Publications

2013

• Transmission eigenvalues
  
  • Cakoni Fioralba
  • Haddar Houssem
  Inverse Problems, IOP Publishing, 2013, 29 (10), pp.100201. In inverse scattering theory, transmission eigenvalues can be seen as the extension of the notion of resonant frequencies for impenetrable objects to the case of penetrable dielectrics. The transmission eigenvalue problem is a relatively late arrival to the spectral theory of partial differential equations. Its first appearance was in 1986 in a paper by Kirsch who was investigating the denseness of far-field patterns for scattering solutions of the Helmholtz equation or, in more modern terminology, the injectivity of the far-field operator [1]. The paper of Kirsch was soon followed by a more systematic study by Colton and Monk in the context of developing the dual space method for solving the inverse scattering problem for acoustic waves in an inhomogeneous medium [2]. In this paper they showed that for a spherically stratified media transmission eigenvalues existed and formed a discrete set. Numerical examples were also given showing that in principle transmission eigenvalues could be determined from the far-field data. This first period of interest in transmission eigenvalues was concluded with papers by Colton et al in 1989 [3] and Rynne and Sleeman in 1991 [4] showing that for an inhomogeneous medium (not necessarily spherically stratified) transmission eigenvalues, if they existed, formed a discrete set. For the next seventeen years transmission eigenvalues were ignored. This was mainly due to the fact that, with the introduction of various sampling methods to determine the shape of an inhomogeneous medium from far-field data, transmission eigenvalues were something to be avoided and hence the fact that transmission eigenvalues formed at most a discrete set was deemed to be sufficient. In addition, questions related to the existence of transmission eigenvalues or the structure of associated eigenvectors were recognized as being particularly difficult due to the nonlinearity of the eigenvalue problem and the special structure of the associated transmission eigenvalue problem. The need to answer these questions became important after a series of papers by Cakoni et al [5], and Cakoni et al [6] suggesting that these transmission eigenvalues could be used to obtain qualitative information about the material properties of the scattering object from far-field data. The first answer to the existence of transmission eigenvalues in the general case was given in 2008 when Päivärinta and Sylvester showed the existence of transmission eigenvalues for the index of refraction sufficiently large [7] followed in 2010 by the paper of Cakoni et al who removed the size restriction on the index of refraction [8]. More importantly, in the latter it was shown that transmission eigenvalues yielded qualitative information on the material properties of the scattering object and Cakoni et al established in [9] that transmission eigenvalues could be determined from the Tikhonov regularized solution of the far-field equation. Since the appearance of these papers there has been an explosion of interest in the transmission eigenvalue problem (we refer the reader to our recent survey paper [10] for a detailed account of the developments in this field up to 2012) and the papers in this special issue are representative of the myriad directions that this research has taken. Indeed, we are happy to see that many open theoretical and numerical questions raised in [10] have been answered (totally or partially) in the contributions of this special issue: the existence of transmission eigenvalues with minimal assumptions on the contrast, the numerical evaluation of transmission eigenvalues, the inverse spectral problem, applications to non-destructive testing, etc. In addition to these topics, many other new investigations and research directions have been proposed as we shall see in the brief content summary below. A number of papers in this special issue are concerned with the question of existence of transmission eigenvalues and the structure of the associated transmission eigenfunctions. The three papers by respectively Robbiano [11], Blasten and Päivärinta [12], and Lakshtanov and Vainberg [13] provide new complementary results on the existence of transmission eigenvalues for the scalar problem under weak assumptions on the (possibly complex valued) refractive index that mainly stipulates that the contrast does not change sign on the boundary. It is interesting here to see three different new methods to obtain these results. On the other hand, the paper by Bonnet-Ben Dhia and Chesnel [14] addresses the Fredholm properties of the interior transmission problem when the contrast changes sign on the boundary, exhibiting cases where this property fails. Using more standard approaches, the existence and structure of transmission eigenvalues are analyzed in the paper by Delbary [15] for the case of frequency dependent materials in the context of Maxwell's equations, whereas the paper by Vesalainen [16] initiates the study of the transmission eigenvalue problem in unbounded domains by considering the transmission eigenvalues for Schrödinger equation with non-compactly supported potential. The paper by Monk and Selgas [17] addresses the case where the dielectric is mounted on a perfect conductor and provides some numerical examples of the localization of associated eigenvalues using the linear sampling method. A series of papers then addresses the question of localization of transmission eigenvalues and the associated inverse spectral problem for spherically stratified media. More specifically, the paper by Colton and Leung [18] provides new results on complex transmission eigenvalues and a new proof for uniqueness of a solution to the inverse spectral problem, whereas the paper by Sylvester [19] provides sharp results on how to locate all the transmission eigenvalues associated with angular independent eigenfunctions when the index of refraction is constant. The paper by Gintides and Pallikarakis [20] investigates an iterative least square method to identify the spherically stratified index of refraction from transmission eigenvalues. On the characterization of transmission eigenvalues in terms of far-field measurements, a promising new result is obtained by Kirsch and Lechleiter [21] showing how one can identify the transmission eigenvalues using the eigenvalues of the scattering operator which are available in terms of measured scattering data. In the paper by Kleefeld [22], an accurate method for computing transmission eigenvalues based on a surface integral formulation of the interior transmission problem and numerical methods for nonlinear eigenvalue problems is proposed and numerically validated for the scalar problem in three dimensions. On the other hand, the paper by Sun and Xu [23] investigates the computation of transmission eigenvalues for Maxwell's equations using a standard iterative method associated with a variational formulation of the interior transmission problem with an emphasis on the effect of anisotropy on transmission eigenvalues. From the perspective of using transmission eigenvalues in non-destructive testing, the paper by Cakoni and Moskow [24] investigates the asymptotic behavior of transmission eigenvalues with respect to small inhomogeneities. The paper by Nakamura and Wang [25] investigates the linear sampling method for the time dependent heat equation and analyses the interior transmission problem associated with this equation. Finally, in the paper by Finch and Hickmann [26], the spectrum of the interior transmission problem is related to the unique determination of the acoustic properties of a body in thermoacoustic imaging. We hope that this collection of papers will stimulate further research in the rapidly growing area of transmission eigenvalues and inverse scattering theory. (10.1088/0266-5611/29/10/100201)
  DOI : 10.1088/0266-5611/29/10/100201
• First and second order optimality conditions for optimal control problems of state constrained integral equations
  
