Publications

2026

• Separation rates for the detection of synchronization of interacting point processes in a mean field frame. Application to neuroscience.
  
  • Tchouanti Josué
  • Löcherbach Éva
  • Reynaud-Bouret Patricia
  • Tanré Etienne
  Electronic Journal of Statistics, Shaker Heights, OH : Institute of Mathematical Statistics, 2026, 20 (1), pp.560--632. Permutation tests have been proposed by Albert et al. (2015) to detect dependence between point processes, modeling in particular spike trains, that is the time occurrences of action potentials emitted by neurons. Our present work focuses on exhibiting a criterion on the separation rate to ensure that the Type II errors of these tests are controlled non asymptotically. This criterion is then discussed in two major models in neuroscience: the jittering Poisson model and Hawkes processes having \(M\) components interacting in a mean field frame and evolving in stationary regime. For both models, we obtain a lower bound of the size \(n\) of the sample necessary to detect the dependency between two neurons. (10.1214/26-EJS2483)
  DOI : 10.1214/26-EJS2483
• Hybrid FEM/IPDG semi-implicit schemes for time domain electromagnetic wave propagation in non cylindrical coaxial cables
  
  • Beni Hamad Akram
  • Imperiale Sébastien
  • Joly Patrick
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2026. In this work, we develop an efficient numerical method for solving 3D Maxwell's equations in non-cylindrical coaxial cables. The main challenge arises from the elongated geometry of the computational domain, which induces strong anisotropy between the longitudinal direction (along the cable) and the transverse directions (within the cross-sections). This leads to the use of highly anisotropic meshes, where the longitudinal mesh size is much larger than the transverse one.<p>Our objective is to design a numerical scheme that is explicit in the longitudinal direction, with a CFL stability condition depending only on the longitudinal mesh size. In a previous work, we achieved this for cylindrical cables by employing prismatic edge elements, 1D quadrature for longitudinal mass lumping, and a hybrid explicit/implicit time discretization. The present paper extends this approach to non-cylindrical cables, addressing several new difficulties with the following key ingredients: (1) representing the cable as a deformation of a reference cylindrical cable and employing mapping techniques between the physical and reference domains; (2) using an anisotropic space discretization that combines an interior penalty discontinuous Galerkin (IPDG) method in the transverse directions with a conforming finite element method in the longitudinal direction; (3) utilizing prismatic edge elements on a prismatic mesh of the reference cable; and (4) adapting the construction of the hybrid explicit-implicit time discretization to the new structure of the semidiscrete problem. From a theoretical perspective, the main difficulty lies in the stability analysis, which requires extending and adapting standard techniques for DG methods in space and energy methods in time.</p>
• Validated Enclosure of Renormalization Fixed Points Via Chebyshev Series and the DFT
  
  • Breden Maxime
  • Gonzalez Jorge
  • Mireles James Jason D.
  Foundations of Computational Mathematics, Springer Verlag, 2026. This work develops a computational framework for proving existence, uniqueness, isolation, and stability results for degree d, real analytic, unimodal functions fixed by m-th order Feigenbaum-Coullet-Tresser-Cvitanović renormalization operators using Chebyshev series approximation. Here the order m of the operator refers to the number of function compositions involved in its definition, and the degree d of the fixed function is the number of derivatives vanishing at its (unique) critical point. The advantage of Chebyshev series is that they are naturally adapted to spaces of real analytic functions, in the sense that they converge on ellipses containing real intervals rather than on disks in C. This facilitates the development of a functional analytic approach independent of order and degree. The main disadvantage of working with Chebyshev series is that the operations of rescaling and composition, essential to the definition of the renormalization operators, are less straightforward for Chebyshev than for Taylor series. These difficulties are overcome via a combination of a-priori information about decay rates in Banach spaces of rapidly decaying coefficient sequences, with a-posteriori estimates on Chebyshev interpolation errors for analytic functions. We illustrate the utility of the method by proving existence, local uniquness, global non-uniqueness results, and interval enclosures of universal constants for fixed points of varying orders and degrees. (10.1007/s10208-026-09763-9)
  DOI : 10.1007/s10208-026-09763-9
• Finite element modelling for the reproduction of dynamic OCE measurements in the cornea
  
