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532 results on '"[ MATH ] Mathematics [math]"'

1. The attainment set of the $\varphi$-envelope and genericity properties

Author

Lionel Thibault, Dariusz Zagrodny, Alexandre Cabot, Abderrahim Jourani, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Institut Montpelliérain Alexander Grothendieck ( IMAG ), and Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS )

Subjects
[ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC], [ MATH ] Mathematics [math], Convex hull, Discrete mathematics, Infimal convolution, General Mathematics, 010102 general mathematics, Attainment sets, Subderivative, Norm Subdifferential Local Uniform Convexity (NSLUC), 01 natural sciences, Klee envelope, Supremal convolution, $\varphi$-envelope, Legendre-Fenchel conjugate, Norm (mathematics), [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH]Mathematics [math], 0101 mathematics, Subdifferential, Mathematics
Abstract

International audience; The attainment set of the $\varphi-$envelope of a function at a given point is investigated. The inclusion of the attainment set of the $\varphi$-envelope of the closed convex hull of a function into the attainment set of the function is preserved in sufficiently general settings toencompass the case $\varphi $ being a norm in a power not less than $1$. The nonemptiness of the attainment set is guaranteed on genericsubsets of a given space, in several fundamental cases.

Published
2019
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2. Non-parametric statistical analysis of spatially distributed functional data

Author

Dabo-Niang, Sophie, Ternynck, Camille, Thiam, Baba, Yao, Anne-Françoise, Lille économie management - UMR 9221 (LEM), Université d'Artois (UA)-Université catholique de Lille (UCL)-Université de Lille-Centre National de la Recherche Scientifique (CNRS), Economie Quantitative, Intégration, Politiques Publiques et Econométrie (EQUIPPE), Université de Lille, Droit et Santé-PRES Université Lille Nord de France-Université de Lille, Sciences Humaines et Sociales-Université de Lille, Sciences et Technologies, Ramon Giraldo, Jorge Mateu, Celet, Isabelle, MOdel for Data Analysis and Learning ( MODAL ), Laboratoire Paul Painlevé - UMR 8524 ( LPP ), 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 ) -Santé publique : épidémiologie et qualité des soins-EA 2694 ( CERIM ), Université de Lille-Centre Hospitalier Régional Universitaire [Lille] ( CHRU Lille ) -Université de Lille-Centre Hospitalier Régional Universitaire [Lille] ( CHRU Lille ) -Polytech Lille-Université de Lille 1, IUT’A, Lille - Economie et Management ( LEM ), Université catholique de Lille ( UCL ) -Université de Lille-Centre National de la Recherche Scientifique ( CNRS ), Economie Quantitative, Intégration, Politiques Publiques et Econométrie ( EQUIPPE ), Université de Lille, Sciences et Technologies-Université de Lille, Sciences Humaines et Sociales-PRES Université Lille Nord de France-Université de Lille, Droit et Santé, Laboratoire de Mathématiques de Versailles ( LMV ), Université Paris-Saclay-Centre National de la Recherche Scientifique ( CNRS ) -Université de Versailles Saint-Quentin-en-Yvelines ( UVSQ ), Laboratoire de MicrobiologiE de Géochimie et d'Ecologie Marines ( LMGEM ), Université de la Méditerranée - Aix-Marseille 2-Centre National de la Recherche Scientifique ( CNRS ), Ramon Giraldo (éd.), Jorge Mateu (éd.), MOdel for Data Analysis and Learning (MODAL), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille, Sciences et Technologies-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Evaluation des technologies de santé et des pratiques médicales - ULR 2694 (METRICS), Université de Lille-Centre Hospitalier Régional Universitaire [Lille] (CHRU Lille)-Université de Lille-Centre Hospitalier Régional Universitaire [Lille] (CHRU Lille)-École polytechnique universitaire de Lille (Polytech Lille), 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 Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Evaluation des technologies de santé et des pratiques médicales - ULR 2694 (METRICS), and Université de Lille-Centre Hospitalier Régional Universitaire [Lille] (CHRU Lille)-Université de Lille-Centre Hospitalier Régional Universitaire [Lille] (CHRU Lille)-École polytechnique universitaire de Lille (Polytech Lille)-Université de Lille, Sciences et Technologies

Subjects
[ MATH ] Mathematics [math], [SHS.STAT]Humanities and Social Sciences/Methods and statistics, [SHS.STAT] Humanities and Social Sciences/Methods and statistics, [ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST], [SHS] Humanities and Social Sciences, ComputingMilieux_MISCELLANEOUS, [SHS]Humanities and Social Sciences
Abstract

International audience

Published
2021

3. Revisiting the Isoperimetric Graph Partitioning Problem

Author

Sravan Danda, Laurent Najman, Aditya Challa, B. S. Daya Sagar, Systems Science and Informatics Unit, Indian Statistical Institute [Bangalore] (ISI), Laboratoire Algorithmique et Architecture des Systèmes Informatiques (A2SI), École Supérieure d'Ingénieurs en Électronique et Électrotechnique, ANR-14-CE27-0001,GRAPHSIP,Traitement de signaux sur graphes(2014), ANR-10-LABX-0058,Bézout,Models and algorithms: from the discrete to the continuous(2010), INDIAN STATISTICAL INSTITUTE, and Laboratoire Algorithmique et Architecture des Systèmes Informatiques ( A2SI )

Subjects
[ MATH ] Mathematics [math], [ INFO.INFO-TS ] Computer Science [cs]/Signal and Image Processing, General Computer Science, Computer science, Cheeger cut, 02 engineering and technology, [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], [ INFO.INFO-CV ] Computer Science [cs]/Computer Vision and Pattern Recognition [cs.CV], Image Segmentation, [INFO.INFO-AI]Computer Science [cs]/Artificial Intelligence [cs.AI], [INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing, Power Watersheds, [ INFO.INFO-TI ] Computer Science [cs]/Image Processing, 0202 electrical engineering, electronic engineering, information engineering, General Materials Science, [MATH]Mathematics [math], [ INFO.INFO-AI ] Computer Science [cs]/Artificial Intelligence [cs.AI], Cluster analysis, Discrete mathematics, Spanning tree, Spectral Clustering, Heuristic, Isoperimetric Partitioning, General Engineering, Graph partition, [INFO.INFO-CV]Computer Science [cs]/Computer Vision and Pattern Recognition [cs.CV], Approximation algorithm, [ INFO.INFO-DM ] Computer Science [cs]/Discrete Mathematics [cs.DM], Image segmentation, Graph, Spectral clustering, [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV], Graph (abstract data type), 020201 artificial intelligence & image processing, lcsh:Electrical engineering. Electronics. Nuclear engineering, Isoperimetric inequality, lcsh:TK1-9971
Abstract

Isoperimetric graph partitioning, which is also known as the Cheeger cut, is NP-hard in its original form. In the literature, multiple modifications to this problem have been proposed to obtain approximation algorithms for clustering applications. In the context of image segmentation, a heuristic continuous relaxation to this problem introduced by Leo Grady and Eric L. Schwartz has yielded good quality results. This algorithm is based on solving a linear system of equations involving the Laplacian of the image graph. Furthermore, the same algorithm applied to a maximum spanning tree (MST) of the image graph was shown to produce similar results at a much lesser computational cost. However, the data reduction step (i.e., considering an MST, a much sparser graph compared to the original graph) leading to a faster yet useful algorithm has not been analyzed. In this paper, we revisit the isoperimetric graph partitioning problem and rectify a few discrepancies in the simplifications of the heuristic continuous relaxation, leading to a better interpretation of what is really done by this algorithm. We then use the power watershed (PW) framework to show that is enough to solve the relaxed isoperimetric graph partitioning problem on the graph induced by the union of MSTs (UMST) with a seed constraint. The UMST has a lesser number of edges compared to the original graph, thus improving the speed of sparse matrix multiplication. Furthermore, given the interest of PW framework in solving the relaxed seeded isoperimetric partitioning problem, we discuss the links between the PW limit of the discrete isoperimetric graph partitioning and watershed cuts. We then illustrate with experiments, a detailed comparison of solutions to the relaxed seeded isoperimetric partitioning problem on the original graph with the ones on the UMST and an MST. Our study opens many research directions, which are discussed in the conclusion section.

Published
2019
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4. RNLp: Mixing Nonlocal and TV-Lp Methods to Remove Impulse Noise from Images

Author

Agathe Viano, Julie Delon, Agnès Desolneux, Camille Sutour, 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), Université Paris Descartes - Paris 5 (UPD5), Centre National de la Recherche Scientifique (CNRS), Centre de Mathématiques et de Leurs Applications (CMLA), École normale supérieure - Cachan (ENS Cachan)-Centre National de la Recherche Scientifique (CNRS), Deloitte Consulting [Diegem], ANR-14-CE27-0019,MIRIAM,Restauration Multi-Images: des Mathématiques Appliqueés à l'Industrie de l'Imagerie.(2014), Mathématiques Appliquées à Paris 5 ( MAP5 - UMR 8145 ), Université Paris Descartes - Paris 5 ( UPD5 ) -Institut National des Sciences Mathématiques et de leurs Interactions-Centre National de la Recherche Scientifique ( CNRS ), Centre National de la Recherche Scientifique ( CNRS ), Centre de Mathématiques et de Leurs Applications ( CMLA ), École normale supérieure - Cachan ( ENS Cachan ) -Centre National de la Recherche Scientifique ( CNRS ), and Deloitte

Subjects
[ MATH ] Mathematics [math], Statistics and Probability, Mathematical optimization, convex optimization, Noise reduction, variational methods, 02 engineering and technology, Impulse noise, 0202 electrical engineering, electronic engineering, information engineering, [MATH]Mathematics [math], Mathematics, patch-based methods, Pixel, Applied Mathematics, Detector, Total variation denoising, Condensed Matter Physics, Term (time), impulse noise, Noise, Computer Science::Computer Vision and Pattern Recognition, Modeling and Simulation, Image denoising, 020201 artificial intelligence & image processing, Geometry and Topology, Computer Vision and Pattern Recognition, Algorithm, Energy (signal processing)
Abstract

We propose a new variational framework to remove random-valued impulse noise from images. This framework combines, in the same energy, a nonlocal $$L^p$$ data term and a total variation regularization term. The nonlocal $$L^p$$ term is a weighted $$L^p$$ distance between pixels, where the weights depend on a robust distance between patches centered at the pixels. In a first part, we study the theoretical properties of the proposed energy, and we show how it is related to classical denoising models for extreme choices of the parameters. In a second part, after having explained how to numerically find a minimizer of the energy thanks to primal-dual approaches, we show extensive denoising experiments on various images and noise intensities. The denoising performance of the proposed methods is on par with state-of-the-art approaches, and the remarkable fact is that, unlike other successful variational approaches for impulse noise removal, they do not rely on a noise detector.