  • Bonnans J. Frédéric
  • de La Vega Constanza
  • Dupuis Xavier
  Journal of Optimization Theory and Applications, Springer Verlag, 2013, 159 (1), pp.1-40. This paper deals with optimal control problems of integral equations, with initial-final and running state constraints. The order of a running state constraint is defined in the setting of integral dynamics, and we work here with constraints of arbitrary high orders. First and second-order necessary conditions of optimality are obtained, as well as second-order sufficient conditions. (10.1007/s10957-013-0299-3)
  DOI : 10.1007/s10957-013-0299-3
• Ecologie prédictive & changement planétaire
  
  • Austerlitz Frédéric
  • Blum Michael
  • Calba Sarah
  • Chave Jérôme
  • Choisy Marc
  • Coreau Audrey
  • Devictor Vincent
  • Doyen Luc
  • Dray Stéphane
  • Duputié Anne
  • Eveillard Damien
  • Faure Denis
  • Favier Charly
  • Gaggiotti Oscar
  • Galtier Nicolas
  • Garnier Éric
  • Gimenez Olivier
  • Guis Helene
  • Herbreteau Vincent
  • Huneman Philippe
  • Jabot Franck
  • Jarne Philippe
  • Joly Dominique
  • Julliard Romain
  • Kéfi Sonia
  • Kergoat Gael
  • Lacroix Gerard
  • Lagadeuc Yvan
  • Lavorel Sandra
  • Le Gaillard Jean-François
  • Le Gall Line
  • Loreau Michel
  • Maris Virginie
  • Morand Serge
  • Morin Xavier
  • Morlon Hélène
  • Mouquet Nicolas
  • Pinay Gilles
  • Pottier Julien
  • Pradel Roger
  • Ronce Ophélie
  • Schurr Frank
  • Simonet Pascal
  • Teplitsky Céline
  • Thuiller Wilfried
  • Tran Anne-Lise
  • Venner Samuel
  , 2013, hors s\'{e}rie, pp.9-44.
• A Hamilton-Jacobi approach to junction problems and application to traffic flows
  
