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1. Continuous limits of large plant-pollinator random networks and some applications

Author: Sylvain Billiard, Hélène Leman, Thomas Rey, Viet-Chi Tran, Tran, Viet Chi, Laboratoires d'excellence - Centre Européen pour les Mathématiques, la Physique et leurs Interactions - - CEMPI2011 - ANR-11-LABX-0007 - LABX - VALID, Laboratoires d'excellence - Models and algorithms: from the discrete to the continuous - - Bézout2010 - ANR-10-LABX-0058 - LABX - VALID, Modèles multi-échelles et simulation numérique hybride de semi-conducteurs - - MoHyCon2017 - ANR-17-CE40-0027 - AAPG2017 - VALID, APPEL À PROJETS GÉNÉRIQUE 2018 - Modèles statistiques avancés pour les réseaux écologiques - - EcoNet2018 - ANR-18-CE02-0010 - AAPG2018 - VALID, Propagation de processus épidémiques sur des réseaux dynamiques de mouvements d'animaux avec application aux bovins en France - - CADENCE2016 - ANR-16-CE32-0007 - AAPG2016 - VALID, Évolution, Écologie et Paléontologie (Evo-Eco-Paleo) - UMR 8198 (Evo-Eco-Paléo (EEP)), Université de Lille-Centre National de la Recherche Scientifique (CNRS), Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS), Inria Lyon, Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire Paul Painlevé (LPP), Reliable numerical approximations of dissipative systems (RAPSODI ), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Chaire Modélisation Mathématique et Biodiversité (MMB)' of Veolia-Ecole Polytechnique-Museum National d'Histoire Naturelle-Fondation X, ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011), ANR-10-LABX-0058,Bézout,Models and algorithms: from the discrete to the continuous(2010), ANR-17-CE40-0027,MoHyCon,Modèles multi-échelles et simulation numérique hybride de semi-conducteurs(2017), ANR-18-CE02-0010,EcoNet,Modèles statistiques avancés pour les réseaux écologiques(2018), ANR-16-CE32-0007,CADENCE,Propagation de processus épidémiques sur des réseaux dynamiques de mouvements d'animaux avec application aux bovins en France(2016), Évolution, Écologie et Paléontologie (Evo-Eco-Paleo) - UMR 8198 (Evo-Eco-Paléo), École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Paul Painlevé - UMR 8524 (LPP), Évolution, Écologie et Paléontologie (Evo-Eco-Paleo) - UMR 8198 [Evo-Eco-Paléo (EEP)], Institut National de Recherche en Informatique et en Automatique [Inria], Unité de Mathématiques Pures et Appliquées [UMPA-ENSL], Laboratoire Paul Painlevé [LPP], Reliable numerical approximations of dissipative systems [RAPSODI ], and Laboratoire Analyse et Mathématiques Appliquées [LAMA]
Subjects: [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], integro-differential equation, graphon, 92D40, 92D25, 05C90, 60J80, 60F17, 47G20, kinetic limit, Mathematics - Analysis of PDEs, FOS: Mathematics, Quantitative Biology::Populations and Evolution, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Quantitative Biology - Populations and Evolution, ecological mutualistic community, Probability (math.PR), AMS: 92D40, 92D25, 05C90, 60J80, 60F17, 47G20, Populations and Evolution (q-bio.PE), [SDV.EE.IEO] Life Sciences [q-bio]/Ecology, environment/Symbiosis, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], stationary solution, FOS: Biological sciences, interacting particles, limit theorem, birth and death process, Mathematics - Probability, Analysis of PDEs (math.AP), [SDV.EE.IEO]Life Sciences [q-bio]/Ecology, environment/Symbiosis
Abstract: International audience; We study a stochastic individual-based model of interacting plant and pollinator species through a bipartite graph: each species is a node of the graph, an edge representing interactions between a pair of species. The dynamics of the system depends on the between- and within-species interactions: pollination by insects increases plant reproduction rate but has a cost which can increase plant death rate, depending on pollinators density. Pollinators reproduction is increased by the resources harvested on plants. Each species is characterized by a trait corresponding to its degree of generalism. This trait determines the structure of the interactions graph and the quantity of resources exchanged between species. Our model includes in particular nested or modular networks. Deterministic approximations of the stochastic measure-valued process by systems of ordinary differential equations or integro-differential equations are established and studied, when the population is large or when the graph is dense and can be replaced with a graphon. The long-time behaviors of these limits are studied and central limit theorems are established to quantify the difference between the discrete stochastic individual-based model and the deterministic approximations. Finally, studying the continuous limits of the interaction network and the resulting PDEs, we show that nested plant-pollinator communities are expected to collapse towards a coexistence between a single pair of species of plants and pollinators.
Published: 2022

2. Torus stability under Kato bounds on the Ricci curvature

Author: Carron, Gilles, Mondello, Ilaria, Tewodrose, David, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Nantes université - UFR des Sciences et des Techniques (Nantes univ - UFR ST), Nantes Université - pôle Sciences et technologie, Nantes Université (Nantes Univ)-Nantes Université (Nantes Univ)-Nantes Université - pôle Sciences et technologie, Nantes Université (Nantes Univ)-Nantes Université (Nantes Univ), Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, and Carron, Gilles
Subjects: Mathematics - Differential Geometry, Differential Geometry (math.DG), [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], General Mathematics, FOS: Mathematics, Mathematics::Metric Geometry, Mathematics::Differential Geometry, [MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG], Mathematics::Geometric Topology, Mathematics::Symplectic Geometry
Abstract: We show two stability results for a closed Riemannian manifold whose Ricci curvature is small in the Kato sense and whose first Betti number is equal to the dimension. The first one is a geometric stability result stating that such a manifold is Gromov-Hausdorff close to a flat torus. The second one states that, under a stronger assumption, such a manifold is diffeomorphic to a torus: this extends a result by Colding and Cheeger-Colding obtained in the context of a lower bound on the Ricci curvature., 24 pages. Comments are welcome!
Published: 2022
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3. On the weak invariance principle for ortho-martingale in Banach spaces. Application to stationary random fields

Author: Han-Mai Lin, Florence Merlevède, Lin, Han-Mai, Laboratoire Analyse et Mathématiques Appliquées (LAMA), and Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel
Subjects: [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], ortho-martingale approximation, Statistics and Probability, random eld, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Empirical distribution function, Applied Mathematics, Modeling and Simulation, Brownian sheet, Wasserstein distance, Invariance principle, [MATH.MATH-ST] Mathematics [math]/Statistics [math.ST]
Abstract: In this paper, we study the weak invariance principle for stationary ortho-martingales with values in 2-smooth or cotype 2 Banach spaces. Then, with the help of a suitable maximal ortho-martingale approximation, we derive the weak invariance principle for stationary random fields in L 1 under a condition in the spirit of Hannan. As an application, we get an asymptotic result for the L 1-Wasserstein distance between the common distribution function and the corresponding empirical distribution function of stationary random fields.
Published: 2022
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4. Transport-entropy forms of direct and Converseblaschke-Santal{\'o} inequalities

Author: Fradelizi, Matthieu, Gozlan, Nathael, Sadovsky, Shay, Zugmeyer, Simon, Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), School of Mathematical Sciences [Tel Aviv] (TAU), Raymond and Beverly Sackler Faculty of Exact Sciences [Tel Aviv] (TAU), Tel Aviv University (TAU)-Tel Aviv University (TAU), Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS), and The first and second named author are supported by a grant of the Simone and Cino Del Duca foundation.
Subjects: [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Mathematics - Functional Analysis, Mahler conjecture, Optimal Transport, Probability (math.PR), FOS: Mathematics, 52A20, 52A40, 60E15, Talagrand’s transport inequality, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], Mathematics - Probability, Blaschke-Santaló inequality, Functional Analysis (math.FA)
Abstract: We explore alternative functional or transport-entropy formulations of the Blaschke-Santal{\'o} inequality and of its conjectured counterpart due to Mahler. In particular, we obtain new direct and reverse Blaschke-Santal{\'o} inequalities for s-concave functions. We also obtain new sharp symmetrized transport-entropy inequalities for a large class of spherically invariant probability measures, including the uniform measure on the unit Euclidean sphere and generalized Cauchy and Barenblatt distributions. Finally, we show that the Mahler's conjecture is equivalent to some reinforced log-Sobolev type inequality on the sphere.
Published: 2023

