Nos 50 dernières publications
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25 March 2025 hal-05005367
We study the discretization, convergence, and numerical implementation of recent reformulations of the quadratic porous medium equation (multidimensional and anisotropic) and Burgers' equation (one-dimensional, with optional viscosity), as forward in time variants of the Benamou-Brenier formulation of optimal transport. This approach turns those evolution problems into global optimization problems in time and space, of which we introduce a discretization, one of whose originalities lies in the harmonic interpolation of the densities involved. We prove that the resulting schemes are unconditionally stable w.r.t. the space and time steps, and we establish a quadratic convergence rate for the dual PDE solution, under suitable assumptions. We also show that the schemes can be efficiently solved numerically using a proximal splitting method and a global space-time fast Fourier transform, and we illustrate our results with numerical experiments.
Jean-Marie Mirebeau, Erwan Stampfli
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25 March 2025 hal-05004709
We study the second Huber Theorem in dimensions 2 and 4. In dimension 2, we prove the optimal regularity for the conformal factor using Coulomb frames. In dimension 4, we introduce another Coulomb-type condition which is similar to the case of Yang--Mills connections. We obtain a generalization of the two-dimensional case that can be applied to study the singularities of Bach-flat metrics.
Paul Laurain, Dorian Martino
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20 March 2025 hal-04999502
We study the martingale property and moment explosions of a signature volatility model, where the volatility process of the log-price is given by a linear form of the signature of a time-extended Brownian motion. Excluding trivial cases, we demonstrate that the price process is a true martingale if and only if the order of the linear form is odd and a correlation parameter is negative. The proof involves a fine analysis of the explosion time of a signature stochastic differential equation. This result is of key practical relevance, as it highlights that, when used for approximation purposes, the linear combination of signature elements must be taken of odd order to preserve the martingale property. Once martingality is established, we also characterize the existence of higher moments of the price process in terms of a condition on a correlation parameter.
Eduardo Abi Jaber, Paul Gassiat, Dimitri Sotnikov
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19 March 2025 tel-03905418
This thesis studies the persistent homology of real-valued continuous functions f on compact topological spaces X. The introduction of homological indices and homological dimensions allows us to link persistence theory to metric quantities of the compact space X, such as its upper-box dimension. These quantities give a precise framework to the Wasserstein p-stability results known in the literature, but also extend them to Hölder functions on more general spaces (including all compact Riemannian manifolds) with explicit constants and whose regime for p is optimal. In degree zero of homology, a more in-depth study can be made using trees associated to f, which generalize the merge trees which are definable when f is Morse. It is possible to link the dimension of these trees to the persistence index of f and to its barcode. We apply these deterministic results to the stochastic setting to draw consequences about the barcodes of random functions of prescribed regularity. These consequences also allow us to develop distributional discrimination tests for the processes, of which we present a particular example. Finally, we define the zeta-functions associated with a stochastic process and compute these functions and other related quantities for several processes in dimension one, including the Brownian motion and the alpha-stable Lévy processes.
Daniel Perez
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18 March 2025 hal-04996372
This thesis studies the persistent homology of real-valued continuous functions f on compact topological spaces X. The introduction of homological indices and homological dimensions allows us to link persistence theory to metric quantities of the compact space X, such as its upper-box dimension. These quantities give a precise framework to the Wasserstein p-stability results known in the literature, but also extend them to Hölder functions on more general spaces (including all compact Riemannian manifolds) with explicit constants and whose regime for p is optimal. In degree zero of homology, a more in-depth study can be made using trees associated to f, which generalize the merge trees which are definable when f is Morse. It is possible to link the dimension of these trees to the persistence index of f and to its barcode. We apply these deterministic results to the stochastic setting to draw consequences about the barcodes of random functions of prescribed regularity. These consequences also allow us to develop distributional discrimination tests for the processes, of which we present a particular example. Finally, we define the zeta-functions associated with a stochastic process and compute these functions and other related quantities for several processes in dimension one, including the Brownian motion and the alpha-stable Lévy processes.
Tony Jin, João Ferreira, Michel Bauer, Michele Filippone, Thierry Giamarchi
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18 March 2025 hal-04994930
The cutoff phenomenon, conceptualized at the origin for finite Markov chains, states that for a parametric family of evolution equations, started from a point, the distance towards a long time equilibrium may become more and more abrupt for certain choices of initial conditions, when the parameter tends to infinity. This threshold phenomenon can be seen as a critical competition between trend to equilibrium and worst initial condition. In this note, we investigate this phenomenon beyond stochastic processes, in the context of the analysis of nonlinear partial differential equations, by proving cutoff for the fast diffusion and porous medium Fokker-Planck equations on the Euclidean space, when the dimension tends to infinity. We formulate the phenomenon using quadratic Wasserstein distance, as well as using specific relative entropy and Fisher information. Our high dimensional asymptotic analysis uses the exact solvability of the model involving Barenblatt profiles. It includes the Ornstein-Uhlenbeck dynamics as a special linear case.
Djalil Chafaï, Max Fathi, Nikita Simonov
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12 March 2025 hal-04974699
Ce recueil regroupe les traductions en français des textes officiels de présentation des sept problèmes mathématiques du millénaire. Il donne donc un petit aperçu des mathématiques contemporaines.
Andrew Wiles, Pierre Deligne, Charles Fefferman, John Milnor, Stephen Cook, Enrico Bombieri, Arthur Jaffe, Edward Witten, Nicolas Bacaër, Michel Balazard, Gérard Besson, Jean-Louis Colliot-Thélène, Isabelle Gallagher, Catherine Goldstein, Thibaut Lemoine, Sylvain Perifel, Claire Voisin
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10 March 2025 hal-04807947
In this paper, we investigate the link between kinetic equations (including Boltzmann with or without cutoff assumption and Landau equations) and the incompressible Navier-Stokes equation. We work with strong solutions and we treat all the cases in a unified framework. The main purpose of this work is to be as accurate as possible in terms of functional spaces. More precisely, it is well-known that the Navier-Stokes equation can be solved in a lower regularity setting (in the space variable) than kinetic equations. Our main result allows to get a rigorous link between solutions to the Navier-Stokes equation with such low regularity data and kinetic equations.
Kleber Carrapatoso, Isabelle Gallagher, Isabelle Tristani
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5 March 2025 hal-04261339
In the interest of reproducible research, this is exactly the version of the code used in the paper "Abide by the Law and Follow the Flow: Conservation Laws for Gradient Flows" by the same authors, available at https://inria.hal.science/hal-04150576 with its detailed bibliographical notice. Any updates to this code will be available at https://github.com/sibyllema/Conservation_laws.
