Publications

2019

• Local decay for weak interactions with massless particles
  
  • Barbaroux Jean-Marie
  • Faupin Jérémy
  • Guillot Jean-Claude
  Journal of Spectral Theory, European Mathematical Society, 2019, 9 (2), pp.453–512. We consider a mathematical model for the weak decay of the intermediate boson $Z^0$ into neutrinos and antineutrinos. We prove that the total Hamiltonian has a unique ground state in Fock space and we establish a limiting absorption principle, local decay and a property of relaxation to the ground state for initial states and observables suitably localized in energy and position. Our proofs rest, in particular, on Mourre's theory and a low-energy decomposition. (10.4171/JST/253)
  DOI : 10.4171/JST/253
• The Homogenization Method for Topology Optimization of Structures: Old and New
  
  • Allaire Grégoire
  • Cavallina Lorenzo
  • Miyake Nobuhito
  • Oka Tomoyuki
  • Yachimura Toshiaki
  Interdisciplinary Information Sciences, Editorial Committee of the Interdisciplinary Information Sciences, 2019, 25 (2), pp.75-146. (10.4036/iis.2019.B.01)
  DOI : 10.4036/iis.2019.B.01
• Uniform propagation of chaos and creation of chaos for a class of nonlinear diffusions
  
  • del Moral Pierre
  • Tugaut Julian
  Stochastic Analysis and Applications, Taylor & Francis: STM, Behavioural Science and Public Health Titles, 2019, 37 (6), pp.909-935. We are interested in nonlinear diffusions in which the own law intervenes in the drift. This kind of diffusions corresponds to the hydrodynamical limit of some particle system. One also talks about propagation of chaos. It is well known, for McKean-Vlasov diffusions, that such a propagation of chaos holds on finite-time interval. We here aim to establish a uniform propagation of chaos even if the external force is not convex, with a diffusion coefficient sufficiently large. The idea consists in combining the propagation of chaos on a finite-time interval with a functional inequality, already used by Bolley, Gentil and Guillin. Here, we also deal with a case in which the system at time t = 0 is not chaotic and we show under easily checked assumptions that the system becomes chaotic as the number of particles goes to infinity together with the time. This yields the first result of this type for mean field particle diffusion models as far as we know. (10.1080/07362994.2019.1622426)
  DOI : 10.1080/07362994.2019.1622426
• Optimal control problem for viscous systems of conservation laws, with geometric parameter, and application to the Shallow-Water equations
  
  • Court Sébastien
  • Kunisch Karl
  • Pfeiffer Laurent
  Interfaces and Free Boundaries : Mathematical Analysis, Computation and Applications, European Mathematical Society, 2019, 21 (3), pp.273-311. A theoretical framework and numerical techniques to solve optimal control problems with a spatial trace term in the terminal cost and governed by regularized nonlinear hyperbolic conservation laws are provided. Depending on the spatial dimension, the set at which the optimum of the trace term is reached under the action of the control function can be a point, a curve or a hypersurface. The set is determined by geometric parameters. Theoretically the lack of a convenient functional framework in the context of optimal control for hyperbolic systems leads us to consider a parabolic regularization for the state equation, in order to derive optimality conditions. For deriving these conditions, we use a change of variables encoding the sensitivity with respect to the geometric parameters. As illustration, we consider the shallow-water equations with the objective of maximizing the height of the wave at the final time, a wave whose location and shape are optimized via the geometric parameters. Numerical results are obtained in 1D and 2D, using finite difference schemes, combined with an immersed boundary method for iterating the geometric parameters. (10.4171/IFB/424)
  DOI : 10.4171/IFB/424
• A kinetic model of reactive crystal surfaces A Kinetic Model of Reactive Crystal Surfaces
  
  • Aoki Kazuo
  • Giovangigli Vincent
  AIP Conference Proceedings, American Institute of Physics, 2019, 2132. A kinetic model describing chemical reactions on a crystal surface is introduced. The Boltzmann equations involve particles interacting with potentials generated by fixed crystal particles and interacting with a phonon gas describing the fluctuating part of the potentials. Chemical reactions between gas/physisorbed, chemisorbed and crystal species are taken into account. The phonons are assumed to be at equilibrium for the sake of simplicity. A modified kinetic entropy is introduced for the coupled system and the H theorem is established. Using a fluid scaling and the Chapman-Enskog asymptotic method, species fluid boundary conditions involving heterogeneous reactions are recovered at the surface. (10.1063/1.5119623)
  DOI : 10.1063/1.5119623
• Analysis of Langevin Monte Carlo via convex optimization
  