  • Merlini Giulia
  • Imperiale Sébastien
  • Allain Jean-Marc
  Journal of the Mechanics and Physics of Solids, Elsevier, 2026, 206, pp.106363. Recent advances in dynamic elastography, particularly through optical coherence tomography combined with transient excitations have enabled rapid, localized, and non-invasive mechanical data acquisition of the cornea. This dataopens the path to early-detection of pathologies and more accurate treatment. However, the analysis of the wave propagation is a complex mechanical problem: the cornea is a structure under pressure, with non-linear material behavior. Thus, computational analysis are needed to extract mechanical parameters from the data. In this study, we present a time-dependent finite element model for the reproduction of transient shear wave elastographic measurements in the cornea. The mechanical problem consists in a smallamplitude wave propagating in the cornea, largely deformed by intraocular pressure in physiological conditions. The model accounts for anisotropic, hyperelastic, and incompressible behavior of the cornea, as well as its accurate geometry, and the preloaded condition. We have implemented two different numerical approaches to solve first the static non-linear inflation of the cornea and then the linear wave propagation problem to reproduce the measurements. We investigate the impact of material anisotropy and prestress on wave propagation and demonstrate that intraocular pressure critically influences shear wave velocity. Additionally, by introducing a localized mechanical defect to simulate a pathological defect, we show that simulated shear wave can detect and quantify mechanical weaknesses, suggesting potential as a diagnostic tool to assess corneal health. (10.1016/j.jmps.2025.106363)
  DOI : 10.1016/j.jmps.2025.106363
• Both genome instability and replicative senescence stem from the shortest telomere in telomerase-negative cells
  
  • Berardi Prisca
  • Martinez-Fernandez Veronica
  • Rat Anaïs
  • Rosas Bringas Fernando R
  • Jolivet Pascale
  • Langston Rachel
  • Mattarocci Stefano
  • Maes Alexandre
  • Aspert Théo
  • Zeinoun Bechara
  • Casier Karine
  • Kazemier Hinke G
  • Charvin Gilles
  • Doumic Marie
  • Chang Michael
  • Teixeira Maria Teresa
  Nature Communications, Nature Publishing Group, 2026, 17 (1), pp.4271. In the absence of telomerase, telomere shortening triggers replicative senescence, a tumor suppressor mechanism that is also associated with oncogenic genomic instability. Yet, the precise mechanism that connects these seemingly opposing forces remains poorly understood. To directly study the complex interplay between senescence, telomere dynamics, and genomic instability, we develop a system in Saccharomyces cerevisiae to generate and track telomeres of precise length in the absence of telomerase. Using single-telomere and single-cell analyses combined with mathematical modeling, we identify a threshold length at which telomeres switch into dysfunction. A single shortest telomere below the threshold length is necessary and sufficient to trigger the onset of replicative senescence in a majority of cells. At population level, fluctuation assays establish that rare genomic instability arises predominantly in cis to the shortest telomere as Pol32-dependent non-reciprocal translocations that result in re-elongation of the shortest telomere and likely transient escape from senescence. The switch of the shortest telomere into dysfunction and subsequent processing in telomerase-negative cells thus serves as the mechanistic link between replicative senescence onset, genomic instability and the initiation of post-senescence survival. (10.1038/s41467-026-70352-z)
  DOI : 10.1038/s41467-026-70352-z
• Self-interacting approximation to McKean-Vlasov long-time limit: a Markov chain Monte Carlo method
  