Sibylle Marcotte, Rémi Gribonval, Gabriel Peyré
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24 February 2025 hal-04963968
In this paper, we investigate the properties of the Sliced Wasserstein Distance (SW) when employed as an objective functional. The SW metric has gained significant interest in the optimal transport and machine learning literature, due to its ability to capture intricate geometric properties of probability distributions while remaining computationally tractable, making it a valuable tool for various applications, including generative modeling and domain adaptation. Our study aims to provide a rigorous analysis of the critical points arising from the optimization of the SW objective. By computing explicit perturbations, we establish that stable critical points of SW cannot concentrate on segments. This stability analysis is crucial for understanding the behaviour of optimization algorithms for models trained using the SW objective. Furthermore, we investigate the properties of the SW objective, shedding light on the existence and convergence behavior of critical points. We illustrate our theoretical results through numerical experiments.
Christophe Vauthier, Quentin Merigot, Anna Korba
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23 February 2025 hal-04534268
Turbulent cascades characterize the transfer of energy injected by a random force at large scales towards the small scales. We construct a linear equation that mimics the phenomenology of energy cascades when the external force is a statistically homogeneous and stationary stochastic process. In the Fourier variable, this equation can be seen as a wave equation, which corresponds to a wave operator of degree 0 in physical space. Our results give a complete characterization of the solution: it is smooth at any finite time, and, up to smaller order corrections, it converges to a fractional Gaussian field at infinite time. We apply a finite volume method in the Fourier variables formulation in order to reach the invariant measure of the equation.
Geoffrey Beck, Charles-Edouard Bréhier, Laurent Chevillard, Isabelle Gallagher, Ricardo Grande, Wandrille Ruffenach
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21 February 2025 hal-04961416
Dans cette note, nous expliquons la construction de variétés de Hecke p-adiques associées aux groupes unitaires définis d'un corps de nombres CM et décrivons des propriétés de la famille de représentations galoisiennes portée par ces variétés. Nous donnons une application à la construction de certaines représentations galoisiennes qui joue un rôle important dans cette série de livres. Les variétés de Hecke ci-dessus sont aussi utilisées dans [BCh2]. La construction des variétés de Hecke que nous donnons est une combinaison essentiellement triviale des méthodes de [Ch] (cas d'un corps quadratique imaginaire, revisité dans [BCh, §7]) et de [Bu] (cas d'une algèbre de quaternions sur un corps totalement réel, voir aussi [Y]). Elle pourrait aussi se déduire comme cas particulier des travaux d'Emerton [E], et elle est aussi contenue dans les travaux récents de Loeffler [Loe]. En ce qui concerne les propriétés galoisiennes, nous utilisons et étendons certains résultats de [BCh, §7].
Gaëtan Chenevier
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19 February 2025 hal-02144896
We develop a functional analytic approach to the study of the Kramers and kinetic Fokker-Planck equations which parallels the classical $H^1$ theory of uniformly elliptic equations. In particular, we identify a function space analogous to $H^1$ and develop a well-posedness theory for weak solutions in this space. In the case of a conservative force, we identify the weak solution as the minimizer of a uniformly convex functional. We prove new functional inequalities of Poincaré and Hörmander type and combine them with basic energy estimates (analogous to the Caccioppoli inequality) in an iteration procedure to obtain the~$C^\infty$ regularity of weak solutions. We also use the Poincaré-type inequality to give an elementary proof of the exponential convergence to equilibrium for solutions of the kinetic Fokker-Planck equation which mirrors the classic dissipative estimate for the heat equation. Finally, we prove enhanced dissipation in a weakly collisional limit.
Dallas Albritton, Scott Armstrong, Jean-Christophe Mourrat, Matthew Novack
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18 February 2025 tel-04955468
This PhD thesis presents contributions to the field of deep learning. From convolutional ResNets to Transformers, residual connections are ubiquitous in state-of-the-art deep learning models. The continuous depth analogues of residual networks, neural ODEs, have been widely adopted, but the connection between the discrete and continuous models still lacks a solid mathematical foundation. In this manuscript, we will show that for a formal correspondence between residual networks and neural ODEs to hold, the residual functions must be smooth with depth, and we will present an implicit regularization result of deep residual networks towards neural ODEs. We will then present two applications of this analogy to the design and study of new architectures. First, we will introduce a drop-in replacement for any residual network that can be trained with the same accuracy, but with much less memory. Second, by viewing the attention mechanism as an interacting particle system, where the particles are the tokens, we will study the impact of attention map normalization on the Transformer model. Finally, we will present some other contributions to Transformers: how Transformers perform in-context autoregressive learning and how to differentiably route tokens to experts in Sparse Mixture of Experts Transformers.
Michael E. Sander
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18 February 2025 hal-04953521
To a bicomplex one can associate two natural filtrations, the column and row filtrations, and then two associated spectral sequences. This can be generalized to $N$-multicomplexes. We present a family of model category structures on the category of $N$-multicomplexes where the weak equivalences are the morphisms inducing a quasi-isomorphism at a fixed page $r$ of the first spectral sequence and at a fixed page $s$ of the second spectral sequence. Such weak equivalences arise naturally in complex geometry. In particular, the model structures presented here establish a basis for studying homotopy types of almost and generalized complex manifolds.
Joana Cirici, Muriel Livernet, Sarah Whitehouse
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18 February 2025 hal-04953513
We study homotopy theory of the category of spectral sequences with respect to the class of weak equivalences given by maps which are quasi-isomorphisms on a fixed page. We introduce the category of extended spectral sequences and show that this is bicomplete by analysis of a certain linear presheaf category modelled on discs. We endow the category of extended spectral sequences with various model category structures, restricting to give the almost Brown category structures on spectral sequences of our earlier work. One of these has the property that spectral sequences is a homotopically full subcategory. By results of Meier, this exhibits the category of spectral sequences as a fibrant object in the Barwick-Kan model structure on relative categories, that is, it gives a model for an infinity category of spectral sequences. We also use the presheaf approach to define two décalage functors on spectral sequences, left and right adjoint to a shift functor, thereby clarifying prior use of the term décalage in connection with spectral sequences.
Muriel Livernet, Sarah Whitehouse
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11 February 2025 hal-04941531
We show that the momentum, the density, and the electromagnetic field associated with the massive Klein-Gordon-Maxwell equations converge in the semi-classical limit towards their respective equivalents associated with the relativistic Euler-Maxwell equations. The proof relies on a modulated stress-energy method and a compactness argument. We also give a proof of the well-posedness of the relativistic Euler-Maxwell equations and show how this system, and so the semi-classical limit of Klein-Gordon-Maxwell, is related to the relativistic massive Vlasov-Maxwell equations.
