sylvie
  • Tran Viet Chi
  Journal of Mathematical Biology, Springer, 2015, 71 (5), pp.1211-1242. How the neutral diversity is affected by selection and adaptation is investigated in an eco-evolutionary framework. In our model, we study a finite population in continuous time, where each individual is characterized by a trait under selection and a completely linked neutral marker. Population dynamics are driven by births and deaths, mutations at birth, and competition between individuals. Trait values influence ecological processes (demographic events, competition), and competition generates selection on trait variation, thus closing the eco-evolutionary feedback loop. The demographic effects of the trait are also expected to influence the generation and maintenance of neutral variation. We consider a large population limit with rare mutation, under the assumption that the neutral marker mutates faster than the trait under selection. We prove the convergence of the stochastic individual-based process to a new measure-valued diffusive process with jumps that we call Substitution Fleming-Viot Process (SFVP). When restricted to the trait space this process is the Trait Substitution Sequence first introduced by Metz et al. (1996). During the invasion of a favorable mutation, a genetical bottleneck occurs and the marker associated with this favorable mutant is hitchhiked. By rigorously analysing the hitchhiking effect and how the neutral diversity is restored afterwards, we obtain the condition for a time-scale separation; under this condition, we show that the marker distribution is approximated by a Fleming-Viot distribution between two trait substitutions. We discuss the implications of the SFVP for our understanding of the dynamics of neutral variation under eco-evolutionary feedbacks and illustrate the main phenomena with simulations. Our results highlight the joint importance of mutations, ecological parameters, and trait values in the restoration of neutral diversity after a selective sweep. (10.1007/s00285-014-0847-y)
  DOI : 10.1007/s00285-014-0847-y
• Analysis of a periodic optimal control problem connected to microalgae anaerobic digestion
  
  • Bayen Térence
  • Mairet Francis
  • Martinon Pierre
  • Sebbah Matthieu
  Optimal Control Applications and Methods, Wiley, 2015, 36 (6), pp.750-773. In this work, we study the coupling of a culture of microalgae limited by light and an anaerobic digester in a two-tank bioreactor. The model for the reactor combines a periodic day-night light for the culture of microalgae and a classical chemostat model for the digester. We first prove the existence and attraction of periodic solutions of this problem for a 1 day period. Then, we study the optimal control problem of optimizing the production of methane in the digester during a certain timeframe, the control on the system being the dilution rate (the input flow of microalgae in the digester). We apply Pontryagin's Maximum Principle in order to characterize optimal controls, including the computation of singular controls. We present numerical simulations by direct and indirect methods for different light models and compare the optimal 1-day periodic solution to the optimal strategy over larger timeframes. Finally, we also investigate the dependence of the optimal cost with respect to the volume ratio of the two tanks. (10.1002/oca.2127)
  DOI : 10.1002/oca.2127
• Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem
  
  • Cakoni Fioralba
  • Haddar Houssem
  • Harris Isaac
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2015, 9 (4), pp.1025 - 1049. We consider the interior transmission problem associated with the scattering by an inhomogeneous (possibly anisotropic) highly oscillating periodic media. We show that, under appropriate assumptions, the solution of the interior transmission problem converges to the solution of a homogenized problem as the period goes to zero. Furthermore, we prove that the associated real transmission eigenvalues converge to transmission eigenvalues of the homogenized problem. Finally we show how to use the first transmission eigenvalue of the period media, which is measurable from the scattering data, to obtain information about constant effective material properties of the periodic media. The convergence results presented here are not optimal. Such results with rate of convergence involve the analysis of the boundary correction and will be subject of a forthcoming paper. (10.3934/ipi.2015.9.1025)
  DOI : 10.3934/ipi.2015.9.1025
• Maximal Lower Bounds in the Loewner order
  
  • Stott Nikolas
  • Allamigeon Xavier
  • Gaubert Stéphane
  , 2015. We show that the set of maximal lower bounds of two symmetric matrices with respect to Loewner order can be identified to the quotient set O(p,q)/(O(p)×O(q)). Here, (p,q)denotes the inertia of the difference of the two matrices, O(p) is the p-th orthogonal group, and O(p,q) is the indefinite orthogonal group arising from a quadratic form with inertia (p,q). We discuss the application of this result to the synthesis of ellipsoidal invariants of hybrid dynamical systems.
• Tropical bounds for the eigenvalues of block structured matrices
  
  • Akian Marianne
  • Gaubert Stephane
  • Marchesini Andrea
  , 2015. We establish a log-majorization inequality, which relates the moduli of the eigenvalues of a block structured matrix with the tropical eigenvalues of the matrix obtained by replacing every block entry of the original matrix by its norm. This inequality involves combinatorial constants depending on the size and pattern of the matrix. Its proof relies on diagonal scalings, constructed from the optimal dual variables of a parametric optimal assignment problem.
• Hungarian Scaling of Polynomial Eigenproblems
  
  • Akian Marianne
  • Gaubert Stephane
  • Marchesini Andrea
  • Tisseur Françoise
  , 2015. We study the behaviour of the eigenvalues of a parametric matrix polynomial P in a neighbourhood of zero. If we suppose that the entries of P have Puiseux series expansions we can build an auxiliary matrix polynomial Q whose entries are the leading exponents of those of P. We show that preconditioning P via a diagonal scaling based on the tropical eigenvalues of Q can improve conditioning and backward error of the eigenvalues.
• Discrete-valued-pulse optimal control algorithms: Application to spin systems
  
  • Dridi G
  • Lapert M
  • Salomon Julien
  • Sugny D
  • Glaser S. J.
  Physical Review A : Atomic, molecular, and optical physics [1990-2015], American Physical Society, 2015, 92, pp.043417. This article is aimed at extending the framework of optimal control techniques to the situation where the control field values are restricted to a finite set. We propose generalizations of the standard GRAPE algorithm suited to this constraint. We test the validity and the efficiency of this approach for the inversion of an inhomogeneous ensemble of spin systems with different offset frequencies. It is shown that a remarkable efficiency can be achieved even for a very limited number of discrete values. Some applications in nuclear magnetic resonance are discussed. (10.1103/PhysRevA.92.043417)
  DOI : 10.1103/PhysRevA.92.043417
• The Page-Rényi parking process
  
  • Gerin Lucas
  The Electronic Journal of Combinatorics, Open Journal Systems, 2015, 22 (4), pp.P.4.4. In the Page parking (or packing) model on a discrete interval (also known as the discrete Rényi packing problem or the unfriendly seating problem), cars of length two successively park uniformly at random on pairs of adjacent places, until only isolated places remain. We give a probabilistic proof of the (known) fact that the proportion of the interval covered by cars goes to 1-exp(-2) , when the length of the interval goes to infinity. We obtain some new consequences, and also study a version of this process defined on the infinite line.