  • Imbert Cyril
  • Monneau Régis
  • Zidani Hasnaa
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2013, 19 (01), pp.pp 129-166. This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a ''junction'', that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to be new. They are applied to the study of some models arising in traffic flows. The techniques developed in the present article provide new powerful tools for the analysis of such problems. (10.1051/cocv/2012002)
  DOI : 10.1051/cocv/2012002
• A homogenization approach for the motion of motor proteins
  
  • Mirrahimi Sepideh
  • Souganidis Panagiotis E.
  Nonlinear Differential Equations and Applications, Springer Verlag, 2013, 20, pp.129-147. We consider the asymptotic behavior of an evolving weakly coupled Fokker-Planck system of two equations set in a periodic environment. The magnitudes of the diffusion and the coupling are respectively proportional and inversely proportional to the size of the period. We prove that, as the period tends to zero, the solutions of the system either propagate (concentrate) with a fixed constant velocity (determined by the data) or do not move at all. The system arises in the modeling of motor proteins which can take two different states. Our result implies that, in the limit, the molecules either move along a filament with a fixed direction and constant speed or remain immobile. (10.1007/s00030-012-0156-3)
  DOI : 10.1007/s00030-012-0156-3
• Daphnias: from the individual based model to the large population equation
  
  • Metz J.A.J.
  • Tran Viet Chi
  Journal of Mathematical Biology, Springer, 2013, 66 (4-5), pp.915--933. The class of deterministic 'Daphnia' models treated by Diekmann et~al. (J Math Biol 61: 277--318, 2010) has a long history going back to Nisbet and Gurney (Theor Pop Biol 23: 114--135, 1983) and Diekmann et~al. (Nieuw Archief voor Wiskunde 4: 82--109, 1984). In this note, we formulate the individual based models (IBM) supposedly underlying those deterministic models. The models treat the interaction between a general size-structured consumer population ('Daphnia') and an unstructured resource ('algae'). The discrete, size and age-structured Daphnia population changes through births and deaths of its individuals and throught their aging and growth. The birth and death rates depend on the sizes of the individuals and on the concentration of the algae. The latter is supposed to be a continuous variable with a deterministic dynamics that depends on the Daphnia population. In this model setting we prove that when the Daphnia population is large, the stochastic differential equation describing the IBM can be approximated by the delay equation featured in (Diekmann et~al., l.c.). (10.1007/s00285-012-0619-5)
  DOI : 10.1007/s00285-012-0619-5
• On the robust superhedging of measurable claims
  
  • Possamaï Dylan
  • Royer Guillaume
  • Touzi Nizar
  Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2013, 18 (95), pp.1-13. The problem of robust hedging requires to solve the problem of superhedging under a nondominated family of singular measures. Recent progress was achieved by van Handel, Neufeld, and Nutz. We show that the dual formulation of this problem is valid in a context suitable for martingale optimal transportation or, more generally, for optimal transportation under controlled stochastic dynamics (10.1214/ECP.v18-2739)
  DOI : 10.1214/ECP.v18-2739
• On the asymptotics of a Robin eigenvalue problem
  