Tony Salvi
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8 February 2025 hal-04936536
We show that the momentum, the density, and the electromagnetic field associated with the massive Klein-Gordon-Maxwell equations converge in the semi-classical limit towards their respective equivalents associated with the relativistic Euler-Maxwell equations. The proof relies on a modulated stress-energy method and a compactness argument. We also give a proof of the well-posedness of the relativistic Euler-Maxwell equations and show how this system, and so the semi-classical limit of Klein-Gordon-Maxwell, is related to the relativistic massive Vlasov-Maxwell equations.
Emmanuel Franck, Hélène Hivert, Guillaume Latu, Hélène Leman, Bertrand Maury, Michel Mehrenberger, Laurent Navoret
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2 February 2025 hal-04925307
This article studies the problem of estimating the state variable of non-smooth subdifferential dynamics constrained in a bounded convex domain given some real-time observation. On the one hand, we show that the value function of the estimation problem is a viscosity solution of a Hamilton Jacobi Bellman equation whose sub and super solutions have different Neumann type boundary conditions. This intricacy arises from the non-reversibility in time of the non-smooth dynamics, and hinders the derivation of a comparison principle and the uniqueness of the solution in general. Nonetheless, we identify conditions on the drift (including zero drift) coefficient in the non-smooth dynamics that make such a derivation possible. On the other hand, we show in a general situation that the value function appears in the small noise limit of the corresponding stochastic filtering problem by establishing a large deviation result. We also give quantitative approximation results when replacing the non-smooth dynamics with a smooth penalised one.
Louis-Pierre Chaintron, Laurent Mertz, Philippe Moireau, Hasnaa Zidani
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27 January 2025 hal-04915593
Benchmarks are important tools for tracking the rapid advancements in large language model (LLM) capabilities. However, benchmarks are not keeping pace in difficulty: LLMs now achieve over 90% accuracy on popular benchmarks like MMLU, limiting informed measurement of state-of-the-art LLM capabilities. In response, we introduce Humanity's Last Exam (HLE), a multi-modal benchmark at the frontier of human knowledge, designed to be the final closed-ended academic benchmark of its kind with broad subject coverage. HLE consists of 3,000 questions across dozens of subjects, including mathematics, humanities, and the natural sciences. HLE is developed globally by subject-matter experts and consists of multiple-choice and short-answer questions suitable for automated grading. Each question has a known solution that is unambiguous and easily verifiable, but cannot be quickly answered via internet retrieval. State-of-the-art LLMs demonstrate low accuracy and calibration on HLE, highlighting a significant gap between current LLM capabilities and the expert human frontier on closed-ended academic questions. To inform research and policymaking upon a clear understanding of model capabilities, we publicly release HLE at https://lastexam.ai.
Long Phan, Alice Gatti, Ziwen Han, Nathaniel Li, Josephina Hu, Hugh Zhang, Sean Shi, Michael Choi, Anish Agrawal, Arnav Chopra, Adam Khoja, Ryan Kim, Jason Hausenloy, Oliver Zhang, Mantas Mazeika, Daron Anderson, Tung Nguyen, Mobeen Mahmood, Fiona Feng, Steven Y. Feng, Haoran Zhao, Michael Yu, Varun Gangal, Chelsea Zou, Zihan Wang, Jessica P. Wang, Pawan Kumar, Oleksandr Pokutnyi, Robert Gerbicz, Serguei Popov, John-Clark Levin, Mstyslav Kazakov, Johannes Schmitt, Geoff Galgon, Alvaro Sanchez, Yongki Lee, Will Yeadon, Scott Sauers, Marc Roth, Chidozie Agu, Søren Riis, Fabian Giska, Saiteja Utpala, Zachary Giboney, Gashaw M. Goshu, Joan of Arc Xavier, Sarah-Jane Crowson, Mohinder Maheshbhai Naiya, Noah Burns, Lennart Finke, Zerui Cheng, Hyunwoo Park, Francesco Fournier-Facio, John Wydallis, Mark Nandor, Ankit Singh, Tim Gehrunger, Jiaqi Cai, Ben Mccarty, Darling Duclosel, Jungbae Nam, Jennifer Zampese, Ryan G. Hoerr, Aras Bacho, Gautier Abou Loume, Abdallah Galal, Hangrui Cao, Alexis C Garretson, Damien Sileo, Qiuyu Ren, Doru Cojoc, Pavel Arkhipov, Usman Qazi, Lianghui Li, Sumeet Motwani, Christian Schroeder de Witt, Edwin Taylor, Johannes Veith, Eric Singer, Taylor D. Hartman, Paolo Rissone, Jaehyeok Jin, Jack Wei Lun Shi, Chris G. Willcocks, Joshua Robinson, Aleksandar Mikov, Ameya Prabhu, Longke Tang, Xavier Alapont, Justine Leon Uro, Kevin Zhou, Emily de Oliveira Santos, Andrey Pupasov Maksimov, Edward Vendrow, Kengo Zenitani, Julien Guillod, Yuqi Li, Joshua Vendrow, Vladyslav Kuchkin, Ng Ze-An, Pierre Marion, Denis Efremov, Jayson Lynch, Kaiqu Liang, Andrew Gritsevskiy, Dakotah Martinez, Ben Pageler, Nick Crispino, Dimitri Zvonkine, Natanael Wildner Fraga, Saeed Soori, Ori Press, Henry Tang, Julian Salazar, Sean R. Green, Lina Brüssel, Moon Twayana, Aymeric Dieuleveut, T. Ryan Rogers, Wenjin Zhang, Bikun Li, Jinzhou Yang, Arun Rao, Gabriel Loiseau, Mikhail Kalinin, Marco Lukas, Ciprian Manolescu, Subrata Mishra, Ariel Ghislain Kemogne Kamdoum, Tobias Kreiman, Tad Hogg, Alvin Jin, Carlo Bosio, Gongbo Sun, Brian P Coppola, Tim Tarver, Haline Heidinger, Rafael Sayous, Stefan Ivanov, Joseph M