  • Du Kai
  • Ren Zhenjie
  • Suciu Florin
  • Wang Songbo
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2026, 205, pp.103782. For a certain class of McKean-Vlasov processes, we introduce proxy processes that substitute the mean-field interaction with self-interaction, employing a weighted occupation measure. Our study encompasses two key achievements. First, we demonstrate the ergodicity of the self-interacting dynamics, under broad conditions, by applying the reflection coupling method. Second, in scenarios where the drifts are negative intrinsic gradients of convex mean-field potential functionals, we use entropy and functional inequalities to demonstrate that the stationary measures of the self-interacting processes approximate the invariant measures of the corresponding McKean-Vlasov processes. As an application, we show how to learn the optimal weights of a two-layer neural network by training a single neuron. (10.1016/j.matpur.2025.103782)
  DOI : 10.1016/j.matpur.2025.103782
• Counterfactually Fair Regression via Optimal Transport
  
  • Lince Marie Generali
  • Gaucher Solenne
  • Vie Jill-Jênn
  • Loiseau Patrick
  , 2026. We consider the problem of learning a counterfactually fair regressor. We adopt a causal uncertainty view in which counterfactual fairness is defined with resampled noise. We focus on obtaining theoretical fairness guarantees for a new post-processing estimator. We begin by showing that counterfactual fairness is equivalent to satisfying demographic parity conditional on the latent variable. This allows us to provide a closed-form expression of the optimal fair regressor via a barycentric quantile map. In order to handle continuous latent variables, we propose a discretized post-processing method. Then, under mild regularity assumptions, we prove high-probability finite-sample fairness guarantees for our estimator, providing an unfairness decay at rate Õ(n -1/3 ), and establishing a matching risk bound of order Õ(n -1/3 ). We provide a matching lower bound on the excess risk of almost fair predictions. Finally, we extend our results to the setting of relaxed counterfactual fairness. We validate our approach on real-world and synthetic data.
• Positive Markov processes in Laplace duality
  
  • Foucart Clément
  • Vidmar Matija
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2026, 31 (none), pp.1-73. This article develops a general framework for Laplace duality between positive Markov processes in which the one-dimensional Laplace transform of one process can be represented through that of another. We show that a process admits a Laplace dual if and only if it satisfies a certain complete monotonicity condition. Moreover, we analyse how the conventions adopted for the values of 0⋅∞ and ∞⋅0 are reflected in the weak continuity/absorptivity properties of the processes in duality at the boundaries 0 and ∞. A broad class of generators admitting Laplace duals is identified, and we provide sufficient conditions under which the associated martingale problems are well-posed with the solutions being in duality at the level of their semigroups. Laplace duality is shown to furnish a unifying structure for several generalizations of continuous-state branching processes, e.g. those with immigration or evolving in random environments. Along the way, a theorem originally due to Ethier and Kurtz – connecting duality of generators to that of the associated semigroups – is refined, and we provide a concise proof of the Courrège form for the pointwise infinitesimal generator of a positive Markov process whose domain includes the exponential functions. The latter leads naturally to the notion of a Laplace symbol, which is a parsimonious encoding of the infinitesimal dynamics of the process. (10.1214/26-EJP1582)
  DOI : 10.1214/26-EJP1582
• Decompositions of CSBPs via Poissonian Intertwining
  
  • Foucart Clément
  • Hénard Olivier
  ALEA : Latin American Journal of Probability and Mathematical Statistics, Instituto Nacional de Matemática Pura e Aplicada (Rio de Janeiro, Brasil) [2006-....], 2026, 23 (2), pp.1001. (10.30757/ALEA.v23-37)
  DOI : 10.30757/ALEA.v23-37
• Selective excitatory effects of γ‐glutamyl‐glycine, a glutathione metabolite, on GluN1 NMDA and mGlu5 receptors impacting synaptic plasticity
  