  • Cakoni Fioralba
  • Chaulet Nicolas
  • Haddar Houssem
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2013, 351, pp.517-521. The considered Robin problem can formally be seen as a small perturbation of a Dirichlet problem. However, due to the sign of the impedance value, its associated eigenvalues converge point-wise to −∞ as the perturbation goes to zero. We prove in this case that Dirichlet eigenpairs are the only accumulation points of the Robin eigenpairs with normalized eigenvectors. We then provide a criterion to select accumulating sequences of eigenvalues and eigenvectors and exhibit their full asymptotic with respect to the small parameter. (10.1016/j.crma.2013.07.022)
  DOI : 10.1016/j.crma.2013.07.022
• Homogenization of a Conductive, Convective and Radiative Heat Transfer Problem in a Heterogeneous Domain
  
  • Allaire Grégoire
  • Habibi Zakaria
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2013, 45 (3), pp.1136-1178. We are interested in the homogenization of heat transfer in periodic porous media where the fluid part is made of long thin parallel cylinders, the diameter of which is of the same order than the period. The heat is transported by conduction in the solid part of the domain and by conduction, convection and radiative transfer in the fluid part (the cylinders). A non-local boundary condition models the radiative heat transfer on the cylinder walls. To obtain the homogenized problem we first use a formal two-scale asymptotic expansion method. The resulting effective model is a convection-diffusion equation posed in a homogeneous domain with homogenized coefficients evaluated by solving so-called cell problems where radiative transfer is taken into account. In a second step we rigorously justify the homogenization process by using the notion of two-scale convergence. One feature of this work is that it combines homogenization with a 3D to 2D asymptotic analysis since the radiative transfer in the limit cell problem is purely two-dimensional. Eventually, we provide some 3D numerical results in order to show the convergence and the computational advantages of our homogenization method.
• Time-reversal in visco-elastic media.
  
  • Ammari Habib
  • Bretin Elie
  • Garnier Josselin
  • Wahab Abdul
  European Journal of Applied Mathematics, Cambridge University Press (CUP), 2013 (24), pp.565-600. In this paper, we consider the problem of reconstructing sources in a homogeneous viscoelastic medium from wavefield measurements. We first present a modified time-reversal imaging algorithm based on a weighted Helmholtz decomposition and justify mathematically that it provides a better approximation than by simply time reversing the displacement field, where artifacts due to the coupling betwe en the pressure and shear waves appear. Then, we investigate the source inverse problem in an elastic attenuating medium. We provide a regularized time-reversal imagin g which corrects the attenuation effect at the first order. The results of this paper yie ld the fundamental tools for solving imaging problems in elastic media using cross correl ation techniques
• Asymptotic enumeration of Eulerian orientations for graphs with strong mixing properties
  
  • Isaev Mikhail
  • Kseniia Isaeva
  Journal of Applied and Industrial Mathematics / Sibirskii Zhurnal Industrial'noi Matematiki and Diskretnyi Analiz i Issledovanie Operatsii, MAIK Nauka/Interperiodica, 2013, 20 (6), pp.40-58. We prove an asymptotic formula for the number of Eulerian orientations for graphs with strong mixing properties and with vertices having even degrees. The exact value is determined up to the multiplicative error O(n^{-1+epsilon}), where n is the number of vertices.
• Aircraft classification with a low resolution infrared sensor
  