Cavanagh, Jiawei Shen, Joseph Marvin Imperial, Philippe Schwaller, Shaipranesh Senthilkuma, Andres M Bran, Ali Dehghan, Andres Algaba, Brecht Verbeken, David Noever, Ragavendran P V, Lisa Schut, Ilia Sucholutsky, Evgenii Zheltonozhskii, Derek Lim, Richard Stanley, Shankar Sivarajan, Tong Yang, John Maar, Julian Wykowski, Martí Oller, Jennifer Sandlin, Anmol Sahu, Yuzheng Hu, Sara Fish, Nasser Heydari, Archimedes Apronti, Kaivalya Rawal, Tobias Garcia Vilchis, Yuexuan Zu, Martin Lackner, James Koppel, Jeremy Nguyen, Daniil S. Antonenko, Steffi Chern, Bingchen Zhao, Pierrot Arsene, Alan Goldfarb, Sergey Ivanov, Rafał Poświata, Chenguang Wang, Daofeng Li, Donato Crisostomi, Andrea Achilleos, Benjamin Myklebust, Archan Sen, David Perrella, Nurdin Kaparov, Mark H Inlow, Allen Zang, Elliott Thornley, Daniil Orel, Vladislav Poritski, Shalev Ben-David, Zachary Berger, Parker Whitfill, Michael Foster, Daniel Munro, Linh Ho, Dan Bar Hava, Aleksey Kuchkin, Robert Lauff, David Holmes, Frank Sommerhage, Keith Schneider, Zakayo Kazibwe, Nate Stambaugh, Mukhwinder Singh, Ilias Magoulas, Don Clarke, Dae Hyun Kim, Felipe Meneguitti Dias, Veit Elser, Kanu Priya Agarwal, Victor Efren Guadarrama Vilchis, Immo Klose, Christoph Demian, Ujjwala Anantheswaran, Adam Zweiger, Guglielmo Albani, Jeffery Li, Nicolas Daans, Maksim Radionov, Václav Rozhoň, Ziqiao Ma, Christian Stump, Mohammed Berkani, Jacob Platnick, Volodymyr Nevirkovets, Luke Basler, Marco Piccardo, Ferenc Jeanplong, Niv Cohen, Josef Tkadlec, Paul Rosu, Piotr Padlewski, Stanislaw Barzowski, Kyle Montgomery, Aline Menezes, Arkil Patel, Zixuan Wang, Jamie Tucker-Foltz, Jack Stade, Tom Goertzen, Fereshteh Kazemi, Jeremiah Milbauer, John Arnold Ambay, Abhishek Shukla, Yan Carlos Leyva Labrador, Alan Givré, Hew Wolff, Vivien Rossbach, Muhammad Fayez Aziz, Younesse Kaddar, Yanxu Chen, Robin Zhang, Jiayi Pan, Antonio Terpin, Niklas Muennighoff, Hailey Schoelkopf, Eric Zheng, Avishy Carmi, Adam Jones, Jainam Shah, Ethan D. L. Brown, Kelin Zhu, Max Bartolo, Richard Wheeler, Andrew Ho, Shaul Barkan, Jiaqi Wang, Martin Stehberger, Egor Kretov, Kaustubh Sridhar, Zienab El-Wasif, Anji Zhang, Daniel Pyda, Joanna Tam, David M. Cunningham, Vladimir Goryachev, Demosthenes Patramanis, Michael Krause, Andrew Redenti, Daniel Bugas, David Aldous, Jesyin Lai, Shannon Coleman, Mohsen Bahaloo, Jiangnan Xu, Sangwon Lee, Sandy Zhao, Ning Tang, Michael K. Cohen, Micah Carroll, Orr Paradise, Jan Hendrik Kirchner, Stefan Steinerberger, Maksym Ovchynnikov, Jason O. Matos, Adithya Shenoy, Benedito Alves de Oliveira Junior, Michael Wang, Yuzhou Nie, Paolo Giordano, Philipp Petersen, Anna Sztyber-Betley, Priti Shukla, Jonathan Crozier, Antonella Pinto, Shreyas Verma, Prashant Joshi, Zheng-Xin Yong, Allison Tee, Jérémy Andréoletti, Orion Weller, Raghav Singhal, Gang Zhang, Alexander Ivanov, Seri Khoury, Hamid Mostaghimi, Kunvar Thaman, Qijia Chen, Tran Quoc Khánh, Jacob Loader, Stefano Cavalleri, Hannah Szlyk, Zachary Brown, Jonathan Roberts, William Alley, Kunyang Sun, Ryan Stendall, Max Lamparth, Anka Reuel, Ting Wang, Hanmeng Xu, Sreenivas Goud Raparthi, Pablo Hernández-Cámara, Freddie Martin, Dmitry Malishev, Thomas Preu, Tomek Korbak, Marcus Abramovitch, Dominic Williamson, Ziye Chen, Biró Bálint, M Saiful Bari, Peyman Kassani, Zihao Wang, Behzad Ansarinejad, Laxman Prasad Goswami, Yewen Sun, Hossam Elgnainy, Daniel Tordera, George Balabanian, Earth Anderson, Lynna Kvistad, Alejandro José Moyano, Rajat Maheshwari, Ahmad Sakor, Murat Eron, Isaac C. Mcalister, Javier Gimenez, Innocent Enyekwe, Andrew Favre D. O., Shailesh Shah, Xiaoxiang Zhou, Firuz Kamalov, Ronald Clark, Sherwin Abdoli, Tim Santens, Khalida Meer, Harrison K Wang, Kalyan Ramakrishnan, Evan Chen, Alessandro Tomasiello, G. Bruno de Luca, Shi-Zhuo Looi, Vinh-Kha Le, Noam Kolt, Niels Mündler, Avi Semler, Emma Rodman, Jacob Drori, Carl J Fossum, Milind Jagota, Ronak Pradeep, Honglu Fan, Tej Shah, Jonathan Eicher, Michael Chen, Kushal Thaman, William Merrill, Carter Harris, Jason Gross, Ilya Gusev, Asankhaya Sharma, Shashank Agnihotri, Pavel Zhelnov, Siranut Usawasutsakorn, Mohammadreza Mofayezi, Sergei Bogdanov, Alexander Piperski, Marc Carauleanu, David K. Zhang, Dylan Ler, Roman Leventov, Ignat Soroko, Thorben Jansen, Pascal Lauer, Joshua Duersch, Vage Taamazyan, Wiktor Morak, Wenjie Ma, William Held, Tran Đuc Huy, Ruicheng Xian, Armel Randy Zebaze, Mohanad Mohamed, Julian Noah Leser, Michelle X Yuan, Laila Yacar, Johannes Lengler, Hossein Shahrtash, Edson Oliveira, Joseph W. Jackson, Daniel Espinosa Gonzalez, Andy Zou, Muthu Chidambaram, Timothy Manik, Hector Haffenden, Dashiell Stander, Ali Dasouqi, Alexander Shen, Emilien Duc, Bita Golshani, David Stap, Mikalai Uzhou, Alina Borisovna Zhidkovskaya, Lukas Lewark, Mátyás Vincze, Dustin Wehr, Colin Tang, Zaki Hossain, Shaun Phillips, Jiang Muzhen, Fredrik Ekström, Angela Hammon, Oam Patel, Nicolas Remy, Faraz Farhidi, George Medley, Forough Mohammadzadeh, Madellene Peñaflor, Haile Kassahun, Alena Friedrich, Claire Sparrow, Taom Sakal, Omkar Dhamane, Ali Khajegili Mirabadi, Eric Hallman, Mike Battaglia, Mohammad Maghsoudimehrabani, Hieu Hoang, Alon Amit, Dave Hulbert, Roberto Pereira, Simon Weber, Stephen Mensah, Nathan Andre, Anton Peristyy, Chris Harjadi, Himanshu Gupta, Stephen Malina, Samuel Albanie, Will Cai, Mustafa Mehkary, Frank Reidegeld, Anna-Katharina Dick, Cary Friday, Jasdeep Sidhu, Wanyoung Kim, Mariana Costa, Hubeyb