Published
2018
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5. Modeling Temporal Dominance of Sensations with semi-Markov chains

Author

Hervé Cardot, G. Lecuelle, Michel Visalli, Pascal Schlich, Centre des Sciences du Goût et de l'Alimentation [Dijon] (CSGA), Centre National de la Recherche Scientifique (CNRS)-AgroSup Dijon - Institut National Supérieur des Sciences Agronomiques, de l'Alimentation et de l'Environnement-Institut National de la Recherche Agronomique (INRA)-Université de Bourgogne (UB), Université Bourgogne Franche-Comté [COMUE] (UBFC), Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Projet : Thèse Guillaume Lecuelle, Institut National de la Recherche Agronomique (INRA)-Université de Bourgogne (UB)-AgroSup Dijon - Institut National Supérieur des Sciences Agronomiques, de l'Alimentation et de l'Environnement-Centre National de la Recherche Scientifique (CNRS), Université de Bourgogne (UB)-Centre National de la Recherche Scientifique (CNRS), Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Centre des Sciences du Goût et de l'Alimentation [Dijon] ( CSGA ), Institut National de la Recherche Agronomique ( INRA ) -Université de Bourgogne ( UB ) -AgroSup Dijon - Institut National Supérieur des Sciences Agronomiques, de l'Alimentation et de l'Environnement-Centre National de la Recherche Scientifique ( CNRS ), Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), and Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS )

Subjects
[ MATH ] Mathematics [math], 0301 basic medicine, 030109 nutrition & dietetics, Nutrition and Dietetics, Current (mathematics), Markov chain, Generalization, Stochastic process, [ SDV.AEN ] Life Sciences [q-bio]/Food and Nutrition, Linear model, Stochastic modeling, 03 medical and health sciences, Dominance (ethology), Discrete time and continuous time, Statistics, Selection (linguistics), Semi-Markov processes, Statistical physics, [MATH]Mathematics [math], Temporal Dominance of Sensations, [SDV.AEN]Life Sciences [q-bio]/Food and Nutrition, Maximum likelihood, Food Science, Mathematics
Abstract

International audience; Temporal Dominance of Sensations (TDS) data are usually represented by TDS curves of dominance rates and analyzed by linear models of dominance durations. Such approaches do not properly take into account the fact that the selection of a new dominant attribute likely depends on the current dominant attribute.Thus, modeling TDS data with a stochastic process seems natural, as recently proposed by Franczak et al. (2015) who used discrete time Markov chains. This approach gives the probabilities of transition from one dominant attribute to another. However Markov chains present some limitations when applied to TDS data. As an alternative, this paper considers semi-Markov chains (SMC), a generalization of Markov chains, which allow the duration of the dominant attribute to be distributed arbitrarily. Because probabilities of transition from one attribute to another one can also depend on time, SMC are applied on sequences split into time periods with specific durations, with one model per time period.Graphs built upon this stochastic pattern can be plotted to represent chronological main transitions between attributes. Contrarily to the TDS curves which summarize a mean panel overview, these graphs can be interpreted as individual’s most probable paths and contribute to a better understanding of consumer perception.

Published
2018
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6. Hiriart-Urruty–Phelps-Like Formula for the Subdifferential of Integral Sums

Author

Abderrahim Jourani, Abderrahim Hantoute, Centre de modélisation mathématique ( CMM ), UNIV CHILI-Centre National de la Recherche Scientifique ( CNRS ), Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), CONICYT. Grant number: Fondecyt 1151003, Conicyt-Redes. Grant number: 150040, Mathamsud. Grant number: 17-MATH-06, Centre de modélisation mathématique (CMM), Universitad de Chile-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Bourgogne [Dijon] (IMB), 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)-Université de Bourgogne (UB)

Subjects
[ MATH ] Mathematics [math], Mathematics::Functional Analysis, Pure mathematics, 021103 operations research, Convex normal integrands, General Mathematics, 010102 general mathematics, 0211 other engineering and technologies, Regular polygon, Subdifferential calculus, Integral sums, 02 engineering and technology, Subderivative, ε-subdifferential, Type (model theory), 01 natural sciences, MSC: 26B05, 26J25, 49H05, [MATH]Mathematics [math], 0101 mathematics, Mathematics
Abstract

International audience; We provide subdifferential calculus rules for continuous sums parametrized in measurable spaces that use the approximate subdifferentials of the data functions. As in Hiriart-Urruty and Phelps (J. Funct. Anal. 118: 154–166, 1993) given for finite sums, the resulting formulas hold without any conditions of continuity type on the involved functions. All this analysis is done in the setting of locally convex Suslin spaces.

Published
2018
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7. Discrete kinetic theory for 2D modeling of a moving crowd: Application to the evacuation of a non-connected bounded domain

Author

Pierre Argoul, M. El Rhabi, A. Elmoussaoui, Abdelilah Hakim, Laboratoire de Mathématiques Appliquées, Faculté des Sciences et Techniques, Université Cadi Ayyad ( LAMAI ), Sécurité et Durabilité des Ouvrages d'Art ( IFSTTAR/MAST/SDOA ), Institut Français des Sciences et Technologies des Transports, de l'Aménagement et des Réseaux ( IFSTTAR ) -Communauté Université Paris-Est, Département Ingénierie Mathématique et Informatique, Ecole des Ponts ParisTech ( IMI ), Université Cadi Ayyad [Marrakech] (UCA), Département Matériaux et Structures (IFSTTAR/MAST), Institut Français des Sciences et Technologies des Transports, de l'Aménagement et des Réseaux (IFSTTAR)-PRES Université Paris-Est, Laboratoire de Mathématiques Appliquées, Faculté des Sciences et Techniques, Université Cadi Ayyad (LAMAI), Sécurité et Durabilité des Ouvrages d'Art (IFSTTAR/MAST/SDOA), Institut Français des Sciences et Technologies des Transports, de l'Aménagement et des Réseaux (IFSTTAR)-Communauté Université Paris-Est, Département Ingénierie Mathématique et Informatique, École des Ponts ParisTech (IMI), and École des Ponts ParisTech (ENPC)

Subjects
[ MATH ] Mathematics [math], Crowd dynamics, [ INFO.INFO-MO ] Computer Science [cs]/Modeling and Simulation, Mathematical optimization, [ INFO ] Computer Science [cs], Complex system, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], SYSTEME DISCRET, 010103 numerical & computational mathematics, Topology, 01 natural sciences, DYNAMIQUE, Domain (mathematical analysis), PIETON, FOULE, Position (vector), SYSTEME COMPLEXE, [INFO]Computer Science [cs], Uniqueness, [MATH]Mathematics [math], 0101 mathematics, Mathematics, MODELE MATHEMATIQUE, Discrete kinetic theory, Stochastic game, Perspective (graphical), Probabilistic logic, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, 010101 applied mathematics, Splitting scheme, Computational Mathematics, Computational Theory and Mathematics, Modeling and Simulation, Obstacle, Bounded function, Evacuation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], MOUVEMENT
Abstract

International audience; This paper concerns the mathematical modeling of the motion of a crowd in a non connected bounded domain, based on kinetic and stochastic game theories. The proposed model is a mesoscopic probabilistic approach that retains features obtained from both micro-and macro-scale representations; pedestrian interactions with various obstacles being managed from a probabilistic perspective. A proof of the existence and uniqueness of the proposed mathematical model's solution is given for large times. A numerical resolution scheme based on the splitting method is implemented and then applied to crowd evacuation in a non connected bounded domain with one rectangular obstacle. The evacuation time of the room is then calculated by our technique, according to the dimensions and position of a square-shaped obstacle, and finally compared to the time obtained by a deterministic approach by means of randomly varying some of its parameters.

Published
2018
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8. Lenses on very curved zones of a singular foliation of C2

Author

Jean-Claude Sifre, Rémi Langevin, Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Mathématiques de Bourgogne [Dijon] (IMB), 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)-Université de Bourgogne (UB)

Subjects
Isolated singularity, [ MATH ] Mathematics [math], Complex curve, Polynomial, Pure mathematics, 010102 general mathematics, Holomorphic function, Curvature, 01 natural sciences, Complex foliation, Graph, MSC: 14H20, 14B05, 53C65, 53C12, 0103 physical sciences, Foliation (geology), Profile, 010307 mathematical physics, Geometry and Topology, [MATH]Mathematics [math], 0101 mathematics, Mathematics
Abstract

We renormalize, using suitable lenses, small domains of a singular holomorphic foliation of C 2 where the curvature is concentrated. At a proper scale, the leaves are almost translates of a graph that we will call profile. When the leaves of the foliations are levels f = λ , where f is a polynomial in 2 variables, this graph is polynomial. Finally we will indicate how our methods may be adapted to study levels of polynomials and 1-forms in C 3 .