  • Durmus Alain
  • Majewski Szymon
  • Miasojedow Błażej
  Journal of Machine Learning Research, Microtome Publishing, 2019. In this paper, we provide new insights on the Unadjusted Langevin Algorithm. We show that this method can be formulated as a first order optimization algorithm of an objective functional defined on the Wasserstein space of order $2$. Using this interpretation and techniques borrowed from convex optimization, we give a non-asymptotic analysis of this method to sample from logconcave smooth target distribution on $\mathbb{R}^d$. Based on this interpretation, we propose two new methods for sampling from a non-smooth target distribution, which we analyze as well. Besides, these new algorithms are natural extensions of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm, which is a popular extension of the Unadjusted Langevin Algorithm. Similar to SGLD, they only rely on approximations of the gradient of the target log density and can be used for large-scale Bayesian inference.
• Approximating the Volume of Tropical Polytopes is Difficult
  
  • Gaubert Stéphane
  • Maccaig Marie
  International Journal of Algebra and Computation, World Scientific Publishing, 2019, 29 (02), pp.357--389. We investigate the complexity of counting the number of integer points in tropical polytopes, and the complexity of calculating their volume. We study the tropical analogue of the outer parallel body and establish bounds for its volume. We deduce that there is no approximation algorithm of factor $\alpha=2^{\text{poly}(m,n)}$ for the volume of a tropical polytope given by $n$ vertices in a space of dimension $m$, unless P$=$NP. Neither is there such an approximation algorithm for counting the number of integer points in tropical polytopes described by vertices. If follows that approximating these values for tropical polytopes is more difficult than for classical polytopes. Our proofs use a reduction from the problem of calculating the tropical rank. For tropical polytopes described by inequalities we prove that counting the number of integer points and calculating the volume are $\#$P-hard. (10.1142/S0218196718500686)
  DOI : 10.1142/S0218196718500686
• Avis en réponse à la saisine HCB sur le dossier EFSA-GMO-ES-2018-154. Paris, le 5 avril 2019
  
  • Comité Scientifique Du Haut Conseil Des Biotechnologies .
  • Angevin Frédérique
  • Bagnis Claude
  • Bar-Hen Avner
  • Barny Marie-Anne
  • Boireau Pascal
  • Brévault Thierry
  • Chauvel Bruno B.
  • Collonnier Cécile
  • Couvet Denis
  • Dassa Elie
  • de Verneuil Hubert
  • Demeneix Barbara
  • Franche Claudine
  • Guerche Philippe
  • Guillemain Joël
  • Hernandez Raquet Guillermina
  • Khalife Jamal
  • Klonjkowski Bernard
  • Lavielle Marc
  • Le Corre Valérie
  • Lefèvre François
  • Lemaire Olivier
  • Lereclus Didier D.
  • Maximilien Rémy
  • Meurs Eliane
  • Naffakh Nadia
  • Négre Didier
  • Noyer Jean-Louis
  • Ochatt Sergio
  • Pages Jean-Christophe
  • Raynaud Xavier
  • Regnault-Roger Catherine
  • Renard Michel M.
  • Renault Tristan
  • Saindrenan Patrick
  • Simonet Pascal
  • Troadec Marie-Bérengère
  • Vaissière Bernard
  • Vilotte Jean-Luc
  , 2019, pp.14 p..
• Spreading and vanishing for a monostable reaction-diffusion equation with forced speed
  
  • Bouhours Juliette
  • Giletti Thomas
  Journal of Dynamics and Differential Equations, Springer Verlag, 2019, 35, pp.92 - 117. Invasion phenomena for heterogeneous reaction-diffusion equations are contemporary and challenging questions in applied mathematics. In this paper we are interested in the question of spreading for a reaction-diffusion equation when the subdomain where the reaction term is positive is shifting/contracting at a given speed c. This problem arises in particular in the modelling of the impact of climate change on population dynamics. By placing ourselves in the appropriate moving frame, this leads us to consider a reaction-diffusion-advection equation with a heterogeneous in space reaction term, in dimension N ≥ 1. We investigate the behaviour of the solution u depending on the value of the advection constant c, which typically stands for the velocity of climate change. We find that, when the initial datum is compactly supported, there exists precisely three ranges for c leading to drastically different situations. In the lower speed range the solution always spreads, while in the upper range it always vanishes. More surprisingly, we find that that both spreading and vanishing may occur in an intermediate speed range. The threshold between those two outcomes is always sharp, both with respect to c and to the initial condition. We also briefly consider the case of an exponentially decreasing initial condition, where we relate the decreasing rate of the initial condition with the range of values of c such that spreading occurs. (10.1007/s10884-018-9643-5)
  DOI : 10.1007/s10884-018-9643-5
• Kinetic model of adsorption on crystal surfaces
  