5. Lightweight Curvature Estimation on Point Clouds with Randomized Corrected Curvature Measures

Author: Lachaud, Jacques-Olivier, Coeurjolly, David, Labart, Céline, Romon, Pascal, Thibert, Boris, Laboratoire de Mathématiques (LAMA), Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS), Origami (Origami), Laboratoire d'InfoRmatique en Image et Systèmes d'information (LIRIS), Université Lumière - Lyon 2 (UL2)-École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Lumière - Lyon 2 (UL2)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), ANR-22-CE46-0006,StableProxies,Traitement numérique stable de la géométrie et calcul haute-performance sur des données géométriques hétérogènes(2022), and ANR-11-LABX-0025,PERSYVAL-lab,Systemes et Algorithmes Pervasifs au confluent des mondes physique et numérique(2011)
Subjects: Computing methodologies → Shape analysis, [INFO.INFO-CG]Computer Science [cs]/Computational Geometry [cs.CG]
Abstract: International audience; The estimation of differential quantities on oriented point cloud is a classical step for many geometry processing tasks in computer graphics and vision. Even if many solutions exist to estimate such quantities, they usually fail at satisfying both a stable estimation with theoretical guarantee, and the efficiency of the associated algorithm. Relying on the notion of corrected curvature measures [LRT22, LRTC20] designed for surfaces, the method introduced in this paper meets both requirements. Given a point of interest and a few nearest neighbours, our method estimates the whole curvature tensor information by generating random triangles within these neighbours and normalising the corrected curvature measures by the corrected area measure. We provide a stability theorem showing that our pointwise curvatures are accurate and convergent, provided the noise in position and normal information has a variance smaller than the radius of neighbourhood. Experiments and comparisons with the state-of-the-art confirm that our approach is more accurate and much faster than alternatives. The method is fully parallelizable, requires only one nearest neighbour request per point of computation, and is trivial to implement.
Published: 2023
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6. Approximation for the invariant measure with applications for jump processes (convergence in total variation distance)

Author: Bally, Vlad, Qin, Yifeng, Mathematical Risk Handling (MATHRISK), Université Paris-Est Marne-la-Vallée (UPEM)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire Analyse et Mathématiques Appliquées (LAMA), and Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel
Subjects: [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Unadjusted Langevin algorithm, Total variation distance, Malliavin calculus, Probability (math.PR), FOS: Mathematics, Euler scheme with decreasing steps, Regularization lemma, Jump process, Mathematics - Probability, Invariant measure
Abstract: In this paper, we establish an abstract framework for the approximation of the invariant probability measure for a Markov semigroup. Following Pag{\`e}s and Panloup [40] we use an Euler scheme with decreasing step (unadjusted Langevin algorithm). Under some contraction property with exponential rate and some regularization properties, we give an estimate of the error in total variation distance. This abstract framework covers the main results in [40] and [14]. As a specific application we study the convergence in total variation distance to the invariant measure for jump type equations. The main technical difficulty consists in proving the regularzation properties-this is done under an ellipticity condition, using Malliavin calculus for jump processes., Comment: arXiv admin note: text overlap with arXiv:2212.07411
Published: 2023

7. Concentration inequality for U-statistics of order two for uniformly ergodic Markov chains

Author: Duchemin, Quentin, De Castro, Yohann, Lacour, Claire, Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS), This work was supported by grants from Région Ile de France., Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), and Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)
Subjects: FOS: Computer and information sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Statistics and Probability, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Statistics - Machine Learning, Probability (math.PR), FOS: Mathematics, Machine Learning (stat.ML), Mathematics - Statistics Theory, Statistics Theory (math.ST), [STAT.OT]Statistics [stat]/Other Statistics [stat.ML], Mathematics - Probability
Abstract: We prove a new concentration inequality for U-statistics of order two for uniformly ergodic Markov chains. Working with bounded and $\pi$-canonical kernels, we show that we can recover the convergence rate of Arcones and Gin{\'e} who proved a concentration result for U-statistics of independent random variables and canonical kernels. Our result allows for a dependence of the kernels $h_{i,j}$ with the indexes in the sums, which prevents the use of standard blocking tools. Our proof relies on an inductive analysis where we use martingale techniques, uniform ergodicity, Nummelin splitting and Bernstein's type inequality. Assuming further that the Markov chain starts from its invariant distribution, we prove a Bernstein-type concentration inequality that provides sharper convergence rate for small variance terms., Comment: Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, In press
Published: 2023
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8. Unique quasi-stationary distribution, with a possibly stabilizing extinction

Author: Aurélien Velleret, Laboratoire Analyse et Mathématiques Appliquées (LAMA), and Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel
Subjects: Statistics and Probability, Applied Mathematics, Probability (math.PR), diffusion, survival capacity, birth-and-death process, quasi-stationary distribution, 37A30, 60J25, 60J27, 92D15, Harris recurrence, Modeling and Simulation, Q-process, FOS: Mathematics, [MATH]Mathematics [math], Mathematics - Probability
Abstract: We establish sufficient conditions for exponential convergence to a unique quasi-stationary distribution in the total variation norm. These conditions also ensure the existence and exponential ergodicity of the Q-process, the process conditionned upon never being absorbed. The technique relies on a coupling procedure that is related to Harris recurrence (for Markov Chains). It applies to general continuous-time and continuous-space Markov processes. The main novelty is that we modulate each coupling step depending both on a final horizon of time (for survival) and on the initial distribution. By this way, we could notably include in the convergence a dependency on the initial condition. As an illustration, we consider a continuous-time birth-death process with catastrophes and a diffusion process describing a (localized) population adapting to its environment., Comment: 2 figures
Published: 2022
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9. An infinite-dimensional metapopulation SIS model

Author: Jean-François Delmas, Dylan Dronnier, Pierre-André Zitt, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), Laboratoire Analyse et Mathématiques Appliquées (LAMA), and Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel
Subjects: Applied Mathematics, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], [SDV.SPEE]Life Sciences [q-bio]/Santé publique et épidémiologie, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], Analysis, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract: International audience; In this article, we introduce an infinite-dimensional deterministic SIS model which takes into account the heterogeneity of the infections and the social network among a large population. We study the long-time behavior of the dynamic. We identify the basic reproduction number $R_0$ which determines whether there exists a stable endemic steady state (super-critical case: $R_0>1$) or if the only equilibrium is disease-free (critical and sub-critical case: $R_0\leq1$). As an application of this general study, we prove that the so-called ``leaky'' and ``all-or-nothing'' vaccination mechanism have the same effect on $R_0$. This framework is also very natural and intuitive to model lockdown policies and study their impact.
Published: 2022
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10. Local times and capacity for transient branching random walks

Author: Asselah, Amine, Schapira, Bruno, Sousi, Perla, Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), University of Cambridge [UK] (CAM), and ANR-22-CE40-0012,LOCAL,Localisation pour les polymères et marches aléatoires(2022)
Subjects: [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], capacity, Probability (math.PR), FOS: Mathematics, local times, Branching random walk, equilibrium measure, 60G50, 60J80, Mathematics - Probability
Abstract: We consider branching random walks on the Euclidean lattice in dimensions five and higher. In this non-Markovian setting, we first obtain a relationship between the equilibrium measure and Green's function, in the form of an approximate last passage decomposition. Secondly, we obtain exponential moment bounds for functionals of the branching random walk, under optimal condition. As a corollary we obtain an approximate variational characterisation of the branching capacity. We finally derive upper bounds involving the branching capacity for the tail of the time spent in an arbitrary finite collection of balls. This generalises the results of [AHJ] and [AS22] for $d\geq 5$. For random walks, the analogous tail estimates have been instrumental tools for tackling deviations problems on the range, related to folding of the walk., 41 pages, 3 figures
Published: 2023

11. Central limit theorem and almost sure results for the empirical estimator of superquantiles/CVaR in the stationary case

Author: Dedecker, Jérôme, Merlevède, Florence, Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, and Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)
Subjects: [STAT]Statistics [stat], Statistics and Probability, Superquantiles, Expected shortfall, Statistics, Probability and Uncertainty, Stationary sequences, Average value-at-risk, Conditional value-at-risk, Mixing sequences
Abstract: In this paper, we show that the difference between the empirical estimator and the Conditional value-at-risk can be written as a simple partial sum + a residual term. Starting from this decomposition, we prove a central limit theorem and some almost sure results for the empirical estimator, for a large class of stationary sequences. We also construct a confidence interval with asymptotic level 1 − α, and we study its coverage level through two different sets of simulation.
Published: 2022
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12. On null 3-hypergraphs

Author: Giulia Palma, Andrea Frosini, William L. Kocay, Lama Tarsissi, Laboratoire d'Informatique Gaspard-Monge (LIGM), École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Laboratoire Analyse et Mathématiques Appliquées (LAMA), and Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel
Subjects: Hypergraph, self-complementary, Applied Mathematics, null hypergraph, 0211 other engineering and technologies, 3-hypergraph, NP-complete, Null hypergraph, Null labelling, Self-complementary, 021107 urban & regional planning, 0102 computer and information sciences, 02 engineering and technology, [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], 01 natural sciences, Vertex (geometry), Combinatorics, Computer Science::Discrete Mathematics, 010201 computation theory & mathematics, Labelling, null labelling, Enumeration, Discrete Mathematics and Combinatorics, [INFO]Computer Science [cs], Mathematics
Abstract: Given a 3-uniform hypergraph H consisting of a set V of vertices, and T ⊆ V 3 triples, a null labelling is an assignment of ± 1 to the triples such that each vertex is contained in an equal number of triples labelled + 1 and − 1 . Thus, the signed degree of each vertex is zero. A necessary condition for a null labelling is that the degree of every vertex of H is even. The Null Labelling Problem is to determine whether H has a null labelling. It is proved that this problem is NP-complete. Computer enumerations suggest that most hypergraphs which satisfy the necessary condition do have a null labelling. Some constructions are given which produce hypergraphs satisfying the necessary condition, but which do not have a null labelling. A self complementary 3-hypergraph with this property is also constructed.
Published: 2021
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13. Almost sure behavior of the critical points of random polynomials