Published
2018
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9. The pure descent statistic on permutations

Author

Jean-Luc Baril, Sergey Kirgizov, Laboratoire d'Electronique, d'Informatique et d'Image UMR CNRS 6306 ( Le2i ), Université de Technologie de Belfort-Montbeliard ( UTBM ) -Centre National de la Recherche Scientifique ( CNRS ) -École Nationale Supérieure d'Arts et Métiers ( ENSAM ) -Université de Bourgogne ( UB ) -AgroSup Dijon - Institut National Supérieur des Sciences Agronomiques, de l'Alimentation et de l'Environnement, Laboratoire d'Electronique, d'Informatique et d'Image [EA 7508] (Le2i), Université de Technologie de Belfort-Montbeliard (UTBM)-Université de Bourgogne (UB)-École Nationale Supérieure d'Arts et Métiers (ENSAM), Arts et Métiers Sciences et Technologies, HESAM Université (HESAM)-HESAM Université (HESAM)-Arts et Métiers Sciences et Technologies, and HESAM Université (HESAM)-HESAM Université (HESAM)-AgroSup Dijon - Institut National Supérieur des Sciences Agronomiques, de l'Alimentation et de l'Environnement-Centre National de la Recherche Scientifique (CNRS)

Subjects
[ MATH ] Mathematics [math], Golomb–Dickman constant, Distribution (number theory), Permutation, Stirling numbers of the first kind, Stirling number, 0102 computer and information sciences, 01 natural sciences, Theoretical Computer Science, Combinatorics, ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, Discrete Mathematics and Combinatorics, [MATH]Mathematics [math], 0101 mathematics, Patterns, Statistic, Mathematics, Discrete mathematics, Mathematics::Combinatorics, 010102 general mathematics, Descent, Parity of a permutation, Gray Code, 010201 computation theory & mathematics, Random permutation statistics, Dyck path, Popularity Fixed Number
Abstract

International audience; We introduce a new statistic based on permutation descents which has a distribution given by the Stirling numbers of the first kind, i.e., with the same distribution as for the number of cycles in permutations. We study this statistic on the sets of permutations avoiding one pattern of length three by giving bivariate generating functions. As a consequence, new classes of permutations enumerated by the Motzkin numbers are obtained. Finally, we deduce results about the popularity of the pure descents in all these restricted sets. (C) 2017 Elsevier B.V. All rights reserved.

Published
2017
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10. Rational integrability of trigonometric polynomial potentials on the flat torus

Author

Thierry Combot, Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Mathématiques de Bourgogne [Dijon] (IMB), 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)-Université de Bourgogne (UB)

Subjects
[ MATH ] Mathematics [math], Pure mathematics, Integrable system, Trigonometric polynomials, Dynamical Systems (math.DS), 010103 numerical & computational mathematics, Differential Galois theory, Integrability, 37J30, Trigonometric polynomial, 01 natural sciences, symbols.namesake, Mathematics (miscellaneous), Lattice (order), FOS: Mathematics, [MATH]Mathematics [math], Mathematics - Dynamical Systems, 0101 mathematics, Flat torus, Mathematical Physics, Mathematics, Mechanical Engineering, Applied Mathematics, 010102 general mathematics, First integrals, Statistical and Nonlinear Physics, Fourier transform, Modeling and Simulation, MSC: 37J30, symbols, Trigonometry, Toda lattice
Abstract

We consider a lattice $\mathcal{L}\subset \mathbb{R}^n$ and a trigonometric potential $V$ with frequencies $k\in\mathcal{L}$. We then prove a strong integrability condition on $V$, using the support of its Fourrier transform. We then use this condition to prove that a real trigonometric polynomial potential is integrable if and only if it separates up to rotation of the coordinates. Removing the real condition, we also make a classification of integrable potentials in dimension $2$ and $3$, and recover several integrable cases. These potentials after a complex variable change become real, and correspond to generalized Toda integrable potentials. Moreover, along the proof, some of them with high degree first integrals are explicitly integrated., Comment: 29 pages

Published
2017
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11. The walk on moving spheres: A new tool for simulating Brownian motion’s exit time from a domain

Author

Sylvain Maire, Samuel Herrmann, Madalina Deaconu, TO Simulate and CAlibrate stochastic models (TOSCA), Inria Sophia Antipolis - Méditerranée (CRISAM), 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), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Laboratoire Modélisation et Signal (LMS), Université de Toulon (UTLN)-ISITV, Probabilités et statistiques, 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 ), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique ( Inria ), Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire des Sciences de l'Information et des Systèmes ( LSIS ), Aix Marseille Université ( AMU ) -Université de Toulon ( UTLN ) -Arts et Métiers Paristech ENSAM Aix-en-Provence-Centre National de la Recherche Scientifique ( CNRS ), Université de Bourgogne (UB)-Centre National de la Recherche Scientifique (CNRS), TO Simulate and CAlibrate stochastic models ( TOSCA ), Inria Sophia Antipolis - Méditerranée ( CRISAM ), 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 ), Laboratoire Modélisation et Signal ( LMS ), Université de Toulon ( UTLN ) -ISITV, and TOSCA

Subjects
[ MATH ] Mathematics [math], [ INFO ] Computer Science [cs], Brownian hitting time, General Computer Science, Bessel process, Boundary (topology), hitting time, 01 natural sciences, Bessel processes, Theoretical Computer Science, 010104 statistics & probability, symbols.namesake, Position (vector), FOS: Mathematics, 0101 mathematics, Brownian motion, Mathematics, Numerical Analysis, Series (mathematics), Boundary, Applied Mathematics, Probability (math.PR), Mathematical analysis, Hitting time, 65C20, 65C05, Algorithm, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010101 applied mathematics, Distribution (mathematics), Walk on moving spheres method, Modeling and Simulation, 1st-Passage Density, symbols, numerical algorithm, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR], Mathematics - Probability, Bessel function
Abstract

International audience; In this paper we introduce a new method for the simulation of the exit time and exit position of a (delta-dimensional Brownian motion from a domain. The main interest of our method is that it avoids splitting time schemes as well as inversion of complicated series. The method, called walk on moving spheres algorithm, was first introduced for hitting times of Bessel processes. In this study this method is adapted and developed for the first time for the Brownian motion hitting times. The idea is to use the connexion between the (delta-dimensional Bessel process and the (delta-dimensional Brownian motion thanks to an explicit Bessel hitting time distribution associated with a particular curved boundary. This allows to build a fast and accurate numerical scheme for approximating the hitting time. We introduce also an overview of existing methods for the simulation of the Brownian hitting time and perform numerical comparisons with existing methods. (C) 2015 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

Published
2017
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12. Darboux systems with a cusp point and pseudo-abelian integrals

Author

Aymen Braghtha, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Braghtha, Aymen, Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), and Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS )

Subjects
[ MATH ] Mathematics [math], Cusp (singularity), Pure mathematics, Polynomial, Applied Mathematics, [ MATH.MATH-DS ] Mathematics [math]/Dynamical Systems [math.DS], 010102 general mathematics, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Darboux integrability, [MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS], Pseudo-abelian integrals, [MATH] Mathematics [math], 01 natural sciences, 010101 applied mathematics, Limit cycles, Singularity, Uniform boundedness, Point (geometry), First integral, 0101 mathematics, Abelian group, MSC : 34C07, 34C08, [MATH]Mathematics [math], Analysis, Mathematics
Abstract

International audience; We study pseudo-abelian integrals associated with polynomial deformations of Darboux systems having a cuspidal singularity. Under some genericity hypothesis we provide locally uniform boundedness of on the number of their zeros.

Published
2020
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13. On the stability of the Schwartz class under the magnetic Schrödinger flow

Author

Boil, Grégory, Raymond, Nicolas, Vu Ngoc, San, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, 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)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), 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 Agrocampus Ouest, and 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)

Subjects
[ MATH ] Mathematics [math], [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Analysis of PDEs, General Mathematics, Mathematics::Analysis of PDEs, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics::Spectral Theory, [MATH]Mathematics [math], ComputingMilieux_MISCELLANEOUS
Abstract

We prove that the Schwartz class is stable under the magnetic Schr\"odinger flow when the magnetic $2$-form is non-degenerate and does not oscillate too much at infinity.

Published
2020

14. NULL CONTROLLABILITY OF SALMONELLA SPREAD WITHIN AN INDUSTRIAL HEN HOUSE

Author

Mophou, Gisèle, Zongo, Pascal, Dorville, René, Dorville, René, Ecologie des forêts de Guyane ( ECOFOG ), Institut National de la Recherche Agronomique ( INRA ) -Université des Antilles ( UA ) -Centre de Coopération Internationale en Recherche Agronomique pour le Développement ( CIRAD ) -AgroParisTech-Université de Guyane ( UG ) -Centre National de la Recherche Scientifique ( CNRS ), Université des Antilles (UA), Laboratoire de Mathématiques Informatique et Applications (LAMIA), Laboratoire des matériaux et molécules en milieu Agressif [EA 7526] (L3MA), Ecologie des forêts de Guyane (UMR ECOFOG), and Université des Antilles (UA)-Université de Guyane (UG)-Centre National de la Recherche Scientifique (CNRS)-AgroParisTech-Centre de Coopération Internationale en Recherche Agronomique pour le Développement (Cirad)-Institut National de la Recherche Agronomique (INRA)

Subjects
Controllability, [ MATH ] Mathematics [math], [ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC], Heat equation, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH] Mathematics [math], [MATH]Mathematics [math], Carleman inequalities, Optimal control
Abstract

International audience; The main purpose of this paper is to propose a deterministic mathematical model with control variable which describes the spatiotemporal spread of \emph{Salmonella} within a laying flock. This model assume an indirect transmission of the disease within the flock through the bacteria density in the environment. We provide a mathematical analyze of the null controllability of the model: we begin to construct a family of solutions to the linear null-controllability problem using the the Carleman inequality. Then, by a generalized Leray-Shauder fixed point theorem, we show that one can bring the system to rest at final time T.

Published
2020

15. Stochastic modeling for biotechnologies Anaerobic model AM2b

Author

Campillo , Fabien, Chebbi , Mohsen, Toumi , Salwa, Mathématiques pour les Neurosciences (MATHNEURO), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Ecole Nationale d'Ingénieurs de Tunis (ENIT), Université de Tunis El Manar (UTM), and Institut National des Sciences Appliquées et de Technologie - Carthage (INSAT Carthage)

Subjects
[ MATH ] Mathematics [math], pure jump process, modèle AM2b, [ SDV ] Life Sciences [q-bio], [SDV]Life Sciences [q-bio], diffusion approximation, approximation diffusion, ordinary differential equation, stochastic differential equation, AM2b model, [ CHIM ] Chemical Sciences, processus de saut pur, équation différentielle stochastique, équation différentielle ordinaire, [CHIM]Chemical Sciences, [MATH]Mathematics [math]
Abstract

Le modèle AM2b est classiquement représenté par un système d'équations différentielles. Toutefois ce modèle n'est valide qu'en grande population et notre objectif est d'établir plusieurs mo-dèles stochastiques à différentes échelles. À l'échelle microscopique, on propose un modèle sto-chastique de saut pur que l'on peut simuler de fa con exacte. Mais dans la plupart des situations ce genre de simulation n'est pas réaliste, et nous proposons des méthodes de simulation approchées de type poissonnien ou de type diffusif. La méthode de simulation de type diffusif peut être vue comme une discrétisation d'une équation différentielle stochastique. Nous présentons enfin de fa con infor-melle un résultat de type loi des grands nombres/théorème central limite fonctionnelle qui démontre la convergence de ses modèles stochastiques vers le modèles déterministe initial., The model AM2b is conventionally represented by a system of differential equations. However, this model is valid only in a large population context and our objective is to establish several stochastic models at different scales. At a microscopic scale, we propose a pure jump stochastic model that can be simulated exactly. But in most situations this exact simulation is not feasible, and we propose approximate simulation methods of Poisson type and of diffusive type. The diffusive type simulation method can be seen as a discretization of a stochastic differential equation. Finally, we formally present a result of law of large numbers and of functional central limit theorem which demonstrates the convergence of these stochastic models towards the initial deterministic models.