  • Aoki Kazuo
  • Giovangigli Vincent
  Physical Review E, American Physical Society (APS), 2019, 99. A kinetic theory model describing physisorption and chemisorption of gas particles on a crystal surface is introduced. A single kinetic equation is used to model gas and physisorbed particles interacting with a crystal potential and colliding with phonons. The phonons are assumed to be at equilibrium and the physisorbate-gas equation is coupled to similar kinetic equations describing chemisorbed particles and crystal atoms on the surface. A kinetic entropy is introduced for the coupled system and the H theorem is established. Using the Chapman-Enskog method with a fluid scaling, the asymptotic structure of the adsorbate is investigated and fluid boundary conditions are derived from the kinetic model. (10.1103/PhysRevE.99.052137)
  DOI : 10.1103/PhysRevE.99.052137
• THE INCOMPATIBILITY OPERATOR: FROM RIEMANN'S INTRINSIC VIEW OF GEOMETRY TO A NEW MODEL OF ELASTO-PLASTICITY
  
  • Amstutz Samuel
  • van Goethem Nicolas
  , 2019. The mathematical modelling in mechanics has a long-standing history as related to geometry, and significant progresses have often been achieved by the invention of new geometrical tools. Also, it happened that the elucidation of practical issues led to the invention of new scientific concepts, and possibly new paradigms, with potential impact far beyond. One such example is Riemann's intrinsic view in geometry, that offered a radically new insight in the Physics of the early 20-th century. On the other hand, the rather recent intrinsic approaches in elasticity and elasto-plasticity also share this philosophical standpoint of looking from inside, i.e., from the "manifold" point of view. Of course, this approach requires smoothness, and is thus incomplete for an analyst. Nevertheless, its first aim is to highlight the concepts of metric, curvature and torsion; these notions are addressed in the first part of this survey paper. In a second part, they are given a precise functional meaning and their properties are studied systematically. Further, a novel approach to elasto-plasticity constructed upon a model of incompatible elasticity is designed, carrying this intrinsic spirit. The main mathematical object in this theory is the incompatibility operator, i.e., a linearized version of Riemann's curvature tensor. So far, this route not only has led the authors to a new model with a solid functional foundation and proof of existence results, but also to a framework with a minimal amount of ad-hoc assumptions, and complying with both the basic principles of thermodynamics and invariance principles of Physics. The questions arising from this novel approach are complex and intriguing, but we believe that the model is now sufficiently well-posed to be studied simultaneously as a problem of mathematics and of mechanics. Most of the research program remains to be done, and this survey paper is written to present our model, with a particular care to put this approach into a historical perspective.
• A MICROSCOPIC VIEW ON THE FOURIER LAW
  
  • Bodineau Thierry
  • Gallagher Isabelle
  • Saint-Raymond Laure
  Comptes Rendus. Physique, Académie des sciences (Paris), 2019. The Fourier law of heat conduction describes heat diffusion in macroscopic systems. This physical law has been experimentally tested for a large class of physical systems. A natural question is to know whether it can be derived from the microscopic models using the fundamental laws of mechanics.
• A degenerate Cahn‐Hilliard model as constrained Wasserstein gradient flow
  
  • Matthes Daniel
  • Cancès Clément
  • Nabet Flore
  , 2019, 19 (1). (10.1002/pamm.201900158)
  DOI : 10.1002/pamm.201900158
• Option pricing under fast-varying long-memory stochastic volatility
  
  • Garnier Josselin
  • Solna Knut
  Mathematical Finance, Wiley, 2019, 29 (1), pp.39-83. (10.1111/mafi.12186)
  DOI : 10.1111/mafi.12186
• ConvSCCS: convolutional self-controlled case-seris model for lagged adverser event detection
  