Author: Angst, Jürgen, Malicet, Dominique, Poly, Guillaume, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, ANR-17-CE40-0008,UNIRANDOM,Universalité pour les domaines nodaux aléatoires(2017), Poly, guillaume, and Universalité pour les domaines nodaux aléatoires - - UNIRANDOM2017 - ANR-17-CE40-0008 - AAPG2017 - VALID
Subjects: 30C15, 60G57, 60B10, Probability (math.PR), FOS: Mathematics, [MATH] Mathematics [math], [MATH]Mathematics [math], Mathematics - Probability
Abstract: Let $(Z_k)_{k\geq 1}$ be a sequence of independent and identically distributed complex random variables with common distribution $\mu$ and let $P_n(X):=\prod_{k=1}^n (X-Z_k)$ the associated random polynomial in $\mathbb C[X]$. In [Kab15], the author established the conjecture stated by Pemantle and Rivin in [PR13] that the empirical measure $\nu_n$ associated with the critical points of $P_n$ converges weakly in probability to the base measure $\mu$. In this note, we establish that the convergence in fact holds in the almost sure sense. Our result positively answers a question raised by Z. Kabluchko and formalized as a conjecture in the recent paper [MV22]., Comment: 16 pages
Published: 2023

14. A short proof of the strong three dimensional Gaussian product inequality

Author: Herry, Ronan, Malicet, Dominique, Poly, Guillaume, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, ANR-17-CE40-0008,UNIRANDOM,Universalité pour les domaines nodaux aléatoires(2017), Poly, guillaume, and Universalité pour les domaines nodaux aléatoires - - UNIRANDOM2017 - ANR-17-CE40-0008 - AAPG2017 - VALID
Subjects: Probability (math.PR), FOS: Mathematics, [MATH] Mathematics [math], [MATH]Mathematics [math], Mathematics - Probability
Abstract: International audience; We prove the strong form of the Gaussian product conjecture in dimension three. Our purely analytical proof simplifies previously known proofs based on combinatorial methods or computer-assisted methods, and allows us to solve the case of any triple of even positive integers which remained open so far.
Published: 2023

15. A note on the one-dimensional critical points of the Ambrosio-Tortorelli functional

Author: Babadjian, Jean-François, Millot, Vincent, Rodiac, Remy, Babadjian, Jean-François, Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Laboratoire Analyse et Mathématiques Appliquées (LAMA), and Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel
Subjects: [MATH] Mathematics [math], [MATH]Mathematics [math]
Abstract: This note addresses the question of convergence of critical points of the Ambrosio-Tortorelli functional in the one-dimensional case under pure Dirichlet boundary conditions. An asymptotic analysis argument shows the convergence to two possible limits points: either a globally affine function or a step function with a single jump at the middle point of the space interval, which are both critical points of the one-dimensional Mumford-Shah functional under a Dirichlet boundary condition. As a byproduct, non minimizing critical points of the Ambrosio-Tortorelli functional satisfying the energy convergence assumption as in \cite{BMR} are proved to exist.
Published: 2023

16. Asymptotics of the Inertia Moments and the Variance Conjecture in Schatten Balls

Author: B. Dadoun, M. Fradelizi, O. Guédon, P.-A. Zitt, Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), ANR-17-CE40-0017,ASPAG,Analyse et Simulation Probabilistes des Algorithmes Géométriques(2017), and Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM)
Subjects: 60B20, 46B09, Probability (math.PR), Mathematics::Spectral Theory, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], November 15, Functional Analysis (math.FA), Mathematics - Functional Analysis, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], FOS: Mathematics, 82B05 Asymptotic convex geometry, 2021. 2020 Mathematics Subject Classification. 52A23, inertia moments, 46B07, Schatten balls, variance conjecture, Analysis, Mathematics - Probability
Abstract: We study the limit, as the dimension goes to infinity, of the moments of the Hilbert-Schmidt norm of a uniformly distributed matrix in the p-Schatten ball, with entries in the real, complex or quaternionic field. We also consider the restriction to the space of self-adjoint matrices. We build on the connection with spectral asymptotics of β-ensembles to adapt some fluctuation results due to Bekerman, Leblé and Serfaty [8]. When p > 3, this allows us to obtain the next asymptotic order for ratios of q-inertia moments of p-Schatten balls of self-adjoint matrices, and to establish a strong version of the variance conjecture for these families of convex bodies.
Published: 2023
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17. Acoustic and gravity waves in the ocean: a new derivation of a linear model from the compressible Euler equation

Author: Dubois, Juliette, Imperiale, Sébastien, Mangeney, Anne, Bouchut, François, Sainte-Marie, Jacques, Dubois, Juliette, Numerical Analysis, Geophysics and Ecology (ANGE), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut de Physique du Globe de Paris (IPG Paris), Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, and This work was supported by grants from Région Ile-de-France and the project ERC-CG-2013-PE10-617472 SLIDEQUAKES
Subjects: [SDU.OCEAN]Sciences of the Universe [physics]/Ocean, Atmosphere, [SDU.OCEAN] Sciences of the Universe [physics]/Ocean, Atmosphere, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [PHYS.MECA.MEFL] Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP]
Abstract: In this paper we construct an accurate linear model describing the propagation of both acoustic and gravity waves in water. This original model is obtainedby the linearization of the compressible Euler equations, written in Lagrangian coordinates. The system is studied in the isentropic case, with a free surface, an arbitrary bathymetry, and vertical variations of the background temperature and density. We show that our model is an extension of some models from the litterature to the case of a non-barotropic fluid with a variable sound speed.Other models from the literature are recovered from our model through two asymptotic analysis, one for the incompressible regime and one for the acoustic regime. We also propose a method to write the model in Eulerian coordinates. Our model includes many physical properties, such as the existence of internal gravity waves or the variation of the sound speed with depth.
Published: 2023

18. Evolving genealogies for branching populations under selection and competition

Author: Blancas, Airam, Gufler, Stephan, Kliem, Sandra, Tran, Viet-Chi, Wakolbinger, Anton, Instituto Tecnológico Autónomo de México (ITAM), Department of mathematics, Goethe-university, Goethe-University Frankfurt am Main, Mathematisches Institut der Universität Leipzig, Universität Leipzig [Leipzig], Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, CONACyT, Zeff Fellowship, a postdoctoral fellowship of the Minerva Foundation, Israel Science Foundation (ISF) grant No. 1382/17, Binational Science Foundation (BSF) award 2018330, Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) research grant contract number 2337/1-1, project 432176920, DFG - Project number 393092071, Chaire 'Modélisation Mathématique et Biodiversité' of Veolia Environnement-Ecole Polytechnique-Museum National d'Histoire Naturelle-Fondation X, DFG project WA 967/4-2 in the SPP 1590, Institute for Mathematical Sciences, National University of Singapore, European Union (ERC-AdG SINGER-101054787), ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011), ANR-10-LABX-0058,Bézout,Models and algorithms: from the discrete to the continuous(2010), Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) research grant contract number 2337/1-1, Tran, Viet Chi, Laboratoires d'excellence - Centre Européen pour les Mathématiques, la Physique et leurs Interactions - - CEMPI2011 - ANR-11-LABX-0007 - LABX - VALID, and Laboratoires d'excellence - Models and algorithms: from the discrete to the continuous - - Bézout2010 - ANR-10-LABX-0058 - LABX - VALID
Subjects: [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], tree-valued processes, lookdown process, Probability (math.PR), selection, fluctuating population size, 60J80, 60K35, 92D10, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Birth-death particle system, FOS: Mathematics, genealogy, Mathematics - Probability, selective lookdown space, density-dependent competition
Abstract: For a continuous state branching process with two types of individuals which are subject to selection and density dependent competition, we characterize the joint evolution of population size, type configurations and genealogies as the unique strong solution of a system of SDE's. Our construction is achieved in the lookdown framework and provides a synthesis as well as a generalization of cases considered separately in two seminal papers by Donnelly and Kurtz (1999), namely fluctuating population sizes under neutrality, and selection with constant population size. As a conceptual core in our approach we introduce the selective lookdown space which is obtained from its neutral counterpart through a state-dependent thinning of ``potential'' selection/competition events whose rates interact with the evolution of the type densities. The updates of the genealogical distance matrix at the ``active'' selection/competition events are obtained through an appropriate sampling from the selective lookdown space. The solution of the above mentioned system of SDE's is then mapped into the joint evolution of population size and symmetrized type configurations and genealogies, i.e. marked distance matrix distributions. By means of Kurtz' Markov mapping theorem, we characterize the latter process as the unique solution of a martingale problem. For the sake of transparency we restrict the main part of our presentation to a prototypical example with two types, which contains the essential features. In the final section we outline an extension to processes with multiple types including mutation., This version takes the suggestions of two anonymous referees into account, which led to an improvement in the presentation and to some corrections in the proofs in Section 5
Published: 2022