Gurdogan, Brian Weber, Harsh Kumar, Tong Jiang, Arunim Agarwal, Chiara Ceconello, Warren S. Vaz, Chao Zhuang, Haon Park, Andrew R. Tawfeek, Daattavya Aggarwal, Michael Kirchhof, Linjie Dai, Evan Kim, Johan Ferret, Yuzhou Wang, Minghao Yan, Krzysztof Burdzy, Lixin Zhang, Antonio Franca, Diana T. Pham, Kang Yong Loh, Joshua Robinson, Shreen Gul, Gunjan Chhablani, Zhehang Du, Adrian Cosma, Colin White, Robin Riblet, Prajvi Saxena, Jacob Votava, Vladimir Vinnikov, Ethan Delaney, Shiv Halasyamani, Syed M. Shahid, Jean-Christophe Mourrat, Lavr Vetoshkin, Renas Bacho, Vincent Ginis, Aleksandr Maksapetyan, Florencia de la Rosa, Xiuyu Li, Guillaume Malod, Leon Lang, Julien Laurendeau, Fatimah Adesanya, Julien Portier, Lawrence Hollom, Victor Souza, Yuchen Anna Zhou, Yiğit Yalın, Gbenga Daniel Obikoya, Luca Arnaboldi, Filippo Bigi, Kaniuar Bacho, Pierre Clavier, Gabriel Recchia, Mara Popescu, Nikita Shulga, Ngefor Mildred Tanwie, Thomas C. H. Lux, Ben Rank, Colin Ni, Alesia Yakimchyk, Huanxu Liu, Olle Häggström, Emil Verkama, Himanshu Narayan, Hans Gundlach, Leonor Brito-Santana, Brian Amaro, Vivek Vajipey, Rynaa Grover, Yiyang Fan, Gabriel Poesia Reis E Silva, Linwei Xin, Yosi Kratish, Jakub Łucki, Wen-Ding Li, Justin Xu, Kevin Joseph Scaria, Freddie Vargus, Farzad Habibi, Emanuele Rodolà, Jules Robins, Vincent Cheng, Declan Grabb, Ida Bosio, Tony Fruhauff, Ido Akov, Eve J. Y. Lo, Hao Qi, Xi Jiang, Ben Segev, Jingxuan Fan, Sarah Martinson, Erik Y. Wang, Kaylie Hausknecht, Michael P. Brenner, Mao Mao, Yibo Jiang, Xinyu Zhang, David Avagian, Eshawn Jessica Scipio, Muhammad Rehan Siddiqi, Alon Ragoler, Justin Tan, Deepakkumar Patil, Rebeka Plecnik, Aaron Kirtland, Roselynn Grace Montecillo, Stephane Durand, Omer Faruk Bodur, Zahra Adoul, Mohamed Zekry, Guillaume Douville, Ali Karakoc, Tania C. B. Santos, Samir Shamseldeen, Loukmane Karim, Anna Liakhovitskaia, Nate Resman, Nicholas Farina, Juan Carlos Gonzalez, Gabe Maayan, Sarah Hoback, Rodrigo de Oliveira Pena, Glen Sherman, Hodjat Mariji, Rasoul Pouriamanesh, Wentao Wu, Gözdenur Demir, Sandra Mendoza, Ismail Alarab, Joshua Cole, Danyelle Ferreira, Bryan Johnson, Hsiaoyun Milliron, Mohammad Safdari, Liangti Dai, Siriphan Arthornthurasuk, Alexey Pronin, Jing Fan, Angel Ramirez-Trinidad, Ashley Cartwright, Daphiny Pottmaier, Omid Taheri, David Outevsky, Stanley Stepanic, Samuel Perry, Luke Askew, Raúl Adrián Huerta Rodríguez, Abdelkader Dendane, Sam Ali, Ricardo Lorena, Krishnamurthy Iyer, Sk Md Salauddin, Murat Islam, Juan Gonzalez, Josh Ducey, Russell Campbell, Maja Somrak, Vasilios Mavroudis, Eric Vergo, Juehang Qin, Benjámin Borbás, Eric Chu, Jack Lindsey, Anil Radhakrishnan, Antoine Jallon, I. M. J. Mcinnis, Alex Hoover, Sören Möller, Song Bian, John Lai, Tejal Patwardhan, Summer Yue, Alexandr Wang, Dan Hendrycks
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23 January 2025 hal-04907674
Python Programming for Mathematics focuses on the practical use of the Python language in a range of different areas of mathematics. Through fifty-five exercises of increasing difficulty, the book provides an expansive overview of the power of using programming to solve complex mathematical problems. This book is intended for undergraduate and graduate students who already have learned the basics of Python programming and would like to learn how to apply that programming skill in mathematics.
Julien Guillod
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23 January 2025 hal-04907643
Python est un langage de programmation phare dans le monde scientifique. Il est parfaitement adapté pour programmer des problèmes mathématiques. Cet ouvrage est consacré à l’utilisation pratique du langage Python dans différents domaines des mathématiques : les suites, l’algèbre linéaire, l’intégration, la théorie des graphes, le calcul symbolique, la recherche de zéros de fonctions, les probabilités, les statistiques, les équations différentielles, la science des données et la théorie des nombres.Cette nouvelle édition augmentée de chapitres sur la science des données et l’intelligence artificielle dresse un panorama des applications de la programmation en Python dans les mathématiques. À travers 55 exercices de difficulté croissante et corrigés en détail, vous pourrez acquérir les compétences nécessaires pour résoudre des problèmes complexes. Les codes sources de l’ouvrage sont disponibles en ligne.
Julien Guillod
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21 January 2025 tel-04904446
In this thesis, we study the problem of extending a Morse function on a closed manifold to a Lefschetz fibration on the cotangent bundle of said manifold, as well as certain Floer-theoretic questions related to this fibration. In the first chapter, we present the extension theorem due to E. Giroux, we extend it to the real-analytic setting and prove a theorem in the local model on the result of parallel transport along the unit semi-circle. In chapters 2 and 3, we re-examine a conjecture of P. Seidel in this context, which establishes a relation between the flow category of the Morse function and the directed Donaldson-Fukaya category of the Lefschetz fibration, particularly in dimension 2, resp. 3. In the last chapter, we give a negative answer to a natural question of E. Giroux, on a relation between the Heegaard-Floer homology of a 3-manifold and the Lefschetz fibration which extends a Morse function which induces a Heegaard diagram.