  • Morel Maryan
  • Bacry Emmanuel
  • Gaïffas Stéphane
  • Guilloux Agathe
  • Leroy Fanny
  Biostatistics, Oxford University Press (OUP), 2019. With the increased availability of large electronic health records databases comes the chance of enhancing health risks screening. Most post-marketing detection of adverse drug reaction (ADR) relies on physicians' spontaneous reports, leading to under-reporting. To take up this challenge, we develop a scalable model to estimate the effect of multiple longitudinal features (drug exposures) on a rare longitudinal outcome. Our procedure is based on a conditional Poisson regression model also known as self-controlled case series (SCCS). To overcome the need of precise risk periods specification, we model the intensity of outcomes using a convolution between exposures and step functions, which are penalized using a combination of group-Lasso and total-variation. Up to our knowledge, this is the first SCCS model with flexible intensity able to handle multiple longitudinal features in a single model. We show that this approach improves the state-of-the-art in terms of mean absolute error and computation time for the estimation of relative risks on simulated data. We apply this method on an ADR detection problem, using a cohort of diabetic patients extracted from the large French national health insurance database (SNIIRAM), a claims database containing medical reimbursements of more than 53 million people. This work has been done in the context of a research partnership between Ecole Polytechnique and CNAMTS (in charge of SNIIRAM). (10.1093/biostatistics/kxz003)
  DOI : 10.1093/biostatistics/kxz003
• Avis en réponse à la saisine HCB- dossier 2019-159. Paris, le 16 décembre 2019
  
  • Comité Scientifique Du Haut Conseil Des Biotechnologies .
  • Angevin Frédérique
  • Bagnis Claude
  • Bar-Hen Avner
  • Barny Marie-Anne
  • Boireau Pascal
  • Brévault Thierry
  • Chauvel Bruno B.
  • Collonnier Cécile
  • Couvet Denis
  • Dassa Elie
  • Demeneix Barbara
  • Franche Claudine
  • Guerche Philippe
  • Guillemain Joël
  • Hernandez Raquet Guillermina
  • Khalife Jamal
  • Klonjkowski Bernard
  • Lavielle Marc
  • Le Corre Valérie
  • Lefèvre François
  • Lemaire Olivier
  • Lereclus Didier D.
  • Maximilien Rémy
  • Meurs Eliane
  • Naffakh Nadia
  • Négre Didier
  • Noyer Jean-Louis
  • Ochatt Sergio
  • Pages Jean-Christophe
  • Raynaud Xavier
  • Regnault-Roger Catherine
  • Renard Michel M.
  • Renault Tristan
  • Saindrenan Patrick
  • Simonet Pascal
  • Troadec Marie-Bérengère
  • Vaissière Bernard
  • de Verneuil Hubert
  • Vilotte Jean-Luc
  , 2019, pp.34 p.. Le Haut Conseil des biotechnologies (HCB) a été saisi sur le fondement du règlement (CE) n° 1829/2003 d’une demande d’avis relative au dossier EFSA-GMO-NL-2019-159 dans le but de proposer des commentaires à destination de l’EFSA en contribution à l’évaluation européenne du dossier, et d’éclairer les autorités compétentes françaises dans une étape intermédiaire en amont du vote à la Commission européenne. Déposé par la société Pioneer Hi-Bred International, Inc., ce dossier est une demande d’autorisation de mise sur le marché du maïs génétiquement modifié DP202216 à des fins d’importation, de transformation et d’alimentation humaine et animale dans l’Union européenne.
• Scaling limits of population and evolution processes in random environment
  
  • Bansaye Vincent
  • Caballero Maria-Emilia
  • Méléard Sylvie
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2019, 95 (5), pp.749-784. Our motivation comes from the large population approximation of individual based models in population dynamics and population genetics. We propose a general method to investigate scaling limits of finite dimensional population size Markov chains to diffusion with jumps. The statements of tightness, identification and convergence in law are based on the convergence of suitable characteristics of the transition of the chain and strongly exploit the structure of the population processes defined recursively as sums of independent random variables. These results allow to reduce the convergence of characteristics of semimartingales to analytically tractable functional spaces. We develop two main applications. First, we extend the classical Wright-Fisher diffusion approximation to independent and identically distributed random environment. Second, we obtain the convergence in law of generalized Galton-Watson processes with interactions and random environment to the solution of stochastic differential equations with jumps. (10.1214/19-EJP262)
  DOI : 10.1214/19-EJP262
• Optimal inventory management and order book modeling
  