19. LIMITING ABSORPTION PRINCIPLE FOR DIRAC OPERATORS WITH A PERTURBED WIGNER-VON NEUMANN POTENTIAL

Author: MBAREK, AIMAN, Ben Hafsia, Sana, Analyse, Géométrie et Modélisation (AGM - UMR 8088), Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY), Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, and MBAREK, AIMAN
Subjects: [MATH] Mathematics [math], [MATH]Mathematics [math]
Abstract: Making use of the localised Putnam theory developed in [GJ1], applied in [JM],..., we show the limiting absorption principle for Dirac operators with a perturbed Wigner-Von Neumann potential on appropriate energy intervals. Furthermore we can allow longrange and short-range components.
Published: 2022

20. Transport Proofs Of Some Discrete Variants Of The Pr{\'e}Kopa-leindler Inequality

Author: Gozlan, Nathael, Roberto, Cyril, Samson, Paul-Marie, Tetali, Prasad, Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Modélisation aléatoire de Paris X (MODAL'X), Université Paris Nanterre (UPN), Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Carnegie Mellon University [Pittsburgh] (CMU), and Roberto, Cyril
Subjects: Mathematics - Functional Analysis, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Mathematics::Metric Geometry, Mathematics - Probability
Abstract: We give a transport proof of a discrete version of the displacement convexity of entropy on integers (Z), and get, as a consequence, two discrete forms of the Pr{\'e}kopa-Leindler Inequality : the Four Functions Theorem of Ahlswede and Daykin on the discrete hypercube [1] and a recent result on Z due to Klartag and Lehec [16].
Published: 2022

21. Fast Evaluation of Real and Complex Polynomials

Author: Anton, Ramona, Mihalache, Nicolae, Vigneron, François, Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Laboratoire de Mathématiques de Reims (LMR), and Université de Reims Champagne-Ardenne (URCA)-Centre National de la Recherche Scientifique (CNRS)
Subjects: FOS: Computer and information sciences, [INFO.INFO-CC]Computer Science [cs]/Computational Complexity [cs.CC], [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Numerical Analysis (math.NA), Dynamical Systems (math.DS), [INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA], Polynomials, Fast Evaluation, FOS: Mathematics, Computer Science - Mathematical Software, FPE / FastPolyEval library, Mathematics - Numerical Analysis, Mathematics - Dynamical Systems, Mathematical Software (cs.MS), Algorithms, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], [INFO.INFO-MS]Computer Science [cs]/Mathematical Software [cs.MS]
Abstract: We propose an algorithm for quickly evaluating polynomials. It pre-conditions a complex polynomial $P$ of degree $d$ in time $O(d\log d)$, with a low multiplicative constant independent of the precision. Subsequent evaluations of $P$ computed with a fixed precision of $p$ bits are performed in average arithmetic complexity $O\big(\sqrt{d(p+\log d)}\big)$ and memory $O(dp)$. The average complexity is computed with respect to points $z \in \mathbb{C}$, weighted by the spherical area of $\overline{\mathbb{C}}$. The worst case does not exceed the complexity of H{\"o}rner's scheme. In particular, our algorithm performs asymptotically as $O(\sqrt{d\log d})$ per evaluation. For many classes of polynomials, in particular those with random coefficients in a bounded region of $\mathbb{C}$, or for sparse polynomials, our algorithm performs much better than this upper bound, without any modification or parameterization.The article contains a detailed analysis of the complexity and a full error analysis, which guarantees that the algorithm performs as well as H\''orner's scheme, only faster. Our algorithm is implemented in a companion library, written in standard C and released as an open-source project [MV22].Our claims regarding complexity and accuracy are confirmed in practice by a set of comprehensive benchmarks.
Published: 2022

22. La densité critique du modèle des marches aléatoires activées est toujours inférieure à 1

Author: Asselah, Amine, Forien, Nicolas, Gaudillière, Alexandre, Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Dipartimento di Matematica 'Guido Castelnuovo' [Roma I] (Sapienza University of Rome), Università degli Studi di Roma 'La Sapienza' = Sapienza University [Rome] (UNIROMA), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), and ANR-17-CE40-0032,SWiWS,Gruyère et Saucisse de Wiener(2017)
Subjects: [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 60K35, 82B26, Probability (math.PR), FOS: Mathematics, Mathematics - Probability
Abstract: Activated Random Walks, on $\mathbb{Z}^d$ for any $d\geqslant 1$, is an interacting particle system, where particles can be in either of two states: active or frozen. Each active particle performs a continuous-time simple random walk during an exponential time of parameter $\lambda$, after which it stays still in the frozen state, until another active particle shares its location, and turns it instantaneously back into activity. This model is known to have a phase transition, and we show that the critical density, controlling the phase transition, is less than one in any dimension and for any value of the sleep rate $\lambda$. We provide upper bounds for the critical density in both the small $\lambda$ and large $\lambda$ regimes., Comment: 48 pages, 1 figure
Published: 2022

23. The Periodic Unfolding Method for countably many scales

Author: Damlamian, Alain, Griso, Georges, Cioranescu, Doina, Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and Griso, Georges
Subjects: [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: The periodic unfolding method, originally introduced in[4], is extended to the case of finitely or countably many distinct microscopic scales using the model diffusion problem as an application. This generalization can readily be applied to numerous other linear and non-linear problems.
Published: 2022

24. A review of univariate and multivariate multifractal analysis illustrated by the analysis of marathon runners physiological data

Author: Jaard, Stéphane, Saës, Guillaume, Nasr, Wejdene, Palacin, Florent, Billat, Véronique, Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, University of Mons [Belgium] (UMONS), ULB Neuroscience Institute [Brussels] (UNI), Université libre de Bruxelles (ULB), and Université Paris-Saclay
Subjects: Hölder regularity, Signal Processing (eess.SP), Marathon races, physiological data, [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], Multifractal Spectrum, Scaling, [INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing, Hausdor dimension, [SDV.MHEP.PHY]Life Sciences [q-bio]/Human health and pathology/Tissues and Organs [q-bio.TO], Classical Analysis and ODEs (math.CA), FOS: Electrical engineering, electronic engineering, information engineering, FOS: Mathematics, Electrical Engineering and Systems Science - Signal Processing, Wavelet Leader, Tissues and Organs (q-bio.TO), Scale Invariance, heartbeat frequency, Quantitative Biology - Tissues and Organs, Multifractal, Functional Analysis (math.FA), Mathematics - Functional Analysis, Multivariate analysis, Mathematics - Classical Analysis and ODEs, FOS: Biological sciences, p-leader, Fractal, Wavelet, [PHYS.PHYS.PHYS-DATA-AN]Physics [physics]/Physics [physics]/Data Analysis, Statistics and Probability [physics.data-an]
Abstract: We review the central results concerning wavelet methods in multifractal analysis, which consists in analysis of the pointwise singularities of a signal, and we describe its recent extension to multivariate multifractal analysis, which deals with the joint analysis of several signals; we focus on the mathematical questions that this new techniques motivate. We illustrate these methods by an application to data recorded on marathon runners.; Nous passons en revue les résultats centraux concernant les méthodes d'ondelettes dans l'analyse multifractale, qui consiste à analyser les singularités ponctuelles d'un signal, et nous décrivons sa récente extension à l'analyse multifractale multivariée, qui traite de l'analyse conjointe de plusieurs signaux ; nous nous concentrons sur les questions mathématiques que ces nouvelles techniques motivent. Nous illustrons ces méthodes par une application aux données enregistrées sur des marathoniens.
Published: 2022