Published
2019

16. A coherent framework for learning spatiotemporal piecewise- geodesic trajectories from longitudinal manifold-valued data

Author

Chevallier, Juliette, Debavelaere, Vianney, Allassonnière, Stéphanie, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Centre de Recherche des Cordeliers (CRC), Université Paris Diderot - Paris 7 (UPD7)-École pratique des hautes études (EPHE)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Descartes - Paris 5 (UPD5)-Institut National de la Santé et de la Recherche Médicale (INSERM), Ce travail bénéficie d'un financement public Investissement d'avenir, réference ANR-11-LABX-0056-LMH. This work was supported by a public grant as part of the Investissement d'avenir, project reference ANR-11-LABX-0056-LMH., Travail réalisé dans le cadre d'un projet financé par la Fondation de la Recherche Médicale, 'DBI20131228564'. Work performed as a part of a project funded by the Fondation of Medical Research, grant number 'DBI20131228564'., Department of Medical Oncology, Hôpital Européen Georges Pompidou [APHP] (HEGP), Assistance publique - Hôpitaux de Paris (AP-HP) (APHP)-Hôpitaux Universitaires Paris Ouest - Hôpitaux Universitaires Île de France Ouest (HUPO)-Assistance publique - Hôpitaux de Paris (AP-HP) (APHP)-Hôpitaux Universitaires Paris Ouest - Hôpitaux Universitaires Île de France Ouest (HUPO), Centre de Mathématiques Appliquées - Ecole Polytechnique ( CMAP ), École polytechnique ( X ) -Centre National de la Recherche Scientifique ( CNRS ), Hôpital Européen Georges Pompidou [APHP] ( HEGP ), Centre de Recherche des Cordeliers ( CRC ), and Université Paris Diderot - Paris 7 ( UPD7 ) -École pratique des hautes études ( EPHE ) -Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Descartes - Paris 5 ( UPD5 ) -Institut National de la Santé et de la Recherche Médicale ( INSERM )

Subjects
[ MATH ] Mathematics [math], [ INFO.INFO-MO ] Computer Science [cs]/Modeling and Simulation, [STAT.AP]Statistics [stat]/Applications [stat.AP], Longitudinal data, Bayesian estimation, Spatio-temporal analysis, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, MCMC methods, [ STAT.ME ] Statistics [stat]/Methodology [stat.ME], [STAT.ML]Statistics [stat]/Machine Learning [stat.ML], [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], EM like algorithm, [ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST], [MATH]Mathematics [math], [STAT.ME]Statistics [stat]/Methodology [stat.ME], Nonlinear mixed-effects model
Abstract

This paper provides a coherent framework for studying longitudinal manifold-valued data. We introduce a Bayesian mixed-effects model which allows to estimate both a group-representative piecewise-geodesic trajectory in the Riemannian space of shape and inter-individual variability. We prove the existence of the maximum a posteriori estimate and its asymptotic consistency under reasonable assumptions. Due to the non-linearity of the proposed model, we use a stochastic version of Expectation-Maximization algorithm to estimate the model parameters. Our simulations show that our model is not noise-sensitive and succeed in explaining various paths of progression.

Published
2019

17. The radical of the differential ideal generated by $XY$ in the ring of two variable differential polynomials is not differentially finitely generated

Author

Julien Sebag, David Bourqui, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, 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)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), 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 Agrocampus Ouest, 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), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), and Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS )

Subjects
Differential ideal, [ MATH ] Mathematics [math], Pure mathematics, Ring (mathematics), 14B05, Mathematics::Commutative Algebra, 010102 general mathematics, 0102 computer and information sciences, Type (model theory), Differential algebra, 01 natural sciences, Differential field, curve singularity, 010201 computation theory & mathematics, Finitely-generated abelian group, 0101 mathematics, [MATH]Mathematics [math], 12H05, Differential (mathematics), ComputingMilieux_MISCELLANEOUS, Mathematics, Variable (mathematics)
Abstract

Let $(k,\delta )$ be a differential field containing $\mathbf {Q}$. We prove that, in general, a radical differential ideal of the differential ring $k\{X,Y\}$ is not a differential ideal of finite type.

Published
2019
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18. The tennis racket effect in a three-dimensional rigid body

Author

Pavao Mardešić, Dominique Sugny, Léo Van Damme, Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB), Université de Technologie de Belfort-Montbeliard (UTBM)-Université de Bourgogne (UB)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Bourgogne [Dijon] (IMB), Université de Bourgogne (UB)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Inst Adv Study, Tech Univ Munich, Technische Universitat Munchen Institute for Advanced Study - German Excellence Initiative, ANR-14-CE35-0013,Explosys,Exploring the physical limits of spin systems: A challenge in medical imaging(2014), European Project: 291763, European Union Seventh Framework Programme under grant agreement, Laboratoire Interdisciplinaire Carnot de Bourgogne [Dijon] ( LICB ), Université de Technologie de Belfort-Montbeliard ( UTBM ) -Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), ANR-14-CE35-0013,Explosys,Exploring the physical limits of spin systems: A challenge in medical imaging ( 2014 ), European Project : 291763, European Union Seventh Framework Programme under grant agreement, Laboratoire Interdisciplinaire Carnot de Bourgogne [Dijon] (LICB), Université de Bourgogne (UB)-Université de Technologie de Belfort-Montbeliard (UTBM)-Centre National de la Recherche Scientifique (CNRS), 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)-Université de Bourgogne (UB)

Subjects
[ MATH ] Mathematics [math], media_common.quotation_subject, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Euler angles, FOS: Physical sciences, Physics - Classical Physics, Inertia, Rotation, 01 natural sciences, 010305 fluids & plasmas, symbols.namesake, Simple (abstract algebra), 0103 physical sciences, Racket, Classical mechanics, [MATH]Mathematics [math], 010306 general physics, media_common, Mathematics, computer.programming_language, [PHYS]Physics [physics], [ PHYS ] Physics [physics], Dynamics (mechanics), Classical Physics (physics.class-ph), Statistical and Nonlinear Physics, Moment of inertia, Condensed Matter Physics, Rigid body, Geometric effect, symbols, [ PHYS.MPHY ] Physics [physics]/Mathematical Physics [math-ph], computer
Abstract

We propose a complete theoretical description of the tennis racket effect, which occurs in the free rotation of a three-dimensional rigid body. This effect is characterized by a flip ($\pi$- rotation) of the head of the racket when a full ($2\pi$) rotation around the unstable inertia axis is considered. We describe the asymptotics of the phenomenon and conclude about the robustness of this effect with respect to the values of the moments of inertia and the initial conditions of the dynamics. This shows the generality of this geometric property which can be found in a variety of rigid bodies. A simple analytical formula is derived to estimate the twisting effect in the general case. Different examples are discussed., Comment: 20 pages, 11 figures, submitted to Physica D (2016)

Published
2017
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19. Puits magnétiques en dimension trois

Author

Yuri A. Kordyukov, Nicolas Raymond, San Vu Ngoc, Bernard Helffer, Laboratoire de Mathématiques Jean Leray ( LMJL ), Université de Nantes ( UN ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques d'Orsay ( LM-Orsay ), Université Paris-Sud - Paris 11 ( UP11 ) -Centre National de la Recherche Scientifique ( CNRS ), Institute of Mathematics, Russian Academy of Sciences, Ufa, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, 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)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)

Subjects
[ MATH ] Mathematics [math], 70H15, FOS: Physical sciences, Semiclassical physics, 01 natural sciences, Mathematics - Spectral Theory, FOS: Mathematics, 81Q20, Birkhoff normal forms, [MATH]Mathematics [math], 0101 mathematics, Spectral Theory (math.SP), Mathematical Physics, Eigenvalues and eigenvectors, Mathematical physics, Physics, Numerical Analysis, Applied Mathematics, Operator (physics), 010102 general mathematics, Degenerate energy levels, Magnetic confinement fusion, 37G05, Mathematical Physics (math-ph), Mathematics::Spectral Theory, 16. Peace & justice, Magnetic field, 010101 applied mathematics, microlocal analysis, Magnetic fields, 81Q20, 35P15, 37G05, 70H15, Asymptotic expansion, Laplace operator, Analysis, 35P15
Abstract

International audience; This paper deals with semiclassical asymptotics of the three-dimensional magnetic Laplacian in presence of magnetic confinement. Using generic assumptions on the geometry of the confinement, we exhibit three semiclassical scales and their corresponding effective quantum Hamiltonians, by means of three microlocal normal forms à la Birkhoff. As a consequence, when the magnetic field admits a unique and non degenerate minimum, we are able to reduce the spectral analysis of the low-lying eigenvalues to a one-dimensional pseudo-differential operator whose Weyl's symbol admits an asymptotic expansion in powers of the square root of the semiclassical parameter.