  • Baradel Nicolas
  • Bouchard Bruno
  • Evangelista David
  • Mounjid Othmane
  ESAIM: Proceedings and Surveys, EDP Sciences, 2019, 65, pp.145-181. We model the behavior of three agent classes acting dynamically in a limit order book of a financial asset. Namely, we consider market makers (MM), high-frequency trading (HFT) firms, and institutional brokers (IB). Given a prior dynamic of the order book, similar to the one considered in the Queue-Reactive models [14, 20, 21], the MM and the HFT define their trading strategy by optimizing the expected utility of terminal wealth, while the IB has a prescheduled task to sell or buy many shares of the considered asset. We derive the variational partial differential equations that characterize the value functions of the MM and HFT and explain how almost optimal control can be deduced from them. We then provide a first illustration of the interactions that can take place between these different market participants by simulating the dynamic of an order book in which each of them plays his own (optimal) strategy.
• Approximation of functions with small jump sets and existence of strong minimizers of Griffith's energy
  
  • Chambolle Antonin
  • Conti Sergio
  • Iurlano Flaviana
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2019, 128 (9), pp.119--139. We prove that special functions of bounded deformation with small jump set are close in energy to functions which are smooth in a slightly smaller domain. This permits to generalize the decay estimate by De Giorgi, Carriero, and Leaci to the linearized context in dimension n and to establish the closedness of the jump set for local minimizers of the Griffith energy. (10.1016/j.matpur.2019.02.001)
  DOI : 10.1016/j.matpur.2019.02.001
• Avis en réponse à la saisine HCB – dossier RX-016. Paris, le 21 février 2019
  
  • Comité Scientifique Du Haut Conseil Des Biotechnologies .
  • Angevin Frédérique
  • Bagnis Claude
  • Bar-Hen Avner
  • Barny Marie-Anne
  • Boireau Pascal
  • Brévault Thierry
  • Chauvel Bruno B.
  • Collonnier Cécile
  • Couvet Denis
  • Dassa Elie
  • Demeneix Barbara
  • Franche Claudine
  • Guerche Philippe
  • Guillemain Joël
  • Hernandez Raquet Guillermina
  • Khalife Jamal
  • Klonjkowski Bernard
  • Lavielle Marc
  • Le Corre Valérie
  • Lefèvre François
  • Lemaire Olivier
  • Lereclus Didier D.
  • Maximilien Rémy
  • Meurs Eliane
  • Naffakh Nadia
  • Négre Didier
  • Noyer Jean-Louis
  • Ochatt Sergio
  • Pages Jean-Christophe
  • Raynaud Xavier
  • Regnault-Roger Catherine
  • Renard Michel M.
  • Renault Tristan
  • Saindrenan Patrick
  • Simonet Pascal
  • Troadec Marie-Bérengère
  • Vaissière Bernard
  • de Verneuil Hubert
  • Vilotte Jean-Luc
  , 2019, pp.23 p.. Le Haut Conseil des biotechnologies (HCB) a été saisi sur le fondement du règlement (CE) n° 1829/2003 d’une demande d’avis relative au dossier EFSA-GMO-RX-016 dans le but de proposer des commentaires à destination de l’EFSA en contribution à l’évaluation européenne du dossier, et d’éclairer les autorités compétentes françaises dans une étape intermédiaire en amont du vote à la Commission européenne. Déposé par la société Syngenta Crop Protection NV/SA, ce dossier est une demande de renouvellement d’autorisation de mise sur le marché du maïs génétiquement modifié Bt11 à des fins d’importation, transformation, et alimentation humaine et animale dans l’Union européenne.
• Curvature: a variational approach
  
  • Agrachev Andrei
  • Barilari Davide
  • Rizzi Luca
  Memoirs of the American Mathematical Society, American Mathematical Society, 2019, 256 (1225). The curvature discussed in this paper is a rather far going generalization of the Riemann sectional curvature. We define it for a wide class of optimal control problems: a unified framework including geometric structures such as Riemannian, sub-Riemannian, Finsler and sub-Finsler structures; a special attention is paid to the sub-Riemannian (or Carnot-Caratheodory) metric spaces. Our construction of the curvature is direct and naive, and it is similar to the original approach of Riemann. Surprisingly, it works in a very general setting and, in particular, for all sub-Riemannian spaces. (10.1090/memo/1225)
  DOI : 10.1090/memo/1225
• New preconditioners for Laplace and Helmholtz integral equations on open curves
  