  • Lefebvre Sidonie
  • Allassonniere Sidonie
  • Jakubowicz Jérémie
  • Lasne Thomas
  • Moulines Éric
  Machine Vision and Applications, Springer Verlag, 2013, 24 (1), pp.175-186. Existing computer simulations of aircraft infrared signature (IRS) do not account for dispersion induced by uncertainty on input parameters, such as aircraft aspect angles and meteorological conditions. As a result, they are of little use to quantify the detection performance of IR optronic systems: in this case, the scenario encompasses a lot of possible situations that must indeed be considered, but cannot be individually simulated. In this paper, we focus on low resolution infrared sensors and we propose a methodological approach for predicting simulated IRS dispersion of an aircraft, and performing a classification of different aircraft on the resulting set of low resolution infrared images. It is based on a quasi-Monte Carlo survey of the code output dispersion, and on a maximum likelihood classification taking advantage of Bayesian dense deformable template models estimation. This method is illustrated in a typical scenario, i.e., a daylight air-to-ground full-frontal attack by a generic combat aircraft flying at low altitude, over a database of 30,000 simulated aircraft images. Assuming a spatially white noise background model, classification performance is very promising, and appears to be more accurate than more classical state of the art techniques (such as kernel-based support vector classifiers). (10.1007/s00138-012-0437-1)
  DOI : 10.1007/s00138-012-0437-1
• On the extinction of Continuous State Branching Processes with catastrophes
  
  • Bansaye Vincent
  • Pardo Millan Juan Carlos
  • Smadi Charline
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2013, 18, pp.1-31. We consider continuous state branching processes (CSBP) with additional multiplicative jumps modeling dramatic events in a random environment. These jumps are described by a Lévy process with bounded variation paths. We construct a process of this class as the unique solution of a stochastic differential equation. The quenched branching property of the process allows us to derive quenched and annealed results and to observe new asymptotic behaviors. We characterize the Laplace exponent of the process as the solution of a backward ordinary differential equation and establish the probability of extinction. Restricting our attention to the critical and subcritical cases, we show that four regimes arise for the speed of extinction, as in the case of branching processes in random environment in discrete time and space. The proofs are based on the precise asymptotic behavior of exponential functionals of Lévy processes. Finally, we apply these results to a cell infection model and determine the mean speed of propagation of the infection. (10.1214/EJP.v18-2774)
  DOI : 10.1214/EJP.v18-2774
• Semi-infinite paths of the two dimensional radial spanning tree
  
  • Baccelli François
  • Coupier David
  • Tran Viet Chi
  Advances in Applied Probability, Applied Probability Trust, 2013, 45 (4), pp.895-916. We study semi-infinite paths of the radial spanning tree (RST) of a Poisson point process in the plane. We first show that the expectation of the number of intersection points between semi-infinite paths and the sphere with radius $r$ grows sublinearly with $r$. Then, we prove that in each (deterministic) direction, there exists with probability one a unique semi-infinite path, framed by an infinite number of other semi-infinite paths of close asymptotic directions. The set of (random) directions in which there are more than one semi-infinite paths is dense in $[0,2\pi)$. It corresponds to possible asymptotic directions of competition interfaces. We show that the RST can be decomposed in at most five infinite subtrees directly connected to the root. The interfaces separating these subtrees are studied and simulations are provided. (10.1239/aap/1386857849)
  DOI : 10.1239/aap/1386857849
• Energy and regularity dependent stability estimates for near-field inverse scattering in multidimensions
  
  • Isaev Mikhail
  Journal of Mathematics, Hindawi Publishing Corp., 2013, pp.DOI:10.1155/2013/318154. We prove new global Hölder-logarithmic stability estimates for the near-field inverse scattering problem in dimension $d\geq 3$. Our estimates are given in uniform norm for coefficient difference and related stability efficiently increases with increasing energy and/or coefficient regularity. In addition, a global logarithmic stability estimate for this inverse problem in dimension $d=2$ is also given. (10.1155/2013/318154)
  DOI : 10.1155/2013/318154
• Optimally swimming stokesian robots
  
  • Alouges François
  • Desimone Antonio
  • Heltai Luca
  • Lefebvre-Lepot Aline
  • Merlet Benoît
  Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences, 2013, 18 (5), pp.1189-1215. We study self-propelled stokesian robots composed of assemblies of balls, in dimensions 2 and 3, and prove that they are able to control their position and orientation. This is a result of controllability, and its proof relies on applying Chow's theorem in an analytic framework, similar to what has been done in [3] for an axisymmetric system swimming along the axis of symmetry. We generalize the analyticity result given in [3] to the situation where the swimmers can move either in a plane or in three-dimensional space, hence experiencing also rotations. We then focus our attention on energetically optimal strokes, which we are able to compute numerically. Some examples of computed optimal strokes are discussed in detail. (10.3934/dcdsb.2013.18.1189)
  DOI : 10.3934/dcdsb.2013.18.1189
• Path Planning and Ground Control Station Simulator for UAV
  