Published
2016
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20. Solutions of the Ablowitz–Kaup–Newell–Segur hierarchy equations of the 'rogue wave' type: A unified approach

Author

Vladimir V. Matveev, A. O. Smirnov, Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Saint-Petersburg State University of Aerospace Instrumentation ( SUAI ), Russian Foundation for Basic Research. Grant number: 14-01-00589_a, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), and Saint-Petersburg State University of Aerospace Instrumentation (SUAI)

Subjects
[ MATH ] Mathematics [math], AKNS hierarchy, Integrable system, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Structure (category theory), Type (model theory), 01 natural sciences, 010305 fluids & plasmas, symbols.namesake, 0103 physical sciences, Nonlinear Schrödinger equation, [MATH]Mathematics [math], Rogue wave, 010306 general physics, Mathematical Physics, Mathematical physics, Mathematics, [PHYS]Physics [physics], [ PHYS ] Physics [physics], Hierarchy (mathematics), Nonlinear optics, Statistical and Nonlinear Physics, Freak wave, Nonlinear Sciences::Exactly Solvable and Integrable Systems, Hirota equation, symbols, [ PHYS.MPHY ] Physics [physics]/Mathematical Physics [math-ph]
Abstract

International audience; We describe a unified structure of solutions for all equations of the Ablowitz–Kaup–Newell–Segur hierarchy and their combinations. We give examples of solutions that satisfy different equations for different parameter values. In particular, we consider a rank-2 quasirational solution that can be used to investigate many integrable models in nonlinear optics. An advantage of our approach is the possibility to investigate changes in the behavior of a solution resulting from changing the model.

Published
2016
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21. The Zeta Mahler measure of $(z^n − 1)/(z − 1)$

Author

Arunabha Biswas and M. Ram Murty

Subjects
[ MATH ] Mathematics [math], Sequence, Polynomial, Conjecture, Mathematics::Number Theory, Laurent polynomial, Generating function, higher Mahler measure 2010 Mathematics Subject Classification 11R06, Mathematics::Geometric Topology, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Combinatorics, Unit circle, Mahler measure, Computer Science::Symbolic Computation, 11M99, Mathematics
Abstract

We consider the k-higher Mahler measure $m_k (P) $ of a Laurent polynomial $P$ as the integral of ${\log}^k |P | $ over the complex unit circle and zeta Mahler measure as the generating function of the sequence ${m_k (P)}$. In this paper we derive a few properties of the zeta Mahler measure of the polynomial $P_n (z) := (z^N − 1)/(z − 1) $ and propose a conjecture.

Published
2019
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22. Set Equidistribution of subsets of (Z/nZ) *

Author

Veekesh Kumar, Jaitra Chattopadhyay, Ravindranathan Thangadurai, and Harish-Chandra Research Institute

Subjects
[ MATH ] Mathematics [math], Class (set theory), Modulo, Mathematics::Number Theory, Cyclic group, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Combinatorics, Set (abstract data type), [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Order (group theory), Coset, [MATH]Mathematics [math], Mathematics, Complement (set theory)
Abstract

In 2010, Murty and Thangadurai [MuTh10] provided a criterion for the set equidistribution of residue classes of subgroups in (Z/nZ) *. In this article, using similar methods, we study set equidistribution for some class of subsets of (Z/nZ) *. In particular, we study the set equidistribution modulo 1 of cosets, complement of subgroups of the cyclic group (Z/nZ) * and the subset of elements of fixed order, whenever the size of the subset is sufficiently large.

Published
2019

23. A note on some congruences involving arithmetic functions

Author

József Sándor

Subjects
[ MATH ] Mathematics [math], Divisor function, 11A07, Euler's totient function, Function (mathematics), Congruence relation, 11D45, Euler's totient, Combinatorics, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], symbols.namesake, congruences 2010 Mathematics Subject Classification 11A25, 11N05, primality, symbols, Congruence (manifolds), Arithmetic function, Dual polyhedron, Dedekind cut, Dedekind's arithmetical function, number of divisors, Mathematics
Abstract

We consider some congruences involving arithmetical functions. For example, we study the congruences nψ(n) ≡ 2 (mod ϕ(n)), nϕ(n) ≡ 2 (mod ψ(n)), ψ(n)d(n) − 2 ≡ 0 (mod n), where ϕ(n), ψ(n), d(n) denote Euler's totient, Dedekind's function, and the number of divisors of n, respectively. Two duals of the Lehmer congruence n − 1 ≡ 0 (mod ϕ(n)) are also considered.

Published
2019
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24. On the Wintner-Ingham-Segal summability method

Author

Yoshio Tanigawa, Shigeru Kanemitsu, T Kuzumaki, Kyushu Institute of Technology, Faculty of Engineering, Gifu University, Graduate School of Mathematics [Nagoya], and Nagoya University

Subjects
[ MATH ] Mathematics [math], Pure mathematics, Stieltjes integral, Mathematics::Classical Analysis and ODEs, Riemann–Stieltjes integral, (Ingham) summability, Term (logic), [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Mathematics::K-Theory and Homology, Subject (grammar), weak Riemann hypothesis 2010 Mathematics Subject Classification 40D99, [MATH]Mathematics [math], 11M99, zero-free region, Mathematics, Prime number theorem
Abstract

The aim of this note is to establish a subclass of $\mathcal{F}$ considered by Segal if functions for which the Ingham-Wintner summability implies $\mathcal{F}$-summability as wide as possible. The subclass is subject to the estimate for the error term of the prime number theorem. We shall make good use of Stieltjes integration which elucidates previous results obtained by Segal.

Published
2019
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25. On-Regular Bipartitions Modulo $m$

Author

K N Vidya and D D Somashekara

Subjects
[ MATH ] Mathematics [math], Modulo, regular partition, Congruence relation, regular bipartition, partition, Combinatorics, Congruence, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], 05A17, theta functions 2010 Mathematics Subject Classification 11P83, Partition (number theory), Congruence (manifolds), Mathematics
Abstract

Let $b_l (n)$ denote the number of $l$-regular partitions of $n$ and $B_l (n)$ denote the number of $l$-regular bipartitions of $n$. In this paper, we establish several infinite families of congruences satisfied by $B_l (n)$ for $l \in {2, 4, 7}$. We also establish a relation between $b_9 (2n)$ and $B_3 (n)$.

Published
2019
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26. Hybrid level aspect subconvexity for GL(2) × GL(1) Rankin-Selberg L-Functions

Author

Yeongseong Jo, Kevin Nowland, and Keshav Aggarwal

Subjects
[ MATH ] Mathematics [math], Coprime integers, Mathematics::Number Theory, 010102 general mathematics, Prime number, Holomorphic function, 11F67, Square-free integer, 01 natural sciences, Cusp form, Dirichlet character, Combinatorics, 11L05, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Special values of L-functions, 0103 physical sciences, δ-method 2010 Mathematics Subject Classification 11F11, Second moment method, 010307 mathematical physics, 0101 mathematics, subconvexity, Mathematics, Rankin-Selberg convolution
Abstract

Let $M$ be a squarefree positive integer and $P$ a prime number coprime to $M$ such that $P \sim M^{\eta}$ with $0 < \eta < 2/5$. We simplify the proof of subconvexity bounds for $L(\frac{1]{2}, f \otimes \chi)$ when $f$ is a primitive holomorphic cusp form of level $P$ and $\chi$ is a primitive Dirichlet character modulo $M$. These bounds are attained through an unamplified second moment method using a modified version of the delta method due to R. Munshi. The technique is similar to that used by Duke-Friedlander-Iwaniec save for the modification of the delta method.

Published
2019
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27. Integral points on circles

Author

Mariusz Skałba, Andrzej Schinzel, Institut of Mathematics - Polish Academy of Sciences (PAN), Polska Akademia Nauk = Polish Academy of Sciences (PAN), and University of Warsaw (UW)

Subjects
[ MATH ] Mathematics [math], [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Combinatorics, 11D09, Fermat's theorem on sums of two squares, Gaussian integers 2010 Mathematics Subject Classification 11D25, Square-free integer, Radius, [MATH]Mathematics [math], sums of two squares, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Mathematics
Abstract

Sixty years ago the first named author gave an example \cite{sch} of a circle passing through an arbitrary number of integral points. Now we shall prove: {\it The number $N$ of integral points on the circle $(x-a)^2+(y-b)^2=r^2$ with radius $r=\frac{1}{n}\sqrt{m}$, where $m,n\in\mathbb Z$, $m,n>0$, $\gcd(m,n^2)$ squarefree and $a,b\in\mathbb Q$ does not exceed $r(m)/4$, where $r(m)$ is the number of representations of $m$ as the sum of two squares, unless $n|2$ and $n\cdot (a,b)\in\mathbb Z^2$; then $ N\leq r(m)$}.

Published
2019

28. Two applications of number theory to discrete tomography

Author

Robert Tijdeman

Subjects
Structure (mathematical logic), [ MATH ] Mathematics [math], 15A06, business.industry, Discrete tomography, Too quickly, Object (computer science), switching components 2010 Mathematics Subject Classification 94A08, sums of two squares, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Number theory, Computer vision, Artificial intelligence, Tomography, business, Mathematics
Abstract

Tomography is the theory behind scans, e.g. MRI-scans. Most common is continuous tomography where an object is reconstructed from numerous projections. In some cases this is not applicable, because the object changes too quickly or is damaged by making hundreds of projections (by X-rays). In such cases discrete tomography may apply where only few projections are made. The present paper shows how number theory helps to provide insight in the application and structure of discrete tomography.

Published
2019
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29. A Theorem of Fermat on Congruent Number Curves

Author

Lorenz Halbeisen and Norbert Hungerbühler

Subjects
Congruent numbers, [ MATH ] Mathematics [math], Fermat's Last Theorem, 2010 Mathematics Subject Classification. primary 11G05, secondary 11D25, Mathematics - Number Theory, Mathematics::General Mathematics, Mathematics::Number Theory, 010102 general mathematics, Order (ring theory), 010103 numerical & computational mathematics, Pythagorean triple, 01 natural sciences, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Combinatorics, 11G05, 11D25, Integer, Rational point, Elementary proof, FOS: Mathematics, Mathematics::Metric Geometry, Number Theory (math.NT), 0101 mathematics, Mathematics, Congruent number
Abstract

A positive integer $A$ is called a congruent number if $A$ is the area of a right-angled triangle with three rational sides. Equivalently, $A$ is a congruent number if and only if the congruent number curve $y^2=x^3-A^2x$ has a rational point $(x,y)\in\mathbb Q^2$ with $y\neq 0$. Using a theorem of Fermat, we give an elementary proof for the fact that congruent number curves do not contain rational points of finite order., Comment: 8 pages, one figure

Published
2019
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30. On Pillai's problem with Pell numbers and powers of 2

Author

Alain Togbé, Florian Luca, Mohand Ouamar Hernane, and Salah Eddine Rihane

Subjects
010101 applied mathematics, Discrete mathematics, reduction method 2010 Mathematics Subject Classification primary 11B39, secondary 11J86, [ MATH ] Mathematics [math], [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], 010102 general mathematics, 0101 mathematics, Pell numbers, 01 natural sciences, Mathematics, Pell number, Linear form in logarithms
Abstract

In this paper, we find all integers c having at least two representations as a difference between a Pell number and a power of 2.