25. Roughness and intermittency within metropolitan regions - Application in three French conurbations

Author: Janka Lengyel, Stéphane Roux, François Sémécurbe, Stéphane Jaffard, Patrice Abry, Laboratoire Ville, Mobilité, Transport (LVMT), École des Ponts ParisTech (ENPC)-Université Gustave Eiffel, Département de Physique [ENS Lyon], École normale supérieure - Lyon (ENS Lyon)-Université de Lyon, Théoriser et modéliser pour aménager (UMR 6049) (ThéMA), Université de Bourgogne (UB)-Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC), Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, and École normale supérieure de Lyon (ENS de Lyon)-Université de Lyon
Subjects: Urban Studies, Geography, Planning and Development, Architecture, [SHS.GEO]Humanities and Social Sciences/Geography, Management, Monitoring, Policy and Law, Nature and Landscape Conservation
Abstract: International audience; Even though the past three decades have seen numerous crucial investigations on interurban scaling characteristics, there has been less focus on revealing multiscale properties within municipal or metropolitan structures. We demonstrate how a newly developed methodology, the Geographically Weighted Multiscale Analysis (GWMSA) stemming from the theory of multifractal systems, can be used to analyze small-scale urban environments with respect to their intermittency and roughness simultaneously. To this end, apart from the widely used sand-box method, we introduce wavelet coefficients in the multiscale analysis of urban systems. In more detail, the spatially continuous scanning of the three largest French conurbations—Paris, Marseille, and Lyon—over their territories and at length scales ranging from parcel to neighborhood level will allow to derive and compare globally and locally characteristic scaling exponents. Depending on the feature under analysis, the exponents reveal qualitatively distinct structural properties, whereby the viability of our findings is further verified on four exemplary typologies of multiscale behavior in urban systems. To introduce GWMSA, this paper focuses primarily on morphological characteristics and findings provide a compelling alternative to how we capture and define district-scale spatial organization and interdependencies within urban settlements.
Published: 2022
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26. Corrected Curvature Measures

Author: Jacques-Olivier Lachaud, Pascal Romon, Boris Thibert, Laboratoire de Mathématiques (LAMA), Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS), Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), ANR-15-CE40-0006,CoMeDiC,Métriques convergentes pour le calcul digital(2015), Centre National de la Recherche Scientifique (CNRS)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry]), Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), Calcul des Variations, Géométrie, Image (CVGI ), Laboratoire Jean Kuntzmann (LJK ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), Calcul des Variations, Géométrie, Image (CVGI), and Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)
Subjects: Computational Theory and Mathematics, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Curvature estimators, Digital geometry, Discrete geometry, Discrete Mathematics and Combinatorics, Geometric measure theory, Geometry and Topology, MSC 52B70, 65D18, 49Q15, Normal cycle, [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], [MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG], Theoretical Computer Science
Abstract: International audience; This paper proposes a new mathematical and computational tool for infering the geometry of shapes known only through approximations such as triangulated or digital surfaces. The main idea is to decouple the position of the shape boundary from its normal vector field. To do so, we extend a classical tool of geometric measure theory, the normal cycle, so that it takes as input not only a surface but also a normal vector field. We formalize it as a current in the oriented Grassmann bundle R^3 × S^2. By choosing adequate differential forms, we define geometric measures like area, mean and Gaussian curvatures. We then show the stability of these measures when both position and normal input data are approximations of the underlying continuous shape. As a byproduct, our tool is able to correctly estimate curvatures over polyhedral approximations of shapes with explicit bounds, even when their natural normal are not correct, as long as an external convergent normal vector field is provided. Finally, the accuracy, convergence and stability under noise perturbation is evaluated experimentally onto digital surfaces.
Published: 2022
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27. Foreign Exchange Multivariate Multifractal Analysis

Author: Abry, P., yannick malevergne, Wendt, H., Jaffard, S., Senneret, M., Jaffrès, L., Laboratoire de Physique de l'ENS Lyon (Phys-ENS), École normale supérieure - Lyon (ENS Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), Pôle de recherche interdisciplinaire en sciences du management (PRISM Sorbonne), Université Paris 1 Panthéon-Sorbonne (UP1), CoMputational imagINg anD viSion (IRIT-MINDS), Institut de recherche en informatique de Toulouse (IRIT), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Centre National de la Recherche Scientifique (CNRS), Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Vivienne Investissement, European Association for Signal Processing (EURASIP), Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), École normale supérieure de Lyon (ENS de Lyon)-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), and Wendt, Herwig
Subjects: Foreign exchange, [INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing, [QFIN]Quantitative Finance [q-fin], wavelet leaders, [INFO.INFO-TS] Computer Science [cs]/Signal and Image Processing, Financial times series, multivariate multifractal analysis, basket properties, [INFO]Computer Science [cs], [QFIN] Quantitative Finance [q-fin]
Abstract: International audience; After Mandelbrot’s seminal work, scale-free and multifractal temporal dynamics have been recognized as classical stylized facts for financial time series and massively documented. Multifractal analysis in finance has however mainly remained univariate (one time series at a time) when multivariate (or basket) properties are critical for financial applications. This is mostly due to a lack of theoretical foundations and practical tools for multivariate multifractal analysis. Expanding on a theoretically-grounded recently proposed multivariate multifractal formalism, the present work performs an original multivariate analysis for a basket of six Foreign Exchange rate time series. Beyond confirming multifractality for each component independently, the definition of cross-multifractalities amongst components is introduced, assessing cross-dependencies in temporal dynamics not already accounted for by cross-correlations. The key practical outcome is to show that, essentially, one same multifractal time governs jointly the temporal dynamics of all the Foreign Exchange time series studied here.
Published: 2022

28. On the size of stable minimal surfaces in $${\mathbb {R}}^4$$

Author: Ari Aiolfi, Marc Soret, Marina Ville, Universidade Federal de Santa Maria = Federal University of Santa Maria [Santa Maria, RS, Brazil] (UFSM), Institut Denis Poisson (IDP), Université d'Orléans (UO)-Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Analyse et Mathématiques Appliquées (LAMA), and Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel
Subjects: Grassmannian, General Mathematics, Gauss map, minimal surface, stability, [MATH]Mathematics [math]
Abstract: International audience; The Gauss map g of a surface ⌃ in R 4 takes its values in the Grassmannian of oriented 2-planes of R 4 : G + (2, 4). We give geometric criteria of stability for minimal surfaces in R 4 in terms of g. We show in particular that if the spherical area of the Gauss map |g(⌃)| of a minimal surface is smaller than 2⇡ then the surface is stable by deformations which fix the boundary of the surface. This answers the question of [BDC3] in R 4 .
Published: 2022
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29. Weighted Erdős–Kac theorem in short intervals

Author: Jie Wu, Kui Liu, Qingdao University, Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, and Qingdao University (QDU)
Subjects: Algebra and Number Theory, Mathematics::Operator Algebras, 010102 general mathematics, 0102 computer and information sciences, 01 natural sciences, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Combinatorics, High Energy Physics::Theory, symbols.namesake, Number theory, Mathematics::K-Theory and Homology, 010201 computation theory & mathematics, Fourier analysis, Mathematics::Quantum Algebra, symbols, Erdős–Kac theorem, 0101 mathematics, Mathematics::Representation Theory, Mathematics
Abstract: International audience; In this paper, we generalize Elliott's weighted Erdős-Kac theorem to the case of short intervals.
Published: 2021
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30. CONVERGENCE OF THE FULLY DISCRETE INCREMENTAL PROJECTION SCHEME FOR INCOMPRESSIBLE FLOWS

Author: Gallouët, Thierry, Herbin, Raphaèle, Latché, Jean-Claude, Maltese, David, Aix Marseille Université (AMU), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Service des Agressions Internes et des risques Industriels (IRSN/PSN-RES/SA2I), Institut de Radioprotection et de Sûreté Nucléaire (IRSN), Laboratoire Analyse et Mathématiques Appliquées (LAMA), and Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel
Subjects: convergence, analysis, projection scheme, MAC scheme, 76M12 Incompressible Navier-Stokes equations, FOS: Mathematics, 65N12, staggered finite volumes, Mathematics - Numerical Analysis, Numerical Analysis (math.NA), 2000 Mathematics Subject Classification. 35Q30, 65M08, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: The present paper addresses the convergence of a first order in time incremental projection scheme for the time-dependent incompressible Navier-Stokes equations to a weak solution, without any assumption of existence or regularity assumptions on the exact solution. We prove the convergence of the approximate solutions obtained by the semi-discrete scheme and a fully discrete scheme using a staggered finite volume scheme on non uniform rectangular meshes. Some first a priori estimates on the approximate solutions yield the existence. Compactness arguments, relying on these estimates, together with some estimates on the translates of the discrete time derivatives, are then developed to obtain convergence (up to the extraction of a subsequence), when the time step tends to zero in the semi-discrete scheme and when the space and time steps tend to zero in the fully discrete scheme; the approximate solutions are thus shown to converge to a limit function which is then shown to be a weak solution to the continuous problem by passing to the limit in these schemes.
Published: 2022

31. A transverse index theorem in the calculus of filtered manifolds

Author: Cren, Clément, Cren, Clément, Laboratoire Analyse et Mathématiques Appliquées (LAMA), and Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel
Subjects: Mathematics - Differential Geometry, Lie Groupoids, [MATH.MATH-OA]Mathematics [math]/Operator Algebras [math.OA], Mathematics - Operator Algebras, K-Theory and Homology (math.KT), Bivariant K-theory, Foliations, [MATH.MATH-KT] Mathematics [math]/K-Theory and Homology [math.KT], Noncommutative Geometry, Differential Geometry (math.DG), [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Mathematics - K-Theory and Homology, [MATH.MATH-KT]Mathematics [math]/K-Theory and Homology [math.KT], FOS: Mathematics, Index Theory, Pseudodifferential operators, [MATH.MATH-OA] Mathematics [math]/Operator Algebras [math.OA], [MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG], Operator Algebras (math.OA), Global analysis
Abstract: We use filtrations of the tangent bundle of a manifold starting with an integrable subbundle to define transverse symbols to the corresponding foliation, define a condition of transversally Rockland and prove that transversally Rockland operators yield a K-homology class. We construct an equivariant KK-class for transversally Rockland transverse symbols and show a Poincare duality type result linking the class of an operator and its symbol.
Published: 2022