  • Ajami Alain
  • Balmat Jean-François
  • Gauthier Jean-Paul
  • Maillot Thibault
  , 2013, pp.1-13. no abstract
• Asymptotic enumeration of Eulerian circuits for graphs with strong mixing properties
  
  • Isaev Mikhail
  Izvestiya RAN. Serya Matematicheskaya, 2013, 77 (6), pp.45-70. We prove an asymptotic formula for the number of Eulerian circuits for graphs with strong mixing properties and with vertices having even degrees. The exact value is determined up to the multiplicative error $O(n^{-1/2+\varepsilon})$, where $n$ is the number of vertices
• Analysis of the factorization method for a general class of boundary conditions
  
  • Chamaillard Mathieu
  • Chaulet Nicolas
  • Haddar Houssem
  Journal of Inverse and Ill-posed Problems, De Gruyter, 2013. We analyze the factorization method (introduced by Kirsch in 1998 to solve inverse scattering problems at fixed frequency from the farfield operator) for a general class of boundary conditions that generalizes impedance boundary conditions. For instance, when the surface impedance operator is of pseudo-differential type, our main result stipulates that the factorization method works if the order of this operator is different from one and the operator is Fredholm of index zero with non negative imaginary part. We also provide some validating numerical examples for boundary operators of second order with discussion on the choice of the testing function. (10.1515/jip-2013-0013)
  DOI : 10.1515/jip-2013-0013
• A Formula for Popp’s Volume in Sub-Riemannian Geometry
  
  • Barilari Davide
  • Rizzi Luca
  Analysis and Geometry in Metric Spaces, Versita, 2013, 1. For an equiregular sub-Riemannian manifold M, Popp's volume is a smooth volume which is canonically associated with the sub-Riemannian structure, and it is a natural generalization of the Riemannian one. In this paper we prove a general formula for Popp's volume, written in terms of a frame adapted to the sub-Riemannian distribution. As a first application of this result, we prove an explicit formula for the canonical sub-Laplacian, namely the one associated with Popp's volume. Finally, we discuss sub-Riemannian isometries, and we prove that they preserve Popp's volume. We also show that, under some hypotheses on the action of the isometry group of M, Popp's volume is essentially the unique volume with such a property. (10.2478/agms-2012-0004)
  DOI : 10.2478/agms-2012-0004
• Spatiotemporal Dynamic Simulation of Acute Perfusion/Diffusion Ischemic Stroke Lesions Evolution: A Pilot Study Derived from Longitudinal MR Patient Data
  
  • Rekik Islem
  • Allassonnière Stéphanie
  • Durrleman Stanley
  • Carpenter Trevor
  • Wardlaw Joanna M
  Computational and Mathematical Methods in Medicine, Hindawi Publishing Corporation, 2013. The spatiotemporal evolution of stroke lesions, from acute injury to final tissue damage, is complex. Diffusion-weighted (DWI) and perfusion-weighted (PWI) imaging is commonly used to detect early ischemic changes and attempts to distinguish between permanently damaged and salvageable tissues. To date, 2D and 3D measures of diffusion/perfusion regions at individual timepoints have been widely used but may underestimate the true lesion spatio-temporal dynamics. Currently there is no spatio-temporal 4D dynamic model that simulates the continuous evolution of ischemic stroke from MR images. We determined whether a 4D current-based diffeomorphic model, developed in the field of statistical modeling for measuring the variability of anatomical surfaces, could estimate patient-specific spatio-temporal continuous evolution for MR PWI (measured as mean transit time, (MTT)) and DWI lesions. In our representative pilot sample, the model fitted the data well. Our dynamic analysis of lesion evolution showed different patterns; for example, some DWI/PWI dynamic changes corresponded with DWI lesion expansion into PWI lesions, but other patterns were much more complex and diverse. There was wide variation in the time when the final tissue damage was reached after stroke for DWI and MTT (10.1155/2013/283593)
  DOI : 10.1155/2013/283593
• Partition-Based Conditional Density Estimation
  