Matija Sreckovic
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20 January 2025 hal-04901397
We study the Maximum Zero-Sum Partition problem (or MZSP), defined as follows: given a multiset S={a1,a2,…,an} of integers ai∈Z⁎ (where Z⁎ denotes the set of non-zero integers) such that ∑i=1nai=0, find a maximum cardinality partition {S1,S2,…,Sk} of S such that, for every 1≤i≤k, ∑aj∈Siaj=0. Solving MZSP is useful in genomics for computing evolutionary distances between pairs of species. Our contributions are a series of algorithmic results concerning MZSP, in terms of complexity, (in)approximability, with a particular focus on the fixed-parameter tractability of MZSP with respect to either (i) the size k of the solution, (ii) the number of negative (resp. positive) values in S and (iii) the largest integer in S.
Guillaume Fertin, Oscar Fontaine, Géraldine Jean, Stéphane Vialette
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10 January 2025 hal-04878445
We consider Riesz energy problems with radial external fields. We study the question of whether or not the equilibrium is the uniform distribution on a sphere. We develop general necessary as well as general sufficient conditions on the external field that apply to powers of the Euclidean norm as well as certain Lennard--Jones type fields. Additionally, in the former case, we completely characterize the values of the power for which dimension reduction occurs in the sense that the support of the equilibrium measure becomes a sphere. We also briefly discuss the relation between these problems and certain constrained optimization problems. Our approach involves the Frostman characterization, the Funk--Hecke formula, and the calculus of hypergeometric functions.
Djalil Chafaï, Ryan W. Matzke, Edward B. Saff, Minh Quan H. Vu, Robert S. Womersley
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9 January 2025 hal-04877983
It is known that a cutoff phenomenon occurs in high dimension for certain positively curved overdamped Langevin diffusions in the Euclidean space, including the Ornstein-Uhlenbeck process and its Dyson version. In this note, we provide a structural explanation of this phenomenon, and we extend the result to a wide class of non-Gaussian and non-product models with a convex interaction. The key observation is a relation to a spectral rigidity result of Cheng and Zhou, linked to the presence of a Gaussian factor. We formulate the phenomenon using a Wasserstein coupling distance, and we deduce from it the formulation for total variation distance and relative entropy divergence. Furthermore, we discuss a natural extension to Riemannian manifolds, and ask about a possible extension or stability by perturbation.
Max Fathi, Djalil Chafai
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7 January 2025 hal-04871261
This work is concerned with the appearance of decay bounds in the velocity variable for solutions of the space-inhomogeneous Boltzmann equation without cutoff posed in a domain in the case of hard and moderately soft potentials. Such bounds are derived for general non-negative suitable weak subsolutions. These estimates hold true as long as mass, energy and entropy density functions are under control. The following boundary conditions are treated: in-flow, bounce-back, specular reflection, diffuse reflection and Maxwell reflection. The proof relies on a family of Truncated Convex Inequalities that is inspired by the one recently introduced by F. Golse, L. Silvestre and the first author (2023). To the best of our knowledge, the generation of arbitrary polynomial decay in the velocity variable for the Boltzmann equation without cutoff is new in the case of soft potentials, even for classical solutions.
Cyril Imbert, Amélie Loher
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6 January 2025 hal-02869757
We prove an identity in integral geometry, showing that if $P_x$ and $Q_x$ are two polynomials, $\int dx \, \delta(P_x) \otimes \delta(Q_x)$ is proportional to $\delta(R)$ where $R$ is the resultant of $P_x$ and $Q_x$.
Michel Bauer, Jean-Bernard Zuber
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5 January 2025 hal-04864913
We establish quantitative stability bounds for the quadratic optimal transport map $T_\mu$ between a fixed probability density $\rho$ and a probability measure $\mu$ on $\mathbb{R}^d$. Under general assumptions on $\rho$, we prove that the map $\mu\mapsto T_\mu$ is bi-Hölder continuous, with dimension-free Hölder exponents. The linearized optimal transport metric $W_{2,\rho}(\mu,\nu)=\|T_\mu-T_\nu\|_{L^2(\rho)}$ is therefore bi-Hölder equivalent to the $2$-Wasserstein distance, which justifies its use in applications. We show this property in the following cases: (i) for any log-concave density $\rho$ with full support in $\mathbb{R}^d$, and any log-bounded perturbation thereof; (ii) for $\rho$ bounded away from $0$ and $+\infty$ on a John domain (e.g., on a bounded Lipschitz domain), while the only previously known result of this type assumed convexity of the domain; (iii) for some important families of probability densities on bounded domains which decay or blow-up polynomially near the boundary. Concerning the sharpness of point (ii), we also provide examples of non-John domains for which the Brenier potentials do not satisfy any Hölder stability estimate. Our proofs rely on local variance inequalities for the Brenier potentials in small convex subsets of the support of $\rho$, which are glued together to deduce a global variance inequality. This gluing argument is based on two different strategies of independent interest: one of them leverages the properties of the Whitney decomposition in bounded domains, the other one relies on spectral graph theory.
Cyril Letrouit, Quentin Mérigot
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1 January 2025 hal-04758079
We consider a system of diffusion processes interacting through their empirical distribution. Assuming that the empirical average of a given observable can be observed at any time, we derive regularity and quantitative stability results for the optimal solutions in the associated version of the Gibbs conditioning principle. The proofs rely on the analysis of a McKean-Vlasov control problem with distributional constraints. Some new estimates are derived for Hamilton-Jacobi-Bellman equations and the Hessian of the log-density of diffusion processes, which are of independent interest.
Louis-Pierre Chaintron, Giovanni Conforti
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25 December 2024 hal-04855491
We consider a system of diffusion processes interacting through their empirical distribution. Assuming that the empirical average of a given observable can be observed at any time, we derive regularity and quantitative stability results for the optimal solutions in the associated version of the Gibbs conditioning principle. The proofs rely on the analysis of a McKean-Vlasov control problem with distributional constraints. Some new estimates are derived for Hamilton-Jacobi-Bellman equations and the Hessian of the log-density of diffusion processes, which are of independent interest.
Raphaël Côte, Radu Ignat, Stefan Le Coz, Mihai Mariş, Vincent Millot, Didier Smets
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23 December 2024 hal-04854262
We study and classify the emergence of protected edge modes at the junction of one-dimensional materials. Using symmetries of Lagrangian planes in boundary symplectic spaces, we present a novel proof of the periodic table of topological insulators in one dimension. We show that edge modes necessarily arise at the junction of two materials having different topological indices. Our approach provides a systematic framework for understanding symmetry-protected modes in one-dimension. It does not rely on periodic nor ergodicity and covers a wide range of operators which includes both continuous and discrete models.