32. A NOTE ON STARSHAPED HYPERSURFACES WITH ALMOST CONSTANT MEAN CURVATURE IN SPACE FORMS

Author: Roth, Julien, Upadhyay, Abhitosh, Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, INDIAN INSTITUTE OF TECHNOLOGY GOA (IIT GOA), and Roth, Julien
Subjects: Mathematics - Differential Geometry, Differential Geometry (math.DG), [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], FOS: Mathematics, Mathematics::Differential Geometry, [MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG], Mathematics::Symplectic Geometry
Abstract: We show that closed starshaped hypersurfaces of space forms with almost constant mean curvature or almost constant higher order mean curvature are closed to geodesic spheres., 9 pages. arXiv admin note: text overlap with arXiv:2206.06727
Published: 2022

33. Growth dynamics of large networks using hidden Markov chains

Author: Duchemin, Quentin, Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Université Gustave Eiffel, Claire Lacour, STAR, ABES, Financement de la Région Ile de France (DIM Math Innov), Université Gustave Eiffeil, and Yohann De Castro
Subjects: Inférence post-sélection de modèle, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Measure concentration, Markov chains, Markov chain, Post-Selection inference, Latent Space, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], Graphes aléatoires, Chaines de Markov cachees, [STAT.OT]Statistics [stat]/Other Statistics [stat.ML], Chaîne de Markov, Estimation non paramétrique, Inférence post sélection, Espace latent, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Random Graphs, Inference post model-selection, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Concentration de la mesure, Concentration of measure, Non-Parametric estimation
Abstract: The first part of this thesis aims at introducing new models of random graphs that account for the temporal evolution of networks. More precisely, we focus on growth models where at each instant a new node is added to the existing graph. We attribute to this new entrant properties that characterize its connectivity to the rest of the network and these properties depend only on the previously introduced node. Our random graph models are thus governed by a latent Markovian dynamic characterizing the sequence of nodes in the graph. We are particularly interested in the Stochastic Block Model and in Random Geometric Graphs for which we propose algorithms to estimate the unknown parameters or functions defining the model. We then show how these estimates allow us to solve link prediction or collaborative filtering problems in networks.The theoretical analysis of the above-mentioned algorithms requires advanced probabilistic tools. In particular, one of our proof is relying on a concentration inequality for U-statistics in a dependent framework. Few papers have addressed this thorny question and existing works consider sets of assumptions that do not meet our needs. Therefore, the second part of this manuscript will be devoted to the proof of a concentration inequality for U-statistics of order two for uniformly ergodic Markov chains. In Chapter 5, we exploit this concentration result for U-statistics to make new contributions to three very active areas of Statistics and Machine Learning.Still motivated by link prediction problems in graphs, we study post-selection inference procedures in the framework of logistic regression with. penalty. We prove a central limit theorem under the distribution conditional on the selection event and derive asymptotically valid testing procedures and confidence intervals, La première partie de cette thèse vise à introduire de nouveaux modèles de graphes aléatoires rendant compte de l'évolution temporelle des réseaux. Plus précisément, nous nous concentrons sur des modèles de croissance où à chaque instant un nouveau noeud s'ajoute au graphe existant. Nous attribuons à ce nouvel entrant des propriétés qui caractérisent son pouvoir de connectivité au reste du réseau et celles-ci dépendent uniquement du noeud précédemment introduit. Nos modèles de graphes aléatoires sont donc régis par une dynamique markovienne latente caractérisant la séquence de noeuds du graphe. Nous nous intéresserons particulièrement au Stochastic Block Model et aux Graphes Aléatoires Géométriques pour lesquels nous proposons des algorithmes permettant d'estimer les paramètres du modèle. Nous montrons ensuite comment ce travail d'estimation nous permet de résoudre des problèmes de prédiction de lien ou de filtrage collaboratif dans les graphes.L'étude théorique des algorithmes précédemment décrits mobilisent des résultats probabilistes poussés. Nous avons notamment dû recourir à une inégalité de concentration pour les U-statistiques dans un cadre dépendant. Peu nombreux sont les travaux ayant abordé cette épineuse question et l'existant considère des jeux d'hypothèses ne répondant pas à nos besoins. Aussi, la deuxième partie de ce manuscrit sera consacrée à la preuve d'une inégalité de concentration pour les U-statistiques d'ordre deux pour des chaînes de Markov uniformément ergodique. Dans le Chapitre 5, nous exploitons notre résultat de concentration pour les U-statistiques pour apporter de nouvelles contributions à trois domaines très actifs des Statistiques et du Machine Learning.Toujours motivés par des problèmes de prédictions liens dans les graphes, nous nous intéressons dans un dernier chapitre aux procédures d'inférence post-sélection dans le cadre de la régression logistique avec pénalité. Nous prouvons un théorème central limite sous la distribution conditionnelle à l'événement de sélection et nous en déduisons des procédures de test et des intervalles de confiance asymptotiquement valides
Published: 2022

34. Decimations for One- and Two-dimensional Ising and Rotator Models II: Continuous versus Discrete Symmetries

Author: Matteo D’Achille, Aernout C. D. van Enter, Arnaud Le Ny, Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Bernoulli Institute for Mathematics for Mathematics, Computer Science and Artificial Intelligence Groningen, and University of Groningen [Groningen]
Subjects: [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 60K35 (Primary), 82B20 (Secondary), Probability (math.PR), [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Mathematics, FOS: Physical sciences, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), Mathematics - Probability, Mathematical Physics
Abstract: We show how decimated Gibbs measures which have an unbroken continuous symmetry due to the Mermin-Wagner theorem, although their discrete equivalents have a phase transition, still can become non-Gibbsian. The mechanism rests on the occurrence of a spin-flop transition with a broken discrete symmetry, once the model is constrained by the decimated spins in a suitably chosen "bad" configuration., Comment: 16 pages, 1 figure (updated)
Published: 2022

35. A NEW CONVERGENCE PROOF FOR APPROXIMATIONS OF THE STEFAN PROBLEM

Author: Eymard, Robert, Gallouët, Thierry, Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Institut de Mathématiques de Marseille (I2M), and Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS)
Subjects: FOS: Mathematics, Mathematics::Analysis of PDEs, Numerical Analysis (math.NA), Mathematics - Numerical Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: International audience; We consider the Stefan problem, firstly with regular data and secondly with irregular data. In both cases is given a proof for the convergence of an approximation obtained by regularising the problem. These proofs are based on weak formulations and on compactness results in some Sobolev spaces with negative exponents.
Published: 2022
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36. Uniform Deconvolution for Poisson Point Processes

Author: Anna Bonnet, Claire Lacour, Franck Picard, Vincent Rivoirard, Laboratoire de Probabilités, Statistique et Modélisation (LPSM (UMR_8001)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Département PEGASE [LBBE] (PEGASE), Laboratoire de Biométrie et Biologie Evolutive - UMR 5558 (LBBE), Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-VetAgro Sup - Institut national d'enseignement supérieur et de recherche en alimentation, santé animale, sciences agronomiques et de l'environnement (VAS)-Centre National de la Recherche Scientifique (CNRS)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-VetAgro Sup - Institut national d'enseignement supérieur et de recherche en alimentation, santé animale, sciences agronomiques et de l'environnement (VAS)-Centre National de la Recherche Scientifique (CNRS), Statistique en grande dimension pour la génomique, Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-VetAgro Sup - Institut national d'enseignement supérieur et de recherche en alimentation, santé animale, sciences agronomiques et de l'environnement (VAS)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Biométrie et Biologie Evolutive - UMR 5558 (LBBE), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Probabilités, Statistiques et Modélisations (LPSM (UMR_8001)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPC), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)
Subjects: FOS: Computer and information sciences, Statistics::Theory, [STAT.AP]Statistics [stat]/Applications [stat.AP], Poisson Point Process, Poisson process, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Convolution, Methodology (stat.ME), Adaptive estimation, [INFO.INFO-LG]Computer Science [cs]/Machine Learning [cs.LG], [STAT.ML]Statistics [stat]/Machine Learning [stat.ML], [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], [SDV.BBM.GTP]Life Sciences [q-bio]/Biochemistry, Molecular Biology/Genomics [q-bio.GN], [INFO.INFO-BI]Computer Science [cs]/Bioinformatics [q-bio.QM], [STAT.ME]Statistics [stat]/Methodology [stat.ME], Statistics - Methodology, [INFO.INFO-MS]Computer Science [cs]/Mathematical Software [cs.MS]
Abstract: International audience; We focus on the estimation of the intensity of a Poisson process in the presence of a uniform noise. We propose a kernel-based procedure fully calibrated in theory and practice. We show that our adaptive estimator is optimal from the oracle and minimax points of view, and provide new lower bounds when the intensity belongs to a Sobolev ball. By developing the Goldenshluger-Lepski methodology in the case of deconvolution for Pois-son processes, we propose an optimal data-driven selection of the kernel's bandwidth, and we provide a heuristic framework to calibrate the estimator in practice. Our method is illustrated on the spatial repartition of replication origins along the human genome.
Published: 2022