Published
2019
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31. An elementary property of correlations

Author

Giovanni Coppola, Università degli Studi di Salermo, and Università degli Studi di Salerno (UNISA)

Subjects
[ MATH ] Mathematics [math], Pure mathematics, Mathematics::General Mathematics, Mathematics::Number Theory, Mathematics::Classical Analysis and ODEs, Lambda, Convolution, Ramanujan's sum, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], symbols.namesake, 11N37, 11N05, 11P32, 11N37, FOS: Mathematics, Number Theory (math.NT), [MATH]Mathematics [math], 11P32, Mathematics, Conjecture, Von Mangoldt function, Mathematics - Number Theory, Mathematics::History and Overview, Elementary property, shift Ramanujan expansion, 2k-twin primes 2010 Mathematics Subject Classification 11N05, Shift factor, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], correlation, symbols
Abstract

We study the shift-Ramanujan expansion (see 1705.07193) of general $f,g$ satisfying Ramanujan Conjecture, in order to get formulae, for their shifted convolution sum, say $C_{f,g}(N,a)$, of length $N$ and shift $a$ (so, the Ramanujan expansion is with respect to a>0). We prove that, assuming Delange Hypothesis (DH) for the expansion, we get say Ramnujan exact explicit formula (R.e.e.f.). A noteworthy case, of course, is $f=g=\Lambda$, the von Mangoldt function, so $C_{\Lambda,\Lambda}(N,2k)$, for natural $k$, regards $2k-$twin primes; assuming $(DH)$ for them, we get (from corresponding R.e.e.f.) the proof, easily, of Hardy-Littlewood Conjecture for them., Comment: Assuming Delange Hypothesis(DH), we prove the "Ramanujan exact explicit formula" for $f,g$ correlation; for 2k-twin primes, assuming $(DH)$ we prove Hardy-Littlewood Conjecture

Published
2019

32. On certain sums over ordinates of zeta-zeros II

Author

Kristian Seip, Eero Saksman, Aleksandar Ivić, Andriy Bondarenko, Norwegian University of Science and Technology [Trondheim] (NTNU), Norwegian University of Science and Technology (NTNU), and Serbian Academy of Sciences and Arts (SASA)

Subjects
[ MATH ] Mathematics [math], Mathematics - Number Theory, Laurent series, Analytic continuation, Second moment of area, Function (mathematics), second moment, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Combinatorics, Riemann hypothesis, symbols.namesake, analytic continuation, 11M06, Critical line, symbols, FOS: Mathematics, Number Theory (math.NT), Laurent expansion, [MATH]Mathematics [math], 2010 Mathematics Subject Classification 11M06 Keywords Riemann zeta-function, Mathematics
Abstract

Let $\gamma$ denote the imaginary parts of complex zeros $\rho = \beta+i\gamma$ of $\zeta(s)$. The problem of analytic continuation of the function $G(s) := \sum\limits_{\gamma > 0}\gamma^{-s}$ to the left of the line $\Re s = -1$ is investigated, and its Laurent expansion at the pole $s=1$ is obtained. Estimates for the second moment on the critical line $\int_1^T|G(1/2+it)|^2\,dt$ are revisited. This paper is a continuation of work begun by the second author in 2001., Comment: 12 pages

Published
2019

33. A remark on cube-free numbers in Segal-Piatestki-Shapiro sequences

Author

Jean-Marc Deshouillers, Institut de Mathématiques de Bordeaux (IMB), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
[ MATH ] Mathematics [math], Segal-Piatetski-Shapiro sequences, media_common.quotation_subject, 010102 general mathematics, Cube (algebra), 010103 numerical & computational mathematics, Infinity, 01 natural sciences, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Combinatorics, 11N37, 11L07, discrepancy 2010 Mathematics Subject Classification 11B75, cube-free numbers, estimation of trigonometric sums, 0101 mathematics, [MATH]Mathematics [math], 11N56, 11L03, Mathematics, media_common
Abstract

Using a method due to G. J. Rieger, we show that for $1 < c < 2 $ one has, as $x$ tends to infinity $$\textrm{Card}{n \leq x : \lfloor{n^c}\rfloor} \ \textrm{ is cube-free} } = \frac{x}{\zeta(3)} + O (x^{ (c+1)/3} \log x)$$ , thus improving on a recent result by Zhang Min and Li Jinjiang.

Published
2019

34. Explicit abc-conjecture and its applications

Author

Chi Chim Kwok, Saranya G. Nair, and Tarlok Nath Shorey

Subjects
Discrete mathematics, [ MATH ] Mathematics [math], Factorial, Conjecture, factorials, abc conjecture, State (functional analysis), Astrophysics::Cosmology and Extragalactic Astrophysics, explicit conjecture, 11B25, 11Z05, Statistics::Computation, Primes, 11D41, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Product (mathematics), Diophantine equations 2010 Mathematics Subject Classification 11A41, abc-conjecture, High Energy Physics::Experiment, Mathematics, 11N13
Abstract

We state well-known abc-conjecture of Masser-Oesterlé and its explicit version, popularly known as the explicit abc-conjecture, due to Baker. Laishram and Shorey derived from the explicit abc-conjecture that (1.1) implies that $c < N^{1.75}$. We give a survey on improvements of this result and its consequences. Finally we prove that $c < N^{1.7}$ and apply this estimate on an equation related to a conjecture of Hickerson that a factorial is not a product of factorials non-trivially.

Published
2019
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35. Fluctuation of the primitive of Hardy's function

Author

Matti Jutila

Subjects
[ MATH ] Mathematics [math], Mathematics::Functional Analysis, S function, Riemann's zeta-function, Mathematics::Classical Analysis and ODEs, Function (mathematics), Statistics::Computation, Gibbs phenomenon, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], symbols.namesake, symbols, the Gibbs phenomenon 2010 Mathematics Subject Classification 11M06, Hardy's function, Mathematics, Mathematical physics
Abstract

We consider oscillatory properties of the primitive of Hardy's function including a certain Gibbs phenomenon due to M. A. Korolev.

Published
2019
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36. The Barban-Vehov Theorem in Arithmetic Progressions

Author

V. Kumar Murty

Subjects
[ MATH ] Mathematics [math], Mean square, Discrete mathematics, Sequence, Sieve (category theory), Upper and lower bounds, Brun-Titchmarsh theorem, 11N69, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Secondary 11B25, Arithmetic progression, Asymptotic formula, arithmetic progressions 2010 Mathematics Subject Classification Primary 11N37, Selberg's sieve, 11N13, 11N35, Brun–Titchmarsh theorem, Mathematics
Abstract

A result of Barban-Vehov (and independently Motohashi) gives an estimate for the mean square of a sequence related to Selberg's sieve. This upper bound was refined to an asymptotic formula by S. Graham in 1978. In 1992, I made the observation that Graham's method can be used to obtain an asymptotic formula when the sum is restricted to an arithmetic progression. This formula immediately gives a version of the Brun-Titchmarsh theorem. I am taking the occasion of a volume in honour of my friend S. Srinivasan to revisit and publish this observation in the hope that it might still be of interest.

Published
2019
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37. On an identity of Ramanujan

Author

Yoichi Motohashi

Subjects
[ MATH ] Mathematics [math], Pure mathematics, Ramanujan's identity, Mathematics::General Mathematics, Mathematics::Number Theory, media_common.quotation_subject, Selberg's ${\Lambda}^2$-sieve, Contrast (music), Mathematical proof, Ramanujan's sum, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], symbols.namesake, Identity (philosophy), symbols, Euler's formula, Mobius inversion, Möbius inversion, Mathematics, media_common
Abstract

Proofs published so far in articles and books, of the Ramanujan identity presented in this note, which depend on Euler products, are essentially the same as Ramanujan's original proof. In contrast, the proof given here is short and independent of the use of Euler products.

Published
2019
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38. ISOTROPIC DIFFEOMORPHISMS: SOLUTIONS TO A DIFFERENTIAL SYSTEM FOR A DEFORMED RANDOM FIELDS STUDY

Author

Briant , Marc, Fournier , Julie, Briant, Marc, 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), Fournier, Julie, Mathématiques Appliquées à Paris 5 ( MAP5 - UMR 8145 ), and Université Paris Descartes - Paris 5 ( UPD5 ) -Institut National des Sciences Mathématiques et de leurs Interactions-Centre National de la Recherche Scientifique ( CNRS )

Subjects
[ MATH ] Mathematics [math], Partial differential equation, [MATH] Mathematics [math], Isotropy, Diffeomorphisms of the plane, Deformed random fields, Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH]Mathematics [math], [MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Analysis of PDEs (math.AP)
Abstract

This Note presents the resolution of a differential system on the plane that translates a geometrical problem about isotropic deformations of area and length. The system stems from a probability study on deformed random fields [J.Fournier '17], which are the composition of a random field with invariance properties defined on the plane with a deterministic diffeomorphism. The explicit resolution of the differential system allows to prove that a weak notion of isotropy of the deformed field, linked to its excursion sets, in fact coincides with the strong notion of isotropy. The present Note first introduces the probability framework that gave rise to the geometrical issue and then proposes its resolution., 8 pages

Published
2019

39. Compressible Navier-Stokes system with general inflow-outflow boundary data

Author

Tongkeun Chang, Antonín Novotný, Bum Ja Jin, Yonsei University, Institut de Mathématiques de Toulon - EA 2134 (IMATH), Université de Toulon (UTLN), Institut de Mathématiques de Toulon - EA 2134 ( IMATH ), and Université de Toulon ( UTLN )

Subjects
[ MATH ] Mathematics [math], Astrophysics::High Energy Astrophysical Phenomena, Mathematics::Analysis of PDEs, Inflow, 01 natural sciences, Physics::Fluid Dynamics, Compressible Navier–Stokes system, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], large inflow, Barotropic fluid, weak solutions, inhomogeneous boundary conditions, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Navier stokes, large outflow, 0101 mathematics, [MATH]Mathematics [math], Adiabatic process, Mathematics, renormalized continuity equation, Applied Mathematics, Mathematical analysis, 010101 applied mathematics, Computational Mathematics, Compressibility, Outflow boundary, Analysis
Abstract

International audience; We prove existence of weak solutions to the compressible Navier-Stokes equations in barotropic regime (adiabatic coefficient γ > 3/2, in three dimensions, γ > 1 in two dimensions) with large velocity prescribed at the boundary and large density prescribed at the inflow boundary of a bounded sufficiently smooth domain, without any restriction neither on the shape of the inflow/outflow boundaries nor on the shape of the domain. The result applies also to pressure laws that are non monotone on a compact portion of interval [0, ∞).