  • Averseng Martin
  , 2019. This paper is the second part of a work on Laplace and Helmholtz integral equations in 2 space dimensions on open curves. A new Galerkin method in weighted L 2 spaces together with new preconditioners for the weighted layer potentials are studied. This second part provides the theoretical analysis needed to establish the results announced in the first part. The main novelty is the introduction of a pseudo-differential calculus on open curves that allows to build parametrices for the weighted layer potentials. Contrarily to more classical approaches where the Mellin transform is used, this new approach is well-suited to the specific singularities that appear in the problem.
• Galton–Watson and branching process representations of the normalized Perron–Frobenius eigenvector
  
  • Cerf Raphaël
  • Dalmau Joseba
  ESAIM: Probability and Statistics, EDP Sciences, 2019, 23, pp.797-802. Let A be a primitive matrix and let λ be its Perron–Frobenius eigenvalue. We give formulas expressing the associated normalized Perron–Frobenius eigenvector as a simple functional of a multitype Galton–Watson process whose mean matrix is A , as well as of a multitype branching process with mean matrix e ( A − I ) t . These formulas are generalizations of the classical formula for the invariant probability measure of a Markov chain. (10.1051/ps/2019007)
  DOI : 10.1051/ps/2019007
• Self-Exclusion among Online Poker Gamblers: Effects on Expenditure in Time and Money as Compared to Matched Controls
  
  • Luquiens Amandine
  • Dugravot Aline
  • Panjo Henri
  • Benyamina Amine
  • Gaïffas Stéphane
  • Bacry Emmanuel
  International Journal of Environmental Research and Public Health, MDPI, 2019, 16 (22), pp.4399. Background: No comparative data is available to report on the effect of online self-exclusion. The aim of this study was to assess the effect of self-exclusion in online poker gambling as compared to matched controls, after the end of the self-exclusion period. Methods: We included all gamblers who were first-time self-excluders over a 7-year period (n = 4887) on a poker website, and gamblers matched for gender, age and account duration (n = 4451). We report the effects over time of self-exclusion after it ended, on money (net losses) and time spent (session duration) using an analysis of variance procedure between mixed models with and without the interaction of time and self-exclusion. Analyzes were performed on the whole sample, on the sub-groups that were the most heavily involved in terms of time or money (higher quartiles) and among short-duration self-excluders (<3 months). Results: Significant effects of self-exclusion and short-duration self-exclusion were found for money and time spent over 12 months. Among the gamblers that were the most heavily involved financially, no significant effect on the amount spent was found. Among the gamblers who were the most heavily involved in terms of time, a significant effect was found on time spent. Short-duration self-exclusions showed no significant effect on the most heavily involved gamblers. Conclusions: Self-exclusion seems efficient in the long term. However, the effect on money spent of self-exclusions and of short-duration self-exclusions should be further explored among the most heavily involved gamblers. (10.3390/ijerph16224399)
  DOI : 10.3390/ijerph16224399
• A breakdown of injectivity for weighted ray transforms in multidimensions
  
  • Goncharov Fedor O
  • Novikov Roman G
  Arkiv för Matematik, Royal Swedish Academy of Sciences, Institut Mittag-Leffler, 2019, 57, pp.333–371. We consider weighted ray-transforms $P_W$ (weighted Radon transforms along straight lines) in $\mathbb{R}^d, \, d\geq 2,$ with strictly positive weights $W$. We construct an example of such a transform with non-trivial kernel in the space of infinitely smooth compactly supported functions on $\mathbb{R}^d$. In addition, the constructed weight $W$ is rotation-invariant continuous and is infinitely smooth almost everywhere on $\mathbb{R}^d \times \mathbb{S}^{d-1}$. In particular, by this construction we give counterexamples to some well-known injectivity results for weighted ray transforms for the case when the regularity of $W$ is slightly relaxed. We also give examples of continous strictly positive $W$ such that $\dim \ker P_W \geq n$ in the space of infinitely smooth compactly supported functions on $\mathbb{R}^d$ for arbitrary $n\in \mathbb{N}\cup \{\infty\}$, where $W$ are infinitely smooth for $d=2$ and infinitely smooth almost everywhere for $d\geq 3$. (10.4310/ARKIV.2019.v57.n2.a5)
  DOI : 10.4310/ARKIV.2019.v57.n2.a5