David Gontier, Clément Tauber
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20 December 2024 hal-04363393
We consider Markov processes with generator of the form $γ\mathcal{L}_{1} + \mathcal{L}_{0}$, in which $\mathcal{L}_{1}$ generates a so-called dominant process that converges at large times towards a random point in a fixed subset called the effective state space. Using the usual characterization through martingales problems, we give general conditions under which homogenization holds true: the original process converges, when $γ$ is large and for the Meyer-Zheng pseudo-path topology and for finite-dimensional time marginals, towards an identified effective Markov process on the effective space. Few simple model examples for diffusions are studied.
Dimitri Faure, Mathias Rousset
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30 November 2024 hal-04519657
We show the existence and uniqueness of fundamental solution operators to Kolmo\-gorov-Fokker-Planck equations with rough (measurable) coefficients and local or integral diffusion on finite and infinite time strips. In the local case, that is to say when the diffusion operator is of differential type, we prove $\L^2$ decay using Davies' method and the conservation property. We also prove that the existence of a generalized fundamental solution with the expected pointwise Gaussian upper bound is equivalent to Moser's $\L^2-\L^\infty$ estimates for local weak solutions to the equation and its adjoint. When coefficients are real, this gives the existence and uniqueness of such a generalized fundamental solution and a new and natural way to obtain pointwise decay.
Pascal Auscher, Cyril Imbert, Lukas Niebel
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25 November 2024 hal-04801633
In this paper we present a unifying framework of residual domination for henselian valued fields of equicharacteristic zero, encompassing some valued fields with operators. We introduce a general definition for residual domination and show that it is well behaved. For instance, we prove a change of base theorem for residual domination over algebraically closed sets. We show that the class of residually dominated types coincides with the types that are orthogonal to the value group, and with the class of types whose reduct to $\ACVF$ (the theory of algebraically closed valued fields with a non-trivial valuation) are generically stable. If the residue field is stable, we show that an $\A$-invariant type concentrated on the field sort is orthogonal to the value group if and only if it is generically stable. If the residue field is simple, the theory of the valued field is $\NTP_{2}$ and algebraically closed sets of imaginary parameters are extension basis, we show that an $\A$-invariant type concentrated on the field sort is orthogonal to the value group if and only if it is generically simple. Examples of the simple case, among others, include the limit theory $\VFA_{0}$ of the Frobenius automorphism acting on an algebraically closed valued field of characteristic $\p$ (where $\p$ tends to infinity), as well as non-principal ultraproducts of the $\p$-adics.
Pablo Cubides Kovacsics, Silvain Rideau-Kikuchi, Mariana Vicaría
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25 November 2024 hal-04801609
In this paper we present a unifying framework of residual domination for henselian valued fields of equicharacteristic zero, encompassing some valued fields with operators. We introduce a general definition for residual domination and show that it is well behaved. For instance, we prove a change of base theorem for residual domination over algebraically closed sets. We show that the class of residually dominated types coincides with the types that are orthogonal to the value group, and with the class of types whose reduct to $\ACVF$ (the theory of algebraically closed valued fields with a non-trivial valuation) are generically stable. If the residue field is stable, we show that an $\A$-invariant type concentrated on the field sort is orthogonal to the value group if and only if it is generically stable. If the residue field is simple, the theory of the valued field is $\NTP_{2}$ and algebraically closed sets of imaginary parameters are extension basis, we show that an $\A$-invariant type concentrated on the field sort is orthogonal to the value group if and only if it is generically simple. Examples of the simple case, among others, include the limit theory $\VFA_{0}$ of the Frobenius automorphism acting on an algebraically closed valued field of characteristic $\p$ (where $\p$ tends to infinity), as well as non-principal ultraproducts of the $\p$-adics.
Silvain Rideau-Kikuchi
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25 November 2024 hal-04801057
In this note, the classical problem of two-dimensional flow in a cylindrical domain, driven by a non-uniform tangential velocity imposed at the boundary, is reconsidered in straightforward manner. When the boundary velocity is a pure rotation Ω plus a small perturbation η Ωf (θ) and when the Reynolds number based on Ω is large (Re ≫ 1), this flow is of 'Prandtl-Batchelor' type, namely, a flow of uniform vorticity ωc in a core region inside a viscous boundary layer of thickness O(Re) -1/2 . The O(η 2 ) contribution to ωc is here determined by asymptotic analysis up to O(Re -1 ). The result is in good agreement with numerical computation for Re ≳ 400.
Emmanuel Dormy, H. Keith Moffatt
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24 November 2024 hal-04800980
The low-Reynolds-number Stokes flow driven by rotation of two parallel cylinders of equal unit radius is investigated by both analytical and numerical techniques. In Part I, the case of counterrotating cylinders is considered. A numerical (finite-element) solution is obtained by enclosing the system in an outer cylinder of radius R0 >>1, on which the no-slip condition is imposed. A model problem with the same symmetries is first solved exactly, and the limit of validity of the Stokes approximation is determined; this model has some relevance for ciliary propulsion. For the twocylinder problem, attention is focused on the small-gap situation ε ≪ 1. An exact analytic solution is obtained in the contact limit ε = 0, and a net force Fc acting on the pair of cylinders in this contact limit is identified; this contributes to the torque that each cylinder experiences about its axis. The far-field torque doublet ('torquelet') is also identified. Part II treats the case of co-rotating cylinders, for which again a finite-element numerical solution is obtained for R0 >>1. The theory of Watson (1995, Mathematika, 42, 105-126) is elucidated and shown to agree well with the numerical solution. In contrast to the counter-rotating case, inertia effects are negligible throughout the fluid domain, however large, provided Re << 1. In the concluding section, the main results for both cases are summarised, and the situation when the fluid is unbounded (R0 = ∞) is discussed. If the cylinders are free to move (while rotating about their axes), in the counter-rotating case they will then translate relative to the fluid at infinity with constant velocity, the drag force exactly compensating the self-induced force due to the counterrotation. In the co-rotating case, if the cylinders are free to move, they will rotate as a pair relative to the fluid at infinity and the net torque on the cylinder pair is zero; the flow relative to the fluid at infinity is identified as a 'radial quadrupole'. If, on the other hand, the cylinder axes are held fixed, the Stokes flow in the counter-rotating case extends only for a distance r ∼ Re-1 log [Re-1] from the cylinders, and it is argued that the cylinders then experience a (dimensionless) force Fy ∼ 1/ log [Re-1 log [Re-1]]; in the co-rotating case, the cylinder pair experiences a (dimensionless) torque T , which tends to 17.2587 as ε ↓ 0; this torque is associated with a vortex-type flow ∼ r-1 that is established in the far field. Situations that can be described by the condition ε < 0 are treated for both counter-and corotating cases in the supplementary material.