  • Andreu Sandro
  • Gizzi Victoria Sonia
  • Pasco Ugo
  • Sebih Fatiha
  • Zitouni Yattou
  • Jimenez Pablo
  • Rolland Valérie
  • Chavanieu Alain
  • Estaran Sébastien
  • Rousset Matthieu
  • Cens Thierry
  • Cazals Guillaume
  • Charnet Pierre
  • Guiramand Janique
  • Cohen‐solal Catherine
  • Ménard Claudine
  • de Jésus Ferreira Marie‐céleste
  • Chevallier Nathalie
  • Cantel Sonia
  • Vignes Michel
  • Roussel Julien
  British Journal of Pharmacology, Wiley, 2026. Background and Purpose: Gamma‐glutamyl dipeptides are essentially produced after glutathione (GSH) degradation. While several γ‐glutamyl dipeptides have been detected in different organs and body fluids, their biological activity remains elusive. This is the case for γ‐Glutamyl‐Glycine (γ‐Glu‐Gly) which has been found in different brain areas. γ‐Glutamyl‐glycine may bind to ionotropic glutamate receptors. Here, we have investigated whether γ‐Glu‐Gly could exhibit modulatory actions on excitatory synaptic transmission and plasticity by performing its functional characterization on glutamate, GABA and glycine receptors. Experimental Approach: Electrophysiological and calcium imaging experiments were performed on cells heterologously expressing either Glutamate, GABA or Glycine receptors and on cultured hippocampal neurons and acute hippocampal slices. The γ‐Glu‐Gly production by cultured cells and hippocampal slices was measured by LC‐MS analysis, following incubation with modulators of the glutathione metabolic cycle. Key Results: γ‐Glu‐Gly exerted partial agonist effects on both NMDA GluN1 and mGlu5 receptors, but without any binding to the glutamate binding domain on both AMPA and NMDA GluN2A receptors. γ‐Glu‐Gly was devoid of any effect on GABA, glycine and GluN3A receptors. γ‐Glu‐Gly was able to trigger a long‐term increase in synaptic transmission in the CA1 area of mouse hippocampus. Conclusion and implications γ‐Glu‐Gly partially retains the excitatory actions of glutamate on specific receptor sub‐types and may trigger plastic events in the hippocampus. Its accumulation under pathological conditions associated with oxidative stress and a high GSH consumption could thus interfere with endogenous synaptic transmission and disrupt natural plasticity, which is a hallmark of neurodegenerative diseases. (10.1111/bph.70650)
  DOI : 10.1111/bph.70650
• Évaluation de l'équilibre électrique du scénario négawatt 2022 à l'aide du modèle open source eoles
  
  • Cédiey Hippolyte
  • Quirion Philippe
  • Baratgin Laure
  • Bustarret Quentin
  • de Oliveira-Gill Nilam
  • Perrier Quentin
  • Escribe Célia
  • Letz Thomas
  • Salomon Thierry
  • Shirizadeh Behrang
  Revue de l'OFCE, Presses de Sciences Po, 2026. Nous présentons la dernière version du modèle d'optimisation du système énergétique open source Eoles et nous l'utilisons pour évaluer dans quelle mesure le mix énergétique du scénario négaWatt 2022 peut satisfaire la demande d'énergie en France à l'horizon 2050, pour 19 années météorologiques. Nous obtenons que même sans recours aux interconnexions, la demande d'électricité n'excède la production que 3 à 4 heures par an en moyenne, ce qui ne dépasse que de très peu les critères de défaillance du Code de l'énergie. Pour éliminer toute heure de défaillance et assurer les besoins de réserves, une puissance supplémentaire de technologies pilotables de 13,8 GW est nécessaire, soit une augmentation de 39 % par rapport au scénario négaWatt. Nous étudions l'ajout de trois technologies pilotables : turbines à gaz (méthane ou hydrogène) et batteries, qui sont toutes proches en termes de coût total du système énergétique. Par ailleurs, l'équilibre électrique peut être atteint même en réduisant la capacité photovoltaïque sur toitures par rapport au scénario négaWatt. Le gain associé (3,4 Md€/an) est plus élevé que le surcoût entraîné par les capacités pilotables mentionnées ci-dessus (environ 1 Md€/an).
• Dense and nondense limits for uniform random intersection graphs
  