37. Existence of SRB measures for hyperbolic maps with weak regularity

Author: Boukhecham, Houssam, Laboratoire Analyse et Mathématiques Appliquées (LAMA), and Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel
Subjects: Mathematics::Dynamical Systems, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], FOS: Mathematics, Mathematics::Classical Analysis and ODEs, Dynamical Systems (math.DS), Mathematics - Dynamical Systems
Abstract: We prove that a $C^1$ hyperbolic map whose differential is regular enough has an SRB measure. The precise regularity condition is weaker than H{\"o}lder and was mentionned by various authors through the developement of expanding and uniformly hyperbolic dynamics.
Published: 2022

38. SRB measures for hyperbolic attractor in low regularity

Author: Boukhecham, Houssam, Laboratoire Analyse et Mathématiques Appliquées (LAMA), and Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel
Subjects: Mathematics::Dynamical Systems, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS]
Abstract: We consider a $C^1$ hyperbolic attractor, and prove the existence of a physic measure provided that the differential satisfies some summability condition which is weaker than Hölder continuity.
Published: 2022

39. Weighted p--Laplace approximation of linear and quasi-linear elliptic problems with measure data

Author: Robert Eymard, David Maltese, Alain Prignet, Laboratoire Analyse et Mathématiques Appliquées (LAMA), and Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel
Subjects: Mathematics - Classical Analysis and ODEs, Applied Mathematics, Classical Analysis and ODEs (math.CA), FOS: Mathematics, [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], Analysis
Abstract: International audience; We approximate the solution to some linear and degenerate quasi-linear problem involving a linear elliptic operator (like the semi-discrete in time implicit Euler approximation of Richards and Stefan equations) with measure right-hand side and heterogeneous anisotropic diffusion matrix. This approximation is obtained through the addition of a weighted p−Laplace term. A well chosen diffeomorphism between R and (−1, 1) is used for the estimates of the approximated solution, and is involved in the above weight. We show that this approximation converges to a weak sense of the problem for general right-hand-side, and to the entropy solution in the case where the right-hand-side is in L 1 .
Published: 2022

40. Injectivity of Lipschitz operators

Author: Luis C. García-Lirola, Colin Petitjean, Antonín Procházka, Departamento de matemáticas, Universidad de Zaragoza, Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), and Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)
Subjects: Mathematics - Functional Analysis, Mathematics - Metric Geometry, General Mathematics, Lipschitz function, Injectivity, FOS: Mathematics, Lipschitz-free space, Metric Geometry (math.MG), Support, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], [MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG], Functional Analysis (math.FA)
Abstract: Any Lipschitz map $f\colon M \to N$ between metric spaces can be ``linearised'' in such a way that it becomes a bounded linear operator $\widehat{f}\colon \mathcal F(M) \to \mathcal F(N)$ between the Lipschitz-free spaces over $M$ and $N$. The purpose of this note is to explore the connections between the injectivity of $f$ and the injectivity of $\widehat{f}$. While it is obvious that if $\widehat{f}$ is injective then so is $f$, the converse is less clear. Indeed, we pin down some cases where this implication does not hold but we also prove that, for some classes of metric spaces $M$, any injective Lipschitz map $f\colon M \to N$ (for any $N$) admits an injective linearisation. Along our way, we study how Lipschitz maps carry the support of elements in free spaces and also we provide stronger conditions on $f$ which ensure that $\widehat{f}$ is injective.
Published: 2022

41. Filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations

Author: Champagnat, Nicolas, Méléard, Sylvie, Mirrahimi, Sepideh, Tran, Viet-Chi, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Biology, genetics and statistics (BIGS), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Institut Universitaire de France (IUF), Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.), Institut Montpelliérain Alexander Grothendieck (IMAG), Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM), Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Chaire 'Modélisation Mathématique et Biodiversité' de Veolia Environnement-Ecole Polytechnique-Museum National d’Histoire Naturelle-Fondation X, ANR-10-LABX-0058,Bézout,Models and algorithms: from the discrete to the continuous(2010), ANR-20-CE40-0011,DEEV,Modèles intégro-différentiels venant de la biologie évolutive(2020), and Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)
Subjects: viscosity solution, [SDV.BID.EVO]Life Sciences [q-bio]/Biodiversity/Populations and Evolution [q-bio.PE], Probability (math.PR), Populations and Evolution (q-bio.PE), selection, stochastic birth death models, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Mathematics - Analysis of PDEs, maximum principle, large population approximation, FOS: Biological sciences, FOS: Mathematics, Quantitative Biology::Populations and Evolution, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 92D25, 92D15, 60J80, 60F99, 35F21, mutation, Quantitative Biology - Populations and Evolution, Mathematics - Probability, Analysis of PDEs (math.AP), MSC 2000 : 92D25, 92D15, 60J80, 60F99, 35F21
Abstract: We consider a stochastic model for the evolution of a discrete population structured by a trait with values on a finite grid of the torus, and with mutation and selection. Traits are vertically inherited unless a mutation occurs, and influence the birth and death rates. We focus on a parameter scaling where population is large, individual mutations are small but not rare, and the grid mesh for the trait values is much smaller than the size of mutation steps. When considering the evolution of the population in a long time scale, the contribution of small sub-populations may strongly influence the dynamics. Our main result quantifies the asymptotic dynamics of sub-population sizes on a logarithmic scale. We establish that under the parameter scaling the logarithm of the stochastic population size process, conveniently normalized, converges to the unique viscosity solution of a Hamilton-Jacobi equation. Such Hamilton-Jacobi equations have already been derived from parabolic integro-differential equations and have been widely developed in the study of adaptation of quantitative traits. Our work provides a justification of this framework directly from a stochastic individual based model, leading to a better understanding of the results obtained within this approach. The proof makes use of almost sure maximum principles and careful controls of the martingale parts.
Published: 2022

42. Séminaire de Probabilités LI

Author: Karandikar, R. L., Rao, B. V., Leuridan, Christophe, Vidmar, Matija, Miclo, Laurent, Patie, Pierre, Bernou, Armand, Caballero, Maria-Emilia, Rouault, Alain, Dedecker, Jérôme, Merlevède, Florence, Rio, Emmanuel, Brosset, Fabien, Klein, Thierry, Lagnoux, Agnès, Petit, Pierre, Marinelli, Carlo, Scarpa, Luca, Castaing, Charles, Marie, Nicolas, Fitte, Paul Raynaud De, Attal, Stéphane, Deschamps, Julien, Pellegrini, Clément, Eisenbaum, Nathalie, Chennai Mathematical Institute [Inde], Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Department of Mathematics (University of Ljubljana), University of Ljubljana, Toulouse School of Economics (TSE-R), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), School of Operations Research and Information Engineering (ORIE), Cornell University [New York], Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Instituto de Matematicas (UNAM), Universidad Nacional Autónoma de México = National Autonomous University of Mexico (UNAM), Laboratoire de Mathématiques de Versailles (LMV), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Université Paris Descartes - Paris 5 (UPD5)-Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Ecole Nationale de l'Aviation Civile (ENAC), University College of London [London] (UCL), Faculty of Mathematics [Vienna], University of Vienna [Vienna], Université de Montpellier (UM), Modélisation aléatoire de Paris X (MODAL'X), Université Paris Nanterre (UPN), Laboratoire de Mathématiques Raphaël Salem (LMRS), Université de Rouen Normandie (UNIROUEN), Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS), Università degli studi di Genova = University of Genoa (UniGe), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Donati-Martin, Catherine, Lejay, Antoine, Rouault, Alain, Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Équations aux dérivées partielles, analyse (EDPA), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL)
Subjects: [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], ComputingMilieux_MISCELLANEOUS
Abstract: International audience
Published: 2022
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43. Lattices with finite renormalized coulombian interaction energy in the plane

Author: Yuxin Ge, Etienne Sandier, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), and Institut National des Sciences Appliquées (INSA)-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3)
Subjects: Physics, Condensed matter physics, Plane (geometry), General Mathematics, 010102 general mathematics, 0103 physical sciences, Interaction energy, [MATH]Mathematics [math], 0101 mathematics, 010306 general physics, 01 natural sciences, ComputingMilieux_MISCELLANEOUS
Abstract: International audience
Published: 2021
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44. The arcsine law on divisors in arithmetic progressions modulo prime powers