E. Dormy, H. Moffatt
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20 November 2024 hal-04464740
We consider the nonlinear Schrödinger equation with double power nonlinearity. We extend the scattering result in [17] for all L 2-supercritical powers, specially, our results adapt to the cases of energy-supercritical nonlinearity.
Thomas Duyckaerts, Phan van Tin
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20 November 2024 hal-02374277
We consider the nonlinear Schrödinger equation with double power nonlinearity. We extend the scattering result in [17] for all L 2-supercritical powers, specially, our results adapt to the cases of energy-supercritical nonlinearity.
Raf Cluckers, Julia Gordon, Immanuel Halupczok
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20 November 2024 hal-03332241
We extend the results about Hensel minimality to include also the mixed characteristic case. This completes our axiomatic framework for tame non-archimedean geometry over Henselian valued fields of characteristic zero. We prove a Jacobian property, a strong form of Taylor approximations of definable functions, resplendency results and cell decomposition, all under 1-h-minimality. We obtain a diophantine application of counting rational points of bounded height on Hensel minimal curves.
Raf Cluckers, Immanuel Halupczok, Silvain Rideau-Kikuchi, Floris Vermeulen
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20 November 2024 hal-03017168
We present a framework for tame geometry on Henselian valued fields which we call Hensel minimality. In the spirit of o-minimality, which is key to real geometry and several diophantine applications, we develop geometric results and applications for Hensel minimal structures that were previously known only under stronger, less axiomatic assumptions. We show existence of t-stratifications in Hensel minimal structures and Taylor approximation results which are key to non-archimedean versions of Pila-Wilkie point counting, Yomdin's parameterization results and to motivic integration. In this first paper we work in equi-characteristic zero; in the sequel paper, we develop the mixed characteristic case and a diophantine application.
Raf Cluckers, Immanuel Halupczok, Silvain Rideau-Kikuchi
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17 November 2024 hal-04787256
Abstract The existence of bound states for the magnetic Laplacian in unbounded domains can be quite challenging in the case of a homogeneous magnetic field. We provide an affirmative answer for almost flat corners and slightly curved half‐planes when the total curvature of the boundary is positive.
Virginie Bonnaillie-Noël, Søren Fournais, Ayman Kachmar, Nicolas Raymond
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16 November 2024 hal-03952973
We introduce a weak division-like property for noncommutative rings: a nontrivial ring is fadelian if for all nonzero a, x there exist b, c such that x=ab+ca. We prove properties of fadelian rings, and construct examples of such rings which are not division rings, as well as non-Noetherian and non-Ore examples. Some of these results are formalized using the Lean proof assistant.
Robin Khanfir, Béranger Seguin
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14 November 2024 hal-04782997
Most theoretical and numerical results regarding water waves rely on the irrotationality of the flow in order to simplify significantly the mathematical formulation. This assumption is rarely discussed in details. In this work we investigate the well-foundedness of this important hypothesis using numerical simulations of the free-surface Navier-Stokes equation, using a scheme introduced in Riquier and Dormy (2024). We show that, in the presence of an irregular bottom, a gravity wave of non-negligible height can effectively destabilise the bottom boundary layer, stripping off vortices into the main flow. As a vortex approaches the surface the solution of the Navier-Stokes flow in the limit of vanishing viscosity is shown to differ from the inviscid Euler solution.
Alan Riquier, Emmanuel Dormy
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12 November 2024 hal-04777710
We prove a formula which gives the number of occurrences of certain labels and local configurations inside two-step puzzles introduced by Buch, Kresch, Purbhoo and Tamvakis from the work of Knutson. Puzzles are tilings of the triangular lattice by edge labeled tiles and are known to compute the Schubert structure constants of the cohomology of two-step flag varieties. The formula that we obtain depends only on the boundary conditions of the puzzle. The proof is based on the study of color maps which are tilings of the triangular lattice by edge labeled tiles obtained from puzzles.
Quentin François
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6 November 2024 hal-04769424
We prove a formula which gives the number of occurrences of certain labels and local configurations inside two-step puzzles introduced by Buch, Kresch, Purbhoo and Tamvakis from the work of Knutson. Puzzles are tilings of the triangular lattice by edge labeled tiles and are known to compute the Schubert structure constants of the cohomology of two-step flag varieties. The formula that we obtain depends only on the boundary conditions of the puzzle. The proof is based on the study of color maps which are tilings of the triangular lattice by edge labeled tiles obtained from puzzles.
Fanny Kassel, Nicolas Tholozan
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4 November 2024 hal-04765491
Pursuing ideas in a recent work of the second author, we determine the isometry classes of unimodular lattices of rank 28, as well as the isometry classes of unimodular lattices of rank 29 without nonzero vectors of norm <=2.
Bill Allombert, Gaëtan Chenevier
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4 November 2024 hal-04765423
We develop a method initiated by Bacher and Venkov, and based on a study of the Kneser neighbors of the standard lattice Z^n , which allows to classify the integral unimodular Euclidean lattices of rank n. As an application, of computational flavour, we determine the isometry classes of unimodular lattices of rank 26 and 27.
Gaëtan Chenevier
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3 November 2024 hal-02922416
In this paper, we provide the first rigorous derivation of hydrodynamic equations from the Boltzmann equation for inelastic hard spheres with small inelasticity. The hydrodynamic system that we obtain is an incompressible Navier-Stokes-Fourier system with self-consistent forcing terms and, to our knowledge, it is thus the first hydrodynamic system that properly describes rapid granular flows consistent with the kinetic formulation. To this end, we write our Boltzmann equation in a non dimensional form using the dimensionless Knudsen number which is intended to be sent to $0$. There are several difficulties in such derivation, the first one coming from the fact that the original Boltzmann equation is free-cooling and, thus, requires a self-similar change of variables to introduce an homogeneous steady state. Such a homogeneous state is not explicit and is heavy-tailed, which is a major obstacle to adapting energy estimates and spectral analysis. Additionally, a central challenge is to understand the relation between the restitution coefficient, which quantifies the energy loss at the microscopic level, and the Knudsen number. This is achieved by identifying the correct nearly elastic regime to capture nontrivial hydrodynamic behavior. We are, then, able to prove exponential stability uniformly with respect to the Knudsen number for solutions of the rescaled Boltzmann equation in a close to equilibrium regime. Finally, we prove that solutions to the Boltzmann equation converge in a specific weak sense towards a hydrodynamic limit which depends on time and space variables only through macroscopic quantities. Such macroscopic quantities are solutions to a suitable modification of the incompressible Navier-Stokes-Fourier system which appears to be new in this context.
Ricardo J. Alonso, Bertrand Lods, Isabelle Tristani