  • Bassino Frédérique
  • Bouvel Mathilde
  • Féray Valentin
  • Gerin Lucas
  • Pierrot Adeline
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2026, 31 (none). We obtain the scaling limits of random graphs drawn uniformly in three families of intersection graphs: permutation graphs, circle graphs, and unit interval graphs. The two first families typically generate dense graphs, in these cases we prove a.s. convergence to an explicit deterministic graphon. Uniform unit interval graphs are nondense and we prove convergence in the sense of Gromov-Prokhorov after normalization of the distances: the limiting object is the interval [0,1] endowed with a random metric defined through a Brownian excursion. Asymptotic results for the number of cliques of size k (k fixed) in a uniform random graph in each of these three families are also given. In all three cases, an important ingredient of the proof is that, for indecomposable graphs in each class (where the notion of indecomposability depends on the class), the combinatorial object defining the graph (permutation, matching, or intervals) is essentially unique. (10.1214/26-EJP1570)
  DOI : 10.1214/26-EJP1570
• Covariance-Informed Subspace: an Adaptive Gradient-Free Input Dimension Reduction Method for Bayesian Inference
  
  • Polette Nadège
  • Le Maître Olivier
  • Sochala Pierre
  • Gesret Alexandrine
  , 2026. This paper addresses the challenge of dimension reduction (DR) in Bayesian inference of high-resolution two-or three-dimensional fields, where a priori parametrizations require a large number of terms. The underlying idea is common to state-of-the-art methods in which the parameter space is decomposed into two subspaces, one informed by the likelihood and one constrained by the prior. DR techniques generally use gradient information from the log-likelihood to derive the corresponding subspaces. However, the gradient may be unavailable or expensive to compute accurately, for instance in the case of simulation-based inference. Inspired by approaches based on likelihood-informed subspaces, we develop a new DR method tailored for settings where gradient computation is not feasible. More specifically, we propose a gradient-free indicator for determining whether a direction is informed by the data. This indicator is derived from the posterior-to-prior covariance ratio introduced in Spantini et al. (2015). We show that, in the linear Gaussian case, this indicator combined with an approximate likelihood leads to a better posterior approximation. The method is then extended to nonlinear cases, and strategies to approximate the posterior covariance are detailed. We demonstrate the effectiveness of this DR through two high-dimensional inference problems arising from groundwater and atmospheric applications.
• A REMARK ON SELF-ADJOINT PROBLEMS IN THE OPTIMIZATION OF NON-LINEAR MODELS
  
  • Égoire Allaire G R
  • Cherrière Théodore
  • Gauthey Thomas
  • Hage Hassan Maya
  • Mininger Xavier
  Journal of Optimization Theory and Applications, Springer Verlag, 2026, 208, pp.100. This article considers optimization problems under nonlinear partial differential equation (p.d.e.) constraints. It is assumed that the p.d.e. arises from minimizing a convex energy. We prove that the optimization problem is self-adjoint when the objective function is the dual energy. In other words, the differential of the objective function with respect to the optimization variable does not involve any adjoint state. This result generalizes the well-known fact that the so-called compliance is self-adjoint in the linear case. We also prove that in a large class of objective functions the dual energy is the only one which is self-adjoint.
• Spatio-temporal equilibrium thermodynamics of guided optical waves at positive and negative temperatures
  