  • Cohen Serge X.
  • Le Pennec Erwan
  ESAIM: Probability and Statistics, EDP Sciences, 2013, 17, pp.672--697. (10.1051/ps/2012017)
  DOI : 10.1051/ps/2012017
• New global stability estimates for monochromatic inverse acoustic scattering
  
  • Isaev Mikhail
  • Novikov Roman
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2013, 45 (3), pp.1495-1504. We give new global stability estimates for monochromatic inverse acoustic scattering. These estimates essentially improve estimates of [P. Hahner, T. Hohage, SIAM J. Math. Anal., 33(3), 2001, 670-685] and can be considered as a solution of an open problem formulated in the aforementioned work.
• Extinction probabilities for a distylous plant population modeled by an inhomogeneous random walk on the positive quadrant
  
  • Lafitte-Godillon Pauline
  • Raschel Kilian
  • Tran Viet Chi
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2013, 73 (2), pp.700-722. In this paper, we study a flower population in which self-reproduction is not permitted. Individuals are diploid, {that is, each cell contains two sets of chromosomes}, and {distylous, that is, two alleles, A and a, can be found at the considered locus S}. Pollen and ovules of flowers with the same genotype at locus S cannot mate. This prevents the pollen of a given flower to fecundate its {own} stigmata. Only genotypes AA and Aa can be maintained in the population, so that the latter can be described by a random walk in the positive quadrant whose components are the number of individuals of each genotype. This random walk is not homogeneous and its transitions depend on the location of the process. We are interested in the computation of the extinction probabilities, {as} extinction happens when one of the axis is reached by the process. These extinction probabilities, which depend on the initial condition, satisfy a doubly-indexed recurrence equation that cannot be solved directly. {Our contribution is twofold : on the one hand, we obtain an explicit, though intricate, solution through the study of the PDE solved by the associated generating function. On the other hand, we provide numerical results comparing stochastic and deterministic approximations of the extinction probabilities. (10.1137/120864258)
  DOI : 10.1137/120864258
• Normal forms and invariants for 2-dimensional almost-Riemannian structures
  
  • Boscain Ugo
  • Charlot Grégoire
  • Ghezzi Roberta
  Differential Geometry and its Applications, Elsevier, 2013, 31 (1), pp.41-62. 2-Dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, there are three types of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not, and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket. In this paper we consider the problem of finding normal forms and functional invariants at each type of point. We also require that functional invariants are "complete" in the sense that they permit to recognize locally isometric structures. The problem happens to be equivalent to the one of finding a smooth canonical parameterized curve passing through the point and being transversal to the distribution. For Riemannian points such that the gradient of the Gaussian curvature K is different from zero, we use the level set of K as support of the parameterized curve. For Riemannian points such that the gradient of the curvature vanishes (and under additional generic conditions), we use a curve which is found by looking for crests and valleys of the curvature. For Grushin points we use the set where the vector fields are parallel. Tangency points are the most complicated to deal with. The cut locus from the tangency point is not a good candidate as canonical parameterized curve since it is known to be non-smooth. Thus, we analyse the cut locus from the singular set and we prove that it is not smooth either. A good candidate appears to be a curve which is found by looking for crests and valleys of the Gaussian curvature. We prove that the support of such a curve is uniquely determined and has a canonical parametrization (10.1016/j.difgeo.2012.10.001)
  DOI : 10.1016/j.difgeo.2012.10.001