Published
2019

40. Incorporation of a deformation prior in image reconstruction

Author

Barbara Gris, Gris, Barbara, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), 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)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), The work by Barbara Gris was supported by the Swedish Foundation for Strategic Research grant AM13-0049., Centre de Mathématiques et de Leurs Applications ( CMLA ), and École normale supérieure - Cachan ( ENS Cachan ) -Centre National de la Recherche Scientifique ( CNRS )

Subjects
[ MATH ] Mathematics [math], Statistics and Probability, [ INFO.INFO-TS ] Computer Science [cs]/Signal and Image Processing, Computer science, [INFO.INFO-TS] Computer Science [cs]/Signal and Image Processing, [INFO.INFO-IM] Computer Science [cs]/Medical Imaging, 02 engineering and technology, Iterative reconstruction, [MATH] Mathematics [math], [INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing, 0202 electrical engineering, electronic engineering, information engineering, [INFO.INFO-IM]Computer Science [cs]/Medical Imaging, [MATH]Mathematics [math], [ INFO.INFO-IM ] Computer Science [cs]/Medical Imaging, business.industry, Applied Mathematics, Modular design, Condensed Matter Physics, Reconstruction method, Formalism (philosophy of mathematics), Modeling and Simulation, 020201 artificial intelligence & image processing, Geometry and Topology, Computer Vision and Pattern Recognition, Diffeomorphism, business, Algorithm
Abstract

International audience; This article presents a method to incorporate a deformation prior in image reconstruction via the formalism of deformation modules. The framework of deformation modules allows to build diffeomorphic deformations that satisfy a given structure. The idea is to register a template image against the indirectly observed data via a modular deformation, incorporating this way the deformation prior in the reconstruction method. We show that this is a well-defined regularization method (proving existence, stability and convergence) and present numerical examples of reconstruction from 2-D tomographic simulations and partially-observed images.

Published
2019

41. Directed polymers in heavy-tail random environment

Author

Quentin Berger, Niccolò Torri, Laboratoire de Probabilités, Statistiques et Modélisations (LPSM (UMR_8001)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), The authors acknowledge the support of PEPS grant from CNRS. Q. Berger was supported by a publicgrant overseen by the French National Research Agency ANR-17-CE40- 0032-02. N. Torri was supported by apublic grant overseen by the French National Research Agency (ANR) as part of the 'Investissements d’Avenir'program (ANR-11-LABX-0020-01 and ANR-10-LABX-0098)., Laboratoire de Probabilités et Modèles Aléatoires ( LPMA ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ), and Laboratoire de Probabilités, Statistique et Modélisation ( LPSM UMR 8001 )

Subjects
Statistics and Probability, 82D60, [ MATH ] Mathematics [math], 60G70, Dimension (graph theory), 01 natural sciences, Combinatorics, 010104 statistics & probability, Transversal (geometry), 2010 Mathematics Subject Classification: 60F05, 82D60, 60K37, 60G70, Intermediate disorder, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 60F05, FOS: Mathematics, Beta (velocity), Directed polymer, 0101 mathematics, [MATH]Mathematics [math], Scaling, Mathematics, Conjecture, 010102 general mathematics, Probability (math.PR), Super-diffusivity, [ MATH.MATH-MP ] Mathematics [math]/Mathematical Physics [math-ph], last-passage percolation, superdiffusivity, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Last Passage Percolation, Scaling limit, 60K37, Heavy-tailed distribution, Heavy-tail distributions, Exponent, 60F05, 82D60, 60K37, 60G70, Weak-coupling limit, Statistics, Probability and Uncertainty, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR], Mathematics - Probability
Abstract

We study the directed polymer model in dimension ${1+1}$ when the environment is heavy-tailed, with a decay exponent $\alpha\in(0,2)$. We give all possible scaling limits of the model in the weak-coupling regime, i.e., when the inverse temperature temperature $\beta=\beta_n$ vanishes as the size of the system $n$ goes to infinity. When $\alpha\in(1/2,2)$, we show that all possible transversal fluctuations $\sqrt{n} \leq h_n \leq n$ can be achieved by tuning properly $\beta_n$, allowing to interpolate between all super-diffusive scales. Moreover, we determine the scaling limit of the model, answering a conjecture by Dey and Zygouras [cf:DZ] - we actually identify five different regimes. On the other hand, when $\alpha, Comment: 41 pages, 1 figure

Published
2019

42. Asymptotic behaviour for operators of Grushin type: Invariant measure and singular perturbations

Author

Claudio Marchi, Paola Mannucci, Nicoletta Tchou, Department of Mathematics, University of Padova, Italy, Department of Information Engineering [Padova] ( DEI ), Universita degli Studi di Padova = University of Padua = Université de Padoue, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), ANR-16-CE40-0015,MFG,Mean Field Games ( 2016 ), Department of Information Engineering [Padova] (DEI), Universita degli Studi di Padova, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, 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)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), ANR-16-CE40-0015,MFG,Jeux Champs Moyen(2016), Università degli Studi di Padova = University of Padua (Unipd), 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 Agrocampus Ouest, and 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)

Subjects
[ MATH ] Mathematics [math], Singular perturbation, Subelliptic equations, Grushin vector fields, invariant measure, singular perturbations, viscosity solutions, degenerate elliptic equations, Perturbation (astronomy), 01 natural sciences, Operator (computer programming), [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Analysis of PDEs, FOS: Mathematics, Discrete Mathematics and Combinatorics, Ergodic theory, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Infinitesimal generator, 0101 mathematics, [MATH]Mathematics [math], Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Logarithmic growth, 010101 applied mathematics, Invariant measure, Viscosity solution, Analysis, Analysis of PDEs (math.AP)
Abstract

International audience; This paper concerns singular perturbation problems where the dynamics of the fast variable evolve in the whole space according to an operator whose infinitesimal generator is formed by a Grushin type second order part and a Ornstein-Uhlenbeck first order part. We prove that the dynamics of the fast variables admits an invariant measure and that the associated ergodic problem has a viscosity solution which is also regular and with logarithmic growth at infinity. These properties play a crucial role in the main theorem which establishes that the value functions of the starting perturbation problems converge to the solution of an effective problem whose operator and initial datum are given in terms of the associated invariant measure.

Published
2019
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43. Greedy optimal control for elliptic problems and its application to turnpike problems

Author

Víctor Hernández-Santamaría, Martin Lazar, Enrique Zuazua, Universidad de Deusto [Bilbao] (DEUSTO), Universidad de Deusto (DEUSTO), University of Dubrovnik, Departamento de Matemáticas [Madrid], Universidad Autonoma de Madrid (UAM), Univeristy of Deusto (DEUSTO), Universidad de Deusto ( DEUSTO ), University of Deusto, Universidad Autonoma de Madrid ( UAM ), Univeristy of Deusto ( DEUSTO ), UAM. Departamento de Matemáticas, and Victor, Hernandez-Santamaria

Subjects
[ MATH ] Mathematics [math], 0209 industrial biotechnology, Mathematical optimization, Matemáticas, Property (programming), Parabolic equations, parameterized PDEs, elliptic equations, Parameterized complexity, [MATH] Mathematics [math], 010103 numerical & computational mathematics, 02 engineering and technology, turnpike property, 01 natural sciences, optimal control, 020901 industrial engineering & automation, greedy algorithms, Convergence (routing), 93C20, 0101 mathematics, [MATH]Mathematics [math], 49K20, Mathematics, 65K10, 49N05, Applied Mathematics, Horizon, Numerical analysis, Turnpike property, Sampling (statistics), Geodetic datum, parabolic equations, Parameterized PDEs, Elliptic equations, Optimal control, Greedy algorithms, Computational Mathematics, parameterized PDEs · optimal control · turnpike property · greedy algorithms · elliptic equations · parabolic equations, parabolic equations MSC2010: 49J20
Abstract

This is a post-peer-review, pre-copyedit version of an article published in Numerische Mathematik. The final authenticated version is available online at: https://doi.org/10.1007/s00211-018-1005-z, We adapt and apply greedy methods to approximate in an efficient way the optimal controls for parameterized elliptic control problems. Our results yield an optimal approximation procedure that, in particular, performs better than simply sampling the parameter-space to compute controls for each parameter value. The same method can be adapted for parabolic control problems, but this leads to greedy selections of the realizations of the parameters that depend on the initial datum under consideration. The turnpike property (which ensures that parabolic optimal control problems behave nearly in a static manner when the control horizon is long enough) allows using the elliptic greedy choice of the parameters in the parabolic setting too. We present various numerical experiments and an extensive discussion of the efficiency of our methodology for parabolic control and indicate a number of open problems arising when analyzing the convergence of the proposed algorithms, This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 694126-DyCon). Part of this research was done while the second author visited DeustoTech and Univesity of Deusto with the support of the DyCon project. The second author was also partially supported by Croatian Science Foundation under ConDyS Project, IP-2016-06-2468. The work of the third author was partially supported by the Grants MTM2014-52347, MTM2017-92996 of MINECO (Spain) and ICON of the French ANR

Published
2019

44. The Artin motives, weights and motivic nearby sheaves

Author

Ivorra, Florian, Sebag, Julien, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, 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)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Sebag, Julien, 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 Agrocampus Ouest, 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), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), and Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS )

Subjects
[ MATH ] Mathematics [math], [MATH] Mathematics [math], [MATH]Mathematics [math], ComputingMilieux_MISCELLANEOUS
Abstract

International audience

Published
2019

45. Asymmetric Exchanges

Author

Bouissou , Caroline, Petitjean , Michel, Molécules Thérapeutiques in silico (MTI), Université Paris Diderot - Paris 7 (UPD7)-Institut National de la Santé et de la Recherche Médicale (INSERM), and Petitjean, Michel

Subjects
[ MATH ] Mathematics [math], arts, mesures d'asymétrie, chirality, [MATH] Mathematics [math], [ SHS.ART ] Humanities and Social Sciences/Art and art history, [SHS.ART]Humanities and Social Sciences/Art and art history, most scalene triangle, asymmetry measures, symétrie, triangle le plus scalène, chiralité, [SHS.ART] Humanities and Social Sciences/Art and art history, [MATH]Mathematics [math], symmetry
Abstract

Proceedings of the First European Asymmetry Symposium, We emphasize the asymmetric character of exchanges between artists and scientists or mathematicians in the framework of multidisciplinary collaborations. As an example, we look at the perception of asymmetry in an artwork presented at the First European Asymmetry Symposium (FEAS), Nice, 15-16 March 2018, and its potential connection with the unsolved mathematical problem of maximal or asymptotically maximal asymmetric figures and distributions., Nous soulignons le caractère asymétrique des échanges entre artistes et scientifiques ou mathématiciens dans le cadre de collaborations multidisciplinaires. Comme exemple, nous considérons la symétrie dans une œuvre présentée au symposium FEAS (First European Asymmetry Symposium), Nice, 15-16 mars 2018, et son lien potentiel avec le problème ouvert des figures et distributions d'asymétrie maximale ou asymptotiquement maximale.

Published
2018
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46. On the convergence of a stochastic approximation method for structured bi-level optimization

Author

Couellan , Nicolas, Wang , Wenjuan, Ecole Nationale de l'Aviation Civile (ENAC), 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), Bournemouth University [Poole] (BU), 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), Ecole Nationale de l'Aviation Civile ( ENAC ), Institut de Mathématiques de Toulouse UMR5219 ( IMT ), Université Toulouse 1 Capitole ( UT1 ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse III - Paul Sabatier ( UPS ), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-PRES Université de Toulouse-Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Centre National de la Recherche Scientifique ( CNRS ), and Bournemouth University [Poole] ( BU )

Subjects
[ MATH ] Mathematics [math], [ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC], [INFO.INFO-LG]Computer Science [cs]/Machine Learning [cs.LG], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [ INFO.INFO-LG ] Computer Science [cs]/Machine Learning [cs.LG]
Abstract

We analyze the convergence of stochastic gradient methods for well structured bi-level optimization problems. We address two specific cases: first when the outer objective function can be expressed as a finite sum of independent terms, and next when both the outer and inner objective functions can be expressed as finite sums of independent terms. We assume Lipschitz continuity and differentiability of both objectives as well as convexity of the inner objective and consider diminishing steps sizes. We show that, under these conditions and some other assumptions on the implicit function and the variance of the gradient errors, both methods converge in expectation to a stationary point of the problem if gradient approximations are chosen so as to satisfy a sufficient decrease condition. We also discuss the satisfaction of our assumptions in machine learning problems where these methods can be nicely applied to automatically tune hyperparameters when the loss functions are very large sums of error terms.

Published
2018

47. Dispersion analysis of compatible Galerkin schemes for the 1D shallow water model

Author

Daniel Y. Le Roux, Christopher Eldred, Mathematics and computing applied to oceanic and atmospheric flows (AIRSEA), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-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])-Centre National de la Recherche Scientifique (CNRS), Modélisation mathématique, calcul scientifique (MMCS), Institut Camille Jordan (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)-École Centrale de Lyon (ECL), 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), ANR-14-CE23-0010,HEAT,Modélisation atmosphérique hautement efficace(2014), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-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)-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), Mathematics and computing applied to oceanic and atmospheric flows ( AIRSEA ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Laboratoire Jean Kuntzmann ( LJK ), 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 ) -Institut National Polytechnique de Grenoble ( INPG ), 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 ] Mathematics [math], Work (thermodynamics), dispersion relationship, 010504 meteorology & atmospheric sciences, Physics and Astronomy (miscellaneous), mimetic Galerkin differences, 010103 numerical & computational mathematics, fluid dynamics, 01 natural sciences, geophysical, geophysical fluid dynamics, Fluid dynamics, dynamical core, mixed finite elements, 0101 mathematics, [MATH]Mathematics [math], Dispersion (water waves), Galerkin method, Shallow water equations, 0105 earth and related environmental sciences, Physics, Numerical Analysis, shallow water equations, Applied Mathematics, Mathematical analysis, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], exterior calculus, Mass matrix, Finite element method, Computer Science Applications, Quadrature (mathematics), Computational Mathematics, finite element, Modeling and Simulation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; In this work, we study the dispersion properties of two compatible Galerkin schemes for the 1D linearized shallow water equations: the P C n −P DG n−1 and the GD n −DGD n element pairs. Compatible Galerkin methods have many desirable properties, including energy conservation, steady geostrophic modes and the absence of spurious stationary modes, such as pressure modes. However, this does not guarantee good wave dispersion properties. Previous work on the P C 2 − P DG 1 pair has indeed indicated the presence of spectral gaps, and it is extended in this paper to the study of the P C n − P DG n−1 pair for arbitrary n. Additionally, an alternative element pair is introduced, the GD n − DGD n pair, that is free of spectral gaps while benefiting from the desirable properties of compatible elements. Asymptotic convergence rates are established for both element pairs, including the use of inexact quadrature (which di-agonalizes the velocity mass matrix) for the P C n − P DG n−1 pair and reduced quadrature for the GD n − DGD n pair. Plots of the dispersion relationship and group velocities for a wide range of n and Rossby radii are shown. A brief investigation into the utility of mass lumping to remove the spectral gaps for the P C 3 − P DG 2 pair is performed. Finally, a pair of numerical simulations are run to investigate the consequences of the spectral gaps and highlight the main differences between the two elements.

Published
2018
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48. STOCHASTIC CUCKER-SMALE MODELS: OLD AND NEW

Author

Fanny Delebecque, Laure Pédèches, Patrick Cattiaux, Institut de Mathématiques de Toulouse UMR5219 (IMT), 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), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), 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), Institut de Mathématiques de Toulouse UMR5219 ( IMT ), Université Toulouse 1 Capitole ( UT1 ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse III - Paul Sabatier ( UPS ), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-PRES Université de Toulouse-Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), and Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Centre National de la Recherche Scientifique ( CNRS )

Subjects
Statistics and Probability, Self-organization, [ MATH ] Mathematics [math], flocking, Stochastic modelling, Flocking (behavior), 010102 general mathematics, Cucker-Smale dynamics, 01 natural sciences, 92C17, Cucker–Smale dynamics, 010101 applied mathematics, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], stochastic interacting particles, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Applied mathematics, 60H10, 0101 mathematics, Statistics, Probability and Uncertainty, 82C22, [MATH]Mathematics [math], ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

International audience; In this paper we revisit and generalize various stochastic models extending the deterministic Cucker-Smale model for self organization. We study flocking and swarming properties. We show how these properties strongly depend on the structure and on the variance of the noise.

Published
2018
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49. Nontrivial dynamics beyond the logarithmic shift in two-dimensional Fisher-KPP equations

Author

Violaine Roussier-Michon, Jean-Michel Roquejoffre, Mathématiques pour l'Industrie et la Physique ( MIP ), Université Toulouse 1 Capitole ( UT1 ) -Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Toulouse III - Paul Sabatier ( UPS ), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques, Université de Lille, Sciences Humaines et Sociales, Institut de Mathématiques de Toulouse UMR5219 (IMT), 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 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), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
[ MATH ] Mathematics [math], Logarithm, media_common.quotation_subject, logarithmic shift, General Physics and Astronomy, 01 natural sciences, symbols.namesake, Mathematics - Analysis of PDEs, Planar, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Dimension (vector space), Convergence (routing), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Initial value problem, 0101 mathematics, Mathematical Physics, ComputingMilieux_MISCELLANEOUS, Mathematics, media_common, Heaviside step function, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Statistical and Nonlinear Physics, KPP equations, Infinity, MSC: 35K57, 35B40, 35B35, 35C07, 010101 applied mathematics, symbols, nontrivial dynamics, Analysis of PDEs (math.AP), Reference frame
Abstract

International audience; We study the asymptotic behaviour, as time goes to infinity, of the Fisher-KPP equation \partial_t u = ∆u + u − u^2 in spatial dimension 2, when the initial condition looks like a Heaviside function. Thus the solution is, asymptotically in time, trapped between two planar critical waves whose positions are corrected by the Bramson logarithmic shift. The issue is whether, in this reference frame, the solutions will converge to a travelling wave, or will exhibit more complex behaviours. We prove here that both convergence and nonconvergence may happen: the solution may converge towards one translate of the planar wave, or oscillate between two of its translates. This relies on the behaviour of the initial condition at infinity in the transverse direction.

Published
2018
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50. Torsion of a finite base locus

Author

Bignalet-Cazalet , Rémi, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Bignalet-Cazalet, Rémi, Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), and Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS )

Subjects
[ MATH ] Mathematics [math], [MATH.MATH-AC] Mathematics [math]/Commutative Algebra [math.AC], Mathematics - Algebraic Geometry, 13D02, 14E05, 14B05, 14H20, [MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC], [MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG], [MATH] Mathematics [math], [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], [MATH]Mathematics [math], Mathematics - Commutative Algebra, [ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG], [ MATH.MATH-AC ] Mathematics [math]/Commutative Algebra [math.AC]
Abstract

We interpret geometrically the torsion of the symmetric algebra of the ideal sheaf of a zero-dimensional scheme Z defined by $n+1$ equations in an $n$-dimensional variety. This leads us to generalise a formula of A.Dimca and S.Papadima in positive characteristic for a rational transformation with finite base locus. Among other applications, we construct an explicit example of a homaloidal curve of degree $5$ in characteristic $3$, answering negatively a question of A.V.D\'oria, S.H.Hassanzadeh and A.Simis., Comment: 19 pages

Published
2018

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