  • Zanaglia Lucas
  • Garnier Josselin
  • Michel Claire
  • Doya Valérie
  • Ferraro Mario
  • Wabnitz Stefan
  • Carusotto Iacopo
  • Picozzi Antonio
  Physical Review A, American Physical Society, 2026, 113, pp.063533. Optical thermalization has been recently studied theoretically and experimentally in the 2D spatial evolution of (quasi-)monochromatic light waves propagating in multimode fibers. In this work, we investigate the spatio-temporal equilibrium properties of incoherent multimode optical waves through the analysis of the (2+1)D Bose-Einstein thermal distribution and the corresponding classical Rayleigh-Jeans approximation. In the classical regime, we perform numerical simulations of the nonlinear Schr\"odinger equation and demonstrate relaxation toward the spatio-temporal Rayleigh–Jeans equilibrium state, as described by the corresponding wave turbulence kinetic equation. Remarkable adiabatic cooling phenomena stemming from the high-frequency tails of the Rayleigh-Jeans distribution are discussed and the consequent limitations of the classical approximation are highlighted. To overcome these issues and properly include quantum effects, we make use of a quantum version of the nonlinear Schr\"odinger equation that is obtained from the general quantum theory of light propagating in nonlinear waveguides. The associated kinetic equation describes relaxation toward the spatio-temporal Bose-Einstein equilibrium distribution with a fully regular UV behaviour. The analysis of thermodynamic equilibrium properties reveals a strong dependence on the specific dispersion regime under consideration. In the anomalous dispersion regime, the system relaxes to positive-temperatures equilibrium states: as the number of modes of the waveguide increases, the fundamental spatial mode becomes macroscopically populated, while its temporal spectrum undergoes significant narrowing, ultimately leading to complete (2+1)D spatio-temporal condensation in the thermodynamic limit. By contrast, in the normal dispersion regime the system evolves toward negative-temperature equilibrium states characterized by a hybrid structure: the spatial equilibrium displays an inverted modal population, whereas the temporal spectrum remains peaked around the fundamental (carrier) optical frequency. In this regime, we predict that spatio-temporal light waves exhibit a phase transition to Bose-Einstein condensation at negative temperatures, which occurs by increasing the temperature above a negative critical value. Our work opens new avenues for future research, including the possibility for a dual spatio-temporal beam cleaning through full spatio-temporal light condensation, and lay the groundwork for the development of spatio-temporal optical thermodynamics. (10.1103/bf7m-chkd)
  DOI : 10.1103/bf7m-chkd
• Numerical Analysis of differential equations on weighted Sobolev spaces: beyond classical orthogonal polynomials
  
  • Breden Maxime
  • Chu Hugo
  , 2026. We lay mathematical foundations for the Numerical Analysis of differential equations on Sobolev spaces weighted by a Gibbs probability measure $\nu$(dx) = e^-V(x)dx/$\mathcal{Z}$ on the real line. Over recent decades, the Functional Analysis of these spaces has been thoroughly developed to study Schrödinger-type equations and diffusion processes. While such equations should therefore be amenable to a numerical resolution with respect to orthogonal polynomials, this feat has only ever been achieved with respect to classical bases. We bridge this gap by showing that such equations can be solved with respect to suitable bases by factorising their leading linear component. In particular, we propose a new natural notion of Sobolev orthogonal polynomials, simpler and more tractable than those arising from the usual Sobolev inner product. In the case of $V$ being an even polynomial, we further establish quantitative estimates for the compactness of the embedding $H^1(\nu)\hookrightarrow L^2(\nu)$, uncovering a connection with the growth of the Jacobi recurrence coefficients, which are solutions of corresponding Painlevé-type discrete equations. As an application, we rigorously and tightly enclose solutions of the Gross--Pitaevskii equation with sextic potential and rigorously demonstrate the phenomenon of stochastic resonance via a computer-assisted proof.
• Entropic Mirror Monte Carlo
  
  • Cherradi Anas
  • Janati Yazid
  • Durmus Alain
  • Le Corff Sylvain
  • Petetin Yohan
  • Stoehr Julien
  Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research, Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto, 2026, 306, pp.19354--19375. Importance sampling is a Monte Carlo method which designs estimators of expectations under a target distribution using weighted samples from a proposal distribution. When the target distribution is complex, such as multimodal distributions in highdimensional spaces, the efficiency of importance sampling critically depends on the choice of the proposal distribution. In this paper, we propose a novel adaptive scheme for the construction of efficient proposal distributions. Our algorithm promotes efficient exploration of the target distribution by combining global sampling mechanisms with a delayed weighting procedure. The proposed weighting mechanism plays a key role by enabling rapid resampling in regions where the proposal distribution is poorly adapted to the target. Our sampling algorithm is shown to be geometrically convergent under mild assumptions and is illustrated through various numerical experiments.
• SAM-Guide : vers un premier Laser Run à destination de sportifs non-voyants
  