Author: J. Wu, B. Feng, Yangtze Normal University, Laboratoire Analyse et Mathématiques Appliquées (LAMA), and Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel
Subjects: Selberg-Delange method, General Mathematics, Modulo, 010102 general mathematics, 010103 numerical & computational mathematics, arcsine law, 16. Peace & justice, 01 natural sciences, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Prime (order theory), Moduli, arithmetic progressions, Law, Inverse trigonometric functions, 0101 mathematics, Arithmetic, Mathematics
Abstract: Let $$x\to\infty$$ be a parameter. Feng [5] proved that the Deshouillers–Dress–Tenenbaum’s arcsine law on divisors of the integers less than x also holds in arithmetic progressions for ``non-exceptional moduli" $$q\leqslant\exp\{(\frac{1}{4}-\varepsilon)(\log_2 x)^2\}$$ , where $$\varepsilon$$ is an arbitrarily small positive number. We show that in the case of a prime-power modulus ( $$q:=\mathfrak{p}^{\varpi}$$ with $$\mathfrak{p}$$ a fixed odd prime and $$\varpi\in \mathbb{N}$$ ) the arcsine law on divisors holds in arithmetic progressions for $$q\le x^{15/52-\varepsilon}$$ .
Published: 2020
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45. Adaptive estimation of nonparametric geometric graphs

Author: Yohann De Castro, Thanh Mai Pham Ngoc, Claire Lacour, Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Computation, Sphere, Mathematics - Statistics Theory, Statistics Theory (math.ST), Random Networks, 02 engineering and technology, Kernel Matrix, 01 natural sciences, 010104 statistics & probability, [STAT.ML]Statistics [stat]/Machine Learning [stat.ML], Exponential growth, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Spectral Estimation, FOS: Mathematics, 0202 electrical engineering, electronic engineering, information engineering, 62G05, secondary 60C05, 60B15, 0101 mathematics, Time complexity, Mathematics, Discrete mathematics, Graphon, Probability (math.PR), Nonparametric statistics, Spherical harmonics, Spectral density estimation, 020206 networking & telecommunications, Minimax, Metric space, Convolution Operator, [INFO.INFO-IT]Computer Science [cs]/Information Theory [cs.IT], Mathematics - Probability, 62G05 (Primary), 60C05, 60B15 (secondary
Abstract: This article studies the recovery of graphons when they are convolution kernels on compact (symmetric) metric spaces. This case is of particular interest since it covers the situation where the probability of an edge depends only on some unknown nonparametric function of the distance between latent points, referred to as Nonparametric Geometric Graphs (NGG). In this setting, adaptive estimation of NGG is possible using a spectral procedure combined with a Goldenshluger-Lepski adaptation method. The latent spaces covered by our framework encompass (among others) compact symmetric spaces of rank one, namely real spheres and projective spaces. For these latter, explicit computations of the eigen-basis and of the model complexity can be achieved, leading to quantitative non-asymptotic results. The time complexity of our method scales cubicly in the size of the graph and exponentially in the regularity of the graphon. Hence, this paper offers an algorithmically and theoretically efficient procedure to estimate smooth NGG. As a by product, this paper shows a non-asymptotic concentration result on the spectrum of integral operators defined by symmetric kernels (not necessarily positive)., Final version
Published: 2020
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46. A new procedure for Selective Inference with the Generalized Linear Lasso

Author: Duchemin, Quentin, De Castro, Yohann, Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS)
Subjects: [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], [STAT.ME]Statistics [stat]/Methodology [stat.ME]
Abstract: This research report is part of a chapter of a PhD thesis and has not yet been submitted. A new version of this work will come soon with additional numerical experiments, and will be then submitted for review.; This articles investigates the distribution of the solutions of the generalized linear lasso (GLL), conditional on some selection event. In this framework of post-selection inference (PSI), we provide rigorous definitions of the selected and saturated models: two different paradigms that determine the hypothesis being tested. Based on a conditional Maximum Likelihood Estimator (MLE) approach, we give a procedure to obtain asymptotically valid PSI confidence regions and testing procedures for Generalized Linear Models (GLMs). In a second stage, we focus on the sparse logistic regression and we exhibit conditions ensuring that our conditional MLE method is valid. We present extensive numerical simulations supporting our theoretical results.
Published: 2022

47. Random Geometric Graph: Some recent developments and perspectives

Author: Duchemin, Quentin, de Castro, Yohann, Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Social and Information Networks (cs.SI), FOS: Computer and information sciences, Computer Science - Social and Information Networks, Mathematics - Statistics Theory, Statistics Theory (math.ST), [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], [INFO.INFO-SI]Computer Science [cs]/Social and Information Networks [cs.SI], Coupling, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Random Geometric Graphs, Spectral clustering, FOS: Mathematics, Random matrices, Information inequalities, Concentration inequality for U-statistics, Non-parametric estimation
Abstract: The Random Geometric Graph (RGG) is a random graph model for network data with an underlying spatial representation. Geometry endows RGGs with a rich dependence structure and often leads to desirable properties of real-world networks such as the small-world phenomenon and clustering. Originally introduced to model wireless communication networks, RGGs are now very popular with applications ranging from network user profiling to protein-protein interactions in biology. RGGs are also of purely theoretical interest since the underlying geometry gives rise to challenging mathematical questions. Their resolutions involve results from probability, statistics, combinatorics or information theory, placing RGGs at the intersection of a large span of research communities. This paper surveys the recent developments in RGGs from the lens of high dimensional settings and non-parametric inference. We also explain how this model differs from classical community based random graph models and we review recent works that try to take the best of both worlds. As a by-product, we expose the scope of the mathematical tools used in the proofs., This is a research report that is part of a Chapter of a PhD thesis. An updated version will be available soon
Published: 2022

48. Scaling limit of the collision measures of multiple random walks

Author: Nguyen, Dinh-Toan, NGUYEN, Dinh-Toan, Université du Québec à Montréal = University of Québec in Montréal (UQAM), Laboratoire Analyse et Mathématiques Appliquées (LAMA), and Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel
Subjects: [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], 60F05, 60G57, 82C05, Probability (math.PR), Wiener Chaos, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Random Walks, Partition Functions, FOS: Mathematics, Random Environment, U -statistics, Collisions, Scaling Limits, Random Measures, Mathematics - Probability
Abstract: For an integer $k\ge 2$, let $S^{(1)}, S^{(2)}, \dots, S^{(k)}$ be $k$ independent simple symmetric random walks on $\mathbb{Z}$. A pair $(n,z)$ is called a collision event if there are at least two distinct random walks, namely, $S^{(i)},S^{(j)}$ satisfying $S^{(i)}_n= S^{(j)}_n=z$. We show that under the same scaling as in Donsker's theorem, the sequence of random measures representing these collision events converges to a non-trivial random measure on $[0,1]\times \mathbb{R}$. Moreover, the limit random measure can be characterized using Wiener chaos. The proof is inspired by methods from statistical mechanics, especially, by a partition function that has been developed for the study of directed polymers in random environments., 37 pages, 1 figure
Published: 2022

49. POTENTIAL METHOD AND PROJECTION THEOREMS FOR MACROSCOPIC HAUSDORFF DIMENSION

Author: Lara Daw, Stéphane Seuret, University of Luxembourg [Luxembourg], Laboratoire Analyse et Mathématiques Appliquées (LAMA), and Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel
Subjects: Marstrand-Mattila projection theorem, Mathematics - Metric Geometry, Mathematics - Classical Analysis and ODEs, General Mathematics, Macroscopic Hausdorff dimension, Potential theory for dimensions, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Geometric measure theory, Metric Geometry (math.MG), [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], Fine properties of sets, [MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG]
Abstract: The macroscopic Hausdorff dimension Dim H (E) of a set E $\subset$ R d was introduced by Barlow and Taylor to quantify a "fractal at large scales" behavior of unbounded, possibly discrete, sets E. We develop a method based on potential theory in order to estimate this dimension in R d. Then, we apply this method to obtain Marstrand-like projection theorems: given a set E $\subset$ R 2 , for almost every $\theta$ $\in$ [0, 2$\pi$], the projection of E on the straight line passing through 0 with angle $\theta$ has dimension equal to min(Dim H (E) , 1).
Published: 2022

50. Hellinger and total variation distance in approximating Lévy driven SDEs

Author: Emmanuelle Clément, Laboratoire Analyse et Mathématiques Appliquées (LAMA), and Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel
Subjects: Statistics and Probability, 60B10, Hellinger Distance, Lévy Process, 60H07 total variation, Stable Process, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], MSC 2020. Primary: 60H10, total variation, 60H07, Stochastic Differential Equation, Statistics, Probability and Uncertainty, Mathematics - Probability, 60G51
Abstract: In this paper, we get some convergence rates in total variation distance in approximating discretized paths of L{\'e}vy driven stochastic differential equations, assuming that the driving process is locally stable. The particular case of the Euler approximation is studied. Our results are based on sharp local estimates in Hellinger distance obtained using Malliavin calculus for jump processes.
Published: 2022

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