  • Romeo-Pakker Katerin
  • Huet Sylvain
  • Faugloire Elise
  • François Alouges
  • Abdellatif Ennaji
  • Christèle Lecomte
  • Wenqi Luo
  • Pellerin Denis
  • Fons Coline
  • Bruno Mantel
  • Mathieu Thomas
  • Sylvain Ferrand
  • Clément Francqueville
  • Graff Christian
  , 2026. Le projet ANR SAM-Guide vise à rendre le Laser Run praticable par des personnes aveugles et malvoyantes grâce à la combinaison multimodale de systèmes d'aide à la navigation et à l'atteinte de cible. Le dispositif fournit un retour auditif et/ou tactile en temps réel sur la position du corps et l'orientation du pistolet vers une cible virtuelle. Nous décrivons l'architecture du système et rapportons les premiers résultats obtenus auprès de participants avec ou sans capacité visuelle.
• Improved well-posedness for the limit flow of differentiation of roots of polynomials
  
  • Bertucci Charles
  • Pesce Valentin
  , 2026. In this paper, we study the partial differential equation on the circle that was heuristically obtained by Steinerberg [32] on the real line and which represents the evolution of the density of the roots of polynomials under differentiation. After integrating the partial differential equation in question, we observe that it can be treated with the theory of viscosity solutions. This equation at hand is a non linear parabolic integro-differential equation which involves the elliptic operator called the half-Laplacian. Due to the singularity of the equation, we restrict our study to strictly positive initial condition. We obtain a comparison principle for solutions of the primitive equation which yields uniqueness, existence, continuity with respect to initial condition. We also present heuristics to justify that the system of particles indeed approximates the solution of the equation.
• Nonlinear model calibration through bifurcation curves
  
  • Mélot Adrien
  • Denimal Goy Enora
  • Renson Ludovic
  Mechanical Systems and Signal Processing, Elsevier, 2026, 242, pp.113589. Nonlinear systems exhibit a plethora of complex dynamic behaviours that are difficult to model and predict accurately. This difficulty often arises from a lack of knowledge of the physics that induces the nonlinear behaviours and the strong sensitivity of the nonlinear dynamics to parameter variation. We introduce in this paper a methodology to carry out nonlinear model updating based on bifurcations. The proposed approach involves minimising the distance between experimental and numerical bifurcation curves, which are key dynamic features that define stability boundaries and regions of multi-stability. For the model, bifurcation curves are computed via standard numerical bifurcation tracking analyses. In the experiment, we use control-based continuation to obtain the data. The approach is first demonstrated on a Duffing and a beam system using synthetic data, before being applied to experimental data collected on a base-excited energy harvester with magnetic nonlinearity.
• Modelling of relative velocity, velocity fluctuations and their interactions for two-fluid models by Stationary Action Principle
  
  • Haegeman Ward
  • Orlando Giuseppe
  • Kokh Samuel
  • Massot Marc
  ESAIM: Proceedings and Surveys, EDP Sciences, 2026. The objective of this contribution is the derivation of a two-fluid model including a relative velocity between the two phases and velocity fluctuations, describing pseudo-turbulent effects, as internal variables based on Stationary Action Principle. The variational derivation, used to obtain the model, relies on the variation of a single trajectory related to the mass-weighted average velocity under the barotropic assumption. The model is hyperbolic, satisfies a second principle of thermodynamics, and admits either linearly degenerate or genuinely nonlinear characteristic fields. Moreover, the variational approach yields a fully closed model and its non-conservative products are uniquely defined for weak solutions in 1D, i.e. jump conditions can be derived. In the laminar case, when velocity fluctuations are negligible, we recover previously derived multi-fluid models which have been analyzed in several contributions. As such, the present framework allows for an original extension of the existing models to include velocity fluctuations of each phase for pseudo-turbulent flows, their coupling with the relative velocity between phases, as well as dissipative effects compatible with the thermodynamics of irreversible processes. Eventually, we provide a discussion of the limitations of the proposed model, especially regarding the extension to the open problem of non-barotropic flows.
• A stochastic use of the Kurdyka-Lojasiewicz property: Investigation of optimization algorithms behaviours in a non-convex differentiable framework
  
  • Fest Jean-Baptiste
  • Repetti Audrey
  • Chouzenoux Emilie
  Foundations of Data Science, American Institute of Mathematical Sciences, 2026, 9, pp.164-191. Asymptotic analysis of generic stochastic algorithms often relies on descent conditions. In a convex setting, some technical shortcuts can be considered to establish asymptotic convergence guarantees of the associated scheme. However, in a non-convex setting, obtaining similar guarantees is usually more complicated, and relies on the use of the Kurdyka-Łojasiewicz (KŁ) property. While this tool has become popular in the field of deterministic optimization, it is much less widespread in the stochastic context and the few works making use of it are essentially based on trajectory-by-trajectory approaches. In this paper, we propose a new framework for using the KŁ property in a non-convex stochastic setting based on conditioning theory. We show that this framework allows for deeper asymptotic investigations on stochastic schemes verifying some generic descent conditions. We further show that our methodology can be used to prove convergence of generic stochastic gradient descent (SGD) schemes, and unifies conditions investigated in multiple articles of the literature. (10.3934/fods.2025016)
  DOI : 10.3934/fods.2025016
• torchsom: The Reference PyTorch Library for Self-Organizing Maps
  
  • Berthier Louis
  • Shokry Ahmed
  • Moreaud Maxime
  • Ramelet Guillaume
  • Moulines Eric
  Journal of Machine Learning Research, Microtome Publishing, 2026, 27 (190), pp.1–17.
• Stability analysis of a new curl-based full field reconstruction method in 2D isotropic nearly-incompressible elasticity
  
  • Chibli Nagham
  • Genet Martin
  • Imperiale Sébastien
  Inverse Problems, IOP Publishing, 2026, 42 (6), pp.065001. In time-harmonic elastography, the shear modulus is typically inferred from full field displacement data by solving an inverse problem based on the time-harmonic elastodynamic equation. In this paper, we focus on nearly incompressible media, which pose robustness challenges, especially in the presence of noisy data. Restricting ourselves to 2D and considering an isotropic, linearly deforming medium, we reformulate the problem as a non-autonomous hyperbolic system and, through theoretical analysis, establish existence, uniqueness, and stability of the inverse problem. To ensure robustness with noisy data, we propose a least-squares approach with regularization. The convergence properties of the method are verified numerically using in silico data. (10.1088/1361-6420/ae6d2e)
  DOI : 10.1088/1361-6420/ae6d2e
• Spatio-temporal thermalization and adiabatic cooling of guided light waves
  
  • Zanaglia Lucas
  • Garnier Josselin
  • Carusotto Iacopo
  • Doya Valérie
  • Michel Claire
  • Picozzi Antonio
  Physical Review Letters, American Physical Society, 2026, 136, pp.053802. We propose and theoretically characterize three-dimensional spatio-temporal thermalization of a continuous-wave classical light beam propagating along a multi-mode optical waveguide. By combining a non-equilibrium kinetic approach based on the wave turbulence theory and numerical simulations of the field equations, we anticipate that thermalizing scattering events are dramatically accelerated by the combination of strong transverse confinement with the continuous nature of the temporal degrees of freedom. In connection with the blackbody catastrophe, the thermalization of the classical field in the continuous temporal direction provides a novel intrinsic mechanism for adiabatic cooling and spatial beam condensation. This process of adiabatic cooling is distinct from other mechanisms of thermalization and provides new insights into the dynamics of far-from-equilibrium closed systems and their route to thermalization. (10.1103/mqzh-w2gh)
  DOI : 10.1103/mqzh-w2gh